Open Access

Unconditional convergence of difference equations

Boundary Value Problems20132013:63

DOI: 10.1186/1687-2770-2013-63

Received: 19 December 2012

Accepted: 20 March 2013

Published: 28 March 2013

Abstract

We put forward the notion of unconditional convergence to an equilibrium of a difference equation. Roughly speaking, it means that can be constructed a wide family of higher order difference equations, which inherit the asymptotic behavior of the original difference equation. We present a sufficient condition for guaranteeing that a second-order difference equation possesses an unconditional stable attractor. Finally, we show how our results can be applied to two families of difference equations recently considered in the literature.

MSC:39A11.

Keywords

difference equations global asymptotic stability

1 Introduction

It is somewhat frequent that the global asymptotic stability of a family of difference equations can be extended to some higher-order ones (see, for example, [14]). Consider the following simple example. If φ is the map φ ( x , y ) = 1 + ( a x / y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq1_HTML.gif, the sequence y n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq2_HTML.gif defined by y n = φ ( x , y n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq3_HTML.gif, that is,
y n = 1 + a x y n 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equa_HTML.gif
with y 1 , a , x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq4_HTML.gif, converges to F φ ( x ) = ( 1 + 1 + 4 a x ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq5_HTML.gif for any y 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq6_HTML.gif. Observe that F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif is the function satisfying φ ( x , F φ ( x ) ) = F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq8_HTML.gif. Obviously, the second-order difference equation
y n = 1 + a x y n 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equb_HTML.gif
also converges to F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq9_HTML.gif for any y 1 , y 2 , a , x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq10_HTML.gif. Let us continue to add complexity, by considering the second-order difference equations
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ1_HTML.gif
(1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ2_HTML.gif
(2)

For all y 1 , y 2 , a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq11_HTML.gif, the sequence defined by Equation (1) converges to the unique fixed point μ φ = a + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq12_HTML.gif of the function F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif. However, the behavior of Equation (2) depends on the parameter a:

  • For a 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq13_HTML.gif, the odd and even index terms converge respectively to some limits, μ 1 [ 1 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq14_HTML.gif and μ 1 / ( μ 1 1 ) [ 1 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq15_HTML.gif, where μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq16_HTML.gif may depend on y 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq6_HTML.gif, y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq17_HTML.gif (for a = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq18_HTML.gif).

  • For 0 < a < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq19_HTML.gif, it converges to μ φ = a + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq12_HTML.gif, whatever the choice of y 1 , y 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq20_HTML.gif one makes.

No sophisticated tools are needed to reach those conclusions: It suffices to note that the set
A = { n : ( y n + 3 y n + 1 ) ( y n + 2 y n ) 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equc_HTML.gif
must be either finite or equal to . As the sequences y 2 n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq21_HTML.gif and y 2 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq22_HTML.gif are then both eventually monotone, they converge in [ 1 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq23_HTML.gif to some limits, say μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq16_HTML.gif and μ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq24_HTML.gif, satisfying
μ 1 = 1 + a μ 1 μ 2 and μ 2 = 1 + a μ 2 μ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equd_HTML.gif

Therefore, one of the following statements holds: μ 2 = μ 1 / ( μ 1 1 ) [ 1 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq25_HTML.gif, with a = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq18_HTML.gif, or { μ 1 } { μ 2 } { { 1 , + } , { 1 + a } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq26_HTML.gif.

If { μ 1 } { μ 2 } = { 1 , + } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq27_HTML.gif, then that of the sequences, y 2 n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq21_HTML.gif or y 2 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq22_HTML.gif, which converges to +∞, has to be nondecreasing. Just look at Equation (2) to conclude that a 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq13_HTML.gif whenever { μ 1 } { μ 2 } = { 1 , + } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq27_HTML.gif.

The case we are interested in is 0 < a < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq19_HTML.gif and we will say that μ φ = 1 + a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq28_HTML.gif is an unconditional attractor for the map φ, that is, we would consider φ ( x , y ) = 1 + ( a x / y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq29_HTML.gif with 0 < a < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq19_HTML.gif to observe that, not only (1) and (2), but all the following recursive sequences converge to μ φ = 1 + a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq28_HTML.gif, whatever the choice of y 1 , , y max { k , m } > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq30_HTML.gif we make:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Eque_HTML.gif

In this paper, we proceed as follows. The next section is dedicated to notation and a technical result of independent interest. In Section 3, we introduce the main definition and the main result in this paper, unconditional convergence and a sufficient condition for having it in a general framework. We conclude, in Section 4, showing how the later theorem can be applied to provide short proofs for some recent convergence results on two families of difference equations and to improve them.

