Abstract
We study the existence and multiplicity of positive solutions for a system of nonlinear RiemannLiouville fractional differential equations, subject to integral boundary conditions. The nonsingular and singular cases for the nonlinearities are investigated.
MSC: 34A08, 45G15.
Keywords:
RiemannLiouville fractional differential equation; integral boundary conditions; positive solutions1 Introduction
We consider the system of nonlinear ordinary fractional differential equations
with the integral boundary conditions
where , , and denote the RiemannLiouville derivatives of orders α and β, respectively, and the integrals from (BC) are RiemannStieltjes integrals.
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 GuoKrasnosel’skii fixed point theorem (see [1]) and some theorems from the fixed point index theory (from [2] and [3]). 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, multipoint boundary conditions) has been investigated in [4] (the nonsingular case) and [5] (the singular case). In [6], the authors give sufficient conditions for λ, μ, f, and g such that the system
with the boundary conditions (BC) with H and K scale functions, has positive solutions (, for all , and ).
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 [713]).
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.
Definition 2.1 The (leftsided) fractional integral of order of a function is given by
provided the righthand side is pointwise defined on , where is the Euler gamma function defined by , .
Definition 2.2 The RiemannLiouville fractional derivative of order for a function is given by
where , provided that the righthand side is pointwise defined on .
The notation stands for the largest integer not greater than α. We also denote the RiemannLiouville fractional derivative of f by . If then for , and if then for .
Lemma 2.1 ([10])
Letandforandfor; that is, nis the smallest integer greater than or equal toα. Then the solutions of the fractional differential equation, , are
whereare arbitrary real constants.
Let, nbe the smallest integer greater than or equal toα () and. The solutions of the fractional equation, , are
whereare arbitrary real constants.
We consider now the fractional differential equation
with the integral boundary conditions
where , , and is a function of the bounded variation.
Lemma 2.3Ifis a function of bounded variation, and, then the solution of problem (1)(2) is given by
Proof By Lemma 2.2, the solutions of equation (1) are
where . By using the conditions , we obtain . Then we conclude
or
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
where
Proof By Lemma 2.3 and relation (3), we conclude
where and are given in (5) and (4), respectively. Hence for all . □
Lemma 2.5 ([6])
The functiongiven by (5) has the properties:
(a) is a continuous function andfor all.
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.
Proof By using the assumptions of this lemma, we have for all , and so for all . □
Lemma 2.7Assume thatis a nondecreasing function and. Then the Green’s functionof the problem (1)(2) satisfies the inequalities:
Proof The first inequality (a) is evident. For part (b), for and , , we deduce
Therefore, we obtain the inequalities (b) of this lemma. □
Lemma 2.8Assume thatis a nondecreasing function and, , and, for all. Then the solution, of problem (1)(2) satisfies the inequality.
Then we deduce the conclusion of this lemma. □
We can also formulate similar results as Lemmas 2.32.8 above for the fractional differential equation
with the integral boundary conditions
where , , is a nondecreasing function and . We denote by , , , , , and the corresponding constants and functions for the problem (6)(7) defined in a similar manner as , , , , , and , respectively.
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.
(H1) are nondecreasing functions, , .
(H2) The functions are continuous and for all .
A pair of functions is a solution for our problem (S)(BC) if and only if is a solution for the nonlinear integral system
We consider the Banach space with supremum norm and define the cone by .
We also define the operators by
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:
(H3) There exist positive constantsandsuch that
(H4) There exists a positive constantsuch that
then the problem (S)(BC) has at least one positive solution, .
Proof Because the proof of the theorem is similar to that of Theorem 3.1 from [4], we will sketch some parts of it. From assumption (i) of (H3), we deduce that there exist such that
Then for , by using (8), Lemma 2.6, and Lemma 2.7, we obtain after some computations
For c given in (H3), we define the cone , where . From our assumptions and Lemma 2.8, for any , we can easily show that and , that is, and .
We now consider the function , , with for all . We define the set
We will show that and M is a bounded subset of X. If , then there exists such that , . From the definition of , we have
where is defined by . Therefore, , and from the definition of , we get
From (ii) of assumption (H3), we conclude that for there exists such that
For and , by using Lemma 2.7 and the relations (9) and (11), it follows that
Now from relations (10) and (12), one obtains , for all , that is, M is a bounded subset of X.
Besides, there exists a sufficiently large such that
From [2], we deduce that the fixed point index of the operator over with respect to P is
Next, from assumption (H4), we conclude that there exist and such that
where , , . Hence, for any and , we obtain
Therefore, by (14) and (15), we deduce that for any and
This implies that for all . From [2], we conclude that the fixed point index of the operator over with respect to P is
Combining (13) and (16), we obtain
We deduce that has at least one fixed point , that is, .
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 [4], we also obtain the following results for our problem (S)(BC).
Theorem 3.2Assume that (H1)(H2) hold. If the functionsfandgalso satisfy the conditions:
(H5) There exists a positive constantsuch that
then the problem (S)(BC) has at least one positive solution, .
Theorem 3.3Assume that (H1)(H3), and (H6) hold. If the functionsfandgalso satisfy the condition:
(H7) For each, andare nondecreasing with respect tou, and there exists a constantsuch that
where, , , and, are defined in Section 2, then the problem (S)(BC) has at least two positive solutions, , .
4 The singular case
In this section, we investigate the existence of positive solutions for our problem (S)(BC) under various assumptions on functions f and g which may be singular at and/or .
The basic assumptions used here are the following.
() The functions and there exist , , , with , , , such that
We consider the Banach space with supremum norm and define the cone by . We also define the operator by
Lemma 4.1Assume that ()() hold. Thenis completely continuous.
Proof We denote by and . Using (), we deduce that and . By Lemma 2.6 and the corresponding lemma for , we see that maps P into P.
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
Therefore, for any , we obtain
We denote
For the integral of the function h, by exchanging the order of integration, we obtain
For the integral of the function μ, we have
We deduce that . Thus for any given with and , by (17), we conclude
From (18), (19), and the absolute continuity of the integral function, we find that is equicontinuous. By the AscoliArzelà 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. □
Theorem 4.1Assume that ()() hold. If the functionsfandgalso satisfy the conditions:
() There existwithandsuch that
then the problem (S)(BC) has at least one positive solution, .
Proof Because the proof of this theorem is similar to that of Theorem 3 in [5], 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 using (20) and (), for any , we conclude
By the definition of , we can choose sufficiently large such that
From (), we deduce that there exist positive constants , , and such that
where and . From the assumption and the continuity of , we conclude that there exists sufficiently small such that for all , where . Therefore for any and , we have
By (22), (23), Lemma 2.7, and Lemma 2.8, for any and , we obtain
Therefore
By (21), (24), and the GuoKrasnosel’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 [5] (see also [14] for a particular case of the problem studied in [5]), we also obtain the following result for our problem (S)(BC).
Theorem 4.2Assume that ()() hold. If the functionsfandgalso satisfy the conditions:
() There existwithandsuch that
then the problem (S)(BC) has at least one positive solution, .
5 Examples
We consider the system of fractional differential equations
with the boundary conditions
Then we obtain , . We also deduce
For the functions and , we obtain
and
Example 1 We consider the functions
where , , , , . We have , . Then . The functions and are nondecreasing with respect to u, for any , and for and the assumptions (H3) and (H6) are satisfied; indeed we obtain
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
In (), for , , , and , we have
For example, if , , , , , , the above conditions are satisfied. Then, by Theorem 4.2, we deduce that the problem ()() has at least one positive solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to this paper. Both authors read and approved the final manuscript.
Acknowledgements
The work of R Luca was supported by the CNCS grant PNIIIDPCE201130557, Romania.
References

Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, New York (1988)

Amann, H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev.. 18, 620–709 (1976). Publisher Full Text

Zhou, Y, Xu, Y: Positive solutions of threepoint boundary value problems for systems of nonlinear second order ordinary differential equations. J. Math. Anal. Appl.. 320, 578–590 (2006). Publisher Full Text

Henderson, J, Luca, R: Existence and multiplicity for positive solutions of a system of higherorder multipoint boundary value problems. Nonlinear Differ. Equ. Appl.. 20(3), 1035–1054 (2013). Publisher Full Text

Henderson, J, Luca, R: Positive solutions for singular systems of higherorder multipoint boundary value problems. Math. Model. Anal.. 18(3), 309–324 (2013). Publisher Full Text

Henderson, J, Luca, R: Positive solutions for a system of nonlocal fractional boundary value problems. Fract. Calc. Appl. Anal.. 16(4), 985–1008 (2013)

Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods, World Scientific, Boston (2012)

Das, S: Functional Fractional Calculus for System Identification and Controls, Springer, New York (2008)

Graef, JR, Kong, L, Kong, Q, Wang, M: Uniqueness of positive solutions of fractional boundary value problems with nonhomogeneous integral boundary conditions. Fract. Calc. Appl. Anal.. 15(3), 509–528 (2012)

Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

Podlubny, I: Fractional Differential Equations, Academic Press, San Diego (1999)

Sabatier J, Agrawal OP, Machado JAT (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht (2007)

Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, Yverdon (1993)

Liu, B, Liu, L, Wu, Y: Positive solutions for singular systems of threepoint boundary value problems. Comput. Math. Appl.. 53, 1429–1438 (2007). Publisher Full Text