Existence of positive solutions for a kind of periodic boundary value problem at resonance

  • Mirosława Zima1Email author and

    Affiliated with

    • Piotr Drygaś1

      Affiliated with

      Boundary Value Problems20132013:19

      DOI: 10.1186/1687-2770-2013-19

      Received: 20 December 2012

      Accepted: 21 January 2013

      Published: 11 February 2013

      Abstract

      In the paper we provide sufficient conditions for the existence of positive solutions for some second-order differential equation subject to periodic boundary conditions. Our method employs a Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. Two examples are given to illustrate the main result of the paper.

      Keywords

      periodic boundary value problem positive solution coincidence equation

      1 Introduction

      In the paper we are interested in the existence of positive solutions for the periodic boundary value problem (PBVP)
      { x ( t ) + h ( t ) x ( t ) + f ( t , x ( t ) , x ( t ) ) = 0 , t [ 0 , T ] , x ( 0 ) = x ( T ) , x ( 0 ) = x ( T ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ1_HTML.gif
      (1)
      where f : [ 0 , T ] × [ 0 , ) × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq1_HTML.gif and h : [ 0 , T ] ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq2_HTML.gif are continuous functions. Our study is motivated by current activity of many researchers in the area of theory and applications of PVBPs; see, for example, [14] and references therein. In particular, in a recent paper [1], Chu, Fan and Torres have studied the existence of positive periodic solutions for the singular damped differential equation
      x ( t ) + h ( t ) x ( t ) + a ( t ) x ( t ) = f ( t , x ( t ) , x ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equa_HTML.gif
      by combining the properties of the Green’s function of the PBVP
      { x ( t ) + h ( t ) x ( t ) + a ( t ) x ( t ) = 0 , t [ 0 , T ] , x ( 0 ) = x ( T ) , x ( 0 ) = x ( T ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ2_HTML.gif
      (2)
      with a nonlinear alternative of Leray-Schauder type (see, for example, [5]). It should be noted that a 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq3_HTML.gif was the key assumption used in [1]. If a 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq4_HTML.gif, then PBVP (2) has nontrivial solutions, which means that the problem we are concerned with here, that is, PBVP (1), is at resonance. There are several methods to deal with the resonant PBVPs. For example, in [6], Torres studied the existence of a positive solution for the PBVP
      { x ( t ) = f ( t , x ( t ) ) , t ( 0 , 2 π ) , x ( 0 ) = x ( 2 π ) , x ( 0 ) = x ( 2 π ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equb_HTML.gif
      by considering the equivalent problem
      { x ( t ) + a ( t ) x ( t ) = f ( t , x ( t ) ) + a ( t ) x ( t ) , t ( 0 , 2 π ) , x ( 0 ) = x ( 2 π ) , x ( 0 ) = x ( 2 π ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equc_HTML.gif
      via Krasnoselskii’s theorem on cone expansion and compression. Further results in this direction can be found in [7] and [8]. In [9] Rachůnková, Tvrdý and Vrkoč applied the method of upper and lower solutions and topological degree arguments to establish the existence of nonnegative and nonpositive solutions for the PBVP
      { x ( t ) = f ( t , x ( t ) ) , t ( 0 , 1 ) , x ( 0 ) = x ( 1 ) , x ( 0 ) = x ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ3_HTML.gif
      (3)

      The same PBVP was studied by Wang, Zhang and Wang in [10]. Their existence and multiplicity results on positive solutions are based on the theory of a fixed point index for A-proper semilinear operators on cones developed by Cremins [11].

      The goal of our paper is to provide sufficient conditions that ensure the existence of positive solutions of (1) with the function h positive on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq5_HTML.gif. Our general result is illustrated by two examples. The method we use in the paper is to rewrite BVP (1) as a coincidence equation L x = N x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq6_HTML.gif, where L is a Fredholm operator of index zero and N is a nonlinear operator, and to apply the Leggett-Williams norm-type theorem for coincidences obtained by O’Regan and Zima [12]. We would like to emphasize that the idea of results of [11] and [12], as well as these of [1315], goes back to the celebrated Mawhin’s coincidence degree theory [16]. For more details on this significant tool, its modifications and wide applications, we refer the reader to [1722] and references therein.

