Positive Solutions for Systems of Coupled Fractional Boundary Value Problems

Johnny Henderson¹, Rodica Luca², Alexandru Tudorache³

¹Department of Mathematics, Baylor University, Waco, Texas, USA.
²Department of Mathematics, Gh. Asachi Technical University, Iasi, Romania.
³Faculty of Computer Engineering and Automatic Control, Gh. Asachi Technical University, Iasi, Romania.

DOI: 10.4236/ojapps.2015.510059                

Abstract

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with coupled integral boundary conditions which contain some positive constants.

Keywords

Riemann-Liouville Fractional Differential Equations, Coupled Integral Boundary Conditions, Positive Solutions

1. Introduction

Fractional differential equations describe many phenomena in various fields of engineering and scientific dis- ciplines such as physics, biophysics, chemistry, biology (such as blood flow phenomena), economics, control theory, signal and image processing, aerodynamics, viscoelasticity, electromagnetics, and so on (see [1] -[6] ). For some recent developments on the topic, which can be seen in [7] - [19] and the references therein.

In this paper, we consider the system of nonlinear ordinary fractional differential equations

(S)

with the coupled integral boundary conditions

(BC)

where, , , , and denote the Riemann-Liouville derivatives of orders and, respectively, the integrals from (BC) are Riemann-Stieltjes integrals, and are positive constants.

Under some assumptions on the functions f and g, we shall prove the existence of positive solutions of problem (S)-(BC). By a positive solution of (S)-(BC), we mean a pair of functions satisfying (S) and (BC) with, for all. We shall also give sufficient conditions for

the nonexistence of positive solutions for this problem. Some systems of fractional equations with parameters subject to coupled integral boundary conditions were studied in [20] by using the Guo-Krasnosel’skii fixed point theorem. We also mentioned the paper [21] , where we investigated the existence and multiplicity of positive

solutions for the system, , with the integral boundary conditions (BC) with by using some theorems from the fixed point index theory and the

Guo-Krasnosel’skii fixed point theorem. In [21] , the nonlinearities f and g may be nonsingular or singular in and/or. Some systems of Riemann-Liouville fractional equations with or without parameters subject to uncoupled boundary conditions are studied in the papers [22] - [25] , and the book [26] .

In Section 2, we present some auxiliary results which investigate a system of Riemann-Liouville fractional equations subject to coupled integral boundary conditions. In Section 3, we prove our main results, and an example which supports the obtained results is finally presented in Section 4. In the proof of our existence result, we shall use the Schauder fixed point theorem which we present now.

Theorem 1. Let X be a Banach space and a nonempty, bounded, convex and closed subset. If the operator is completely continuous, then A has at least one fixed point.

2. Auxiliary Results

We present here the definitions of the fractional integral and Riemann-Liouville fractional derivative of a function, and some auxiliary results from [20] and [22] that will be used to prove our main theorems.

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. If then for, and if then for.

We consider now the fractional differential system

(1)

with the coupled integral boundary conditions

(2)

where, and are functions of bounded variation.

Lemma 1. ( [20] ) If are functions of bounded variations, and, then the unique solution of problem (1)-(2) is given by

(3)

where

(4)

and

(5)

Lemma 2. ( [22] ) The functions and given by (5) have the properties

a) are continuous functions and, for all ;

b), for all;

c) For any, we have

for all, where, , and

Lemma 3. ( [20] ) If are nondecreasing functions, and, then, given by (4) are continuous functions on and satisfy for all,. Moreover, if satisfy, for all, then the solution of problem (1)-(2) satisfies for all.

Lemma 4. ( [20] ) Assume that are nondecreasing functions and. Then the functions satisfy the inequalities

a₁), where

a₂) For every, we have

b₁), where

b₂) For every, we have

c₁), where

c₂) For every, we have

d₁), where

d₂) For every, we have

Lemma 5. ( [20] ) Assume that are nondecreasing functions, , and, , for all. Then the solution, of problem (1)-(2) (given by (3)) satisfies the inequalities

3. Main Results

We present first the assumptions that we shall use in the sequel.

(J₁) are nondecreasing functions and.

(J₂) The functions are continuous and there exist such that,.

(J₃) are continuous functions and there exists such that

, for all,

where and are de- fined in Lemma 4.

(J₄) are continuous functions and satisfy the conditions

By assumption (J₂) we deduce that, , and , that is, the constant L from (J₃) is positive.

Our first theorem is the following existence result for problem (S)-(BC).

Theorem 2. Assume that assumptions (J₁)-(J₃) hold. Then problem (S)-(BC) has at least one positive solution for and sufficiently small.

Proof. We consider the system of ordinary fractional differential equations

(6)

with the coupled integral boundary conditions

(7)

with and.

The above problem (6)-(7) has the solution

(8)

where is defined in (J₁). By assumption (J₁) we obtain and for all.

We define the functions and, by

where is a solution of (S)-(BC). Then (S)-(BC) can be equivalently written as

(9)

with the boundary conditions

(10)

Using the Green’s functions, from Lemma 1, a pair is a solution of problem (9)-(10) if and only if is a solution for the nonlinear integral equations

(11)

where and, are given in (8).

We consider the Banach space with the supremum norm, the space with the norm, and we define the set

We also define the operators and by

for all, and.

For sufficiently small and, by (J3), we deduce

Then, by using Lemma 3, we obtain, for all and. By Lemma 4, for all, we have

and

Therefore.

Using standard arguments, we deduce that S is completely continuous. By Theorem 1, we conclude that S has a fixed point, which represents a solution for problem (9)-(10). This shows that our problem (S)-(BC) has a positive solution with for sufficiently small and.

In what follows, we present sufficient conditions for the nonexistence of positive solutions of (S)-(BC).

Theorem 3. Assume that assumptions (J₁), (J₂) and (J₄) hold. Then problem (S)-(BC) has no positive solution for and sufficiently large.

Proof. We suppose that is a positive solution of (S)-(BC). Then with, is a solution for problem (9)-(10), where is the solution of problem (6)-(7) (given by (8)). By (J₂) there exists such that, and then, , ,. Now by using Lemma 3, we have, for all, and by Lemma 5 we obtain and.

Using now (8), we deduce that and. Therefore, we obtain and.

We now consider. By using (J₄), for R defined above, we conclude that there exists such that, for all. We consider and sufficiently large such that and. By (J₂), (9), (10) and the above inequalities, we deduce that and.

Now by using Lemma 4 and the above considerations, we have

Therefore, we obtain, which is a contradiction, because. Then, for and sufficiently large, our problem (S)-(BC) has no positive solution.

4. An Example

We consider, for all, , , , for all, then and. We also consider the functions, , , for all, with. We have.

Therefore, we consider the system of fractional differential equations

(S₀)

with the boundary conditions

(BC₀)

Then we obtain

We also deduce

, for all. For the functions, , we obtain

Then we deduce that assumptions (J₁), (J₂) and (J₄) are satisfied. In addition, by using the above functions, , we obtain, ,

, , and then. We choose and if we select, then we conclude that, for all. For example, if and, then the above conditions for f and g are satisfied. So,

assumption (J₃) is also satisfied. By Theorems 2 and 3 we deduce that problem (S₀)-(BC₀) has at least one positive solution for sufficiently small and, and no positive solution for sufficiently large and.

Acknowledgements

The work of R. Luca and A. Tudorache was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0557.

Conflicts of Interest

The authors declare no conflicts of interest.

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