2 Preliminaries

This section is mainly devoted to the notation we employ. In the first part, we establish some operations between subsets of real numbers and we clarify how we identify a function with a multifunction. We noticed that set-valued difference equations are not concerned with us in this paper. The reason for dealing with those set operations and notation is because it allows us to manage unboundedness and singular situations in a homogeneous way. In the second part, we introduce the families of maps Λ m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq31_HTML.gif (a kind of averages of their variables) that we shall employ in the definition of unconditional convergence. We finish the section with a technical result on monotone sequences converging to the fixed point of a monotone continuous function.

2.1 Basic notations

We consider the two points compactification R ¯ = [ , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq32_HTML.gif of endowed with the usual order and compact topology.

2.1.1 Operations and preorder in 2 R ¯ { } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq33_HTML.gif

We define the operations ‘+’, ‘−’, ‘’ and ‘/’ in 2 R ¯ { } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq34_HTML.gif by
A B = { lim sup ( a n b n ) : a n , b n , a n b n R  for all  n N  and  lim a n A , lim b n B } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equf_HTML.gif

where stands for ‘+’, ‘−’, ‘’ or ‘/’. We also agree to write A = A = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq35_HTML.gif.

Remark 1 We introduce the above notation in order to manage unboundedness and singular situations, but we point out that these are natural set-valued extensions for the arithmetic operations. Let X, Y be compact (Hausdorff) spaces, U a dense subset of X and f : U Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq36_HTML.gif. The closure Gr ( f ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq37_HTML.gif of the graph of f in X × Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq38_HTML.gif defines an upper semicontinuous compact-valued map f ¯ : X 2 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq39_HTML.gif by f ¯ ( x ) = { y Y : ( x , y ) Gr ( f ) ¯ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq40_HTML.gif, that is, by Gr ( f ¯ ) = Gr ( f ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq41_HTML.gif (see [5]). Furthermore, as usual, one writes f ¯ ( A ) = x A f ¯ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq42_HTML.gif for A 2 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq43_HTML.gif, thereby obtaining a map f ¯ : 2 X 2 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq44_HTML.gif.

To extend arithmetic operations, consider X = R ¯ × R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq45_HTML.gif, Y = R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq46_HTML.gif and U = R × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq47_HTML.gif, when f denotes addition, substraction or multiplication, and U = R × ( R { 0 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq48_HTML.gif, when f denotes division.

Also define A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq49_HTML.gif (respectively A < B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq50_HTML.gif) to be true if and only if A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq51_HTML.gif, B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq52_HTML.gif and a b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq53_HTML.gif (respectively a < b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq54_HTML.gif) for all a A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq55_HTML.gif, b B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq56_HTML.gif. Here A , B 2 R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq57_HTML.gif.

Notice that both relations ≤ and < are transitive but neither reflexive nor symmetric.

2.1.2 Canonical injections

When no confusion is likely to arise, we identify a R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq58_HTML.gif with { a } 2 R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq59_HTML.gif, that is, in the sequel we consider the fixed injection a { a } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq60_HTML.gif of R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq61_HTML.gif into 2 R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq62_HTML.gif and we identify R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq61_HTML.gif with its image. We must point out that, under this convention, when a is expected to be subset of A, we understand ‘ a A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq55_HTML.gif’ as ‘there is b A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq63_HTML.gif with a = { b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq64_HTML.gif’. For instance, one has 0 ( + ) = R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq65_HTML.gif, 1 / 0 = { , + } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq66_HTML.gif.

2.1.3 Extension of a function as a multifunction

Consider a map h : R ¯ m 2 R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq67_HTML.gif and denote by D ( h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq68_HTML.gif the set formed by those x R ¯ m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq69_HTML.gif for which there is b R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq70_HTML.gif with h ( x ) = { b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq71_HTML.gif.

If A 2 ( R ¯ m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq72_HTML.gif, then h ( A ) 2 R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq73_HTML.gif is defined to be a A h ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq74_HTML.gif. Also, if B ( 2 R ¯ ) m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq75_HTML.gif, then h ( B ) 2 R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq76_HTML.gif is defined to be h ( B ) = h ( B 1 × × B m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq77_HTML.gif.

For each function φ : U R ¯ m R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq78_HTML.gif, let φ ˆ : R ¯ m 2 R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq79_HTML.gif be defined by
φ ˆ ( x ) = { lim sup φ ( x n ) : x n U  for all  n N  and  lim x n = x } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equg_HTML.gif
It is obvious that φ ˆ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq80_HTML.gif if and only if x is in the closure U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq81_HTML.gif of U in R ¯ m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq82_HTML.gif. Also notice that
U D ( φ ˆ ) U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equh_HTML.gif
when φ is continuous. In this case, and if no confusion is likely to arise, we agree to denote also by φ the map φ ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq83_HTML.gif. For example, we write
φ ( 0 ) = [ 1 , 1 ] ; φ ( ) = φ ( + ) = 0 ; D ( φ ) = R ¯ { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equi_HTML.gif

when U = R { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq84_HTML.gif and φ ( x ) = sin ( 1 / x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq85_HTML.gif.