      In this paper, for the first time, the existence theorem from [12] is used for studying the boundary value problem with the nonlinearity f depending also on the derivative. In general, the presence of x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq7_HTML.gif in f makes the problem much harder to handle. We point out that, to the best of our knowledge, there are only a few papers on PBVPs that discuss such a nonlinearity; we refer the reader to [15, 2325] for some results of that type. We also complement several results in the literature, for example, in [1, 26] and [27]. It is evident that the existence theorems for PBVP (1) can be established by the shift method used in [6], that is, one can employ the results of [1] to the periodic problem we study here. However, the conditions imposed on f in [1] are not comparable with ours.

      2 Coincidence equation

      For the convenience of the reader, we begin this section by providing some background on cone theory and Fredholm operators in Banach spaces.

      Definition 1 A nonempty subset C, C { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq8_HTML.gif, of a real Banach space X is called a cone if C is closed, convex and
      1. (i)

        λ x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq9_HTML.gif for all x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq10_HTML.gif and λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq11_HTML.gif,

         
      2. (ii)

        x, x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq12_HTML.gif implies x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq13_HTML.gif.

         
      Every cone induces a partial ordering in X as follows: for x , y X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq14_HTML.gif, we say that
      x y if and only if y x C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equd_HTML.gif

      The following property holds for every cone in a Banach space.

      Lemma 1 [28]For every u C { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq15_HTML.gif, there exists a positive number σ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq16_HTML.gif such that
      x + u σ ( u ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Eque_HTML.gif

      for all x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq10_HTML.gif.

      Consider a linear mapping L : dom L X Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq17_HTML.gif and a nonlinear operator N : X Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq18_HTML.gif, where X and Y are Banach spaces. If L is a Fredholm operator of index zero, that is, ImL is closed and dim Ker L = codim Im L < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq19_HTML.gif, then there exist continuous projections P : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq20_HTML.gif and Q : Y Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq21_HTML.gif such that Im P = Ker L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq22_HTML.gif and Ker Q = Im L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq23_HTML.gif (see, for example, [14, 16]). Moreover, since dim Im Q = codim Im L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq24_HTML.gif, there exists an isomorphism J : Im Q Ker L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq25_HTML.gif. Denote by L P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq26_HTML.gif the restriction of L to Ker P dom L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq27_HTML.gif. Then L P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq26_HTML.gif is an isomorphism from Ker P dom L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq28_HTML.gif to ImL and its inverse
      K P : Im L Ker P dom L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equf_HTML.gif

      is defined.

      As a result, the coincidence equation L x = N x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq6_HTML.gif is equivalent to x = Ψ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq29_HTML.gif, where
      Ψ = P + J Q N + K P ( I Q ) N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equg_HTML.gif
      Let ρ : X C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq30_HTML.gif be a retraction, that is, a continuous mapping such that ρ ( x ) = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq31_HTML.gif for all x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq10_HTML.gif. Put
      Ψ ρ = Ψ ρ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equh_HTML.gif

      Let Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq32_HTML.gif, Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq33_HTML.gif be open bounded subsets of X with Ω ¯ 1 Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq34_HTML.gif and C ( Ω ¯ 2 Ω 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq35_HTML.gif. Assume that

      1 L is a Fredholm operator of index zero,

      2 Q N : X Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq36_HTML.gif is continuous and bounded and K P ( I Q ) N : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq37_HTML.gif is compact on every bounded subset of X,

      3 L x λ N x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq38_HTML.gif for all x C Ω 2 dom L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq39_HTML.gif and λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq40_HTML.gif,

      4 ρ maps subsets of Ω ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq41_HTML.gif into bounded subsets of C,

      5 d B ( [ I ( P + J Q N ) ρ ] | Ker L , Ker L Ω 2 , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq42_HTML.gif, where d B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq43_HTML.gif stands for the Brouwer degree,

