Open Access

Noncommutativity of mappings in hybrid fixed point results

Boundary Value Problems20132013:145

DOI: 10.1186/1687-2770-2013-145

Received: 10 December 2012

Accepted: 25 May 2013

Published: 11 June 2013

Abstract

In this note, some coincidence and common fixed points of nonlinear hybrid mappings have been obtained under certain noncommutativity conditions of mappings. Our results improve several known results in the field of hybrid fixed point theory.

MSC:54H25, 47H10, 54C60.

Keywords

coincidence point fixed point occasionally coincidentally idempotent multi-valued mappings

Introduction

As a generalization of the Banach fixed point theorem, Nadler’s contraction principle has lead to an excellent fixed point result in the area of nonlinear analysis. Some other works focused on fixed point results for multi-valued mappings are, for instance, [15]. Coincidence and common fixed points of nonlinear hybrid contractions (i.e., contractions involving single-valued and multi-valued mappings) have been recently studied by many authors. To mention some of the achievements, we cite, for example, [612].

The concept of commutativity of single-valued mappings [13] was extended in [14] to the setting of a single-valued mapping and a multi-valued mapping on a metric space. This concept of commutativity has been further generalized by different authors, viz weakly commuting [15], compatible [16], weakly compatible [8]. It is interesting to note that in all the results obtained so far concerning common fixed points of hybrid mappings the (single-valued and multi-valued) mappings under consideration satisfy either the commutativity condition or one of its generalizations (see, for instance, [610]). In this note, we show the existence of fixed points of hybrid contractions which do not satisfy any of the commutativity conditions or its above-mentioned generalizations. Our result extends and improves several well-known results in the field of hybrid fixed point theory. Some other recent related references are [17, 18], where common fixed point theorems for hybrid mappings on a symmetric space are proved under the assumptions of weak compatibility and occasional weak compatibility. Some analogous results for the case of contractivity conditions of integral type are presented in [1921] and generalized contractive hybrid pairs are considered in [22]. Finally, in [23], fixed point results are proved in topological vector space valued cone metric spaces (with nonnormal cones).

Preliminaries

For a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq1_HTML.gif, let ( C B ( X ) , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq2_HTML.gif and ( C L ( X ) , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq3_HTML.gif denote respectively the hyper-space of non-empty closed bounded and non-empty closed subsets of X, where H is the Hausdorff metric induced by d. For f : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq4_HTML.gif and T : X C L ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq5_HTML.gif, we shall use the following notations:
L ( x , y ) = max { d ( f x , f y ) , d ( f x , T x ) , d ( f y , T y ) , 1 2 ( d ( f x , T y ) + d ( f y , T x ) ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equa_HTML.gif
and
N ( x , y ) = [ max { d 2 ( f x , f y ) , d ( f x , T x ) d ( f y , T y ) , d ( f x , T y ) d ( f y , T x ) , 1 2 d ( f x , T x ) d ( f y , T x ) , 1 2 d ( f x , T y ) d ( f y , T y ) } ] 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equb_HTML.gif

We recall some definitions.

Definition 1 Mappings f and T are said to be commuting at a point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq6_HTML.gif if f T x T f x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq7_HTML.gif. The mappings f and T are said to be commuting on X if f T x T f x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq8_HTML.gif for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq9_HTML.gif.

Definition 2 Mappings f and T are said to be weakly commuting at a point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq6_HTML.gif if
H ( f T x , T f x ) d ( f x , T x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equc_HTML.gif
The mappings f and T are said to be weakly commuting on X if
H ( f T x , T f x ) d ( f x , T x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equd_HTML.gif

for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq9_HTML.gif.

Definition 3 The mappings f and T are said to be compatible if f T x C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq10_HTML.gif for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq6_HTML.gif and lim n + H ( T f x n , f T x n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq11_HTML.gif, whenever { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif is a sequence in X such that T x n M C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq13_HTML.gif and f x n t M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq14_HTML.gif, as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq15_HTML.gif.

Definition 4 The mappings f and T are said to be f-weak compatible if f T x C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq10_HTML.gif for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq16_HTML.gif and the following limits exist and satisfy the inequalities:
  1. (i)

    lim n H ( T f x n , f T x n ) lim n H ( T f x n , T x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq17_HTML.gif,

     
  2. (ii)

    lim n d ( f T x n , f x n ) lim n H ( T f x n , T x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq18_HTML.gif,

     

whenever { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq19_HTML.gif is a sequence in X such that T x n M C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq20_HTML.gif and f x n t M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq21_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq22_HTML.gif.

Let C ( T , f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq23_HTML.gif denote the set of all coincidence points of the mappings f and T, that is, C ( T , f ) = { u : f u T u } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq24_HTML.gif.

Definition 5 The mappings f and T are said to be coincidentally commuting if they commute at their coincidence points.

Definition 6 Mappings f and T are said to be coincidentally idempotent if f f u = f u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq25_HTML.gif for every u C ( T , f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq26_HTML.gif, that is, if f is idempotent at the coincidence points of f and T.

Definition 7 Mappings f and T are said to be occasionally coincidentally idempotent (or, in brief, oci) if f f u = f u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq25_HTML.gif for some u C ( T , f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq27_HTML.gif.

It should be remarked that coincidentally idempotent pairs of mappings are occasionally coincidentally idempotent, but the converse is not necessarily true as shown in Example 18 of this note.

Main results

We recall the following lemma.

Lemma 8 [8]

Let T : Y C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq28_HTML.gif and f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq29_HTML.gif be f-weak compatible. If { f w } = T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq30_HTML.gif for some w Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq31_HTML.gif and H ( T x , T y ) h ( a L ( x , y ) + ( 1 a ) N ( x , y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq32_HTML.gif for all x, y in Y, where 0 < h < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq33_HTML.gif, 0 a 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq34_HTML.gif, then f T w = T f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq35_HTML.gif.

We remark that the above-mentioned lemma has been used in [8, 9] and [10] to prove the existence of fixed points of hybrid mappings. However, we have noticed some typos in its original statement which have been rectified in the above statement without altering the proof.

Next, we prove a fixed point result for hybrid mappings under a general integral-type contractivity condition. In contrast to [20], we avoid the complete character of the base space X, and we introduce hybrid mappings. With respect to the study in [21], we consider here occasionally coincidentally idempotent mappings.

Theorem 9 Let Y be an arbitrary non-empty set, ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq1_HTML.gif be a metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq36_HTML.gif and T : Y C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq37_HTML.gif be such that
T ( Y ) f ( Y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ1_HTML.gif
(1)
that is, y Y T ( y ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq38_HTML.gif,
there exists q ( 0 , 1 ) such that 0 H ( T x , T y ) ψ ( t ) d t q 0 L ( x , y ) ψ ( t ) d t for all x , y in Y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ2_HTML.gif
(2)
f ( Y ) is complete , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ3_HTML.gif
(3)
ψ : R + R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq39_HTML.gif is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that
ψ ( x ) > 0 , x > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ4_HTML.gif
(4)
which trivially implies that
0 ϵ ψ ( t ) d t > 0 for each ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ5_HTML.gif
(5)
and
0 ϵ ψ ( t ) d t < 0 ϵ ˜ ψ ( t ) d t for each 0 < ϵ < ϵ ˜ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ6_HTML.gif
(6)
Suppose also that
0 μ ϵ ψ ( t ) d t γ ( μ ) 0 ϵ ψ ( t ) d t for each μ > 1 and ϵ > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ7_HTML.gif
(7)
where γ : ( 1 , + ) R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq40_HTML.gif is such that
0 < γ ( q 1 / 2 ) q < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ8_HTML.gif
(8)
and
γ ( q 1 / 2 ) q γ ( 1 γ ( q 1 / 2 ) q ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ9_HTML.gif
(9)

Then T and f have a coincidence point. Further, if f and T are occasionally coincidentally idempotent, then f and T have a common fixed point.