2.2 The maps in Λ m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq31_HTML.gif and Λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq86_HTML.gif

As we have announced, the unconditional convergence of a difference equation guarantees that there exists a family of difference equations that inherit its asymptotic behavior. Here, we define the set of functions that we employ to construct that family of difference equations.

For k , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq87_HTML.gif, let Λ m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq31_HTML.gif be the set formed by the maps λ : R ¯ m R ¯ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq88_HTML.gif such that
min 1 j m x j λ i ( x ) max 1 j m x j for all  x R ¯ m , 1 i k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ3_HTML.gif
(3)
Notice that λ γ Λ m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq89_HTML.gif whenever λ Λ r k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq90_HTML.gif, γ Λ m r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq91_HTML.gif. Let Λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq86_HTML.gif be defined as follows:
Λ k = m N Λ m k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equj_HTML.gif

We note that the functions in Λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq86_HTML.gif satisfy that their behavior is enveloped by the maximum and minimum functions of its variables, which is a common hypothesis in studying higher order nonlinear difference equations.

Some trivial examples of functions in Λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq92_HTML.gif are:

  • λ ( x 1 , , x m ) = j = 1 m α j x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq93_HTML.gif, with α j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq94_HTML.gif for j = 1 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq95_HTML.gif, j = 1 m α j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq96_HTML.gif.

An important particular case is α j 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq97_HTML.gif, α j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq98_HTML.gif for j j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq99_HTML.gif.

  • λ ( x 1 , , x m ) = { j = 1 m x j α j if  min 1 j m x j > 0 , min 1 j m x j if  min 1 j m x j 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq100_HTML.gif with α j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq94_HTML.gif for j = 1 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq101_HTML.gif, j = 1 m α j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq102_HTML.gif.

We refer to this function simply as λ ( x 1 , , x m ) = j = 1 m x j α j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq103_HTML.gif, when it is assumed that λ Λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq104_HTML.gif.

  • λ ( x 1 , , x m ) = max j J x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq105_HTML.gif, where J is a nonempty subset of { 1 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq106_HTML.gif.

  • λ ( x 1 , , x m ) = min j J x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq107_HTML.gif, where J is a nonempty subset of { 1 , , m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq106_HTML.gif.

2.3 A technical result

Assume a < b + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq108_HTML.gif, in the rest of this section. Recall that a continuous non-increasing function F : [ a , b ] [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq109_HTML.gif has a unique fixed point μ [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq110_HTML.gif, that is, { μ } = Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq111_HTML.gif.

Lemma 1 Let F : [ a , b ] [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq112_HTML.gif be a continuous non-increasing function, { μ } = Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq111_HTML.gif and ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq113_HTML.gif. Define F ( x ) = F ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq114_HTML.gif for x < a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq115_HTML.gif, F ( x ) = F ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq116_HTML.gif for x > b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq117_HTML.gif and
a 0 = a ; b 0 = F ( a 0 ) ; a k = F ( b k 1 + ϵ k ) ; b k = F ( a k ϵ k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equk_HTML.gif

for k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq118_HTML.gif.

Then ( a k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq119_HTML.gif and ( b k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq120_HTML.gif are, respectively, a nondecreasing and a nonincreasing sequence in [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq121_HTML.gif. Furthermore, a k μ b k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq122_HTML.gif for all k and { lim a k , μ , lim b k } Fix ( F F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq123_HTML.gif.

Proof Since the map F is nonincreasing and taking into account the hypothesis a 0 a 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq124_HTML.gif, we see that ( a k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq119_HTML.gif is a nondecreasing sequence. Assume a k 1 a k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq125_HTML.gif and a k > a k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq126_HTML.gif to reach a contradiction
a k > a k + 1 F ( b k 1 + ϵ k ) > F ( b k + ϵ k + 1 ) b k 1 + ϵ k < b k + ϵ k + 1 b k 1 < b k F ( a k 1 ϵ k 1 ) < F ( a k ϵ k ) a k 1 ϵ k 1 > a k ϵ k a k 1 > a k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equl_HTML.gif

Therefore, ( a k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq119_HTML.gif is a nondecreasing sequence, so by definition, ( b k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq120_HTML.gif is nonincreasing.

