A coupled system of fractional differential equations with nonlocal integral boundary conditions

Abstract

In this paper, we prove the existence and uniqueness of solutions for a system of fractional differential equations with Riemann-Liouville integral boundary conditions of different order. Our results are based on the nonlinear alternative of Leray-Schauder type and Banachâ[euro](TM)s fixed-point theorem. An illustrative example is also presented.

MSC: 34A08, 34A12, 34B15.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

A coupled system of fractional differential equations with nonlocal integral boundary conditions

Sotiris K. Ntouyas1 and Mustafa Obaid2*

*Correspondence: mailto:[email protected]

Web End [email protected]

2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80111, Jeddah, 21589, Saudi ArabiaFull list of author information is available at the end of the article

Abstract

In this paper, we prove the existence and uniqueness of solutions for a system of

fractional dierential equations with Riemann-Liouville integral boundary conditions

of dierent order. Our results are based on the nonlinear alternative of Leray-Schauder

type and Banachs xed-point theorem. An illustrative example is also presented.

MSC: 34A08; 34A12; 34B15Keywords: Caputo fractional derivative; fractional dierential systems; integral

boundary conditions; xed-point theorems

1 Introduction

In this paper, we investigate a boundary value problem of rst-order fractional dierential equations with Riemann-Liouville integral boundary conditions of dierent order given by

cD+u(t) = f (t, u(t), v(t)), t [, ],

cD+v(t) = g(t, u(t), v(t)), t [, ], u() = Ipu() =

(.)

where cD+, cD+ denote the Caputo fractional derivatives, < , , f , g C([, ]

R, R), and p, q, ,

R.

(s)p

(p) u(s) ds, < < , v() = Iqv() =

(s)q

(q) v(s) ds, < < ,

Fractional dierential equations have recently been addressed by several researchers for a variety of problems. Fractional dierential equations arise in many engineering and scientic disciplines as the mathematical modeling of systems and processes in the elds of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, economics, control theory, signal and image processing, biophysics, blood ow phenomena, etc. []. Fractional-order dierential equations are also regarded as a better tool for the description of hereditary properties of various materials and processes than the corresponding integer order dierential equations. With this advantage, fractional-order models become more realistic and practical than the classical integer-order models, in which such eects are not taken into account. For some recent development on the topic, see [], and the references therein. The study of a coupled system of fractional

2012 Ntouyas and Obaid; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons

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order is also very signicant because this kind of system can often occur in applications. The reader is referred to the papers [], and the references cited therein.

This paper is organized as follows: In Sect. , we present some basic materials needed to prove our main results. In Sect. , we prove the existence and uniqueness of solutions for the system (.) by applying some standard xed-point principles.

2 Preliminaries

Let us introduce the space X = {u(t)|u(t) C([, ])} endowed with the norm u =

max{|u(t)|, t [, ]}. Obviously, (X, ) is a Banach space. Also, let Y = {v(t)|v(t)

C([, ])} endowed with the norm v = max{|v(t)|, t [, ]}. The product space (X

Y, (u, v) ) is also a Banach space with norm (u, v) = u + v .

For the convenience of the readers, we now present some useful denitions and fundamental facts of fractional calculus [, ].

Denition . For at least n-times continuously dierentiable function g : [, )

the Caputo derivative of fractional order q is dened as

cDqg(t) =

(n q)

t (t s)nqg(n)(s) ds, n < q < n, n = [q] + , where [q] denotes the integer part of the real number q.

Denition . The Riemann-Liouville fractional integral of order q is dened as

Iqg(t) =

(q)

t

dierential equation

cDx(t) = g(t), < (.) subject to the boundary condition

x() = Ipx() =

R,

g(s)(t s)q ds, q > ,

provided the integral exists.

The following lemmas gives some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative [].

Lemma . Let p, q , f L[a, b]. Then IpIqf (t) = Ip+qf (t) and cDqIqf (t) = f (t), for all

t [a, b].

Lemma . Let > > , f L[a, b]. Then cDIf (t) = If (t), for all t [a, b].

To dene the solution of the boundary value problem (.), we need the following lemma, which deals with a linear variant of the problem (.).

Lemma . Let = (p+)p. Then for a given g C([, ],

R), the solution of the fractional

( s)p

(p) x(s) ds, <

< (.)

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is given by

x(t) =

()

t (t s)g(s) ds

(p + )

+ (p + ) p

( s)p+ (p + ) g(s) ds, t [, ]. (.)