Proof In view of (1) and Nadler’s remark in [24], given the point x 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq41_HTML.gif, we can construct two sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq42_HTML.gif in Y and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq43_HTML.gif in X such that, for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq44_HTML.gif,
y n = f x n T x n 1 and d ( y n , y n + 1 ) q 1 / 2 H ( T x n 1 , T x n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Eque_HTML.gif

Indeed, since T x 0 f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq45_HTML.gif, there exists x 1 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq46_HTML.gif such that f x 1 = y 1 T x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq47_HTML.gif. Besides, given y 1 T x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq48_HTML.gif, by Nadler’s remark in [24] and using that q 1 / 2 > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq49_HTML.gif, we can choose y 2 T x 1 f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq50_HTML.gif such that d ( y 1 , y 2 ) q 1 / 2 H ( T x 0 , T x 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq51_HTML.gif and y 2 = f x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq52_HTML.gif for a certain x 2 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq53_HTML.gif. The continuation of this process allows to construct the two above-mentioned sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq42_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq43_HTML.gif inductively.

We claim that { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq54_HTML.gif is a Cauchy sequence. Using the inequality in (2) and also property (7), which is trivially valid for ϵ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq55_HTML.gif, it follows, for n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq56_HTML.gif, that
0 d ( f x n 1 , f x n ) ψ ( t ) d t 0 q 1 / 2 H ( T x n 2 , T x n 1 ) ψ ( t ) d t γ ( q 1 / 2 ) 0 H ( T x n 2 , T x n 1 ) ψ ( t ) d t γ ( q 1 / 2 ) q 0 L ( x n 2 , x n 1 ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equf_HTML.gif
where
L ( x n 2 , x n 1 ) = max { d ( f x n 2 , f x n 1 ) , d ( f x n 2 , T x n 2 ) , d ( f x n 1 , T x n 1 ) , 1 2 ( d ( f x n 2 , T x n 1 ) + d ( f x n 1 , T x n 2 ) ) } max { d ( f x n 2 , f x n 1 ) , d ( f x n 2 , f x n 1 ) , d ( f x n 1 , f x n ) , 1 2 d ( f x n 2 , f x n ) } max { d ( f x n 2 , f x n 1 ) , d ( f x n 1 , f x n ) , 1 2 ( d ( f x n 2 , f x n 1 ) + d ( f x n 1 , f x n ) ) } = max { d ( f x n 2 , f x n 1 ) , d ( f x n 1 , f x n ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equg_HTML.gif
Suppose that
d ( f x n 1 , f x n ) > λ d ( f x n 2 , f x n 1 ) for some  n N  with  n 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equh_HTML.gif
where λ = γ ( q 1 / 2 ) q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq57_HTML.gif, hence d ( f x n 1 , f x n ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq58_HTML.gif and
0 < max { d ( f x n 2 , f x n 1 ) , d ( f x n 1 , f x n ) } < 1 λ d ( f x n 1 , f x n ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equi_HTML.gif
so that
0 d ( f x n 1 , f x n ) ψ ( t ) d t γ ( q 1 / 2 ) q 0 max { d ( f x n 2 , f x n 1 ) , d ( f x n 1 , f x n ) } ψ ( t ) d t < γ ( q 1 / 2 ) q 0 1 λ d ( f x n 1 , f x n ) ψ ( t ) d t γ ( q 1 / 2 ) q γ ( 1 λ ) 0 d ( f x n 1 , f x n ) ψ ( t ) d t 0 d ( f x n 1 , f x n ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equj_HTML.gif
where we have also used (6) (a consequence of (4)), (7), (8) and (9). The previous inequalities imply that
0 d ( f x n 1 , f x n ) ψ ( t ) d t < 0 d ( f x n 1 , f x n ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equk_HTML.gif
which is a contradiction. In consequence,
d ( f x n 1 , f x n ) λ d ( f x n 2 , f x n 1 ) , for every  n N , n 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equl_HTML.gif
where λ = γ ( q 1 / 2 ) q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq59_HTML.gif, by hypothesis, and hence { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq60_HTML.gif is a Cauchy sequence in f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif. This is clear from the following inequality, valid for n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq62_HTML.gif, n > m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq63_HTML.gif,
d ( f x n , f x m ) j = m + 1 n d ( f x j , f x j 1 ) j = m + 1 n λ j 1 d ( f x 1 , f x 0 ) = λ m λ n 1 λ d ( f x 1 , f x 0 ) λ m 1 λ d ( f x 1 , f x 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equm_HTML.gif

which tends to zero as m + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq64_HTML.gif.

Since f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif is complete, then the sequence { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq60_HTML.gif has a limit in f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif, say u. Let w f 1 ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq65_HTML.gif and prove that f w T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq66_HTML.gif.

Suppose that f w T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq67_HTML.gif, then, by (2), we have
0 d ( f x n + 1 , T w ) ψ ( t ) d t 0 H ( T x n , T w ) ψ ( t ) d t q 0 L ( x n , w ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equn_HTML.gif
where
L ( x n , w ) = max { d ( f x n , f w ) , d ( f x n , T x n ) , d ( f w , T w ) , 1 2 ( d ( f x n , T w ) + d ( f w , T x n ) ) } = d ( f w , T w ) for  n  large . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equo_HTML.gif
Here, we have used that d ( f x n , f w ) = d ( f x n , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq68_HTML.gif, as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq69_HTML.gif, d ( f x n , T x n ) d ( f x n , f x n + 1 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq70_HTML.gif, as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq69_HTML.gif, d ( f w , T w ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq71_HTML.gif due to f w T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq72_HTML.gif and Tw closed, and
1 2 ( d ( f x n , T w ) + d ( f w , T x n ) ) 1 2 ( 2 d ( f x n , f w ) + d ( f w , T w ) + d ( f x n , T x n ) ) 1 2 d ( f w , T w ) , as  n + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equp_HTML.gif
Hence, for n large enough, we have
0 d ( f x n + 1 , T w ) ψ ( t ) d t q 0 d ( f w , T w ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equq_HTML.gif
Making n tend to +∞ in the previous inequality, we have
0 d ( f w , T w ) ψ ( t ) d t q 0 d ( f w , T w ) ψ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equr_HTML.gif

and, therefore, since q < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq73_HTML.gif and d ( f w , T w ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq71_HTML.gif, we get 0 d ( f w , T w ) ψ ( t ) d t < 0 d ( f w , T w ) ψ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq74_HTML.gif, which is a contradiction. Hence f w T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq75_HTML.gif, that is, w is a coincidence point for T and f.