      6 there exists u 0 C { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq44_HTML.gif such that x σ ( u 0 ) Ψ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq45_HTML.gif for x C ( u 0 ) Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq46_HTML.gif, where
      C ( u 0 ) = { x C : μ u 0 x  for some  μ > 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equi_HTML.gif

      and σ ( u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq47_HTML.gif is such that x + u 0 σ ( u 0 ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq48_HTML.gif for every x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq10_HTML.gif,

      7 ( P + J Q N ) ρ ( Ω 2 ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq49_HTML.gif and Ψ ρ ( Ω ¯ 2 Ω 1 ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq50_HTML.gif.

      Theorem 1 [12]

      Under the assumptions 1-7 the equation L x = N x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq6_HTML.gif has a solution in the set C ( Ω ¯ 2 Ω 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq51_HTML.gif.

      In the next section, we use Theorem 1 to prove the existence of a positive solution for PBVP (1). For applications of Theorem 1 to nonlocal boundary value problems at resonance, we refer the reader to [22], [29] and [30].

      3 Periodic boundary value problem

      We now provide sufficient conditions for the existence of positive solutions for PBVP (1). For convenience and ease of exposition, we make use of the following notation:
      e ( t ) = exp ( 0 t h ( τ ) d τ ) , φ ( t ) = 0 t e ( τ ) d τ , Φ ( t ) = 0 t φ ( τ ) d τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ4_HTML.gif
      (4)
      and
      ψ ( t ) = 1 e ( t ) ( 1 1 e ( T ) φ ( t ) φ ( T ) ) , t [ 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ5_HTML.gif
      (5)
      We observe that 0 < ψ ( t ) < 1 e ( T ) ( 1 e ( T ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq52_HTML.gif on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq5_HTML.gif. Moreover, we put
      k ( t , s ) = 1 T e ( s ) { φ ( s ) φ ( T ) [ φ ( T ) s T φ ( t ) + Φ ( T ) ] Φ ( s ) , 0 s t T , φ ( s ) φ ( T ) [ φ ( T ) ( s T ) T φ ( t ) + Φ ( T ) ] + T φ ( t ) Φ ( s ) , 0 t s T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ6_HTML.gif
      (6)
      and
      K ( t , s ) = k ( t , s ) + M 0 T k ( t , τ ) d τ 0 T ψ ( τ ) d τ ψ ( s ) , t , s [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ7_HTML.gif
      (7)

      where M is a positive constant.

      We assume that

      (H1) f : [ 0 , T ] × [ 0 , ) × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq53_HTML.gif and h : [ 0 , T ] ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq2_HTML.gif are continuous functions.

      We also assume that there exist R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq54_HTML.gif, 0 < α β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq55_HTML.gif, 0 < M e ( T ) ( 1 e ( T ) ) 0 T ψ ( τ ) d τ α T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq56_HTML.gif, r ( 0 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq57_HTML.gif, m ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq58_HTML.gif, η [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq59_HTML.gif and a continuous function g : [ 0 , T ] [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq60_HTML.gif such that

      (H2) f ( t , x , y ) > α x + β | y | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq61_HTML.gif for ( t , x , y ) [ 0 , T ] × [ 0 , R ] × [ R , R ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq62_HTML.gif,

      (H3) f ( t , R , 0 ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq63_HTML.gif for t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq64_HTML.gif,

      (H4) f ( 0 , x , R ) = f ( T , x , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq65_HTML.gif and f ( 0 , x , R ) = f ( T , x , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq66_HTML.gif for x [ 0 , R ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq67_HTML.gif,

      (H5) f ( t , x , R ) h ( t ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq68_HTML.gif for t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq64_HTML.gif and x [ 0 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq69_HTML.gif,

      (H6) f ( t , x , y ) g ( t ) ( x + | y | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq70_HTML.gif for ( t , x , y ) [ 0 , T ] × ( 0 , r ] × [ r , r ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq71_HTML.gif,

      (H7) 1 α T K ( t , s ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq72_HTML.gif for t , s [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq73_HTML.gif and m 0 T K ( η , s ) g ( s ) d s 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq74_HTML.gif.

