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.
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 is a given function, , and denotes the Caputo's fractional derivative. Our results allow the function to depend on the fractional derivative 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.
In , El-Shahed considered the following nonlinear fractional boundary value problem where and 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  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 and is the Riemann-Liouville fractional derivative.
In  Bai and Lü investigated the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem: where and 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.
2. Preliminaries and Lemmas
In this section, we present some lemmas and definitions from fractional calculus theory which will be needed later.
Definition 2.1. If and , then the Riemann-Liouville fractional integral is defined by
Definition 2.2. Let , . If then the Caputo fractional derivative of order of defined by exists almost everywhere on ( is the entire part of ).
Lemma 2.3 (see ). Let and , then the following relations hold: , and , .
Lemma 2.4 (see ). For , , the homogenous fractional differential equation has a solution where, , , and .
Denote by the Banach space of Lebesgue integrable functions from into with the norm .
The following Lemmas gives some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative.
Lemma 2.5 (see ). Let , . Then and , for all .
Lemma 2.6 (see ). Let . Then the formula , holds almost everywhere on , for and it is valid at any point if .
Now we start by solving an auxiliary problem.
Lemma 2.7. Let , and . The unique solution of the fractional boundary value problem is given by where
Proof. Applying Lemmas 2.4 and 2.5 to (2.5) we get Differentiating both sides of (2.8) and using Lemma 2.6 it yields The first condition in (2.5) implies , the second one gives . Substituting by its value in (2.8), we obtain that can be written as that is equivalent to where is defined by (2.7). The proof is complete.
3. Existence and Uniqueness Results
In this section we prove the existence and uniqueness of solutions in the Banach space of all functions into , with the norm . We know that , , see . Denote by . Throughout this section, we suppose that . Define the integral operator by
Lemma 3.1. The function is solution of the fractional boundary value problem (P1) if and only if , for all .
Proof. Let be solution of (P1) and . In view of (2.10) we have With the help of Lemma 2.6 we obtain It is clear that 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 such that for all and , one has where Then the fractional boundary value problem (P1) has a unique solution in .
To prove Theorem 3.2, we use the following property of Riemann-Liouville fractional integrals.
Lemma 3.3. Let , . Then, for all we have
Proof. Let , then
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 has a fixed point in . Now we will prove that is a contraction. Let , in view of (2.10) we get
with the help of (3.4) we obtain
Lemma 3.3 implies
In view of (3.5) it yields
On the other hand we have
Therefore Applying hypothesis (3.4) we get Let us estimate the term . We have Consequently (3.16) becomes
With the help of hypothesis (3.5) it yields
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 and there exist nonnegative functions , nondecreasing on and , such that where , , and are defined as in Theorem 3.2 and Then the fractional boundary value problem (P1) has at least one nontrivial solution .
To prove this Theorem, we apply Leray-Schauder nonlinear alternative.
Lemma 3.5 (see ). Let be a Banach space and a bounded open subset of , . be a completely continuous operator. Then, either there exists , such that , or there exists a fixed point .
Proof. First let us prove that is completely continuous. It is clear that is continuous since and are continuous. Let be a bounded subset in . We shall prove that is relatively compact.
(i) For and using (3.21) we get Since and are nondecreasing then (3.24) implies using similar techniques as to get (3.12) it yields Hence
Moreover, we have Using (3.17) we obtain From (3.27) and (3.29) we get then is uniformly bounded.
(ii) is equicontinuous. Indeed for all , , , , let , , therefore that implies
Let us consider the function , we see that is decreasing on , consequently , from which we deduce Some computations give On the other hand we have Using (3.17) and (3.28) it yields then when , in (3.34) and (3.37) then and tend to 0. Consequently is equicontinuous. From Arzelá-Ascoli Theorem we deduce that is completely continuous operator.
Now we apply Leray Schauder nonlinear alternative to prove that has at least one nontrivial solution in .
Letting , for any , such that , , we get, with the help of (3.27), Taking into account (3.29) we obtain From (3.38), (3.39), and (3.22) we deduce that this contradicts the fact that . Lemma 3.5 allows us to conclude that the operator has a fixed point and then the fractional boundary value problem (P1) has a nontrivial solution . The proof is complete.
4. 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.(H1) where and .(H2).
Now we give the properties of the Green function.
Lemma 4.1. Let be the function defined by (2.7). If then has the following properties: (i), , for all , .(ii)If , , then
Proof. (i) It is obvious that , moreover, we have
which is positive if . Hence is nonnegative for all .
(ii) Let , it is easy to see that , then we have Now we look for lower bounds of Finally, since is nonnegative we obtain .
We recall the definition of positive of solution.
Definition 4.2. A function is called positive solution of the fractional boundary value problem (P1) if , for all .
Lemma 4.3. If and , then the solution of the fractional boundary value problem (P1) is positive and satisfies
Proof. First let us remark that under the assumptions on and , the function is nonnegative. From Lemma 3.1 we have
Applying the right-hand side of inequality (4.1) we get
Moreover, (4.1) gives
Combining (4.7) and (4.8) yields
In view of the-left hand side of (4.1), we obtain for all
on the other hand we have
From (4.11) and (4.12) we get
with the help of (4.10) we deduce
The proof is complete.
Define the quantities and by The case and is called superlinear case and the case and 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.
Theorem 4.5 (see ). Let be a Banach space, and let , be a cone. Assume and are open subsets of with , and let be a completely continuous operator such that (i), , and , , or(ii), , and , .Then has a fixed point in .
Proof. To prove Theorem 4.4 we define the cone by
It is easy to check that is a nonempty closed and convex subset of , hence it is a cone. Using Lemma 4.3 we see that . From the prove of Theorem 3.4, we know that is completely continuous in .
Let us prove the superlinear case. First, since , for any , there exists , such that for . Letting , for any , it yields Moreover, we have From (4.19) and (4.20) we conclude
In view of hypothesis (H2), one can choose such that The inequalities (4.21) and (4.22) imply that , for all . Second, in view of , then for any , there exists , such that for . Let and denote by the open set . If then Using the left-hand side of (4.1) and Lemma 4.3, we obtain Moreover, we get with the help of (4.12) In view of (4.26) and (4.24) we can write Let us choose such that then we get . Hence, The first part of Theorem 4.5 implies that has a fixed point in such that . To prove the sublinear case we apply similar techniques. The proof is complete.
In order to illustrate our results, we give the following examples.
Example 4.6. The fractional boundary value problem has a unique solution in .
Example 4.7. The fractional boundary value problem has at least one nontrivial solution in .
Proof. We apply Theorem 3.4 to prove that the fractional boundary value problem (4.32) has at least one nontrivial solution. We have , , , and where , , , . Let us find such that (3.22) holds, for this we have We see that (3.22) is equivalent to which is negative for .
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