On the other hand, as b 0 = F ( a 0 ) F ( μ ) = μ a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq127_HTML.gif, we see by induction that a k μ b k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq122_HTML.gif for all k,
a k = F ( b k 1 + ϵ k ) F ( μ ) = μ = F ( μ ) F ( a k ϵ k ) = b k for  k 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equm_HTML.gif
Because of the continuity of F, we conclude that
lim a k = F ( lim b k ) = F ( F ( lim a k ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equn_HTML.gif
and
lim b k = F ( lim a k ) = F ( F ( lim b k ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equo_HTML.gif

 □

Remark 2 Suppose F not to be identically equal to +∞ and let x [ a , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq128_HTML.gif. The map
ϵ F ( F ( x ) + ϵ ) ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equp_HTML.gif
is decreasing in the set
{ ϵ 0 : F ( F ( x ) + ϵ ) < + } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equq_HTML.gif

Unless F ( x ) = b < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq129_HTML.gif, the map F verifies F ( F ( x ) ) > x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq130_HTML.gif if and only if there exists ϵ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq131_HTML.gif such that F ( F ( x ) + ϵ ) ϵ > x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq132_HTML.gif for all ϵ [ 0 , ϵ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq133_HTML.gif.

Therefore, if F ( F ( a ) ) > a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq134_HTML.gif, there exists ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq113_HTML.gif such that F ( F ( a ) + ϵ ) ϵ a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq135_HTML.gif and taking a = a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq136_HTML.gif
a F ( F ( a ) + ϵ ) ϵ = a 1 ϵ a k ϵ k μ b k 1 + ϵ k b 0 + ϵ = F ( a ) + ϵ b . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equr_HTML.gif

As a consequence, ( a k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq119_HTML.gif, ( b k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq120_HTML.gif are well defined, without the need of extending F.

3 Unconditional convergence to a point

For a map h : R ¯ k 2 R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq137_HTML.gif, the difference equation
y n = h ( y n 1 , , y n k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ4_HTML.gif
(4)

is always well defined whatever the initial points y 1 , , y k R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq138_HTML.gif are, even though the y n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq2_HTML.gif are subsets of R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq61_HTML.gif, rather than points.

A point μ R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq139_HTML.gif is said to be an equilibrium for the map h if h ( μ , , μ ) = { μ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq140_HTML.gif. The equilibrium μ is said to be stable if, for each neighborhood V of μ in R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq141_HTML.gif, there is a neighborhood W of ( μ , , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq142_HTML.gif in D ( h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq68_HTML.gif such that y n V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq143_HTML.gif for all n, whenever ( y 1 , , y k ) W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq144_HTML.gif.

The equilibrium μ is said to be an attractor in a neighborhood V of μ in R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq141_HTML.gif, if y n R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq145_HTML.gif for all n and y n μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq146_HTML.gif in R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq147_HTML.gif, whenever y n V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq148_HTML.gif for n k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq149_HTML.gif.

Definition 1 The point μ is said to be an unconditional equilibrium of h (respectively, unconditional stable equilibrium, unconditional attractor in V) if it is an equilibrium (respectively, stable equilibrium, attractor in V) of h λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq150_HTML.gif for all λ Λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq151_HTML.gif.

Definition 2 We define the equilibria, stable equilibria, attractors, unconditional equilibria, unconditional stable equilibria and unconditional attractors of a continuous function φ : U R ¯ k R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq152_HTML.gif to be those of φ ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq83_HTML.gif.

3.1 Sufficient condition for unconditional convergence

After giving Definitions 1 and 2 we are going to prove a result guaranteeing that a general second order difference equation as in (4) has an unconditional stable attractor.

Let < c a < b d + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq153_HTML.gif and consider in the sequel a continuous function φ : ( a , b ) × ( c , d ) ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq154_HTML.gif, satisfying the following conditions:

(H1) φ ( x 1 , y ) < φ ( x 2 , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq155_HTML.gif, whenever a < x 2 < x 1 < b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq156_HTML.gif and c < y < d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq157_HTML.gif.

(H2) There exists F φ : [ a , b ] [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq158_HTML.gif such that
F φ ( x ) y φ ( x , y ) y 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equs_HTML.gif

whenever y ( c , d ) { F φ ( x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq159_HTML.gif.

The functions φ ( , y ) : [ a , b ] [ c , d ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq160_HTML.gif and φ ( x , ) : ( c , d ) ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq161_HTML.gif are defined in the obvious way. Notice that φ ( a , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq162_HTML.gif is the limit of a monotone increasing sequence of continuous functions, thus it is lower-semicontinuous, likewise φ ( b , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq163_HTML.gif is an upper-semicontinuous function. Remember that we denote both φ and φ ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq83_HTML.gif by φ.

The next lemma, which we prove at the end of this section, shows that if (H1) and (H2) holds we can get some information about the behavior and properties of φ and F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif.

