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The existence of solutions for a nonlinear mixed problem of singular fractional differential equations
Advances in Difference Equations volume 2013, Article number: 359 (2013)
By using fixed point results on cones, we study the existence of solutions for the singular nonlinear fractional boundary value problem
where is an integer, , , , , f is an -Caratheodory function, and may be singular at value 0 in one dimension of its space variables x, y, z. Here, cD stands for the Caputo fractional derivative.
Fractional differential equations (see, for example, [1–6] and references therein) started to play an important role in several branches of science and engineering. There are some works about existence of solutions for the nonlinear mixed problems of singular fractional boundary value problem (see, for example, [7–11] and ). Also, there are different methods for solving distinct fractional differential equations (see, for example, [13–18] and ). By using fixed point results on cones, we focus on the existence of positive solutions for a nonlinear mixed problem of singular fractional boundary value problem. For the convenience of the reader, we present some necessary definitions from fractional calculus theory (see, for example, ). The Caputo derivative of fractional order α for a function is defined by
Let . As you know, denotes the space of functions, whose q th powers of modulus are integrable on , equipped with the norm . We consider the sup norm
on the space . Also, is the set of absolutely continuous functions on . Let B be a subset of . A function is called an -Caratheodory function whenever the real-valued function on is measurable for all , the function is continuous for almost all , and for each compact set , there exists a function such that for almost all and . Consider the nonlinear fractional boundary value problem
where is an integer, , , , and . We say that the function is a positive solution for the problem whenever on , is a function in , and u satisfies the boundary conditions almost everywhere on . In this paper, we suppose that f is an -Caratheodory function on , where , there exists a positive constant m such that for almost all and , f satisfies the estimate
where are positive and non-increasing, and are positive, w is non-decreasing in all its variables, , , , and . Since we suppose that problem (∗) is singular, that is, may be singular at the value 0 of its space variables x, y, z, we use regularization and sequential techniques for the existence of positive solutions of the problem. In this way, for each natural number n define the function by
for all and , where
It is easy to see that each is an -Caratheodory function on , ,
for almost all and all . In 2012, Agarwal et al. proved the following result.
Lemma 1.1 
Let and . Then we have for all and
2 Main results
Now, we are ready to investigate the problem in regular and singular cases. First, we give the following result.
Lemma 2.1 Let . Then the boundary value problem
is equivalent to the fractional integral equation , where
Proof From and the boundary conditions, we obtain
By properties of the Caputo derivative, we get
By using the boundary conditions and , we get and
This completes the proof. □
Put and . It is easy to check that the Green function G in the last result belongs to , for all ,
for all . Consider the Banach space with the norm and the cone
For each natural number n, define the operator on P by
Now, we prove that is a completely continuous operator (see ).
Lemma 2.2 The operator is a completely continuous operator.
Proof Let . Then, and . Now, define for almost all . Then and for almost all . By using the properties of fractional integral , it is easy to see that , and
for all . This implies that and on . Consequently, maps P into P. In order to prove that is a continuous operator, let be a convergent sequence in P and . Thus, uniformly on for . Since
we get and uniformly on . Also, we have on , and so . Now, put
Then, it is easy to see that for almost all , and there exists such that for almost all and all . Since is an -Caratheodory function, is bounded in , and is bounded in . Therefore, uniformly on . Since is -convergent on ,
uniformly on . Hence, is a continuous operator. Now, we have to show that for each bounded sequence in P, the sequence is relatively compact in . Choose a positive constant k such that and for all m. Note that and for all m, where . But we have
for all and m. This implies that is bounded in . Also, we have
for all , where . Hence, is equicontinuous on . Thus, is relatively compact in by the Arzela-Ascoli theorem. Hence, is a completely continuous operator. □
Lemma 2.3 
Let Y be a Banach space, P a cone in Y and and bounded open balls in Y centered at the origin with . Suppose that is a completely continuous operator such that for all and for all . Then T has a fixed point in .
Theorem 2.4 For each natural number n, problem (∗) has a solution such that , and for all .
Proof Let . It is sufficient to show that has a fixed point in P with the desired conditions. In this way, note that
and so . Put . Then for all . If , then
for all and , because w is non-decreasing in all its variables. Since , , and
where , we have
Hence, , where and . Since
there exists a positive constant L such that
for all . Thus, for all with . Put . Then for all . By using last result, has a fixed point in . But and
for all and . Since ,
for all . This completes the proof. □
Now, we give our last result.
Theorem 2.5 Problem (∗) has a solution u such that , and for all .
Proof By using Theorem 2.4, one gets that for each natural number n, problem (∗) has a solution with the desired conditions. Thus, , and for all and n. Also, we have . Suppose that
for almost all and n. Since , we get
We show that is bounded on . Let . Note that
Thus, for all , where . Also, we have
we get for all n, where , and also . On the other hand, there exists a positive constant L such that for all , and so for all n. Thus, for almost all and all n, we have , where
Note that . We show that is equicontinuous on . Let and . Then
Hence, is equicontinuous on . Since is a bounded sequence in , by using the Arzela-Ascoli theorem, without loss of generality, we can assume that is convergent in . Let . Then, it is easy to see that , and uniformly converges to on . Thus, converges to in . Hence,
for almost all . Since , by using the dominated convergence theorem on the relation
we get for all . This completes the proof. □
2.1 Examples for the problem
Example 2.1 Let , for almost all t in . Suppose that
on , , whenever and whenever , , , and . Then Theorem 2.5 guarantees that problem (∗) has a positive solution.
Example 2.2 Consider the nonlinear mixed problem of singular fractional boundary value problem
via boundary value conditions , and , where . Let f
Then the map f is singular at , and f satisfies the desired conditions, where whenever and whenever , , , , , and . Then Theorem 2.5 guarantees that this problem has a positive solution.
One of the most interesting branches is obtaining solutions of singular fractional differential via boundary value problems. Having these things in mind, we study the existence of solutions for a singular nonlinear fractional boundary value problem. Two illustrative examples illustrate the applicability of the proposed method. It seems that the obtained results could be extended to more general functional spaces. Finally, note that all calculations in proofs of the results depend on the definition of the fractional derivative.
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Research of the second and third authors was supported by Azarbaijan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions, which improved the final version of this paper.
The authors declare that they have no competing interests.
All authors have equal contributions. All authors read and approved the final manuscript.