POSITIVE SOLUTIONS FOR A THREE-POINT FRACTIONAL BOUNDARY VALUE PROBLEMS FOR P-LAPLACIAN WITH A PARAMETER

Abstract

In this paper, we firstly use Krasnosel'skii fixed point theorem to investigate positive solutions for the following three-point boundary value problems for p-Laplacian with a parameter $({\phi}_P(D^{\alpha}_{0}+u(t)))^{\prime}+{\lambda}f(t, u(t))=0$, 0$D^{\alpha}_{0}+u(0)=u(0)=u{\prime}{\prime}(0)=0$, $u^{\prime}(1)={\gamma}u^{\prime}(\eta)$ where ϕ_p(s) = |s|^p−2s, p > 1, $D^{\alpha}_{0^+}$ is the Caputo's derivative, α ∈ (2, 3], η, γ ∈ (0, 1), λ > 0 is a parameter. Then we use Leggett-Williams fixed point theorem to study the existence of three positive solutions for the fractional boundary value problem $({\phi}_P(D^{\alpha}_{0}+u(t)))^{\prime}+f(t, u(t))=0$, 0$D^{\alpha}_{0}+u(0)=u(0)=u{\prime}{\prime}(0)=0$, $u^{\prime}(1)={\gamma}u^{\prime}(\eta)$ where ϕ_p(s) = |s|^p−2s, p > 1, $D^{\alpha}_{0^+}$ is the Caputo's derivative, α ∈ (2, 3], η, γ ∈ (0, 1).

1. Introduction

It is well known that fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes, so the differential equations with fractional-order derivative are more adequate than integer order derivative for some real world problems. Therefore, the fractional differential equations have been of great interest recently, this is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, economics, engineering and biological sciences, etc. see [11,13,17-19,28,32] for example. Some recent investigations have shown that many physical systems can be represented more accurately using fractional derivative formulations [2,3]. Boundary value problems of fractional differential equations have been investigated in many papers (see [1,7,8,14,15,21-25,27,29,33] and references cited therein). The eigenvalue problems of integer differential equations have been studied extensively by many authors. As far as the eigenvalue problems of fractional differential equations are concerned, there are a few results (see [5,10,34]).

Z. Bai [5] studied the eigenvalue intervals for a class of fractional boundary value problem

where 2 < α ≤ 3, is the Caputo fractional derivative, λ > 0 is a parameter.

C. Zhai, L. Xu [25] considered the nonlinear fractional four-point boundary value problem with a parameter

where 1 < α ≤ 2, 0 ≤ ξ ≤ η ≤ 1, 0 ≤ μ1, μ2 ≤ 1, λ > 0 is a parameter.

X. Zhang, L. Liu and Y. Wu [31] investigated the singular eigenvalue problem for a higher order fractional differential equation

where n ≥ 3, n ∈ N, n − 1 < α ≤ n, n − l − 1 < α − μ1 < n − l, l = 1, 2, · · ·, n−2, μ−μn−1 > 0, α−μn−1 ≤ 2, α−μ > 1, aj ∈ [0,+∞), 0 < ξ1 < ξ2 < · · · < ξp-2 < 1, is the Riemann-Liouville fractional derivative.

The equation with a p-Laplacian operator arises in the modeling of different physical and natural phenomena, non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology, nonlinear flow laws, and so on. Recently, the existence of solutions to boundary value problems for fractional differential equation with p-Laplacian operator have been studied extensively in the literatures, (see [6,16,20,26]).

G. Chai [6] investigated the existence and multiplicity of positive solutions for the boundary value problem of fractional differential equation with p-Laplacian operator

where are the standard Riemann-Liouville fractional derivative with 1 < α ≤ 2, 0 < β ≤ 1, 0 < γ ≤ 1, 0 ≤ α − γ − 1, σ is a positive number.

Z. Liu and L. Lu [16] studied the boundary value problem for nonlinear fractional differential equations with p-Laplacian operator

where 0 < α, β ≤ 1, 1 < α + β ≤ 2, μ, λ, k ∈ R, ξ, η ∈ [0, 1], denotes the Caputo fractional derivative of order α. Motivated by the above works, in section 3, we consider the positive solutions for a three-point fractional boundary value problem for p-Laplacian with a parameter

where ϕp(s) = |s|p−2s, p > 1, is the Caputo’s derivative, α ∈ (2, 3], η, γ ∈ (0, 1), f ∈ C([0, 1] × [0,∞), [0,∞)), λ > 0 is a parameter.

