Positive Solution to a Fractional Boundary Value Problem

A fractional boundary value problem is considered. By means of Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function, and Guo-Krasnosel’skii ﬁxed point theorem on cone, some results on the existence, uniqueness, and positivity of solutions are obtained.


Introduction
Fractional differential equations are a natural generalization of ordinary differential equations.In the last few decades many authors pointed out that differential equations of fractional order are suitable for the metallization of various physical phenomena and that they have numerous applications in viscoelasticity, electrochemistry, control and electromagnetic, and so forth, see 1-4 .This work is devoted to the study of the following fractional boundary value problem P1 : where f : 0, 1 × R × R → R is a given function, 2 < q < 3, 1 < σ < 2 and c D q 0 denotes the Caputo's fractional derivative.Our results allow the function f to depend on the fractional International Journal of Differential Equations derivative c D σ 0 u t which leads to extra difficulties.No contributions exist, as far as we know, concerning the existence of positive solutions of the fractional differential equation 1.1 jointly with the nonlocal condition 1.2 .
Our mean objective is to investigate the existence, uniqueness, and existence of positive solutions for the fractional boundary value problem P1 , by using Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cone.
The research in this area has grown significantly and many papers appeared on this subject, using techniques of nonlinear analysis, see 5-14 .
In 6 , El-Shahed considered the following nonlinear fractional boundary value problem D q 0 u t λa t f u t 0, 0 < t < 1, where 2 < q ≤ 3 and D q 0 is the Riemann-Liouville fractional derivative.Using the Krasnoselskii's fixed-point theorem on cone, he proved the existence and nonexistence of positive solutions for the above fractional boundary value problem.
Liang and Zhang in 9 studied the existence and uniqueness of positive solutions by the properties of the Green function, the lower and upper solution method and fixed point theorem for the fractional boundary value problem where 2 < q ≤ 3 and D q 0 is the Riemann-Liouville fractional derivative.In 5 Bai and L ü investigated the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem: where 1 < q ≤ 1 and D q 0 is the Riemann-Liouville fractional derivative.Applying fixed-point theorems on cone, they prove some existence and multiplicity results of positive solutions.
This paper is organized as follows, in the Section 2 we cite some definitions and lemmas needed in our proofs.Section 3 treats the existence and uniqueness of solution by using Banach contraction principle, Leray Schauder nonlinear alternative.Section 4 is devoted to prove the existence of positive solutions with the help of Guo-Krasnoselskii Theorem, then we give some examples illustrating the previous results.

Preliminaries and Lemmas
In this section, we present some lemmas and definitions from fractional calculus theory which will be needed later.
exists almost everywhere on a, b α is the entire part of α .
Denote by L 1 0, 1 , R the Banach space of Lebesgue integrable functions from 0, 1 into R with the norm ||y|| L 1 1 0 |y t |dt.The following Lemmas gives some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative.

International Journal of Differential Equations
Lemma 2.7.Let 2 < q < 3, 1 < σ < 2 and y ∈ C 0, 1 .The unique solution of the fractional boundary value problem is given by where

2.7
Proof.Applying Lemmas 2.4 and 2.5 to 2.5 we get u t I q 0 y t c 1 c 2 t c 3 t 2 .

2.8
Differentiating both sides of 2.8 and using Lemma 2.6 it yields

2.9
The first condition in 2.5 implies c 1 c 3 0, the second one gives c 2 αI q−2 0 y 1 − I q−1 0 y 1 .Substituting c 2 by its value in 2.8 , we obtain that can be written as that is equivalent to where G is defined by 2.7 .The proof is complete.

Existence and Uniqueness Results
In this section we prove the existence and uniqueness of solutions in the Banach space E of all functions u ∈ C 2 0, 1 into R, with the norm ||u|| max t∈ 0,

The function u ∈ E is solution of the fractional boundary value problem (P1) if and only if Tu t u t
, for all t ∈ 0, 1 .
Proof.Let u be solution of P1 and v t In view of 2.10 we have

3.2
With the help of Lemma 2.6 we obtain

3.3
It is clear that v satisfies conditions 1.2 , then it is a solution for the problem P1 .The proof is complete.
Theorem 3.2.Assume that there exist nonnegative functions g, h ∈ L 1 0, 1 , R such that for all x, y ∈ R and t ∈ 0, 1 , one has where

3.6
Then the fractional boundary value problem (P1) has a unique solution u in E.
To prove Theorem 3.2, we use the following property of Riemann-Liouville fractional integrals.