Although this fact is not relevant to the proof, we note that H ( T x n 1 , T x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq76_HTML.gif since
lim n d ( y n 1 , y n ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equs_HTML.gif
Indeed,
0 H ( T x n 1 , T x n ) ψ ( t ) d t q 0 L ( x n 1 , x n ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equt_HTML.gif
where
L ( x n 1 , x n ) max { d ( f x n 1 , f x n ) , d ( f x n , f x n + 1 ) } max { d ( f x n 1 , f x n ) , λ d ( f x n 1 , f x n ) } = d ( f x n 1 , f x n ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equu_HTML.gif
therefore
0 H ( T x n 1 , T x n ) ψ ( t ) d t q 0 d ( f x n 1 , f x n ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equv_HTML.gif
Then lim n + 0 H ( T x n 1 , T x n ) ψ ( t ) d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq77_HTML.gif and, by the properties of ψ, we get H ( T x n 1 , T x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq78_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq79_HTML.gif. From the definition of { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq43_HTML.gif, we deduce that d ( f x n , T x n ) H ( T x n 1 , T x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq80_HTML.gif for every n and, therefore, lim n d ( f x n , T x n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq81_HTML.gif, so that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq82_HTML.gif is asymptotically T-regular with respect to f. However, this property can be deduced directly from the fact that
0 d ( f x n , T x n ) d ( f x n , f x n + 1 ) 0 as  n + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equw_HTML.gif
Now, if f and T are occasionally coincidentally idempotent, then f f w = f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq83_HTML.gif for some w C ( T , f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq84_HTML.gif. Then we have
0 H ( T f w , T w ) ψ ( t ) d t q 0 L ( f w , w ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ10_HTML.gif
(10)
where
L ( f w , w ) = max { d ( f f w , f w ) , d ( f f w , T f w ) , d ( f w , T w ) , 1 2 ( d ( f f w , T w ) + d ( f w , T f w ) ) } = max { d ( f w , f w ) , d ( f w , T f w ) , d ( f w , T w ) , 1 2 ( d ( f w , T w ) + d ( f w , T f w ) ) } = d ( f w , T f w ) H ( T w , T f w ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equx_HTML.gif
If T f w T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq85_HTML.gif, then from inequality (10) and using (5) (which is guaranteed by (4)), we have that
0 H ( T f w , T w ) ψ ( t ) d t q 0 H ( T f w , T w ) ψ ( t ) d t < 0 H ( T f w , T w ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equy_HTML.gif

which is a contradiction. Hence T f w = T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq86_HTML.gif. Thus we have f w = f f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq87_HTML.gif and f w T w = T f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq88_HTML.gif, i.e., fw is a common fixed point of f and T. □

Let Φ denote the family of maps ϕ from the set R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq89_HTML.gif of nonnegative real numbers to itself such that
ϕ ( t ) q t for all  t 0  and for some  q ( 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ11_HTML.gif
(11)
Corollary 10 Let Y be an arbitrary non-empty set, ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq1_HTML.gif be a metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq36_HTML.gif and T : Y C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq37_HTML.gif be such that T ( Y ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq90_HTML.gif,
0 H ( T x , T y ) ψ ( t ) d t ϕ ( 0 L ( x , y ) ψ ( t ) d t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ12_HTML.gif
(12)
for all x, y in Y, where ϕ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq91_HTML.gif (satisfying (11) for a certain q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq92_HTML.gif),
f ( Y ) is complete , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equz_HTML.gif

ψ : R + R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq39_HTML.gif is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Suppose also that (7), (8) and (9) hold for a certain γ : ( 1 , + ) R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq40_HTML.gif and q determined by (11). Then T and f have a coincidence point. Further, if f and T are occasionally coincidentally idempotent, then f and T have a common fixed point.

Proof It is a consequence of Theorem 9 since (11) and (12) imply that
0 H ( T x , T y ) ψ ( t ) d t ϕ ( 0 L ( x , y ) ψ ( t ) d t ) q 0 L ( x , y ) ψ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equaa_HTML.gif

for all x, y in Y and q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq93_HTML.gif. □

Remark 11 The condition
0 μ ϵ ψ ( t ) d t μ 0 ϵ ψ ( t ) d t for each  μ > 1  and  ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ13_HTML.gif
(13)
implies the validity of hypothesis (7) in Theorem 9 for the particular case of γ the identity mapping. Moreover, for 0 < q < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq94_HTML.gif, hypotheses (8) and (9) are trivially satisfied for this choice of γ. Indeed, using that 0 < q < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq94_HTML.gif, we get
0 < γ ( q 1 / 2 ) q = q 1 / 2 q = q 1 / 2 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equab_HTML.gif
and
γ ( q 1 / 2 ) q γ ( 1 γ ( q 1 / 2 ) q ) = q 1 / 2 q 1 q 1 / 2 q = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equac_HTML.gif

Remark 12 Assuming (8), condition (9) is trivially valid if λ γ ( 1 λ ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq95_HTML.gif for every λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq96_HTML.gif or, equivalently, γ ( 1 λ ) 1 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq97_HTML.gif for every λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq98_HTML.gif, that is, γ ( z ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq99_HTML.gif for every z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif. Note that this last condition is trivially valid for γ the identity mapping. Moreover, if γ ( z ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq101_HTML.gif for every z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif, then γ ( z ) < z 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq102_HTML.gif for every z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif and, therefore, if q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq103_HTML.gif, then γ ( q 1 / 2 ) < q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq104_HTML.gif, obtaining (8) if γ ( q 1 / 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq105_HTML.gif.

Remark 13 According to Remark 12, for q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq93_HTML.gif fixed and ψ satisfying (4), an admissible function γ can be obtained by taking
γ ( z ) sup ϵ > 0 0 z ϵ ψ ( t ) d t 0 ϵ ψ ( t ) d t , z > 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equad_HTML.gif

provided that γ ( q 1 / 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq105_HTML.gif and γ ( z ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq101_HTML.gif for every z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif.

Example 14 Taking ψ as the constant function ψ ( t ) = K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq106_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq107_HTML.gif, in the statement of Theorem 9, condition (7) is reduced to
K μ ϵ γ ( μ ) K ϵ for each  μ > 1  and  ϵ > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equae_HTML.gif

so that we must choose γ as a nonnegative function satisfying that γ ( z ) = z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq108_HTML.gif for z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq109_HTML.gif (obviously, γ ( q 1 / 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq105_HTML.gif since q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq93_HTML.gif) in order to guarantee conditions (7), (8) and (9).

Example 15 A simple calculation provides that, for the function ψ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq110_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq107_HTML.gif, condition (7) is written as γ ( z ) z 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq111_HTML.gif for z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq109_HTML.gif and, therefore, in this case condition (8) is never fulfilled. If we take ψ ( t ) = K t m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq112_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq107_HTML.gif, for K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq113_HTML.gif and m > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq114_HTML.gif fixed, then (7) implies that γ ( z ) z m + 1 > z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq115_HTML.gif for z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif.

Example 16 Now, we choose ψ ( t ) = K t m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq116_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq107_HTML.gif, where K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq113_HTML.gif and 1 < m < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq117_HTML.gif are fixed. Note that the case m = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq118_HTML.gif has already been studied in Example 14. In this case 1 < m < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq117_HTML.gif, condition (7) is reduced to
K ( μ ϵ ) m + 1 m + 1 γ ( μ ) K ϵ m + 1 m + 1 for  μ > 1  and  ϵ > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equaf_HTML.gif
which is equivalent to γ ( z ) z m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq119_HTML.gif for z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq109_HTML.gif. Note that this inequality implies, for 0 < q < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq94_HTML.gif, that γ ( q 1 / 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq105_HTML.gif. If we add the hypothesis γ ( z ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq120_HTML.gif for z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif, then we guarantee the validity of conditions (8) and (9) due to Remark 12. Hence, we can take any nonnegative function γ satisfying that
z m + 1 γ ( z ) z for  z > 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equag_HTML.gif

Of course, γ ( z ) = z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq108_HTML.gif and γ ( z ) = z m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq121_HTML.gif are valid choices.

