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

## 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

where *α, β, γ, δ* are nonnegative constants, *ρ* = *αγ* + *αδ* + *βγ* > 0, *q* ≥ 1; Open image in new window , Open image in new window 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, 4, 5, 6, 7, 8, 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 Open image in new window 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;

*h*(

*t*) does not vanish identically on any subinterval of (0, 1) and satisfies

*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, Open image in new window . 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 Open image in new window be completely continuous and

*α*be a nonnegative continuous concave functional on

*P*such that

*α*(

*x*)

*=*||

*x*|| for all Open image in new window . 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 Open image in new window , and Open image in new window 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}.

Then, *A* has a fixed point in Open image in new window .

## Main result

*E*=

*C*[0, 1] endowed norm ||

*u*|| = max

_{0≤t≤1}

*|u|*, and define the cone

*P*⊆

*E*by

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.

*R*

_{1}∈ (0,

*η*), set Ω

_{1}= {

*u*∈

*E*: ||

*u*|| <

*R*

_{1}}. We now prove that

*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 Open image in new window , which is a positive solution of Equation 1. □

**Example**. Let

*q*= 2,

*h*(

*t*) = 1, Φ(

*s*) = 2 +

*s*,

*φ*(

*t*) = 2

*t*, Open image in new window and Open image in new window , namely,

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.

*u*∈

*P*∩ ∂Ω

_{2}, Since min

_{θ≤t≤1-θ}

*u*(

*t*) ≥

*θ*||

*u*|| and ||

*u*|| =

*R*

_{2}, we have

Then, Equation 12 holds.

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

**Example**. Let *q* = 2, *h*(*t*) = *t*, Φ(*s*) = 2 + *s*, *φ*(*t*) = 2*t*, Open image in new window 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:

*a*> 0 such that

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 Open image in new window such that Open image in new window .

*τ*> 0 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*).

*u*∈

*P*(

*θ, a, c*) with ||

*Tu*|| >

*b*, we have

*T*has at least three fixed points

*u*

_{ i }(

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

The proof is complete. □

**Example**. Let

*q*= 2,

*h*(

*t*) =

*t*, Φ(

*s*) = 2 +

*s*,

*φ*(

*t*) = 2

*t*, Open image in new window and, Open image in new window , namely,

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

## Notes

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