Lemma 2 Let φ : ( a , b ) × ( c , d ) ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq164_HTML.gif, where < c a < b d + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq153_HTML.gif, be a continuous function satisfying (H1) and (H2). Then the function F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif in (H2) is unique and it is a continuous nonincreasing map, thus it has a unique fixed point μ φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif. Furthermore,
  1. (i)

    a < φ ( x , y ) < b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq166_HTML.gif for all x , y ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq167_HTML.gif and a φ ( x , y ) b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq168_HTML.gif for all x , y [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq169_HTML.gif.

     
  2. (ii)

    If x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq170_HTML.gif, y ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq171_HTML.gif and φ ( x , y ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq172_HTML.gif, then y = F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq173_HTML.gif.

     
  3. (iii)

    φ ( x , F φ ( x ) ) = F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq8_HTML.gif for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq170_HTML.gif.

     
  4. (iv)

    F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif is decreasing in F φ 1 ( ( a , b ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq174_HTML.gif.

     

We are in conditions of presenting and proving our main result.

Theorem 1 Let φ : ( a , b ) × ( c , d ) ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq164_HTML.gif, where < c a < b d + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq153_HTML.gif, be a continuous function satisfying (H1) and (H2). If μ φ ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq175_HTML.gif and Fix ( F φ F φ ) = Fix ( F φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq176_HTML.gif, then μ φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq177_HTML.gif is an unconditional stable attractor of φ in ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq178_HTML.gif.

Proof of Theorem 1 Consider λ Λ m 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq179_HTML.gif and denote
y n = φ λ ( y n 1 , , y n m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equt_HTML.gif

for some y 1 , , y m ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq180_HTML.gif. Notice that y n ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq181_HTML.gif for all n, as a consequence of (i) in Lemma 2.

We are going to prove first that μ φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif is a stable equilibrium of φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq182_HTML.gif. By (iii) in Lemma 2, as
λ ( μ φ , , μ φ ) = ( μ φ , μ φ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equu_HTML.gif

we see that μ φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif is an equilibrium.

Let ϵ ( 0 , min { μ φ a , b μ φ } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq183_HTML.gif. Because of the continuity of F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif, there is a ( μ φ ϵ , μ φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq184_HTML.gif such that
b F φ ( a ) ( μ φ , μ φ + ϵ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equv_HTML.gif
As Fix ( F φ F φ ) = Fix ( F φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq185_HTML.gif and F φ ( F φ ( a ) ) a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq186_HTML.gif, we have
F φ ( F φ ( x ) ) > x for all  x [ a , μ φ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equw_HTML.gif
If x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq187_HTML.gif, then
F φ ( x ) F φ ( a ) = b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equx_HTML.gif
and
F φ ( x ) F φ ( b ) = F φ ( F φ ( a ) ) > a . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equy_HTML.gif

Therefore, F φ ( [ a , b ] ) [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq188_HTML.gif.

By replacing a, b by a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq189_HTML.gif, b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq190_HTML.gif in Lemma 2(i), we see that
y n ( a , b ) ( μ φ ϵ , μ φ + ϵ ) for all  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equz_HTML.gif

whenever y n ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq191_HTML.gif for n m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq192_HTML.gif, thus μ φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif is an unconditional stable equilibrium of φ.

Now, if we see that
lim F φ ( y n ) = μ φ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaa_HTML.gif
we are done with the whole proof. Indeed, for each accumulation point y ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq193_HTML.gif of ( y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq194_HTML.gif, one would have
F φ ( μ φ ) = μ φ = F φ ( y ¯ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equab_HTML.gif

because of the continuity of F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif. As μ φ ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq175_HTML.gif, this implies y ¯ = μ φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq195_HTML.gif.

Therefore, as a consequence of Lemma 1, it suffices to find an increasing sequence n k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq196_HTML.gif of natural numbers such that
a k F φ ( y n ) b k for all  k 0 , n n k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equac_HTML.gif
Here, a k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq197_HTML.gif and b k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq198_HTML.gif are defined as in Lemma 1, with a 0 = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq199_HTML.gif and ϵ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq200_HTML.gif,
b 0 = F φ ( a ) ; a k = F φ ( b k 1 + 1 k ) ; b k = F φ ( a k 1 k ) for  k 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equad_HTML.gif
Let n 0 = m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq201_HTML.gif, so that
a 0 F φ ( y n ) F φ ( a ) = b 0 for  n n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equae_HTML.gif
Having in mind that λ Λ m 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq202_HTML.gif satisfies (3), we find n k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq196_HTML.gif from n k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq203_HTML.gif as follows. Denote
z k = F φ 1 ( b k 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaf_HTML.gif
and momentarily assume n > n k 1 + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq204_HTML.gif and b k 1 < y n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq205_HTML.gif in such a way that
b k 1 < y n = φ ( λ 1 ( y n 1 , , y n m ) , λ 2 ( y n 1 , , y n m ) ) φ ( min { y n 1 , , y n m } , λ 2 ( y n 1 , , y n m ) ) φ ( z k , λ 2 ( y n 1 , , y n m ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equag_HTML.gif
which implies
φ ( z k , λ 2 ( y n 1 , , y n m ) ) λ 2 ( y n 1 , , y n m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equah_HTML.gif
and then
y n max { b k 1 , λ 2 ( y n 1 , , y n m ) } max { b k 1 , y n 1 , , y n m } w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equai_HTML.gif