      Theorem 2 Under the assumptions (H1)-(H7), PBVP (1) has a positive solution on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq5_HTML.gif.

      Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq75_HTML.gif denote the supremum norm in the space C [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq76_HTML.gif, that is, x = sup t [ 0 , T ] | x ( t ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq77_HTML.gif. Consider the Banach spaces X = C 1 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq78_HTML.gif with the norm x = max { x , x } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq79_HTML.gif, and Y = C [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq80_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq75_HTML.gif.

      We write problem (1) as a coincidence equation
      L x = N x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equj_HTML.gif
      where
      L x ( t ) = x ( t ) h ( t ) x ( t ) , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equk_HTML.gif
      and
      N x ( t ) = f ( t , x ( t ) , x ( t ) ) , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equl_HTML.gif
      with dom L = { x X : x C [ 0 , T ] , x ( 0 ) = x ( T ) , x ( 0 ) = x ( T ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq81_HTML.gif. Then
      Ker L = { x X : x ( t ) = c , t [ 0 , T ] , c R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equm_HTML.gif
      and
      Im L = { y Y : 0 T ψ ( s ) y ( s ) d s = 0 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equn_HTML.gif

      where ψ is given by (5).

      Clearly, ImL is closed and Y = Y 1 + Im L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq82_HTML.gif with
      Y 1 = { y 1 Y : y 1 = 1 0 T ψ ( s ) d s 0 T ψ ( s ) y ( s ) d s , y Y } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equo_HTML.gif

      Since Y 1 Im L = { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq83_HTML.gif, we have Y = Y 1 Im L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq84_HTML.gif. Moreover, dim Y 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq85_HTML.gif, which gives codim Im L = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq86_HTML.gif. Consequently, L is Fredholm of index zero, and the assumption 1 is satisfied.

      Define the projections P : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq20_HTML.gif by
      P x ( t ) = 1 T 0 T x ( s ) d s , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equp_HTML.gif
      and Q : Y Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq21_HTML.gif by
      Q y ( t ) = 1 0 T ψ ( s ) d s 0 T ψ ( s ) y ( s ) d s , t [ 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equq_HTML.gif
      It is a routine matter to show that for y Im L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq87_HTML.gif, the inverse K P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq88_HTML.gif of L P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq26_HTML.gif is given by
      ( K P y ) ( t ) = 0 T k ( t , s ) y ( s ) d s , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equr_HTML.gif
      with the kernel k defined by (6). Clearly, the assumption 2 is satisfied. For y Im Q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq89_HTML.gif, define
      J ( y ) = M y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equs_HTML.gif
      Then J is an isomorphism from ImQ to KerL. Next, consider a cone
      C = { x X : x ( t ) 0  on  [ 0 , T ] } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equt_HTML.gif
      For u 0 ( t ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq90_HTML.gif, we have σ ( u 0 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq91_HTML.gif and
      C ( u 0 ) = { x C : x ( t ) > 0  on  [ 0 , T ] } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equu_HTML.gif
      Let
      Ω 1 = { x X : x < r , | x ( t ) | > m x  and  | x ( t ) | > m x  on  [ 0 , T ] } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equv_HTML.gif
      and
      Ω 2 = { x X : x < R } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equw_HTML.gif

      Obviously, Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq32_HTML.gif and Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq33_HTML.gif are open bounded subsets of X, and Ω ¯ 1 Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq34_HTML.gif.

      To verify 3, suppose that there exist x 0 C Ω 2 dom L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq92_HTML.gif and λ 0 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq93_HTML.gif such that L x 0 = λ 0 N x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq94_HTML.gif. Then x ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq95_HTML.gif on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq5_HTML.gif, x 0 = R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq96_HTML.gif,
      x 0 ( t ) h ( t ) x 0 ( t ) = λ 0 f ( t , x 0 ( t ) , x 0 ( t ) ) , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ8_HTML.gif
      (8)
      and
      x 0 ( 0 ) = x 0 ( T ) , x 0 ( 0 ) = x 0 ( T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ9_HTML.gif
      (9)
      There are two cases to consider.
      1. 1.

