Open Access

Existence and multiplicity of positive solutions for a nonlocal differential equation

Boundary Value Problems20112011:5

DOI: 10.1186/1687-2770-2011-5

Received: 21 February 2011

Accepted: 11 July 2011

Published: 11 July 2011

Abstract

In this paper, the existence and multiplicity results of positive solutions for a nonlocal differential equation are mainly considered.

Keywords

Nonlocal boundary value problems Cone Fixed point theorem

Introduction

In this paper, we are concerned with the existence and multiplicity of positive solutions for the following nonlinear differential equation with nonlocal boundary value condition
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ1_HTML.gif
(1)

where α, β, γ, δ are nonnegative constants, ρ = αγ + αδ + βγ > 0, q ≥ 1; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq2_HTML.gif denote the Riemann-Stieltjes integrals.

Many authors consider the problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ2_HTML.gif
(2)
because of the importance in numerous physical models: system of particles in thermodynamical equilibrium interacting via gravitational potential, 2-D fully turbulent behavior of a real flow, one-dimensional fluid flows with rate of strain proportional to a power of stress multiplied by a function of temperature, etc. In [1, 2], the authors use the Kras-noselskii fixed point theorem to obtain one positive solution for the following nonlocal equation with zero Dirichlet boundary condition
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equa_HTML.gif

when the nonlinearity f is a sublinear or superlinear function in a sense to be established when necessary. Nonlocal BVPs of ordinary differential equations or system arise in a variety of areas of applied mathematics and physics. In recent years, more and more papers were devoted to deal with the existence of positive solutions of nonlocal BVPs (see [39] and references therein). Inspired by the above references, our aim in the present paper is to investigate the existence and multiplicity of positive solutions to Equation 1 using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem.

This paper is organized as follows: In Section 2, some preliminaries are given; In Section 3, we give the existence results.

Preliminaries

Lemma 2.1[3]. Let y(t) C([0, 1]), then the problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equb_HTML.gif
has a unique solution
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equc_HTML.gif
where the Green function G(t, s) is
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equd_HTML.gif
It is easy to see that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Eque_HTML.gif

and there exists a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq3_HTML.gif such that G(t, s) ≥ θ G(s, s), θt ≤ 1 - θ, 0 ≤ s ≤ 1.

For convenience, we assume the following conditions hold throughout this paper:

(H1) f, g, Φ: R+R+ are continuous and nondecreasing functions, and Φ (0) > 0;

(H2) φ(t) is an increasing nonconstant function defined on [0, 1] with φ(0) = 0;

(H3) h(t) does not vanish identically on any subinterval of (0, 1) and satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equf_HTML.gif
Obviously, u C2(0, 1) is a solution of Equation 1 if and only if u C(0, 1) satisfies the following nonlinear integral equation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equg_HTML.gif

At the end of this section, we state the fixed point theorems, which will be used in Section 3.

Let E be a real Banach space with norm || · || and P E be a cone in E, P r = {x P : ||x|| < r}(r > 0). Then, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq4_HTML.gif . A map α is said to be a nonnegative continuous concave functional on P if α: P → [0, +∞) is continuous and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equh_HTML.gif
for all x, y P and t [0, 1]. For numbers a, b such that 0 < a < b and α is a nonnegative continuous concave functional on P, we define the convex set
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equi_HTML.gif
Lemma 2.2[10]. Let https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq5_HTML.gif be completely continuous and α be a nonnegative continuous concave functional on P such that α (x) = ||x|| for all https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq6_HTML.gif . Suppose there exists 0 < d < a < b = c such that
  1. (i)

    {x P (α, a, b): α (x) > a} ≠ and α (Ax) > a for x P (α, a, b);

     
  2. (ii)

    ||Ax|| < d for ||x|| ≤ d;

     

(iii) α(Ax) > a for x P (α, a, c) with ||Ax|| > b.

Then, A has at least three fixed points x1, x2, x3 satisfying
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equj_HTML.gif
Lemma 2.3[10]. Let E be a Banach space, and let P E be a closed, convex cone in E, assume Ω1, Ω2 are bounded open subsets of E with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq7_HTML.gif , and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq8_HTML.gif be a completely continuous operator such that either
  1. (i)

    ||Au|| ≤ ||u||, u P ∩ ∂Ω1 and ||Au|| ≥ ||u||, u P ∩ ∂Ω2; or

     
  2. (ii)

    ||Au|| ≥ ||u||, u P ∩ ∂Ω1 and ||Au|| ≤ ||u||, u P ∩ ∂Ω2.

     

Then, A has a fixed point in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq9_HTML.gif .

Main result

Let E = C[0, 1] endowed norm ||u|| = max0≤t≤1|u|, and define the cone P E by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equk_HTML.gif

Then, it is easy to prove that E is a Banach space and P is a cone in E.

Define the operator T: EE by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equl_HTML.gif

Lemma 3.1. T: EE is completely continuous, and Te now prove thatP P.

Proof. For any u P, then from properties of G(t, s), T (u)(t) ≥ 0, t [0, 1], and it follows from the definition of T that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equm_HTML.gif
Thus, it follows from above that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equn_HTML.gif

From the above, we conclude that TP P. Also, one can verify that T is completely continuous by the Arzela-Ascoli theorem.   □

Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equo_HTML.gif

Then, it is clear to see that 0 < lL < L.

Theorem 3.2. Assume (H1) to (H3) hold. In addition,

(H4)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equp_HTML.gif
(H5) There exists a constant 2 ≤ p1 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equq_HTML.gif
(H6) There exists a constant p2 with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq10_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equr_HTML.gif

Then, problem (Equation 1) has one positive solution.