for all n > n k 1 + m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq204_HTML.gif.

As a consequence, the nonincreasing sequence w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq206_HTML.gif is bounded below by b k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq207_HTML.gif. It cannot be the case that lim w n > b k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq208_HTML.gif, because in such a case there is a subsequence y n j > b k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq209_HTML.gif converging to lim w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq210_HTML.gif and such that λ 2 ( y n j 1 , , y n j m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq211_HTML.gif converges to a point w lim w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq212_HTML.gif.

Since
y n j φ ( z k , λ 2 ( y n j 1 , , y n j m ) ) w n j , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaj_HTML.gif
one has
φ ( z k , w ) = lim w n > b k 1 = F ( z k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equak_HTML.gif
and then
φ ( z k , w ) w > F ( z k ) w . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equal_HTML.gif
By applying (H2), we see that
lim w n = φ ( z k , w ) < w , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equam_HTML.gif

a contradiction.

Therefore, lim w n = b k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq213_HTML.gif and there exists m k n k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq214_HTML.gif such that
y n < b k 1 + 1 k for all  n m k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equan_HTML.gif
that is,
F φ ( y n ) a k for all  n m k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equao_HTML.gif
Analogously, we see that there exists n k m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq215_HTML.gif such that
F φ ( y n ) b k for all  n n k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equap_HTML.gif

 □

Proof of Lemma 2

  • Uniqueness of F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif: Let y 1 < y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq216_HTML.gif and x in [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq217_HTML.gif such that
    y i y φ ( x , y ) y 1 > 0 for  i = 1 , 2 , y ( y 1 , y 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaq_HTML.gif
Then
0 < φ ( x , y ) y < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equar_HTML.gif

a contradiction.

  • (i): It suffices to prove the first assertion, because [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq121_HTML.gif is a closed set and, by definition,
    φ ( x , y ) = { lim sup φ ( x n , y n ) : x n x , y n y , x n ( a , b ) , y n ( c , d ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equas_HTML.gif

for all ( x , y ) [ a , b ] × [ c , d ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq218_HTML.gif. Assume now that ( x , y ) ( a , b ) × ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq219_HTML.gif. We consider the following three possible situations. If φ ( x , y ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq172_HTML.gif, it is obvious that φ ( x , y ) ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq220_HTML.gif.On the other hand, if φ ( x , y ) > y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq221_HTML.gif and x ( a , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq222_HTML.gif, then

a y < φ ( x , y ) < φ ( x , y ) F φ ( x ) b . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equat_HTML.gif
Finally, if φ ( x , y ) < y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq223_HTML.gif and x ( x , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq224_HTML.gif, then
b y > φ ( x , y ) > φ ( x , y ) F φ ( x ) a . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equau_HTML.gif
  • (ii): Suppose y F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq225_HTML.gif. Since φ ( x , y ) = y R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq226_HTML.gif, then φ ( x , y ) y = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq227_HTML.gif or φ ( x , y ) y = R ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq228_HTML.gif. In any event, it cannot be the case that
    F φ ( x ) y φ ( x , y ) y 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equav_HTML.gif

which contradicts hypothesis (H2), thus y = F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq229_HTML.gif.

  • (iii): Since F φ ( [ a , b ] ) [ a , b ] [ c , d ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq230_HTML.gif, it is worth considering the following three cases for each x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq231_HTML.gif: first, x ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq232_HTML.gif, F φ ( x ) ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq233_HTML.gif and then (after probing continuity, monotonicity and statement (iv)), we proceed with the case x { a , b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq234_HTML.gif, F φ ( x ) ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq233_HTML.gif and finally with x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq231_HTML.gif, F φ ( x ) { c , d } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq235_HTML.gif.Case x ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq232_HTML.gif and F φ ( x ) ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq236_HTML.gif: Since φ ( x , y ) > y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq221_HTML.gif when y < F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq237_HTML.gif and φ ( x , y ) < y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq223_HTML.gif when y > F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq238_HTML.gif, we see that
    φ ( x , F φ ( x ) ) = F φ ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaw_HTML.gif

because of the continuity of φ ( x , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq239_HTML.gif.