Example 17 Take ψ ( t ) = e t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq122_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq107_HTML.gif. Condition (7) is equivalent to
1 e μ ϵ γ ( μ ) ( 1 e ϵ ) for  μ > 1  and  ϵ > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equah_HTML.gif
that is,
γ ( μ ) 1 e μ ϵ 1 e ϵ for  μ > 1  and  ϵ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equai_HTML.gif

Now, for each z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif fixed, we calculate sup ϵ > 0 1 e z ϵ 1 e ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq123_HTML.gif, which is obviously positive, and we check that its value is equal to z.

It is easy to prove that for z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif fixed, the function ϵ ( 0 , + ) R z ( ϵ ) = 1 e z ϵ 1 e ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq124_HTML.gif is decreasing on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq125_HTML.gif. Indeed, the sign of its derivative coincides with the sign of the function ν ( ϵ ) = z e z ϵ ( 1 e ϵ ) ( 1 e z ϵ ) e ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq126_HTML.gif and also with the sign of τ ( ϵ ) = z e z ϵ ( e ϵ 1 ) + e z ϵ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq127_HTML.gif for ϵ ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq128_HTML.gif. Now, the function τ is strictly negative on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq125_HTML.gif since τ ( 0 ) = τ ( 0 + ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq129_HTML.gif and τ ( ϵ ) = z ( 1 z ) e z ϵ ( e ϵ 1 ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq130_HTML.gif for ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq131_HTML.gif.

Moreover, lim ϵ 0 + R z ( ϵ ) = z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq132_HTML.gif for each z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif; in consequence, sup ϵ > 0 R z ( ϵ ) = z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq133_HTML.gif for every z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif. Therefore, if γ ( z ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq134_HTML.gif for every z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq135_HTML.gif, then (7) follows. Note also that if q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq93_HTML.gif, then γ ( q 1 / 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq105_HTML.gif. Finally, for q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq93_HTML.gif, if we take γ : ( 1 , + ) R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq40_HTML.gif such that γ ( z ) = z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq136_HTML.gif for z > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq100_HTML.gif, we deduce the validity of (7), (8) and (9).

The following example shows that Theorem 9 is a proper generalization of the fixed point results in [710].

Example 18 Let X = R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq137_HTML.gif be endowed with the Euclidean metric, let f : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq4_HTML.gif and T : X C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq138_HTML.gif be defined by f x = 4 ( x 2 + x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq139_HTML.gif and T x = [ 0 , x 2 + 7 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq140_HTML.gif. Let ϕ : R + R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq141_HTML.gif be defined by ϕ ( t ) = 1 4 t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq142_HTML.gif for all t R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq143_HTML.gif. Then mappings f and T are not commuting and also do not satisfy any of its generalizations, viz weakly commuting, compatibility, weak compatibility. Also the mappings f and T are not coincidentally commuting. Note that f 1 T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq144_HTML.gif, but f f 1 f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq145_HTML.gif and so f and T are not coincidentally idempotent, but f 0 T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq146_HTML.gif and f f 0 = f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq147_HTML.gif thus f and T are occasionally coincidentally idempotent. For all x and y in X, we have
0 H ( T x , T y ) ψ ( t ) d t = 0 | x 2 y 2 | ψ ( t ) d t = 0 ( x + y 4 ) 1 ( x + y + 1 ) ( 4 | x y | ( x + y + 1 ) ) ψ ( t ) d t = 0 ( x + y 4 ) 1 ( x + y + 1 ) ( 4 | x 2 y 2 + x y | ) ψ ( t ) d t 0 1 4 d ( f x , f y ) ψ ( t ) d t 1 4 0 d ( f x , f y ) ψ ( t ) d t 1 4 0 L ( x , y ) ψ ( t ) d t = ϕ ( 0 L ( x , y ) ψ ( t ) d t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equaj_HTML.gif
Note that these inequalities are valid if
0 1 4 d ( f x , f y ) ψ ( t ) d t 1 4 0 d ( f x , f y ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equak_HTML.gif

which is satisfied taking, for instance, the constant function ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq148_HTML.gif. On the other hand, γ is chosen as the identity map and it satisfies (8) and (9).

Note that 0 is a common fixed point of f and T. We remark that the results of [79] and [10] cannot be applied to these mappings f and T.

Theorem 19 In Theorem  9, we can assume, instead of condition (2), one of the inequalities
0 H ( T x , T y ) ψ ( t ) d t q ( a 0 L ( x , y ) ψ ( t ) d t + b 0 N ( x , y ) ψ ( t ) d t ) for all x , y in Y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ14_HTML.gif
(14)
or
0 H ( T x , T y ) ψ ( t ) d t q 0 a L ( x , y ) + b N ( x , y ) ψ ( t ) d t for all x , y in Y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ15_HTML.gif
(15)

where a , b 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq149_HTML.gif, a + b 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq150_HTML.gif and q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq103_HTML.gif.

Similarly, in Corollary  10, we can consider one of the contractivity conditions
0 H ( T x , T y ) ψ ( t ) d t ϕ ( a 0 L ( x , y ) ψ ( t ) d t + b 0 N ( x , y ) ψ ( t ) d t ) for all x , y in Y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ16_HTML.gif
(16)
or
0 H ( T x , T y ) ψ ( t ) d t ϕ ( 0 a L ( x , y ) + b N ( x , y ) ψ ( t ) d t ) for all x , y in Y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ17_HTML.gif
(17)

where a , b 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq149_HTML.gif, a + b 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq150_HTML.gif and ϕ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq151_HTML.gif (satisfying (11) for a certain q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq103_HTML.gif) and the conclusion follows.

Proof It follows from the inequality
N ( x , y ) L ( x , y ) for every  x , y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equal_HTML.gif
and the nonnegative character of a, b and ψ. Indeed, d 2 ( f x , f y ) [ L ( x , y ) ] 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq152_HTML.gif,
d ( f x , T x ) d ( f y , T y ) [ L ( x , y ) ] 2 , d ( f x , T y ) d ( f y , T x ) 1 4 ( d ( f x , T y ) + d ( f y , T x ) ) 2 [ L ( x , y ) ] 2 , 1 2 d ( f x , T x ) d ( f y , T x ) [ max { d ( f x , T x ) , 1 2 ( d ( f x , T y ) + d ( f y , T x ) ) } ] 2 [ L ( x , y ) ] 2 , 1 2 d ( f x , T y ) d ( f y , T y ) [ max { d ( f y , T y ) , 1 2 ( d ( f x , T y ) + d ( f y , T x ) ) } ] 2 [ L ( x , y ) ] 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equam_HTML.gif
hence, for instance,
a 0 L ( x , y ) ψ ( t ) d t + b 0 N ( x , y ) ψ ( t ) d t ( a + b ) 0 L ( x , y ) ψ ( t ) d t 0 L ( x , y ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equan_HTML.gif

Note that, in cases (16) and (17), it is not necessary to assume the nondecreasing character of the function ϕ since, using that ϕ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq91_HTML.gif, we deduce (14) and (15), respectively. □

Of course, the function ϕ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq153_HTML.gif is admissible in the results of this paper.

Note that, taking a = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq154_HTML.gif and b = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq155_HTML.gif in the inequalities of Theorem 19, we obtain the corresponding contractivity conditions of Theorem 9 and Corollary 10. On the other hand, taking a = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq156_HTML.gif and b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq157_HTML.gif in Theorem 19, we have the following results, which are also corollaries of Theorem 9.