In recent years, using Leggett-Williams fixed point theorem, some authors obtained three positive solutions for the fractional boundary value problem.

In [26], Zhang used the Leggett-Williams theorem to show the existence of triple positive solutions to the fractional boundary value problem

In [12], Eric R. Kaufmann and Ebene Mboumi gave sufficient conditions for the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problem

where Dαis the Riemann-Liouville differential operator of order α, f : [0,∞) → [0,∞) is a given continuous function and a(t) is a positive and continuous function on [0, 1].

In [30], X. Zhao, C. Chai, W. Ge considered the existence of three positive solutions of the following fractional boundary value problem

where α is a real number with 1 < α ≤ 2, 0 ≤ ξ ≤ η ≤ 1, 0 ≤ β, γ ≤ 1, f ∈ C([0, 1] × [0,∞) → [0,∞)), is the Caputo fractional derivative.

In [9], M. Jia, X. Liu studied at least three nonnegative solutions for the following fractional differential equation with integral boundary conditions

where CDα is the standard Caputo derivative, α ∈ R and 2 ≤ n = [α] < α < [α]+1, f ∈ C([0, 1]×R+,R+) and gk ∈ C([0, 1],R) (k = 0, 1, 2, · · ·, [α]) are given functions, [α] denotes the integer part of the real number α and R+ = [0,+∞). By means of Leggett-Williams fixed point theorem, some new results on the existence of at least three nonnegative solutions are obtained.

Motivated by the above works, in section 4, by means of Leggett-Williams fixed point theorem, we consider the existence of three positive solutions for the following three-point fractional boundary value problem for p-Laplacian

where ϕp(s) = |s|p−2s, p > 1, is the Caputo’s derivative, α ∈ (2, 3], η, γ ∈ (0, 1), f ∈ C([0, 1] × [0,∞), [0,∞)).

As far as we know, no contribution concerns the above three-point fractional boundary value problem for p-Laplacian with a parameter and the existence of three positive solutions for the three-point fractional boundary value problem for p-Laplacian. The aim of this paper is to fill the gap in the relevant literatures. Such investigations will provide an important platform for gaining a deeper understanding of our environment.

 

2. Preliminaries

Definition 2.1 ([6]). The Riemann-Liouville fractional integral operator of order α > 0 of a function u(t) is given by

provided the right side is point-wise defined on (0,+∞).

Definition 2.2 ([6]). The Caputo fractional derivative of order α > 0 of a continuous function u(t) is given by

where n = [α] + 1, provided the right side is point-wise defined on (0,+∞).

Lemma 2.3 ([20]). The three-point boundary value problem (1), (2) has a unique solution

where

Lemma 2.4 ([20]). Let β ∈ (0, 1) be fixed. The kernel G1(t, s) satisfies the following properties.

(1): 0 ≤ G1(t, s) ≤ G1(1, s) for all s ∈ (0, 1);

(2): for all s ∈ (0, 1).

Lemma 2.5 ([20]). The unique solution u(t) of (1), (2) is nonnegative and satisfies

Theorem 2.6. Suppose E is a real Banach space, K ⊂ E is a cone, let Ω1,Ω2 be two bounded open sets of E such that Let operator T : be completely continuous. Suppose that one of two conditions hold

(i) ∥Tx∥ ≤ ∥x∥, ∀x ∈ K ∩ ∂Ω1, ∥Tx∥ ≥∥x∥, ∀x ∈ K ∩ ∂Ω2;

(ii) ∥Tx∥ ≥ ∥x∥, ∀x ∈ K ∩ ∂Ω1, ∥Tx∥ ≤ ∥x∥, ∀x ∈ K ∩ ∂Ω2,

then T has at least one fixed point in

Define the cone K by

and the operator T : K → E by

Lemma 2.7 ([20]). T is completely continuous and T(K) ⊆ K.

Denote

where β = 0+,∞,

 

3. Main results

Theorem 3.1. Suppose that f∞ > 0, f0 < ∞. Then boundary value problem (1), (2) has at least one positive solution if

Proof. By (6), there exists ε > 0, such that

(i) Fixed ε. By f0 < ∞, there exists H1 > 0, such that for u : 0 < |u| ≤ H1, we have

Define

for u ∈ ∂Ω1, we have

Therefore, ∥Tu∥ ≤ ∥u∥.

