Existence of positive solutions to periodic boundary value problems with sign-changing Green's function

  • Shengren Zhong1 and

    Affiliated with

    • Yukun An2Email author

      Affiliated with

      Boundary Value Problems20112011:8

      DOI: 10.1186/1687-2770-2011-8

      Received: 27 January 2011

      Accepted: 27 July 2011

      Published: 27 July 2011

      Abstract

      This paper deals with the periodic boundary value problems

      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equa_HTML.gif

      where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq1_HTML.gif is a constant and in which case the associated Green's function may changes sign. The existence result of positive solutions is established by using the fixed point index theory of cone mapping.

      Keywords

      periodic boundary value problem positive solution sign-changing Green's function cone fixed point theorem

      1 Introduction

      The periodic boundary value problems
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ1_HTML.gif
      (1)

      where f is a continuous or L1-Caratheodory type function have been extensively studied. A very popular technique to obtain the existence and multiplicity of positive solutions to the problem is Krasnosel'skii's fixed point theorem of cone expansion/compression type, see for example [14], and the references contained therein. In those papers, the following condition is an essential assumptions:

      (A) The Green function G(t, s) associated with problem (1) is positive for all (t, s) ∈ [0, T] × [0, T].

      Under condition (A), Torres get in [4] some existence results for (1) with jumping nonlinearities as well as (1) with a repulsive or attractive singularity, and the authors in [3] obtained the multiplicity results to (1) when f(t, u) has a repulsive singularity near x = 0 and f(t, u) is super-linear near x = +∞. In [2], a special case, a(t) ≡ m2 and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq2_HTML.gif , was considered, the multiplicity results to (1) are obtained when the nonlinear term f(t, u) is singular at u = 0 and is super-linear at u = ∞.

      Recently, in [5], the hypothesis (A) is weakened as

      (B) The Green function G(t, s) associated with problem (1) is nonnegative for all (t, s) ∈ [0, T] × [0, T] but vanish at some interior points.

      By defining a new cone, in order to apply Krasnosel'skii's fixed point theorem, the authors get an existence result when http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq4_HTML.gif is sub-linear at u = 0 and u = ∞ or http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq4_HTML.gif is super-linear at u = 0 and u = ∞ with http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq4_HTML.gif is convex and nondecreasing.

      In [6], the author improve the result of [5] and prove the existence results of at least two positive solutions under conditions weaker than sub- and super-linearity.

      In [7], the author study (1) with f(t, u) = λb(t)f(u) under the following condition:

      (C) The Green function G(t, s) associated with problem (1) changes sign and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq5_HTML.gif where G - is the negative part of G.

      Inspired by those papers, here we study the problem:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ2_HTML.gif
      (2)

      where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq1_HTML.gif is a constant and the associated Green's function may changes sign. The aim is to prove the existence of positive solutions to the problem.

      2 Preliminaries

      Consider the periodic boundary value problem
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ3_HTML.gif
      (3)
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq1_HTML.gif and e(t) is a continuous function on [0, T]. It is well known that the solutions of (3) can be expressed in the following forms
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equb_HTML.gif
      where G(t, s) is Green's function associated to (3) and it can be explicitly expressed
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equc_HTML.gif
      By direct computation, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equd_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Eque_HTML.gif
      for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq6_HTML.gif when http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq7_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equf_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equg_HTML.gif

      where G+ and G- are the positive and negative parts of G.

      We denote
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equh_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equi_HTML.gif

      Let E denote the Banach space C[0, T] with the norm ||u|| = maxt∈[0,T]|u(t)|.

      Define the cone K in E by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equj_HTML.gif

      We know that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq8_HTML.gif and therefore K ≠ ∅. For r > 0, let K r = {uK : ||u|| < r}, and ∂K r = {uK : ||u|| = r}, which is the relative boundary of K r in K.

      To prove our result, we need the following fixed point index theorem of cone mapping.

      Lemma 1 (Guo and Lakshmikantham [8]). Let E be a Banach space and let KE be a closed convex cone in E. Let L : KK be a completely continuous operator and let i(L, K r , K) denote the fixed point index of operator L.