Proof. From (H4), there exists a 0 < η < ∞ such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ3_HTML.gif
(3)
Choosing R1 (0, η), set Ω1 = {u E : ||u|| < R1}. We now prove that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ4_HTML.gif
(4)
Let u P ∩ ∂Ω1. Since minθt≤1-θu(t) ≥ θ ||u|| and ||u|| = R1, from Equation 3, (H1) and (H3), it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equs_HTML.gif

Then, Equation 4 holds.

On the other hand, from (H5), there exists https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq11_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ5_HTML.gif
(5)
From (H6), there exists https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq12_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ6_HTML.gif
(6)
Choosing https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq13_HTML.gif , set Ω2 = {u E : ||u|| < R2}. We now prove that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ7_HTML.gif
(7)
If u P ∩ ∂Ω2, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equt_HTML.gif
From Equations 5, 6, we can prove
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equu_HTML.gif

Then, Equation 7 holds.

Therefore, by Equations 4 and 7 and the second part of Lemma 2.3, T has a fixed point in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq9_HTML.gif , which is a positive solution of Equation 1.   □

Example. Let q = 2, h(t) = 1, Φ(s) = 2 + s, φ(t) = 2t, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq14_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq15_HTML.gif , namely,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equv_HTML.gif
It is easy to see that (H1) to (H3) hold. We also can have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equw_HTML.gif
Take p1 = 2, then it is clear to see that (H4) and (H5) hold. Since
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equx_HTML.gif

then (H6) hold.

Theorem 3.3. Assume (H1) to (H3) hold. In addition,

(H7) There exists a constant 2 ≤ p1 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equy_HTML.gif
(H8) There exists a constant p2 with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq10_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equz_HTML.gif
(H9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equaa_HTML.gif

Then, problem (Equation 1) has one positive solution.

Proof. From (H7), there exists η1 > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ8_HTML.gif
(8)
From (H8), there exists η2 > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ9_HTML.gif
(9)
Choosing https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq16_HTML.gif , set Ω1 = {u E : ||u|| < R1}. We now prove that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ10_HTML.gif
(10)
If u P ∩ ∂Ω1, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equab_HTML.gif
From Equations 8, 9, we can prove
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equac_HTML.gif

Then, Equation 10 holds.

On the other hand, from (H7), there exists https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq11_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ11_HTML.gif
(11)
Choosing https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq17_HTML.gif , set Ω2 = {u E : ||u|| < R2}. We now prove that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ12_HTML.gif
(12)
If u P ∩ ∂Ω2, Since minθt≤1-θu(t) ≥ θ ||u|| and ||u|| = R2, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ13_HTML.gif
(13)
By Equation 11, (H1) and (H3), it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equad_HTML.gif

Then, Equation 12 holds.

Therefore, by Equations 10 and 12 and the first part of Lemma 2.3, T has a fixed point in https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq9_HTML.gif , which is a positive solution of Equation 1.   □

Example. Let q = 2, h(t) = t, Φ(s) = 2 + s, φ(t) = 2t, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq18_HTML.gif and g(s) = s2.

Theorem 3.4. Assume that (H1) to (H3) hold. In addition, φ(1) ≥ 1, and the functions f, g satisfy the following growth conditions:

(H10)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equae_HTML.gif
(H11)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equaf_HTML.gif
(H12) There exists a constant a > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equag_HTML.gif

Then, BVP (Equation 1) has at least three positive solutions.

Proof. For the sake of applying the Leggett-Williams fixed point theorem, define a functional σ(u) on cone P by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equah_HTML.gif

Evidently, σ: PR+ is a nonnegative continuous and concave. Moreover, σ(u) ≤ ||u|| for each u P.

Now, we verify that the assumption of Lemma 2.2 is satisfied.

Firstly, it can verify that there exists a positive number c with https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq19_HTML.gif such that https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq20_HTML.gif .

By (H10), it is easy to see that there exists τ > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equai_HTML.gif
Set
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equaj_HTML.gif
Taking
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equak_HTML.gif
If https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq21_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equal_HTML.gif

by (H1) to (H3) and (H10).

Next, from (H11), there exists d' (0, a) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equam_HTML.gif
Take https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq22_HTML.gif . Then, for each https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq23_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equan_HTML.gif

Finally, we will show that {u P (σ, a, b): σ(u) > a} ≠ and σ(Tu) > a for all u P(σ, a, b).

In fact,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equao_HTML.gif
For u P (σ, a, b), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equap_HTML.gif
for all t [θ, 1 -θ]. Then, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equaq_HTML.gif
by (H1) to (H3), (H12). In addition, for each u P (θ, a, c) with ||Tu|| > b, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equar_HTML.gif
Above all, we know that the conditions of Lemma 2.2 are satisfied. By Lemma 2.2, the operator T has at least three fixed points u i (i = 1, 2, 3) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equas_HTML.gif

The proof is complete.   □

Example. Let q = 2, h(t) = t, Φ(s) = 2 + s, φ(t) = 2t, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq24_HTML.gif and, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq25_HTML.gif , namely,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equat_HTML.gif
From a simple computation, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equau_HTML.gif

Then, it is easy to see that (H1) to (H3) and (H10) to (H11) hold. Especially, take a = 1, by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_IEq26_HTML.gif and (H1), then (H12) holds.

Declarations

Authors’ Affiliations

(1)
College of Aeronautics and Astronautics, Nanjing University of Aeronautics and Astronautics
(2)
College of Science, Hohai University
(3)
Department of Mathematics, Nanjing University of Aeronautics and Astronautics

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Copyright

© Wang et al; 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.