Proof For some constant c

R, we have []

x(t) =

t

(t s)

() g(s) ds c. (.)

Using the Riemann-Liouville integral of order p for (.), we have

Ipx(t) =

t

(t s)p (p)

s

(s r)

() g(r) dr c

ds

tp

(p + ),

where we have used Lemma .. Using the condition (.) in the above expression, we get

c =

(p + )

(p + ) p Ip+g(

= IpIg(t) c tp

(p + ) = Ip+g(t) c

).

Substituting the value of c in (.), we obtain (.).

3 Main results

For the sake of convenience, we set

M =

( + ) +

| |p+ (p + )

(p + q + )| (p + ) p|

, (.)

M =

( + ) +

||q+ (q + )

(q + + )| (q + ) q|

(.)

and

M = min

(Mk + M), (Mk + M)

. (.)

Dene the operator T : X Y X Y by

T(u, v)(t)

T(u, v)(t)

= T(u, v)(t)

t

(t s)f (s, u(s), v(s)) ds +

(p+)

(p+)

(s)p+

(p+) f (s, u(s), v(s)) ds

t

= (t s)g(s, u(s), v(s)) ds +

(q+)

(q+)

(s)q+

(q+) g(s, u(s), v(s)) ds

.

The rst result is based on Leray-Schauder alternative.

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Lemma . (Leray-Schauder alternative, [] p.) Let F : E E be a completely continu

ous operator (i.e., a map that restricted to any bounded set in E is compact). Let

E(F) =

x E : x = F(x) for some < <

.

Then either the set E(F) is unbounded, or F has at least one xed point.

Theorem . Suppose that = (p+)p and = (q+)q. Assume that there exist real constants

ki, i (i = , ) and k > , > such that xi

R (i = , ), we have

f

(t, x, x)

k + k|x| + k|x|,

g(t,

+ |x| + |x|.

In addition, it is assumed that

Mk + M < and Mk + M < ,

where M and M are given by (.) and (.), respectively. Then the boundary value problem (.) has at least one solution.

Proof First, we show that the operator T : X Y X Y is completely continuous. By

continuity of functions f and g, the operator T is continuous.Let X Y be bounded. Then there exist positive constants L and L such that

f t,

u(t), v(t)

x, x)

L,

g t,

u(t), v(t)

L, (u, v) .

Then for any (u, v) , we have

T

(u, v)(t)

()

t (t s) f s,

u(s), v(s)

ds

+ |

| (p + ) | (p + ) p|

( s)p+ (p + )

f s,

u(s), v(s)

ds

L

( + ) +

| |p+ (p + )

(p + q + )| (p + ) p|

= LM.

Similarly, we get

T

(u, v)

L

( + ) +

||q+ (q + )

(q + + )| (q + ) q|

= LM,

Thus, it follows from the above inequalities that the operator T is uniformly bounded.

Next, we show that T is equicontinuous. Let t t . Then we have

T

u(t), v(t)

T

u(t), v(t)

=

t

(t s)

() f

s, u(s), v(s) ds

t

(t s)

() f

s, u(s), v(s) ds

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L

()

t

(t s) (t s) ds +

tt (t s) ds

L

( + )

t t

.

Analogously, we can obtain

T

u(t), v(t)

T

u(t), v(t)

L

()

t

(t s) (t s) ds +

tt (t s) ds

L

( + )

t t

.

Therefore, the operator T(u, v) is equicontinuous, and thus the operator T(u, v) is completely continuous.

Finally, it will be veried that the set E = {(u, v) X Y|(u, v) = T(u, v), } is

bounded. Let (u, v) E, then (u, v) = T(u, v). For any t [, ], we have

u(t) = T(u, v)(t), v(t) = T(u, v)(t).

Then

u(t)

( + ) +

| |p+ (p + )

(p + q + )| (p + ) p|

k + k

u(t)

+ k

v(t)

and

v(t)

( + ) +

||q+ (q + )

(q + + )| (q + ) q|

+

u(t)

+

v(t) .

Hence, we have

u M

k + k u + k v

and

v M

+ u + v

,

which imply that

u + v = (Mk + M) + (Mk + M) u + (Mk + M) v .

Consequently,

(u,

v)

Mk + M

M ,

for any t [, ], where M is dened by (.), which proves that E is bounded. Thus, by

Lemma ., the operator T has at least one xed point. Hence, the boundary value problem (.) has at least one solution. The proof is complete.