Corollary 20 Let Y be an arbitrary non-empty set, ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq158_HTML.gif be a metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq29_HTML.gif and T : Y C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq159_HTML.gif be such that conditions (1), (3) hold and
0 H ( T x , T y ) ψ ( t ) d t q 0 N ( x , y ) ψ ( t ) d t for all x , y Y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ18_HTML.gif
(18)

where 0 < q < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq160_HTML.gif and ψ : R + R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq161_HTML.gif is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Assume also that (7), (8) and (9) are fulfilled for a certain γ : ( 1 , + ) R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq40_HTML.gif. Then f and T have a coincidence point. Further, if  f and T are occasionally coincidentally idempotent, then f and T have a common fixed point.

Corollary 21 Let Y be an arbitrary non-empty set, ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq158_HTML.gif be a metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq29_HTML.gif and T : Y C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq162_HTML.gif be such that conditions (1), (3) hold and
0 H ( T x , T y ) ψ ( t ) d t ϕ ( 0 N ( x , y ) ψ ( t ) d t ) for all x , y Y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ19_HTML.gif
(19)

where ϕ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq91_HTML.gif (satisfying (11) for q ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq163_HTML.gif) and ψ : R + R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq161_HTML.gif is a Lebesgue measurable mapping which is nonnegative, summable on each compact interval and such that (4) holds. Assume also that (7), (8) and (9) are fulfilled for a certain γ : ( 1 , + ) R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq40_HTML.gif. Then f and T have a coincidence point. Further, if  f and T are occasionally coincidentally idempotent, then f and T have a common fixed point.

Let η : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq164_HTML.gif be a function having the following property (see, for instance, [6, 25]):

( P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq165_HTML.gif) For t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq166_HTML.gif, there exist δ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq167_HTML.gif, s ( t ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq168_HTML.gif such that 0 r t < δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq169_HTML.gif implies η ( r ) s ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq170_HTML.gif.

This property obviously holds if η is continuous since η attains its maximum (less than 1) on each compact [ t , t + δ ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq171_HTML.gif.

Definition 22 A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq82_HTML.gif is said to be asymptotically T-regular with respect to f if lim n d ( f x n , T x n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq172_HTML.gif.

The following theorem is related to the main results of Hu [[25], Theorem 2], Jungck [14], Kaneko [26], Nadler [[24], Theorem 5] and Beg and Azam [[6], Theorem 5.4 and Corollary 5.5].

Theorem 23 Let Y be an arbitrary non-empty set, ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq158_HTML.gif be a metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq173_HTML.gif and T : Y C L ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq174_HTML.gif be such that condition (1) holds and
0 H ( T x , T y ) ψ ( t ) d t < η ( d ( f x , f y ) ) 0 d ( f x , f y ) ψ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ20_HTML.gif
(20)

for all x , y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq175_HTML.gif, where η : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq164_HTML.gif satisfies ( P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq165_HTML.gif) and ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq176_HTML.gif is nonincreasing.

Suppose also that Tx is a compact set for every x Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq177_HTML.gif.

If f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif is complete, then
  1. (i)

    there exists an asymptotically T-regular sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq178_HTML.gif with respect to f in Y,

     
  2. (ii)

    f and T have a coincidence point.

     

Further, if  f and T are occasionally coincidentally idempotent, then f and T have a common fixed point.

Proof For some x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq179_HTML.gif in Y, let y 0 = f x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq180_HTML.gif and choose x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq181_HTML.gif in Y such that y 1 = f x 1 T x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq182_HTML.gif. Then, by (20), we have
0 H ( T x 0 , T x 1 ) ψ ( t ) d t < η ( d ( f x 0 , f x 1 ) ) 0 d ( f x 0 , f x 1 ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equao_HTML.gif
Using (1), we can choose x 2 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq53_HTML.gif such that y 2 = f x 2 T x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq183_HTML.gif and satisfying that
d ( y 1 , y 2 ) = d ( f x 1 , y 2 ) = d ( f x 1 , T x 1 ) H ( T x 0 , T x 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equap_HTML.gif
hence
0 d ( y 1 , y 2 ) ψ ( t ) d t = 0 d ( f x 1 , f x 2 ) ψ ( t ) d t < η ( d ( f x 0 , f x 1 ) ) 0 d ( f x 0 , f x 1 ) ψ ( t ) d t 0 d ( f x 0 , f x 1 ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equaq_HTML.gif

Note that, in the previous inequalities, we have used that d ( f x 0 , f x 1 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq184_HTML.gif. If d ( f x 0 , f x 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq185_HTML.gif, then f x 0 = f x 1 T x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq186_HTML.gif and { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif is asymptotically T-regular with respect to f.

By induction, we construct a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq178_HTML.gif in Y and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq187_HTML.gif in f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif such that, for every n,
d ( f x n 1 , y n ) = d ( f x n 1 , T x n 1 ) = min y T x n 1 d ( f x n 1 , y ) H ( T x n 2 , T x n 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equar_HTML.gif

and y n = f x n T x n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq188_HTML.gif.

Also, we have
0 d ( y n + 1 , y n + 2 ) ψ ( t ) d t = 0 d ( f x n + 1 , f x n + 2 ) ψ ( t ) d t = 0 d ( f x n + 1 , T x n + 1 ) ψ ( t ) d t 0 H ( T x n , T x n + 1 ) ψ ( t ) d t η ( d ( f x n , f x n + 1 ) ) 0 d ( f x n , f x n + 1 ) ψ ( t ) d t < 0 d ( f x n , f x n + 1 ) ψ ( t ) d t = 0 d ( y n , y n + 1 ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equas_HTML.gif
It follows that the sequence { d ( y n , y n + 1 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq189_HTML.gif is decreasing and converges to its greatest lower bound, say t. Clearly t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq190_HTML.gif. If t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq107_HTML.gif, then by the property ( P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq165_HTML.gif) of η, there will exist δ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq167_HTML.gif and s ( t ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq168_HTML.gif such that
0 r t < δ ( t ) implies η ( r ) s ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equat_HTML.gif
For this δ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq191_HTML.gif, there exists N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq192_HTML.gif such that 0 d ( y n , y n + 1 ) t < δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq193_HTML.gif, whenever n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq194_HTML.gif. Hence η ( d ( y n , y n + 1 ) ) s ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq195_HTML.gif, whenever n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq194_HTML.gif. Let K = max { η ( d ( y 0 , y 1 ) ) , η ( d ( y 1 , y 2 ) ) , , η ( d ( y N 1 , y N ) ) , s ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq196_HTML.gif. Then for n = 1 , 2 , 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq197_HTML.gif , we have
0 d ( y n , y n + 1 ) ψ ( t ) d t < η ( d ( y n 1 , y n ) ) 0 d ( y n 1 , y n ) ψ ( t ) d t K 0 d ( y n 1 , y n ) ψ ( t ) d t K n 0 d ( y 0 , y 1 ) ψ ( t ) d t 0 as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equau_HTML.gif

which contradicts the assumption that t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq198_HTML.gif. Thus lim n d ( y n , y n + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq199_HTML.gif; i.e., d ( f x n , T x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq200_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq79_HTML.gif. Hence the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq201_HTML.gif is asymptotically T-regular with respect to f.

