# Existence and multiplicity of positive solutions for a nonlocal differential equation

- Yunhai Wang
^{1}Email author, - Fanglei Wang
^{2, 3}and - Yukun An
^{3}

**2011**:5

https://doi.org/10.1186/1687-2770-2011-5

© Wang et al; licensee Springer. 2011

**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

## Introduction

where *α, β, γ, δ* are nonnegative constants, *ρ* = *αγ* + *αδ* + *βγ* > 0, *q* ≥ 1;
,
denote the Riemann-Stieltjes integrals.

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 [3–9] 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

and there exists a
such that *G*(*t, s*) ≥ *θ G*(*s, s*), *θ* ≤ *t* ≤ 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;

*u*∈

*C*

^{2}(0, 1) is a solution of Equation 1 if and only if

*u*∈

*C*(0, 1) satisfies the following nonlinear integral equation

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

*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, . A map

*α*is said to be a nonnegative continuous concave functional on

*P*if

*α*:

*P*→ [0, +∞) is continuous and

*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

**Lemma 2.2**[10]. Let be completely continuous and

*α*be a nonnegative continuous concave functional on

*P*such that

*α*(

*x*)

*=*||

*x*|| for all . Suppose there exists 0 <

*d*<

*a*<

*b*=

*c*such that

- (i)
{

*x*∈*P*(*α, a, b*):*α*(*x*) >*a*} ≠ ∅ and*α*(*Ax*) >*a*for*x*∈*P*(*α, a, b*); - (ii)
||

*Ax*|| <*d*for ||*x*|| ≤*d*;

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

**Lemma 2.3**[10]. Let

*E*be a Banach space, and let

*P*⊂

*E*be a closed, convex cone in

*E*, assume Ω

_{1}, Ω

_{2}are bounded open subsets of

*E*with , and be a completely continuous operator such that either

- (i)
||

*Au*|| ≤ ||*u*||,*u*∈*P*∩ ∂Ω_{1}and ||*Au*|| ≥ ||*u*||,*u*∈*P*∩ ∂Ω_{2}; or - (ii)
||

*Au*|| ≥ ||*u*||,*u*∈*P*∩ ∂Ω_{1}and ||*Au*|| ≤ ||*u*||,*u*∈*P*∩ ∂Ω_{2}.

## Main result

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

**Lemma 3.1**. *T*: *E* → *E* 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

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

Then, it is clear to see that 0 < *l* ≤ *L* < L.

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

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

*u*∈

*P*∩ ∂Ω

_{1}. Since min

_{θ≤t≤1-θ}

*u*(

*t*) ≥

*θ*||

*u*|| and ||

*u*|| =

*R*

_{1}, from Equation 3, (H1) and (H3), it follows that

Then, Equation 4 holds.

Then, Equation 7 holds.

Therefore, by Equations 4 and 7 and the second part of Lemma 2.3, *T* has a fixed point in
, which is a positive solution of Equation 1. □

then (H6) hold.

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

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

Then, Equation 10 holds.

Then, Equation 12 holds.

Therefore, by Equations 10 and 12 and the first part of Lemma 2.3, *T* has a fixed point in
, which is a positive solution of Equation 1. □

**Example**. Let *q* = 2, *h*(*t*) = *t*, Φ(*s*) = 2 + *s*, *φ*(*t*) = 2*t*,
and *g*(*s*) = *s*^{2}.

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

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

Evidently, *σ*: *P* → *R*^{+} 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
such that
.

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

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

*T*has at least three fixed points

*u*

_{ i }(

*i*= 1, 2, 3) such that

The proof is complete. □

Then, it is easy to see that (H1) to (H3) and (H10) to (H11) hold. Especially, take *a* = 1, by
and (H1), then (H12) holds.

## Declarations

## Authors’ Affiliations

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