We study the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to integral boundary conditions. The nonsingular and singular cases for the nonlinearities are investigated.
MSC: 34A08, 45G15.
Keywords:Riemann-Liouville fractional differential equation; integral boundary conditions; positive solutions
We consider the system of nonlinear ordinary fractional differential equations
with the integral boundary conditions
Under sufficient conditions on functions f and g, which can be nonsingular or singular in the points and/or , we study the existence and multiplicity of positive solutions of problem (S)-(BC). We use the Guo-Krasnosel’skii fixed point theorem (see ) and some theorems from the fixed point index theory (from  and ). By a positive solution of problem (S)-(BC) we mean a pair of functions satisfying (S) and (BC) with , for all and , . The system (S) with , and the boundary conditions (BC) where H and K are scale functions (that is, multi-point boundary conditions) has been investigated in  (the nonsingular case) and  (the singular case). In , the authors give sufficient conditions for λ, μ, f, and g such that the system
Fractional differential equations describe many phenomena in various fields of engineering and scientific disciplines such as physics, biophysics, chemistry, biology, economics, control theory, signal and image processing, aerodynamics, viscoelasticity, electromagnetics, and so on (see [7-13]).
In Section 2, we present the necessary definitions and properties from the fractional calculus theory and some auxiliary results dealing with a nonlocal boundary value problem for fractional differential equations. In Section 3, we give some existence and multiplicity results for positive solutions with respect to a cone for our problem (S)-(BC), where f and g are nonsingular functions. The case when f and g are singular at and/or is studied in Section 4. Finally, in Section 5, we present two examples which illustrate our main results.
2 Preliminaries and auxiliary results
We present here the definitions, some lemmas from the theory of fractional calculus and some auxiliary results that will be used to prove our main theorems.
Lemma 2.1 ()
We consider now the fractional differential equation
with the integral boundary conditions
Proof By Lemma 2.2, the solutions of equation (1) are
So, we obtain
Therefore, we get the expression (3) for the solution of problem (1)-(2). □
Lemma 2.4Under the assumptions of Lemma 2.3, the Green’s function for the boundary value problem (1)-(2) is given by
Proof By Lemma 2.3 and relation (3), we conclude
Lemma 2.5 ()
Lemma 2.6Ifis a nondecreasing function and, then the Green’s functionof the problem (1)-(2) is continuous onand satisfiesfor all. Moreover, ifsatisfiesfor all, then the unique solutionuof problem (1)-(2) satisfiesfor all.
Therefore, we obtain the inequalities (b) of this lemma. □
Then we deduce the conclusion of this lemma. □
We can also formulate similar results as Lemmas 2.3-2.8 above for the fractional differential equation
with the integral boundary conditions
3 The nonsingular case
In this section, we investigate the existence and multiplicity of positive solutions for our problem (S)-(BC) under various assumptions on nonsingular functions f and g.
We present the basic assumptions that we shall use in the sequel.
Under the assumptions (H1) and (H2), using also Lemma 2.6, it is easy to see that , ℬ, and are completely continuous from P to P. Thus the existence and multiplicity of positive solutions of the system (S)-(BC) are equivalent to the existence and multiplicity of fixed points of the operator .
Theorem 3.1Assume that (H1)-(H2) hold. If the functionsfandgalso satisfy the conditions:
Proof Because the proof of the theorem is similar to that of Theorem 3.1 from , we will sketch some parts of it. From assumption (i) of (H3), we deduce that there exist such that
From , we deduce that the fixed point index of the operator over with respect to P is
This implies that for all . From , we conclude that the fixed point index of the operator over with respect to P is
Combining (13) and (16), we obtain
Let . Then is a solution of (S)-(BC). In addition . Indeed, if we suppose that , for all , then by using (H2) we have , for all . This implies , for all , which contradicts . The proof of Theorem 3.1 is completed. □
Using similar arguments as those used in the proofs of Theorem 3.2 and Theorem 3.3 in , we also obtain the following results for our problem (S)-(BC).
Theorem 3.2Assume that (H1)-(H2) hold. If the functionsfandgalso satisfy the conditions:
Theorem 3.3Assume that (H1)-(H3), and (H6) hold. If the functionsfandgalso satisfy the condition:
4 The singular case
The basic assumptions used here are the following.
We shall prove that maps bounded sets into relatively compact sets. Suppose is an arbitrary bounded set. Then there exists such that for all . By using () and Lemma 2.7, we obtain for all , where , and . In what follows, we shall prove that is equicontinuous. By using Lemma 2.4, we have
For the integral of the function h, by exchanging the order of integration, we obtain
For the integral of the function μ, we have
From (18), (19), and the absolute continuity of the integral function, we find that is equicontinuous. By the Ascoli-Arzelà theorem, we deduce that is relatively compact. Therefore is a compact operator. Besides, we can easily show that is continuous on P. Hence is completely continuous. □
Proof Because the proof of this theorem is similar to that of Theorem 3 in , we will sketch some parts of it. For c given in (), we consider the cone , where . Under assumptions ()-(), we obtain . By (), we deduce that there exist and such that
By (21), (24), and the Guo-Krasnosel’skii fixed point theorem, we deduce that has at least one fixed point . Then our problem (S)-(BC) has at least one positive solution where . The proof of Theorem 4.1 is completed. □
Using similar arguments as those used in the proof of Theorem 2 in  (see also  for a particular case of the problem studied in ), we also obtain the following result for our problem (S)-(BC).
We consider the system of fractional differential equations
with the boundary conditions
Example 1 We consider the functions
We take and then and . If , then the assumption (H7) is satisfied. For example, if , , , and (e.g.), then the above inequality is satisfied. By Theorem 3.3, we deduce that the problem ()-() has at least two positive solutions.
Example 2 We consider the functions
The authors declare that they have no competing interests.
Both authors contributed equally to this paper. Both authors read and approved the final manuscript.
The work of R Luca was supported by the CNCS grant PN-II-ID-PCE-2011-3-0557, Romania.
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