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In the second result, we prove existence and uniqueness of solutions of the boundary value problem (.) via Banachs contraction principle.

Theorem . Assume that f , g : [, ]

R

R are continuous functions and there exist constants mi, ni, i = , such that for all t [, ] and ui, vi

R, i = , ,

f

(t, u, u) f (t, v, v)

m|u v| + m|u v|

and

g(t,

u, u) g(t, v, v)

n|u v| + n|u v|.

In addition, assume that

M(m + m) + M(n + n) < ,

where M and M are given by (.) and (.), respectively. Then the boundary value problem (.) has a unique solution.

Proof Dene supt[,] f (t, , ) = N < and supt[,] g(t, , ) = N < such that

r

NM + NM

M(m + m) M(n + n).

We show that TBr Br, where Br = {(u, v) X Y : (u, v) r}.

For (u, v) Br, we have

T

(u, v)(t)

()

t (t s) f s,

u(s), v(s)

ds

+ |

| (p + ) | (p + ) p|

( s)p+ (p + )

f s,

u(s), v(s)

ds

()

t (t s) f s,

u(s), v(s)

f (t, , )

+

f

(t, , )

ds

+ |

| (p + ) | (p + ) p|

( s)p+

(p + )

f s,

u(s), v(s)

f (t, , )

+

f

(t, , )

ds

( + ) +

| |p+ (p + )

(p + q + )| (p + ) p|

m u + m v + N

M

(m + m)r + N

.

Hence,

T

(u, v)(t)

M

(m + m)r + N

.

In the same way, we can obtain that

T

(u, v)(t)

M

(n + n)r + N

.

Consequently, T(u, v)(t) r.

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Now for (u, v), (u, v) X Y , and for any t [, ], we get

T

(u, v)(t) T(u, v)(t)

()

t (t s) f s,

u(s), v(s)

f

s, u(s), v(s)

ds

+ |

| (p + ) | (p + ) p|

( s)p+ (p + )

f s,

u(s), v(s)

f

s, u(s), v(s)

ds

( + ) +

| |p+ (p + )

(p + q + )| (p + ) p|

m|u u| + m|v v|

M

m u u + m v v

M(m + m) u u + v v

,

and consequently we obtain

T

(u, v)(t) T(u, v)

M(m + m)

u u + v v

. (.)

Similarly,

T

(u, v)(t) T(u, v)

M(n + n)

u u + v v

. (.)

It follows from (.) and (.) that

T(u

, v)(t) T(u, v)(t)

M(m + m) + M(n + n) u u + v v

.

Since M(m +m)+M(n +n) < , therefore, T is a contraction operator. So, by Banachs xed-point theorem, the operator T has a unique xed point, which is the unique solution of problem (.). This completes the proof.

Example . Consider the following system of fractional boundary value problem:

cD/x(t) =

(t+) |

u(t)|

+|u(t)| + + sin v(t), t [, ],

cD/x(t) =

sin(u(t)) + |

v(t)|

(+|v(t)|) + , t [, ],

u() = I/u(

),

v() = I/v(

).

(.)

Here, = /, = , p = /, = /, = /, = , q = /, = /, and f (t, u, v) =

(t+) |

u|

+|u| + + sin v and g(t, u, v) =

(+|v|) + . Note that = =

(p + )/p = (/)/(/)/ and = = (q + )/q = (/)/(/)/. Furthermore, |f (t, u, u) f (t, v, v)|

|u u| +

sin(u) + |

v|

|v v|, |g(t, u, u) g(t, v, v)|

|u u| +

|v v|, and

M(m + m) + M(n + n) =

+

( )

+

+

( )

. < .

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Thus, all the conditions of Theorem . are satised and consequently, its conclusion applies to the problem (.).

Competing interests

The authors declare that they have no competing interests.

Authors contributions

Each of the authors SKN and MO contributed to each part of this study equaly and read and approved the nal version of the manuscript.

Author details

1Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece. 2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80111, Jeddah, 21589, Saudi Arabia.

Received: 10 June 2012 Accepted: 18 July 2012 Published: 31 July 2012

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doi:10.1186/1687-1847-2012-130Cite this article as: Ntouyas and Obaid: A coupled system of fractional differential equations with nonlocal integral boundary conditions. Advances in Dierence Equations 2012 2012:130.