        If x 0 = x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq97_HTML.gif, then there exists t 0 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq98_HTML.gif such that x ( t 0 ) = R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq99_HTML.gif. For t 0 ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq100_HTML.gif, we get 0 x ( t 0 ) = λ 0 f ( t 0 , R , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq101_HTML.gif, contrary to the assumption (H3). Similarly, if t 0 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq102_HTML.gif or t 0 = T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq103_HTML.gif, BCs (9) imply x ( 0 ) = x ( T ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq104_HTML.gif. Hence, 0 x ( t 0 ) = λ 0 f ( t 0 , R , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq105_HTML.gif which contradicts (H3) again.

         
      2. 2.
        If x 0 = x 0 > x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq106_HTML.gif, then there exists t 0 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq98_HTML.gif such that | x ( t 0 ) | = R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq107_HTML.gif. Observe that (H2) implies f ( t , x , ± R ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq108_HTML.gif for t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq64_HTML.gif and x [ 0 , R ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq67_HTML.gif. Suppose that t 0 ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq100_HTML.gif. If x ( t 0 ) = R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq109_HTML.gif, we get from (8)
        h ( t 0 ) R = λ 0 f ( t 0 , x 0 ( t 0 ) , R ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ10_HTML.gif
        (10)
         
      a contradiction. For x ( t 0 ) = R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq110_HTML.gif, we have
      h ( t 0 ) R = λ 0 f ( t 0 , x 0 ( t 0 ) , R ) < f ( t 0 , x 0 ( t 0 ) , R ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ11_HTML.gif
      (11)

      contrary to (H5). By similar arguments, if t 0 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq102_HTML.gif or t 0 = T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq103_HTML.gif, BCs (9) and (H4) imply either (10) or (11). Thus, 3 is fulfilled.

      Next, for x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq111_HTML.gif, define (see [15])
      ρ x ( t ) = { x ( t ) if  x ( t ) 0  on  [ 0 , T ] , 1 2 ( x ( t ) min { x ( t ) : t [ 0 , T ] } ) if  x ( t ˜ ) < 0  for some  t ˜ [ 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equx_HTML.gif

      Clearly, ρ is a retraction and maps subsets of Ω ¯ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq41_HTML.gif into bounded subsets of C, so 4 holds.

      To verify 5, it is enough to consider, for x Ker L Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq112_HTML.gif and λ [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq113_HTML.gif, the mapping
      H ( x , λ ) ( t ) = x ( t ) λ ( 1 T 0 T ( ρ x ) ( s ) d s + M 0 T ψ ( s ) d s 0 T ψ ( s ) f ( s , ( ρ x ) ( s ) , ( ρ x ) ( s ) ) d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equy_HTML.gif
      Observe that if x Ker L Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq114_HTML.gif, then x ( t ) = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq115_HTML.gif on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq5_HTML.gif and x < R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq116_HTML.gif. Suppose H ( x , λ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq117_HTML.gif for x Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq118_HTML.gif. Then c = ± R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq119_HTML.gif. For c = R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq120_HTML.gif, we have ( ρ x ) ( t ) = x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq121_HTML.gif and in view of (H3), we get
      0 R ( 1 λ ) = λ M 0 T ψ ( s ) d s 0 T ψ ( s ) f ( s , R , 0 ) d s < 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equz_HTML.gif
      which is a contradiction. If c = R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq122_HTML.gif, then ( ρ x ) ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq123_HTML.gif, hence
      R = λ M 0 T ψ ( s ) d s 0 T ψ ( s ) f ( s , 0 , 0 ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equaa_HTML.gif
      which contradicts (H2). Thus, H ( x , λ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq124_HTML.gif for x Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq125_HTML.gif and λ [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq126_HTML.gif. This implies
      d B ( H ( x , 0 ) , Ker L Ω 2 , 0 ) = d B ( H ( x , 1 ) , Ker L Ω 2 , 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equab_HTML.gif
      and
      d B ( [ I ( P + J Q N ) ρ ] | Ker L , Ker L Ω 2 , 0 ) = d B ( H ( c , 1 ) , Ker L Ω 2 , 0 ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equac_HTML.gif
      We next show that 6 holds. Let x C ( u 0 ) Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq127_HTML.gif. Then for t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq64_HTML.gif, we have r x ( t ) m x > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq128_HTML.gif, r | x ( t ) | x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq129_HTML.gif, and by (H6) and (H7), we obtain
      Ψ x ( η ) = 1 T 0 T x ( s ) d s + 0 T K ( η , s ) f ( s , x ( s ) , x ( s ) ) d s 0 T K ( η , s ) g ( s ) [ x ( s ) + | x ( s ) | ] d s m 0 T K ( η , s ) g ( s ) [ x + x ] d s m x 0 T K ( η , s ) g ( s ) d s x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equad_HTML.gif