  • Monotonicity and (iv): Suppose

F φ ( x 1 ) F φ ( x 2 ) for  a x 1 < x 2 b . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equax_HTML.gif
If y [ F φ ( x 1 ) , F φ ( x 2 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq240_HTML.gif, then y = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq241_HTML.gif, y = b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq242_HTML.gif or y φ ( x 2 , y ) < φ ( x 1 , y ) y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq243_HTML.gif, thus
[ F φ ( x 1 ) , F φ ( x 2 ) ] { a , b } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equay_HTML.gif
  • Continuity: If x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq244_HTML.gif and
    w I = ( lim inf z x F φ ( z ) , lim sup y x F φ ( y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaz_HTML.gif
then there exist two sequences y n , z n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq245_HTML.gif with
F φ ( z n ) < w < F φ ( y n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equba_HTML.gif
Thus,
φ ( z n , w ) < w < φ ( y n , w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbb_HTML.gif

and, by (ii), one has φ ( x , w ) = w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq246_HTML.gif. Since w ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq247_HTML.gif, this would imply F φ ( x ) = w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq248_HTML.gif for all w I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq249_HTML.gif, which is impossible.

  • (iii) Case x { a , b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq250_HTML.gif and F φ ( x ) ( c , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq236_HTML.gif: Since
    F φ ( t ) φ ( t , F φ ( a ) ) F φ ( a ) for all  t ( a , b ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbc_HTML.gif
and because of the continuity of F φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif, we have
F φ ( a ) = sup t F φ ( t ) sup t φ ( t , F φ ( a ) ) F φ ( a ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbd_HTML.gif

but

sup t φ ( t , F φ ( a ) ) = φ ( a , F φ ( a ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Eqube_HTML.gif
Analogously, it can be seen that φ ( b , F φ ( b ) ) = F φ ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq251_HTML.gif.
  • (iii) Case F φ ( x ) { c , d } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq252_HTML.gif: First assume F φ ( x ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq253_HTML.gif and recall that, by definition,
    φ ( x , c ) = { lim sup φ ( x n , y n ) : x n x , y n c , x n ( a , b ) , y n ( c , d ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbf_HTML.gif
Suppose φ ( x n , y n ) c > c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq254_HTML.gif for all n. Then
1 F φ ( x n ) y n φ ( x n , y n ) y n F φ ( x n ) y n c y n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbg_HTML.gif

which implies F φ ( x n ) c > c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq255_HTML.gif, eventually for all n.Since F φ ( x n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq256_HTML.gif, we reach a contradiction. Therefore,

φ ( x , F φ ( x ) ) = { c } = { F φ ( x ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbh_HTML.gif
Analogously, we see that φ ( x , F φ ( x ) ) = { d } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq257_HTML.gif when F φ ( x ) = d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq258_HTML.gif. □

4 Examples and applications

4.1 The difference equation y n = A + ( y n k y n q ) p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq259_HTML.gif with 0 < p < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq260_HTML.gif

The paper [6] is devoted to prove that every positive solution to the difference equation
y n = A + ( y n k y n q ) p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbi_HTML.gif
converges to the equilibrium A + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq261_HTML.gif, whenever
A ( 0 , + ) and p ( 0 , min { 1 , ( A + 1 ) / 2 } ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbj_HTML.gif

Here, k , q { 1 , 2 , 3 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq262_HTML.gif are fixed numbers.

Although paper [6] complements [7], where the case p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq263_HTML.gif had been considered, it should be noticed that the case p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq263_HTML.gif, A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq264_HTML.gif is not dealt with in [6]. Furthermore, we cannot assure the global attractivity in this case.

The results in [6] can be easily obtained by applying Theorem 1 above. Furthermore, we slightly improve the results in [6] by establishing the unconditional stability of the equilibrium A + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq261_HTML.gif, whenever A ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq265_HTML.gif, p ( 0 , min { 1 , ( A + 1 ) / 2 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq266_HTML.gif. We may assume without loss of generality that the initial values y 1 , , y m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq267_HTML.gif are greater than A. Here, and in the sequel m = max { k , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq268_HTML.gif.

Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbk_HTML.gif
and
λ ( x 1 , , x m ) = ( x q , x k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbl_HTML.gif
Define F φ ( + ) = A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq269_HTML.gif, consider for the moment a fixed x [ A , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq270_HTML.gif and define F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq9_HTML.gif to be the unique positive zero of the function f x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq271_HTML.gif given by
f x ( y ) = φ ( x , y ) y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbm_HTML.gif

Notice that f x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq271_HTML.gif is concave, f x ( 0 ) = A > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq272_HTML.gif, and f x ( + ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq273_HTML.gif.