3.8
Now we prove Theorem 3.2.
Proof.We transform the fractional boundary value problem to a fixed point problem.By Lemma 3.1, the fractional boundary value problem P1 has a solution if and only if the operator T has a fixed point in E. Now we will prove that T is a contraction.Let u, v ∈ E, in view of 2.10 we get with the help of 3.4 we obtain

3.10
International Journal of Differential Equations 7 Lemma 3.3 implies

3.11
In view of 3.5 it yields |Tu − Tv| < u − v .

3.12
On the other hand we have where

3.15
Applying hypothesis 3.4 we get

3.16
Let us estimate the term 1 0 ∂G s, r /∂s g r dr.We have 3.17 With the help of hypothesis 3.5 it yields

3.19
Taking into account 3.12 -3.19 we obtain from here, the contraction principle ensures the uniqueness of solution for the fractional boundary value problem P1 .This finishes the proof.Now we give an existence result for the fractional boundary value problem P1 .
Theorem 3.4.Assume that f t, 0, 0 / 0 and there exist nonnegative functions k, h, g where C 1 max{C k , C h , C g }, C 2 max{A k , A h , A g }, C h and C g are defined as in Theorem 3.2 and

3.23
Then the fractional boundary value problem (P1) has at least one nontrivial solution u * ∈ E.
To prove this Theorem, we apply Leray-Schauder nonlinear alternative.Proof.First let us prove that T is completely continuous.It is clear that T is continuous since f and G are continuous.Let B r {u ∈ E, u ≤ r} be a bounded subset in E. We shall prove that T B r is relatively compact.
i For u ∈ B r and using 3.21 we get 0 u s g s ds.

3.24
Since ψ and φ are nondecreasing then 3.24 implies 3.25 using similar techniques as to get 3.12 it yields

3.27
Moreover, we have ∂G ∂t t, s g s ds .

3.28
Using 3.17 we obtain

3.29
From 3.27 and 3.29 we get then T B r is uniformly bounded.

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ii T B r is equicontinuous.Indeed for all t 1 , t 2 ∈ 0, 1 ,

3.33
Some computations give

3.34
On the other hand we have

3.38
Taking into account 3.29 we obtain

3.39
From 3.38 , 3.39 , and 3.22 we deduce that 3.40 this contradicts the fact that u ∈ ∂Ω.Lemma 3.5 allows us to conclude that the operator T has a fixed point u * ∈ Ω and then the fractional boundary value problem P1 has a nontrivial solution u * ∈ E. The proof is complete.

Existence of Positive Solutions
In this section we investigate the positivity of solution for the fractional boundary value problem P1 , for this we make the following hypotheses.
International Journal of Differential Equations Now we give the properties of the Green function.
Lemma 4.1.Let G t, s be the function defined by 2.7 .If α ≥ 1 then G t, s has the following properties: which is positive if α ≥ 1. Hence G t, s is nonnegative for all t, s ∈ 0, 1 .
ii Let t, s ∈ τ, 1 , it is easy to see that G s, s / 0, then we have

4.3
Now we look for lower bounds of G t, s Finally, since G s, s is nonnegative we obtain 0 < τG s, s ≤ G t, s ≤ 2/τ G s, s .
We recall the definition of positive of solution.
Definition 4.2.A function u t is called positive solution of the fractional boundary value problem P1 if u t ≥ 0, for all t ∈ 0, 1 .