      This implies x Ψ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq130_HTML.gif for x C ( u 0 ) Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq46_HTML.gif, so 6 is satisfied.

      Finally, we must check if 7 holds. If x Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq131_HTML.gif, then in view of (H2), we get
      ( P + J Q N ) ( ρ x ) ( t ) = 1 T 0 T ( ρ x ) ( s ) d s + M 0 T ψ ( s ) d s 0 T ψ ( s ) f ( s , ( ρ x ) ( s ) , ( ρ x ) ( s ) ) d s 1 T 0 T ( ρ x ) ( s ) d s + M 0 T ψ ( s ) d s 0 T ψ ( s ) [ α ( ρ x ) ( s ) + β | ( ρ x ) ( s ) | ] d s 0 T [ 1 T α M ψ ( s ) 0 T ψ ( τ ) d τ ] ( ρ x ) ( s ) d s 0 T [ 1 T α M e ( T ) ( 1 e ( T ) ) 0 T ψ ( τ ) d τ ] ( ρ x ) ( s ) d s 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equae_HTML.gif
      Moreover, for x Ω ¯ 2 Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq132_HTML.gif, we have from (H2) and (H7)
      Ψ ρ x ( t ) = 1 T 0 T ( ρ x ) ( s ) d s + 0 T K ( t , s ) f ( s , ( ρ x ) ( s ) , ( ρ x ) ( s ) ) d s 1 T 0 T ( ρ x ) ( s ) d s + 0 T K ( t , s ) [ α ( ρ x ) ( s ) + β | ( ρ x ) ( s ) | ] d s 0 T [ 1 T α K ( t , s ) ] ( ρ x ) ( s ) d s 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equaf_HTML.gif

      Thus, 7 is fulfilled and the assertion follows. □

      We now give two examples illustrating Theorem 2. Some calculations have been made with Mathematica. In the first example, the function h is constant, while in the second h ( t ) = 1 / ( 1 + t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq133_HTML.gif and f is independent of t.

      Example 1

      Consider the following PBVP:
      { x ( t ) + x ( t ) + ( t ( 1 t ) + 1 ) ( 2 9 x ( t ) + 3 4 | x ( t ) | + 1 ) = 0 , t [ 0 , 1 ] , x ( 0 ) = x ( 1 ) , x ( 0 ) = x ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ12_HTML.gif
      (12)
      Then e ( t ) = e t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq134_HTML.gif, φ ( t ) = 1 e t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq135_HTML.gif, Φ ( t ) = t + e t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq136_HTML.gif, ψ ( t ) = e e 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq137_HTML.gif, and
      k ( t , s ) = { s + e s t + 1 e 1 t e 1 , 0 s t 1 , s + 1 + e s t e 1 t e 1 , 0 t s 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equag_HTML.gif
      Moreover, (7) with M = 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq138_HTML.gif reads
      K ( t , s ) = { t s + e s t + 1 e 1 , 0 s t 1 , t s + 1 + e s t e 1 , 0 t s 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equah_HTML.gif

      and the assumptions (H2)-(H7) are met with R = 20 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq139_HTML.gif, α = 2 9 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq140_HTML.gif, β = 3 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq141_HTML.gif, r = 36 53 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq142_HTML.gif, m [ 12 ( e 1 ) 17 + 7 e , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq143_HTML.gif, η = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq144_HTML.gif and g ( t ) = t ( 1 t ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq145_HTML.gif. By Theorem 2, problem (12) has a positive solution.