We claim that { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq202_HTML.gif is a Cauchy sequence. Let n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq203_HTML.gif with n < m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq204_HTML.gif, then, by the nonincreasing character of ψ, we get
0 d ( y n , y m ) ψ ( t ) d t 0 d ( y n , y n + 1 ) + d ( y n + 1 , y n + 2 ) + + d ( y m 1 , y m ) ψ ( t ) d t = 0 d ( y n , y n + 1 ) ψ ( t ) d t + d ( y n , y n + 1 ) d ( y n , y n + 1 ) + d ( y n + 1 , y n + 2 ) ψ ( t ) d t + + d ( y n , y n + 1 ) + d ( y n + 1 , y n + 2 ) + + d ( y m 2 , y m 1 ) d ( y n , y n + 1 ) + d ( y n + 1 , y n + 2 ) + + d ( y m 1 , y m ) ψ ( t ) d t 0 d ( y n , y n + 1 ) ψ ( t ) d t + 0 d ( y n + 1 , y n + 2 ) ψ ( t ) d t + + 0 d ( y m 1 , y m ) ψ ( t ) d t = i = n m 1 0 d ( y i , y i + 1 ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equav_HTML.gif
Now, we recall that
0 d ( y n + 1 , y n + 2 ) ψ ( t ) d t η ( d ( y n , y n + 1 ) ) 0 d ( y n , y n + 1 ) ψ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equaw_HTML.gif
for every n, which implies that
0 d ( y n + 2 , y n + 3 ) ψ ( t ) d t η ( d ( y n + 1 , y n + 2 ) ) 0 d ( y n + 1 , y n + 2 ) ψ ( t ) d t η ( d ( y n + 1 , y n + 2 ) ) η ( d ( y n , y n + 1 ) ) 0 d ( y n , y n + 1 ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equax_HTML.gif
Following this procedure, we prove that
0 d ( y j , y j + 1 ) ψ ( t ) d t i = n j 1 η ( d ( y i , y i + 1 ) ) 0 d ( y n , y n + 1 ) ψ ( t ) d t , for every  j = n + 1 , , m 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equay_HTML.gif
Therefore,
0 d ( y n , y m ) ψ ( t ) d t i = n m 1 0 d ( y i , y i + 1 ) ψ ( t ) d t = 0 d ( y n , y n + 1 ) ψ ( t ) d t + i = n + 1 m 1 0 d ( y i , y i + 1 ) ψ ( t ) d t [ 1 + i = n + 1 m 1 l = n i 1 η ( d ( y l , y l + 1 ) ) ] 0 d ( y n , y n + 1 ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equaz_HTML.gif

We check that the right-hand side in the last inequality tends to 0 as n , m + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq205_HTML.gif. Since 0 d ( y n , y n + 1 ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq206_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq79_HTML.gif, it suffices to show that i = n + 1 m 1 l = n i 1 η ( d ( y l , y l + 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq207_HTML.gif is bounded (uniformly on n, m). Indeed, we check that i = n + 1 m 1 l = n i 1 η ( z l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq208_HTML.gif is bounded for any sequence { z l } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq209_HTML.gif with nonnegative terms and tending to 0 as l + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq210_HTML.gif, using the property ( P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq211_HTML.gif) of the function η. Given t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq212_HTML.gif, by ( P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq211_HTML.gif), there exist δ ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq213_HTML.gif, s 0 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq214_HTML.gif such that 0 r < δ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq215_HTML.gif implies η ( r ) s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq216_HTML.gif. Since { z l } 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq217_HTML.gif, given δ ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq218_HTML.gif, there exists l 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq219_HTML.gif such that, for every l l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq220_HTML.gif, we have 0 z l < δ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq221_HTML.gif. This implies that η ( z l ) s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq222_HTML.gif for every l l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq220_HTML.gif.

In consequence, for n l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq223_HTML.gif, we get
0 i = n + 1 m 1 l = n i 1 η ( z l ) i = n + 1 m 1 l = n i 1 s 0 = i = n + 1 m 1 ( s 0 ) i n = s 0 ( s 0 ) m n 1 s 0 < s 0 1 s 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equba_HTML.gif

and this expression is bounded independently of m, n.

Hence { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq224_HTML.gif is a Cauchy sequence in f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif. Since f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif is complete, { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq224_HTML.gif converges to some p in f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif. Let z f 1 ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq225_HTML.gif. Then f z = p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq226_HTML.gif. Next, we have
0 d ( f z , T z ) ψ ( t ) d t 0 d ( f x n + 1 , f z ) + d ( f x n + 1 , T z ) ψ ( t ) d t = 0 d ( f x n + 1 , T z ) ψ ( t ) d t + d ( f x n + 1 , T z ) d ( f x n + 1 , T z ) + d ( f z , f x n + 1 ) ψ ( t ) d t 0 d ( f x n + 1 , T z ) ψ ( t ) d t + 0 d ( f z , f x n + 1 ) ψ ( t ) d t 0 H ( T x n , T z ) ψ ( t ) d t + 0 d ( f z , f x n + 1 ) ψ ( t ) d t η ( d ( f x n , f z ) ) 0 d ( f x n , f z ) ψ ( t ) d t + 0 d ( f z , f x n + 1 ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbb_HTML.gif

Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq227_HTML.gif, we get 0 d ( f z , T z ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq228_HTML.gif. Thus we have d ( f z , T z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq229_HTML.gif. Hence f z T z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq230_HTML.gif.

Now, if f and T are occasionally coincidentally idempotent, then f f w = f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq231_HTML.gif for some w C ( T , f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq232_HTML.gif. Then we have
0 H ( T f w , T w ) ψ ( t ) d t η ( d ( f f w , f w ) ) 0 d ( f f w , f w ) ψ ( t ) d t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbc_HTML.gif

Thus, T f w = T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq86_HTML.gif. It follows that f f w = f w T w = T f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq233_HTML.gif. Hence, fw is a common fixed point of T and f. □

Now we state some fixed point theorems for Kannan-type multi-valued mappings which extend and generalize the corresponding results of Shiau et al. [10] and Beg and Azam [6, 27]. A proper blend of the proof of Theorem 9 and those of [[10], Th. 6, Th. 7, Th. 8 respectively] and [[9], Theorems 3.1, 3.2, 3.3] will complete the proof.

Theorem 24 Let Y be an arbitrary non-empty set, ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq234_HTML.gif be a metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq173_HTML.gif and T : Y C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq235_HTML.gif be such that (1) holds and
0 H r ( T x , T y ) ψ ( t ) d t α 1 ( d ( f x , T x ) ) 0 d r ( f x , T x ) ψ ( t ) d t + α 2 ( d ( f y , T y ) ) 0 d r ( f y , T y ) ψ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ21_HTML.gif
(21)
for all x , y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq236_HTML.gif, where α i : R + [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq237_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq238_HTML.gif) are bounded on bounded sets, r is some fixed positive real number and ψ : R + R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq161_HTML.gif is a Lebesgue measurable mapping which is summable on each compact interval and 0 ϵ ψ ( t ) d t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq239_HTML.gif for each ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq240_HTML.gif. Suppose that there exists an asymptotically T-regular sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif with respect to f in Y. If T ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq241_HTML.gif is complete or
there exists k N such that f x n + k T x n for every n N , and f ( Y ) is complete , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ22_HTML.gif
(22)

then f and T have a coincidence point. Further, if f and T are occasionally coincidentally idempotent, then f and T have a common fixed point.