Lemma 4.3. If u ∈ E and α ≥ 1, then the solution of the fractional boundary value problem (P1) is positive and satisfies
Proof.First let us remark that under the assumptions on u and f, the function c D σ 0 u is nonnegative.From Lemma 3.1 we have Applying the right-hand side of inequality 4.1 we get

4.8
Combining 4.7 and 4.8 yields In view of the-left hand side of 4.1 , we obtain for all t ∈ τ, 1 International Journal of Differential Equations on the other hand we have From 4.11 and 4.12 we get min with the help of 4.10 we deduce The proof is complete.
Define the quantities A 0 and A ∞ by The case A 0 0 and A ∞ ∞ is called superlinear case and the case A 0 ∞ and A ∞ 0 is called sublinear case.
The main result of this section is as follows.
Theorem 4.4.Under the assumption of Lemma 4.3, the fractional boundary value problem (P1) has at least one positive solution in the both cases superlinear as well as sublinear.
To prove Theorem 4.4 we apply the well-known Guo-Krasnosel'skii fixed point theorem on cone.Let us prove the superlinear case.First, since A 0 0, for any ε > 0, there exists R 1 > 0, such that

4.19
Moreover, we have

4.20
From 4.19 and 4.20 we conclude In view of hypothesis H2 , one can choose ε such that

4.23
Using the left-hand side of 4.1 and Lemma 4.3, we obtain G s, s a s ds.

4.24
Moreover, we get with the help of 4.12

4.25
In view of 4.26 and 4.24 we can write

4.26
Let us choose M such that

4.34
We see that 3.22 is equivalent to 1.2664 r 2 /100 ln 1 r 2 /9 1 − r which is negative for r 6.

Definition 2 . 1 .
If g ∈ C a, b and α > 0, then the Riemann-Liouville fractional integral is defined by

Lemma 2 .6 see 15 .
Let β > α > 0. Then the formula c D α , holds almost everywhere on t ∈ a, b , for f ∈ L 1 a, b and it is valid at any point x ∈ a, b if f ∈ C a, b .Now we start by solving an auxiliary problem.

Lemma 3 . 5
see 17 .Let F be a Banach space and Ω a bounded open subset of F, 0 ∈ Ω. T : Ω → F be a completely continuous operator.Then, either there exists x ∈ ∂Ω, λ > 1 such that T x λx, or there exists a fixed point x * ∈ Ω.

Theorem 4 . 5
see18 .Let E be a Banach space, and let K ⊂ E, be a cone.Assume Ω 1 and Ω 2 are open subsets of E with 0 ∈ Ω 1 , Ω 1 ⊂ Ω 2 and let 3.37 when t 1 → t 2 , in 3.34 and 3.37 then |Tu t 1 − Tu t 2 | and | c D σ 0 Tu t 1 − c D σ 0 Tu t 2 | tend to 0. Consequently T B r is equicontinuous.From Arzelá-Ascoli Theorem we deduce that T is completely continuous operator.Now we apply Leray Schauder nonlinear alternative to prove that T has at least one nontrivial solution in E.
The inequalities 4.21 and 4.22 imply that ||Tu|| ≤ ||u||, for all u ∈ K ∩ ∂Ω 1 .Second, in view of A ∞ ∞, then for any M > 0, there exists R 2 > 0, such thatf 1 u, v ≥ M |u| |v| for |u| |v| ≥ R 2 .Let R max{2R 1 , 2R 2 /τ 2 }and denote by Ω 2 the open set {u The first part of Theorem 4.5 implies that T has a fixed point inK ∩ Ω 2 \ Ω 1 such that R 2 ≤ ||u|| ≤ R.To prove the sublinear case we apply similar techniques.The proof is complete.In order to illustrate our results, we give the following examples.Proof.In this case we have f t, x, y t − 1 /10 3 x t 2 y ln t, 2 < q 5/2 < 3, σ 5/4 < 2, α −1/2 and Thus Theorem 3.2 implies that fractional boundary value problem 4.29 has a unique in E.International Journal of Differential EquationsProof.We apply Theorem 3.4 to prove that the fractional boundary value problem 4.32 has at least one nontrivial solution.We have q 7/3, σ 6/5, α 3/2, and /9, f t, 0, 0 / 0. Let us find r such that 3.22 holds, for this we have