      Example 2

      Consider the PBVP
      { x ( t ) + 1 1 + t x ( t ) + 1 10 1 9 x ( t ) + ( x ( t ) ) 4 / 5 = 0 , t [ 0 , 1 2 ] , x ( 0 ) = x ( 1 2 ) , x ( 0 ) = x ( 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equ13_HTML.gif
      (13)
      In this case, we have e ( t ) = 1 1 + t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq146_HTML.gif, φ ( t ) = ln ( 1 + t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq147_HTML.gif, Φ ( t ) = t + ln ( 1 + t ) + t ln ( 1 + t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq148_HTML.gif and
      ψ ( t ) = ( 1 + t ) ( 3 ln ( 1 + t ) ln ( 3 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_Equai_HTML.gif

      The assumptions of Theorem 2 are fulfilled with M = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq149_HTML.gif, R = 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq150_HTML.gif, α = 1 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq151_HTML.gif, β = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq152_HTML.gif, r = 1 100 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq153_HTML.gif, m = 0.9 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq154_HTML.gif, η = 1 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq155_HTML.gif and g ( t ) = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq156_HTML.gif.

      Declarations

      Acknowledgements

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

      Authors’ Affiliations

      (1)
      Institute of Mathematics, University of Rzeszów

      References

      1. Chu J, Fan N, Torres PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 2012, 388: 665-675. 10.1016/j.jmaa.2011.09.061MathSciNetView Article
      2. Cabada A, Cid JÁ: On comparison principles for the periodic Hill’s equation. J. Lond. Math. Soc. 2012, 86: 272-290. 10.1112/jlms/jds001MathSciNetView Article
      3. Graef JR, Kong L, Wang H: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differ. Equ. 2008, 245: 1185-1197. 10.1016/j.jde.2008.06.012MathSciNetView Article
      4. Ma R, Xu J, Han X: Global structure of positive solutions for superlinear second-order periodic boundary value problems. Appl. Math. Comput. 2012, 218: 5982-5988. 10.1016/j.amc.2011.11.079MathSciNetView Article
      5. Meehan M, O’Regan D: Existence theory for nonlinear Volterra integrodifferential and integral equations. Nonlinear Anal. 1998, 31: 317-341. 10.1016/S0362-546X(96)00313-6MathSciNetView Article
      6. Torres PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 2003, 190: 643-662. 10.1016/S0022-0396(02)00152-3View Article
      7. Yao Q: Positive solutions of nonlinear second-order periodic boundary value problems. Appl. Math. Lett. 2007, 20: 583-590. 10.1016/j.aml.2006.08.003MathSciNetView Article
      8. Ma R, Gao C, Chen R: Existence of positive solutions of nonlinear second-order periodic boundary value problems. Bound. Value Probl. 2010., 2010: Article ID 626054. doi:10.1155/2010/626054
      9. Rachůnková I, Tvrdý M, Vrkoč I: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differ. Equ. 2001, 176: 445-469. 10.1006/jdeq.2000.3995View Article
      10. Wang F, Zhang F, Wang F: The existence and multiplicity of positive solutions for second-order periodic boundary value problem. J. Funct. Spaces Appl. 2012., 2012: Article ID 725646. doi:10.1155/2012/725646
      11. Cremins CT: A fixed point index and existence theorems for semilinear equations in cones. Nonlinear Anal. 2001, 46: 789-806. 10.1016/S0362-546X(00)00144-9MathSciNetView Article
      12. O’Regan D, Zima M: Leggett-Williams norm-type theorems for coincidences. Arch. Math. 2006, 87: 233-244. 10.1007/s00013-006-1661-6MathSciNetView Article
