Abstract
We study the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to integral boundary conditions.
MSC:34A08, 45G15.
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1 Introduction
We consider the system of nonlinear ordinary fractional differential equations
with the integral boundary conditions
where , , , and denote the Riemann-Liouville derivatives of orders α and β, respectively, the integrals from (BC) are Riemann-Stieltjes integrals and f, g are sign-changing continuous functions (that is, we have a so-called system of semipositone boundary value problems). These boundary conditions include multi-point and integral boundary conditions and the sum of these in a single framework.
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 [1–7]). Integral boundary conditions arise in thermal conduction problems, semiconductor problems, and hydrodynamic problems.
By using a nonlinear alternative of Leray-Schauder type, we present intervals for parameters λ and μ such that the above problem (S)-(BC) has at least one positive solution. By a positive solution of problem (S)-(BC) we mean a pair of functions satisfying (S) and (BC) with , for all and , for all . In the case when f and g are nonnegative, the above problem (S)-(BC) has been investigated in [8] by using the Guo-Krasnosel’skii fixed point theorem. The system (S) with , and with and replaced by and , respectively, with the boundary conditions (BC), was studied in [9]. In [9], the authors obtained the existence and multiplicity of positive solutions (, for all , , ) by applying some theorems from the fixed point index theory. We would also like to mention the paper [10], where the authors investigated the existence and multiplicity of positive solutions of the semipositone system (S) with and the boundary conditions , , , , , and .
The paper is organized as follows. Section 2 contains some preliminaries and lemmas. The main theorem is presented in Section 3 and, finally, in Section 4, two examples are given to support the new result.
2 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 theorem.
Definition 2.1 The (left-sided) fractional integral of order of a function is given by
provided the right-hand side is pointwise defined on , where is the Euler gamma function defined by , .
Definition 2.2 The Riemann-Liouville fractional derivative of order for a function is given by
where , provided that the right-hand side is pointwise defined on .
The notation stands for the largest integer not greater than α. We also denote the Riemann-Liouville fractional derivative of f by . If then for , and if then for .
Lemma 2.1 ([4])
Let and for and for ; that is, n is the smallest integer greater than or equal to α. Then the solutions of the fractional differential equation , , are
where are arbitrary real constants.
Let , n be the smallest integer greater than or equal to α () and . The solutions of the fractional equation , , are
where are arbitrary real constants.
We consider now the fractional differential equation
with the integral boundary conditions
where , , and is a function of the bounded variation.
By using Lemma 2.2, after some computations, we obtain the following lemma.
Lemma 2.3 ([9])
If is a function of bounded variation, and , then the solution of problem (1)-(2) is , where
Lemma 2.4 The function given by (4) has the properties
-
(a)
is a continuous function, for all and for all .
-
(b)
for all , where .
-
(c)
for all , where
(5)
Proof The first part (a) is evident. For the second part (b), see [11].
For part (c), for , we obtain
If , we have
Therefore, we deduce that , where is defined in (5). □
Lemma 2.5 ([9])
If is a nondecreasing function and , then the Green’s function of problem (1)-(2) given by (3) is continuous on and satisfies for all , for all . Moreover, if satisfies for all , then the unique solution u of problem (1)-(2) satisfies for all .
Lemma 2.6 Assume that is a nondecreasing function and . Then the Green’s function of problem (1)-(2) satisfies the inequalities:
-
(a)
, , where
-
(b)
, , where
Proof (a) We have
(b) For the second inequality, we obtain
We observe that for all , and if , then . □
Lemma 2.7 Assume that is a nondecreasing function, and , for all . Then the solution , of problem (1)-(2) satisfies the inequality for all .
Proof For , we obtain
Therefore, we deduce that for all . □
We can also formulate similar results as Lemmas 2.3-2.7 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 problem (6)-(7) defined in a similar manner as , , , , , , , and , respectively.
In the proof of our main result we shall use the following nonlinear alternative of Leray-Schauder type (see [12]).
Theorem 2.1 Let X be a Banach space with closed and convex. Assume U is a relatively open subset of Ω with , and let be a completely continuous operator (continuous and compact). Then either
-
(1)
S has a fixed point in , or
-
(2)
there exist and such that .
3 Main result
In this section, we investigate the existence of positive solutions for our problem (S)-(BC). We present now the assumptions that we shall use in the sequel.
(H1) are nondecreasing functions, , .
(H2) The functions and there exist functions such that and for any and .
(H3) , for all .
We consider the system of nonlinear fractional differential equations
with the integral boundary conditions
where
and with , is the solution of the system of fractional differential equations
with the integral boundary conditions
By (H2), we have , for all .
We shall prove that there exists a solution for the boundary value problem (8)-(9) with and for all . In this case, the functions and , , represent a nonnegative solution, positive on of the boundary value problem (S)-(BC). Indeed, by (8)-(9) and (10)-(11), we have
and
Therefore, in what follows, we shall investigate the boundary value problem (8)-(9).
By using Lemma 2.3, the system (8)-(9) is equivalent to the system
We consider the Banach space with supremum norm and the Banach space with the norm . We also define the cones
and .
For , we define now the operator by with
Lemma 3.1 If (H1) and (H2) hold, then the operator is a completely continuous operator.
Proof The operators , are well defined. For every , we have
for all , and
for all . Therefore, we obtain
and .
By using standard arguments, we deduce that the operator is a completely continuous operator (a compact operator, that is, it maps bounded sets into relatively compact sets, and it is continuous). □
It is clear that is a solution of problem (8)-(9) if and only if is a fixed point of .
Theorem 3.1 Assume that (H1)-(H3) hold. Then there exist constants and such that for any and , the boundary value problem (S)-(BC) has at least one positive solution.
Proof Let be fixed. From (H3), there exists such that
We define
We will show that for any and , problem (8)-(9) has at least one positive solution.
So, let and be arbitrary, but fixed for the moment. We define the set . We suppose that there exist ( or ) and such that or , .
Because
then, for all , we obtain
Hence and . Then , which is a contradiction.
Therefore, by Theorem 2.1 (with ), we deduce that has a fixed point . That is, and , with and for all .
Moreover, by (12), we obtain
Therefore, and for all .
Let and for all , with , on . Then is a positive solution of the boundary value problem (S)-(BC). □
4 Examples
Let (), (),
and for all . Then and .
We consider the system of fractional differential equations
with the boundary conditions
Then we obtain , .
We also deduce
, , , , , , , , , and .
Example 1 We consider the functions
where , , .
There exists such that , (, ) for all , . Indeed, satisfies the above inequalities.
Let and . Then
for all , . Besides,
Then and . For example, if , , , , , then , , , , , , and .
By Theorem 3.1, for any and , we deduce that problem (S0)-(BC0) has a positive solution , with .
Example 2 We consider the functions
where .
There exists () such that , (, ) for all , .
Let and . Then
Besides,
Then and . For example, if , , , then , , , , and .
By Theorem 3.1, for any and , we deduce that problem (S0)-(BC0) has a positive solution , with .
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Acknowledgements
This work was supported by the CNCS grant PN-II-ID-PCE-2011-3-0557, Romania.
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Luca, R., Tudorache, A. Positive solutions to a system of semipositone fractional boundary value problems. Adv Differ Equ 2014, 179 (2014). https://doi.org/10.1186/1687-1847-2014-179
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DOI: https://doi.org/10.1186/1687-1847-2014-179