Proof By hypotheses,
0 H r ( T x n , T x m ) ψ ( t ) d t α 1 ( d ( f x n , T x n ) ) 0 d r ( f x n , T x n ) ψ ( t ) d t + α 2 ( d ( f x m , T x m ) ) 0 d r ( f x m , T x m ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbd_HTML.gif

Since { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif is asymptotically T-regular with respect to f in Y, then { α 1 ( d ( f x n , T x n ) ) } n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq242_HTML.gif and { α 2 ( d ( f x m , T x m ) ) } m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq243_HTML.gif are bounded sequences and 0 d r ( f x n , T x n ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq244_HTML.gif, 0 d r ( f x m , T x m ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq245_HTML.gif, as n , m + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq246_HTML.gif. This provides the property H ( T x n , T x m ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq247_HTML.gif as n , m + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq248_HTML.gif, so that { T x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq249_HTML.gif is a Cauchy sequence in ( C B ( X ) , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq2_HTML.gif.

If T ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq250_HTML.gif is complete, there exists K T ( Y ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq251_HTML.gif such that H ( T x n , K ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq252_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq79_HTML.gif. Let u Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq253_HTML.gif be such that f ( u ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq254_HTML.gif. Then
0 d r ( f u , T u ) ψ ( t ) d t 0 H r ( K , T u ) ψ ( t ) d t 0 ( H ( K , T x n ) + H ( T x n , T u ) ) r ψ ( t ) d t = 0 H r ( T x n , T u ) ψ ( t ) d t + H r ( T x n , T u ) H r ( T x n , T u ) + t e r m s c o n t a i n i n g H ( K , T x n ) ψ ( t ) d t α 1 ( d ( f x n , T x n ) ) 0 d r ( f x n , T x n ) ψ ( t ) d t + α 2 ( d ( f u , T u ) ) 0 d r ( f u , T u ) ψ ( t ) d t + H r ( T x n , T u ) H r ( T x n , T u ) + t e r m s c o n t a i n i n g H ( K , T x n ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Eqube_HTML.gif
where the number of terms containing H ( K , T x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq255_HTML.gif is a finite number depending on r, and therefore fixed. Calculating the limit as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq79_HTML.gif and taking into account that the length of the intervals in the last integral tends to zero, we get
0 d r ( f u , T u ) ψ ( t ) d t ( 1 α 2 ( d ( f u , T u ) ) ) lim n + α 1 ( d ( f x n , T x n ) ) 0 d r ( f x n , T x n ) ψ ( t ) d t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbf_HTML.gif
Therefore,
0 d r ( f u , T u ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbg_HTML.gif

and, by the properties of ψ, we get d r ( f u , T u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq256_HTML.gif, which implies that f u T u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq257_HTML.gif and u is a coincidence point.

Now, suppose that f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif is complete. Note that T x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq258_HTML.gif is closed and bounded for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq259_HTML.gif. Take k > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq260_HTML.gif fixed. By the results in [24], we can affirm that for every y 1 T x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq261_HTML.gif, there exists y 2 T x m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq262_HTML.gif such that d ( y 1 , y 2 ) k H ( T x n , T x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq263_HTML.gif.

Given n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq264_HTML.gif, we choose y 1 T x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq261_HTML.gif and, for this y 1 T x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq265_HTML.gif fixed, we choose y 2 T x m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq262_HTML.gif such that d ( y 1 , y 2 ) k H ( T x n , T x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq266_HTML.gif. Then
d ( f x n , f x m ) d ( f x n , y 1 ) + d ( y 1 , y 2 ) + d ( y 2 , f x m ) d ( f x n , T x n ) + k H ( T x n , T x m ) + d ( T x m , f x m ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbh_HTML.gif
By the hypothesis on { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif and the Cauchy character of { T x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq249_HTML.gif, we deduce that { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq267_HTML.gif is a Cauchy sequence. Since f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif is complete, there exists f ( u ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq268_HTML.gif such that { f ( x n ) } f ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq269_HTML.gif. By hypotheses, d ( f x n + k , T u ) H ( T x n , T u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq270_HTML.gif for every n, hence
0 d r ( f x n + k , T u ) ψ ( t ) d t 0 H r ( T x n , T u ) ψ ( t ) d t α 1 ( d ( f x n , T x n ) ) 0 d r ( f x n , T x n ) ψ ( t ) d t + α 2 ( d ( f u , T u ) ) 0 d r ( f u , T u ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbi_HTML.gif
and taking the limit as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq271_HTML.gif, we get
0 d r ( f u , T u ) ψ ( t ) d t α 2 ( d ( f u , T u ) ) 0 d r ( f u , T u ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbj_HTML.gif
In this case,
( 1 α 2 ( d ( f u , T u ) ) ) 0 d r ( f u , T u ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbk_HTML.gif
and d ( f u , T u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq272_HTML.gif, which implies that f u T u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq257_HTML.gif. Now, if f and T are coincidentally idempotent, then f f w = f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq273_HTML.gif for some w C ( T , f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq274_HTML.gif. Hence
0 H r ( T f w , T w ) ψ ( t ) d t α 1 ( d ( f f w , T f w ) ) 0 d r ( f f w , T f w ) ψ ( t ) d t + α 2 ( d ( f w , T w ) ) 0 d r ( f w , T w ) ψ ( t ) d t = α 1 ( d ( f w , T f w ) ) 0 d r ( f w , T f w ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbl_HTML.gif
Since f f w = f w T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq275_HTML.gif, we get
0 d r ( f w , T f w ) ψ ( t ) d t 0 H r ( T w , T f w ) ψ ( t ) d t α 1 ( d ( f w , T f w ) ) 0 d r ( f w , T f w ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbm_HTML.gif
Therefore
0 d r ( f w , T f w ) ψ ( t ) d t ( 1 α 1 ( d ( f w , T f w ) ) ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbn_HTML.gif

obtaining d ( f w , T f w ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq276_HTML.gif and f w T f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq277_HTML.gif. Since 0 0 H r ( T f w , T w ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq278_HTML.gif, we deduce that H ( T f w , T w ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq279_HTML.gif and T f w = T w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq280_HTML.gif. In consequence, f f w = f w T w = T f w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq281_HTML.gif and fw is a common fixed point of T and f. □

Remark 25 In the statement of Theorem 24, condition (22) can be replaced by the more general one
f ( Y )  is complete . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbo_HTML.gif

To complete the proof with this more general hypothesis, take into account that for y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq282_HTML.gif, T ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq283_HTML.gif is a closed set in X and T ( Y ) f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq284_HTML.gif. Using that f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif is complete, we deduce that ( C L ( f ( Y ) ) , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq285_HTML.gif is complete. Hence { T x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq249_HTML.gif is a sequence in C L ( f ( Y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq286_HTML.gif and it is a Cauchy sequence in ( C L ( f ( Y ) ) , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq285_HTML.gif. Therefore, there exists K C L ( f ( Y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq287_HTML.gif such that H ( T x n , K ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq288_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq79_HTML.gif. Note also that K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq289_HTML.gif is a closed set in the complete space f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif, then K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq289_HTML.gif is complete and, therefore, a closed set, then K C L ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq290_HTML.gif. Once we have proved that H ( T x n , K ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq252_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq79_HTML.gif in ( C L ( f ( Y ) ) , H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq291_HTML.gif, the proof follows analogously.

Theorem 26 In addition to the hypotheses of Theorem  24, suppose that T x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq258_HTML.gif is compact for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq292_HTML.gif. If f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq293_HTML.gif is a cluster point of { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq60_HTML.gif, then z is a coincidence point of f and T.