      (i) If μLuu for any u ∈ ∂K r and 0 < μ ≤ 1, then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equk_HTML.gif
      (ii) If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq9_HTML.gif and μLuu for any u ∈ ∂K r and μ ≥ 1, then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equl_HTML.gif

      3 Existence result

      We make the following assumptions: (H 1) f : [0, +∞) → [0, +∞) is continuous;

      (H 2) 0 ≤ m = inf u∈[0,+ ∞]f (u) and M = supu∈[0,+ ∞)f (u) ≤ +∞;

      (H 3) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq10_HTML.gif , when m = 0 we define http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq11_HTML.gif .

      To be convenience, we introduce the notations:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equm_HTML.gif

      and suppose that f0, f ∈ [0, ∞].

      Define a mapping L : KE by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equn_HTML.gif

      It can be easily verified that uK is a fixed point of L if and only if u is a positive solution of (2).

      Lemma 2. Suppose that (H1), (H2) and (H3) hold, then L : EE is completely continuous and L(K) ⊆ K.

      Proof Let uK, then in case of γ = +∞, since G(t, s) ≥ 0, we have Lu(t) ≥ 0 on [0, T]; in case of γ < +∞, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equo_HTML.gif
      On the other hand,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equp_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equq_HTML.gif
      for t ∈ [0, T]. Thus,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equr_HTML.gif

      i.e., L(K) ⊆ K. A standard argument can be used to show that L : EE is completely continuous.

      Now we give and prove our existence theorem:

      Theorem 3. Assume that (H1), (H2) and (H3) hold. Furthermore, suppose that f0 > ρ2 and f < ρ2 in case of γ = +∞. Then problem (2) has at least one positive solution.

      Proof Since f0 > ρ2, there exist ε > 0 and ξ > 0 such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ4_HTML.gif
      (4)
      Let r ∈ (0, ξ), then for every u ∈ ∂K r , we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equs_HTML.gif
      Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq9_HTML.gif . Next, we show that μLuu for any u ∈ ∂K r and μ ≥ 1. In fact, if there exist u0 ∈ ∂K r and μ0 ≥ 1 such that μ0Lu0 = u0, then u0(t) satisfies
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ5_HTML.gif
      (5)
      Integrating the first equation in (5) from 0 to T and using the periodicity of u0(t) and (4), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equt_HTML.gif
      Since http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq12_HTML.gif , we see that ρ2 ≥ (ρ2 + ε), which is a contradiction. Hence, by Lemma 1, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ6_HTML.gif
      (6)
      On the other hand, since f < ρ2, there exist ε ∈ (0, ρ2) and ζ > 0 such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equu_HTML.gif
      Set C = max0≤uζ|f (u) - (ρ2 - ε)u| + 1, it is clear that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ7_HTML.gif
      (7)

      If there exist u0K and 0 < μ0 ≤ 1 such that μ0Lu0 = u0, then (5) is valid.

      Integrating again the first equation in (5) from 0 to T, and from (7), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equv_HTML.gif
      Therefore, we obtain that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equw_HTML.gif
      i.e.,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ8_HTML.gif
      (8)
      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq13_HTML.gif , then μLuu for any u ∈ ∂K R and 0 < μ ≤ 1. Therefore, by Lemma 1, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ9_HTML.gif
      (9)
      From (6) and (9) it follows that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equx_HTML.gif

      Hence, L has a fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq14_HTML.gif , which is the positive solution of (2).

      Remark 4. Theorem 3 contains the partial results of [47] obtained in case of positive Green's function, vanishing Green's function and sign-changing Green's function, respectively.

      4 An example

      Let 0 ≠ q < 1 be a constant, h be the function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equy_HTML.gif
      and let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equz_HTML.gif
      By the direct calculation, we get m = 1 and M = γ, and f0 = ∞ and f = 0 in case of γ = +∞. Consider the following problem
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_Equ10_HTML.gif
      (10)

      where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-8/MediaObjects/13661_2011_Article_8_IEq1_HTML.gif is a constant. We know that the conditions of Theorem 3 hold for the problem (10) and therefore, (10) have at least one positive solution from Theorem 3.

      Declarations

      Acknowledgements

      The authors are very grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript.

      Authors’ Affiliations

      (1)
      Department of Engineering Technology, Wuwei Occupational College Wuwei
      (2)
      Department of Mathematics, Nanjing University of Aeronautics and Astronautics Nanjing

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      Copyright

      © Zhong and An; licensee Springer. 2011

      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.