Clearly, F φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq9_HTML.gif is also the unique zero of the increasing function
j x ( y ) = φ ( x , y ) F φ ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbn_HTML.gif
Since
j x ( A ) = A p ( F φ ( x ) ) p x p 0 and j x ( + ) = + > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbo_HTML.gif

we see that condition (H2) holds and μ φ = A + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq274_HTML.gif.

As for condition
Fix ( F φ F φ ) = Fix ( F φ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ5_HTML.gif
(5)
if A x < y < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq275_HTML.gif with
A + ( y x ) p = y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbp_HTML.gif
then y > A + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq276_HTML.gif and
φ ( y , x ) x = A + ( x y ) p x = A + 1 y A y ( y A ) 1 / p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbq_HTML.gif
Since the function
h ( z ) = A + 1 z A z ( z A ) 1 / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbr_HTML.gif

has a unique critical point in ( A , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq277_HTML.gif and h ( A + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq278_HTML.gif, h ( + ) = A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq279_HTML.gif, the necessary and sufficient condition for (5) to hold is that h ( A + 1 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq280_HTML.gif, that is, p ( A + 1 ) / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq281_HTML.gif.

By this reasoning, we also get for free, unconditional stable convergence for several difference equations as, for instance:
y n = A + ( y n q y n r y n s y n t ) p with  0 < p < min { 1 / 2 , ( A + 1 ) / 4 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbs_HTML.gif
or
y n = A + ( y n q + y n r y n s + y n t ) p with  0 < p < min { 1 , ( A + 1 ) / 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbt_HTML.gif
just considering respectively
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbu_HTML.gif

where m = max { q , r , s , t } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq282_HTML.gif.

4.2 The difference equation y n = α + β y n 1 A + B y n 1 + C y n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq283_HTML.gif

Here, α , β , A , B , C , y 0 , y 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq284_HTML.gif. In 2003, three conjectures on the above equation were posed in [8]. In all three cases ( B = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq285_HTML.gif, α , β , A , C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq286_HTML.gif; A = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq287_HTML.gif, α , β , B , C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq288_HTML.gif; and α , β , A , B , C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq289_HTML.gif, respectively) it was postulated the global asymptotic stability of the equilibrium. These conjectures have resulted in several papers since then (see [912]). Let us see when there is unconditional convergence.

Consider a = c = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq290_HTML.gif, b = c = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq291_HTML.gif, and
φ ( x , y ) = α + β y A + D x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbv_HTML.gif
with α , β , A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq292_HTML.gif, D > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq293_HTML.gif. We solve in y the equation y = α + β y A + D x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq294_HTML.gif to obtain
y = α A β + D x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbw_HTML.gif
so we consider A > β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq295_HTML.gif, α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif to define
F φ ( x ) = α A β + D x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbx_HTML.gif
A simple calculation shows that (H2) holds, μ φ ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq175_HTML.gif and Fix ( F φ F φ ) = Fix ( F φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq185_HTML.gif:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equby_HTML.gif

Therefore, μ φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif is an unconditional stable attractor of φ in ( a , b ) = ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq297_HTML.gif whenever A > β 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq298_HTML.gif, D > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq293_HTML.gif and α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif.

If we choose
λ ( x 1 , x 2 ) = ( B x 1 + C x 2 B + C , x 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbz_HTML.gif
and D = B + C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq299_HTML.gif, we obtain unconditional stable convergence for the equation
y n = α + β y n 1 A + B y n 1 + C y n 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equca_HTML.gif

whenever A > β 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq298_HTML.gif, B + C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq300_HTML.gif, C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq301_HTML.gif and α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif.

Other choices of λ result on the unconditional stable convergence of difference equations such as
y n = α + β y n 1 + γ y n 2 A + B y n 1 + C y n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equcb_HTML.gif
with A > β + γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq302_HTML.gif, B + C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq300_HTML.gif, γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq303_HTML.gif, α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif, β 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq304_HTML.gif, C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq301_HTML.gif. Or
y n = α + β max { y n 1 , y n 2 } A + D min { y n 1 , y n 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equcc_HTML.gif

with A > β 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq298_HTML.gif, D > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq293_HTML.gif, α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif.

Declarations

Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.

We are grateful to the anonymous referees for their helpful comments and suggestions. This research was supported in part by the Spanish Ministry of Science and Innovation and FEDER, grant MTM2010-14837.

Authors’ Affiliations

(1)
Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia (UNED)

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© Franco and Peran; licensee Springer. 2013

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