Proof Let y n T x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq294_HTML.gif be such that d ( f x n , y n ) = d ( f x n , T x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq295_HTML.gif, this is possible since T x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq258_HTML.gif is compact. It is obvious that a cluster point of { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq60_HTML.gif is a cluster point of { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq43_HTML.gif. Let f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq293_HTML.gif be a cluster point of { f x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq60_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq43_HTML.gif, then we check that f z T u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq296_HTML.gif, where u is obtained in the proof of Theorem 24. Note that, for every y T u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq297_HTML.gif,
d ( f z , y ) d ( f z , f x n ) + d ( f x n , y n ) + d ( y n , y ) = d ( f z , f x n ) + d ( f x n , T x n ) + d ( y n , y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbp_HTML.gif
hence
d ( f z , T u ) = inf y T u d ( f z , y ) d ( f z , f x n ) + d ( f x n , T x n ) + inf y T u d ( y n , y ) = d ( f z , f x n ) + d ( f x n , T x n ) + d ( y n , T u ) d ( f z , f x n ) + d ( f x n , T x n ) + H ( T x n , T u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbq_HTML.gif
In consequence,
0 d r ( f z , T u ) ψ ( t ) d t 0 ( d ( f z , f x n ) + d ( f x n , T x n ) + H ( T x n , T u ) ) r ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbr_HTML.gif
Using that there exists a subsequence f x n k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq298_HTML.gif converging to fz, the properties of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif and the inequality
0 H r ( T x n k , T u ) ψ ( t ) d t α 1 ( d ( f x n k , T x n k ) ) 0 d r ( f x n k , T x n k ) ψ ( t ) d t + α 2 ( d ( f u , T u ) ) 0 d r ( f u , T u ) ψ ( t ) d t k + 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbs_HTML.gif
then, taking the limit when n k + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq299_HTML.gif, we get 0 d r ( f z , T u ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq300_HTML.gif and f z T u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq296_HTML.gif. To prove that f z T z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq230_HTML.gif, using that f u T u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq301_HTML.gif, we get
0 d r ( f z , T z ) ψ ( t ) d t 0 H r ( T u , T z ) ψ ( t ) d t α 1 ( d ( f u , T u ) ) 0 d r ( f u , T u ) ψ ( t ) d t + α 2 ( d ( f z , T z ) ) 0 d r ( f z , T z ) ψ ( t ) d t = α 2 ( d ( f z , T z ) ) 0 d r ( f z , T z ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbt_HTML.gif
This implies that
( 1 α 2 ( d ( f z , T z ) ) ) 0 d r ( f z , T z ) ψ ( t ) d t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbu_HTML.gif

and, by the properties of α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq302_HTML.gif and ψ, we deduce that d ( f z , T z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq303_HTML.gif, which proves that z is a coincidence point of f and T. □

The following result extends [[10], Theorem 3.3].

Theorem 27 Let Y be an arbitrary non-empty set, ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq234_HTML.gif be a metric space, f : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq173_HTML.gif and T : Y C B ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq235_HTML.gif be such that (1) and (21) hold, where α i : R + [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq304_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq238_HTML.gif) are bounded on bounded sets and such that
α 1 ( d ( f x , T x ) ) + α 2 ( d ( f y , T y ) ) 1 for every x , y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbv_HTML.gif
r is some fixed positive real number and ψ : R + R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq305_HTML.gif is a Lebesgue measurable mapping which is summable on each compact interval, and ψ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq306_HTML.gif for each x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq307_HTML.gif. Suppose that
inf { d ( f z n , T z n ) : n N } = 0 for every sequence { z n } in Y with f z n T z n 1 , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ23_HTML.gif
(23)

If T ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq241_HTML.gif is complete or f ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq61_HTML.gif is complete, then f and T have a coincidence point. Further, if f and T are occasionally coincidentally idempotent, then f and T have a common fixed point.

Proof Using Theorem 24, it suffices to prove that there exists an asymptotically T-regular sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif with respect to f in Y. Let x 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq308_HTML.gif and take { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif in Y such that f x n T x n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq309_HTML.gif for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq44_HTML.gif. Then
0 d r ( f x n , T x n ) ψ ( t ) d t 0 H r ( T x n 1 , T x n ) ψ ( t ) d t α 1 ( d ( f x n 1 , T x n 1 ) ) 0 d r ( f x n 1 , T x n 1 ) ψ ( t ) d t + α 2 ( d ( f x n , T x n ) ) 0 d r ( f x n , T x n ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbw_HTML.gif
Hence,
( 1 α 2 ( d ( f x n , T x n ) ) ) 0 d r ( f x n , T x n ) ψ ( t ) d t α 1 ( d ( f x n 1 , T x n 1 ) ) 0 d r ( f x n 1 , T x n 1 ) ψ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbx_HTML.gif
or also, using the hypothesis on α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq310_HTML.gif and α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq302_HTML.gif,
0 d r ( f x n , T x n ) ψ ( t ) d t α 1 ( d ( f x n 1 , T x n 1 ) ) ( 1 α 2 ( d ( f x n , T x n ) ) ) 0 d r ( f x n 1 , T x n 1 ) ψ ( t ) d t 0 d r ( f x n 1 , T x n 1 ) ψ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equby_HTML.gif
The properties of ψ imply that d r ( f x n , T x n ) d r ( f x n 1 , T x n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq311_HTML.gif for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq259_HTML.gif, and { d ( f x n , T x n ) } n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq312_HTML.gif is nonincreasing and bounded below. Therefore it is convergent to the infimum, that is,
d ( f x n , T x n ) inf { d ( f x n , T x n ) : n N } = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equbz_HTML.gif

and { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif is asymptotically T-regular with respect to f in Y. □

Remark 28 Note that condition (23) in Theorem 27 cannot be replaced by
inf { d ( f x , T x ) : x Y } = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equca_HTML.gif

since the infimum taking the sequence { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq313_HTML.gif could be positive (we calculate the infimum in a smaller set).

Remark 29 In Theorem 27, condition (23) can be replaced by the following:
inf { H ( T z n 1 , T z n ) : n N } = 0 , for every sequence  { z n }  in  Y  with  f z n T z n 1 , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ24_HTML.gif
(24)
Indeed, since
d ( f z n , T z n ) H ( T z n 1 , T z n ) , n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equcb_HTML.gif
then
0 inf { d ( f z n , T z n ) : n N } inf { H ( T z n 1 , T z n ) : n N } = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equcc_HTML.gif

and d ( f z n , T z n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq314_HTML.gif.

Remark 30 In Theorem 27, if we are able to obtain a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_IEq12_HTML.gif with an infinite number of terms which are different, then we can relax condition (23) to the following:
inf { d ( f x , T x ) : x B } = 0 for every infinite set  B  of  Y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-145/MediaObjects/13661_2012_Article_397_Equ25_HTML.gif
(25)

Declarations

Acknowledgements

Dedicated to Professor Jean Mawhin, on the occasion of his seventieth birthday.

We thank the editor, the anonymous referees and also Professor Stojan Radenović for their helpful comments and suggestions. This research was partially supported by the University Grants Commission, New Delhi, India; Ministerio de Economía y Competitividad, project MTM2010-15314, and co-financed by EC fund FEDER.

Authors’ Affiliations

(1)
School of Studies in Mathematics, Pt. Ravishankar Shukla University
(2)
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

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Copyright

© Pathak and Rodríguez-López; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.