      13. Gaines RE, Santanilla J: A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations. Rocky Mt. J. Math. 1982, 12: 669-678. 10.1216/RMJ-1982-12-4-669View Article
      14. Santanilla J: Some coincidence theorems in wedges, cones, and convex sets. J. Math. Anal. Appl. 1985, 105: 357-371. 10.1016/0022-247X(85)90053-8MathSciNetView Article
      15. Santanilla J: Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations. J. Math. Anal. Appl. 1987, 126: 397-408. 10.1016/0022-247X(87)90049-7MathSciNetView Article
      16. Mawhin J: Equivalence theorems for nonlinear operator equations and coincidence degree theory for mappings in locally convex topological vector spaces. J. Differ. Equ. 1972, 12: 610-636. 10.1016/0022-0396(72)90028-9MathSciNetView Article
      17. Gaines RE, Mawhin J Lect. Notes Math. 568. In Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin; 1977.
      18. Webb JRL: Solutions of semilinear equations in cones and wedges. I-IV. In World Congress of Nonlinear Analysts ’92 (Tampa, FL 1992). de Gruyter, Berlin; 1996:137-147.
      19. Feng W, Webb JRL: Solvability of three-point boundary value problems at resonance. Nonlinear Anal. 1997, 30: 3227-3238. 10.1016/S0362-546X(96)00118-6MathSciNetView Article
      20. Liu B: Solvability of multi-point boundary value problems at resonance. IV. Appl. Math. Comput. 2003, 143: 275-299. 10.1016/S0096-3003(02)00361-2MathSciNetView Article
      21. Kosmatov N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 2008, 68: 2158-2171. 10.1016/j.na.2007.01.038MathSciNetView Article
      22. Franco D, Infante G, Zima M: Second order nonlocal boundary value problems at resonance. Math. Nachr. 2011, 284: 875-884. 10.1002/mana.200810841MathSciNetView Article
      23. Cabada A, Pouso R:Existence result for the problem ( ϕ ( u ) ) = f ( t , u , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq157_HTML.gif with periodic and Neumann boundary conditions. Nonlinear Anal. 1997, 30: 1733-1742. 10.1016/S0362-546X(97)00249-6MathSciNetView Article
      24. Sȩdziwy S: Nonlinear periodic boundary value problem for a second order ordinary differential equation. Nonlinear Anal. 1998, 32: 881-890. 10.1016/S0362-546X(97)00533-6MathSciNetView Article
      25. Kiguradze I, Staněk S: On periodic boundary value problem for the equation u = f ( t , u , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq158_HTML.gif with one-sided growth restrictions on f . Nonlinear Anal. 2002, 48: 1065-1075. 10.1016/S0362-546X(00)00235-2MathSciNetView Article
      26. Torres PJ: Existence and stability of periodic solutions of a Duffing equation by using a new maximum principle. Mediterr. J. Math. 2004, 1: 479-486. 10.1007/s00009-004-0025-3MathSciNetView Article
      27. Cheng Z, Ren J: Harmonic and subharmonic solutions for superlinear damped Duffing equation. Nonlinear Anal., Real World Appl. 2013, 14: 1155-1170. 10.1016/j.nonrwa.2012.09.007MathSciNetView Article
      28. Petryshyn WV: On the solvability of x T x + λ F x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-19/MediaObjects/13661_2012_Article_276_IEq159_HTML.gif in quasinormal cones with T and F k -set contractive. Nonlinear Anal. 1981, 5: 585-591. 10.1016/0362-546X(81)90105-XMathSciNetView Article
      29. Infante G, Zima M: Positive solutions of multi-point boundary value problems at resonance. Nonlinear Anal. 2008, 69: 2458-2465. 10.1016/j.na.2007.08.024MathSciNetView Article
      30. Zhang HE, Sun JP: Positive solutions of third-order nonlocal boundary value problems at resonance. Bound. Value Probl. 2012., 2012: Article ID 102

      Copyright

      © Zima and Drygaś; 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.