(ii) By f∞ > 0, there exists > 0, such that for |u| ≥ , we have

Choose

by Lemma 5, for u ∈ ∂Ω2, we have

thus

Therefore, ∥Tu∥ ≥ ∥u∥. So, by Theorem 2.6 (ii), we have T has a fixed point therefore, u is a positive solution of boundary value problem (1), (2). The proof is completed. □

Corollary 3.2. Suppose that f∞ > 0, f0 < ∞. Then boundary value problem (1), (2) has nonnegative solution when

where D1 = {λ > 0}.

Theorem 3.3. Suppose that f0 > 0, f∞ < ∞. Then boundary value problem (1), (2) has at least one positive solution if

Proof. By (8), there exists ε > 0, such that

(i) Fixed ε. By f0 > 0, there exists H1 > 0, such that for u : 0 < |u| ≤ H1, we have

Define

by Lemma 5, for u ∈ ∂Ω1, we have

thus,

Therefore, ∥Tu∥ ≥ ∥u∥.

(ii) By f∞ < ∞, there exists > 0, such that for u : |u| ≥ , we have

We shall consider two cases, case 1, f is bounded. Case 2, f is unbounded.

Case 1. Suppose that f is bounded, there exists L > 0, such that

Define

for u ∈ ∂Ω2, we have

Case 2. Choose H2 > max such that when t ∈ [0, 1] and 0 < |u| ≤ H2, we have f(t, u) ≤ f(t,H2). Let

for u ∈ ∂Ω2, we have

Therefore, ∥Tu∥ ≤ ∥u∥. So, by Theorem 2.6 (ii), we have T has a fixed point therefore, u is a positive solution of boundary value problem (1), (2). The proof is completed. □

Corollary 3.4. Suppose that f0 > 0, f∞ < ∞. Then boundary value problem (1), (2) has nonnegative solution when

where D1 = {λ > 0}.

 

4. Three positive solution of the problem (10), (11)

In this section, we will give the existence of three positive solutions of the following fractional boundary value problem

where ϕp(s) = |s|p−2s, p > 1, is the Caputo’s derivative, α ∈ (2, 3], η, γ ∈ (0, 1), f ∈ C([0, 1] × [0,∞).

The basic space used in this section is a real Banach space E = C[0, 1] with the norm

Definition 4.1. The map α is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that α : P → [0,∞) is continuous and

for all x, y ∈ P and 0 ≤ t ≤ 1.

Definition 4.2. The map γ is a nonnegative continuous convex functional on a cone P of a real Banach space E provided that γ : P → [0,∞) is continuous and

for all x, y ∈ P and 0 ≤ t ≤ 1.

Let α be a nonnegative continuous concave functional on P. Then for positive real numbers 0 < a < b, we define the following convex sets:

The following fixed point theorem is fundamental in the proofs of our main results.

Theorem 4.3 ([30]). Let be a completely continuous operator and let α be a nonnegative continuous concave functional on P such that α(x) ≤ ∥x∥ for all Suppose that there exist positive numbers 0 < a < b < d ≤ c such that

Then A has at least three fixed points x1, x2, x3 such that

Let β ∈ (0, 1) be fixed. Define the cone P by

and the operator A : P → E by

It is obvious that the existence of a positive solution for the problem (10), (11) is equivalent to the existence of nontrivial point of A in P.

We define the nonnegative continuous concave functional on P by

It is clear that α(u) ≤ ∥u∥ for u ∈ P.

Let

Theorem 4.4. Assume that there exist nonnegative numbers a, b, c such that and f(t, u) satisfy the following conditions

Then BVP (10), (11) has at least three positive solutions x1, x2, x3 such that

Proof. We complete the proof by three steps.

Step 1. Show

Firstly, Lemma 2.5 guarantees Secondly, for all we have 0 ≤ u(t) ≤ c and by (A1),

Therefore, ∥Au∥ ≤ c which implies that The operator A is completely continuous by Lemma 2.7. Similarly, Au ∈ Pa for all

Step 2. Show

Let then u ∈ P, and That is, (15) holds.

For we have

then by (A3), we get

Therefore, we have α(Au) > b. Hence, (16) holds.

Step 3. Show α(Au) > b for all u ∈ P(α, b, c) with

If u ∈ P(α, b, c) with by Lemma 2.5, we have Hence, an application of Theorem 4.3 completes the proof. □