Uniqueness results for fully anti-periodic fractional boundary value problems with nonlinearity depending on lower-order derivatives

Abstract

We investigate the uniqueness of solutions for fully anti-periodic fractional boundary value problems of order [InlineEquation not available: see fulltext.] with nonlinearity depending on lower-order fractional derivatives. Our results are based on some standard fixed point theorems. The paper concludes with illustrative examples.

MSC: 34A12, 34A40.

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R E S E A R C H Open Access

Uniqueness results for fully anti-periodic fractional boundary value problems with nonlinearity depending on lower-order derivatives

Ahmed Alsaedi1*, Bashir Ahmad1, Nadia Mohamad1 and Sotiris K Ntouyas2

*Correspondence: [email protected]

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi ArabiaFull list of author information is available at the end of the article

Abstract

We investigate the uniqueness of solutions for fully anti-periodic fractional boundary value problems of order 2 < q 3 with nonlinearity depending on lower-order

fractional derivatives. Our results are based on some standard xed point theorems. The paper concludes with illustrative examples.

MSC: 34A12; 34A40

Keywords: dierential equations of fractional order; anti-periodic fractional boundary conditions; uniqueness; xed point

1 Introduction

In this article, we show the existence of solutions for a fully fractional-order anti-periodic boundary value problem of the form

cDqx(t) = f t, x(t), cDrx(t) , t [, T], T > , < q , < r , (.)

x() = x(T), cDpx() = cDpx(T),

cDp+x() = cDp+x(T), < p < ,

(.)

where cDq denotes the Caputo fractional derivative of order q and f is a given continuous function.

As a second problem, we will discuss the existence of solutions for the following fractional dierential equation with the boundary conditions (.):

cDqx(t) = f t, x(t), cDrx(t), cDr+x(t) , t [, T], T > , < q , < r . (.)

The present work is motivated by a recent paper [] in which the problem (.)-(.) was discussed with the nonlinearity of the type f (t, x). Thus the present paper generalizes the results obtained in [].

In the last few decades, fractional calculus has evolved as an attractive eld of research in view of its extensive applications in basic and technical sciences. Examples can be found

2014 Alsaedi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons At

tribution License (http://creativecommons.org/licenses/by/2.0

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any medium, provided the original work is properly cited.

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in physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood ow phenomena, aerodynamics, tting of experimental data, etc. [].

The subject of boundary value problems of dierential equations, having an enriched history, has been progressing at the same pace as before. In the context of fractional boundary value problems, there has been a much development in the last ten years; for instance, see [] and the references cited therein.

In view of the importance of anti-periodic boundary conditions in the mathematical modeling of a variety of physical processes [], the study of anti-periodic boundary value problems has received considerable attention. Some recent work on anti-periodic boundary value problems of fractional order can be found in a series of papers [] and the references therein.

2 Preliminaries

We begin this section with some basic concepts [, ].

Denition . The Riemann-Liouville fractional integral of order q for a continuous function g is dened as

Iqg(t) =

(q)

t

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

provided the integral exists.

Denition . For a function g ACn([, ),

R), 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.

Lemma . [] For any y C[, T], the unique solution of the linear fractional equation

cDqx(t) = y(t), < t < T, < q with anti-periodic boundary conditions (.) is given by x(t) =

T G(t, s)y(s) ds,where G(t, s) is the Greens function (depending on q and p) given by

G(t, s) =

(ts)q(Ts)q

(q) + (p)(Tt)(Ts)

qp

Tp (qp)

+ ( (p))

Tp(Ts)qp (qp) (p) {(T

t) (p)

(p) T + tT}, s t,

(Ts)

q

(q) + (p)(Tt)(Ts)

qp

Tp (qp)

+ ( (p))

Tp(Ts)qp (qp) (p) {(T

t) (p)

(p) T + tT}, t < s.

3 Uniqueness of solutions

This section is devoted to the uniqueness of solutions for the problems at hand by means of Banachs contraction principle.

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3.1 Uniqueness result for the problem (1.1)-(1.2)

For < r , let us dene a space C = {x : x, cDrx C([, T])}, where C([, T]) denotes the space of all continuous functions dened on [, T]. Note that the space C endowed with the norm dened by x = sup{|x(t)| + |cDrx(t)|, t [, T]} is a Banach space.

In view of Lemma ., let us dene an operator G : C C associated with the problem (.)-(.) as

(Gx)(t) =

t

(q) f s,

x(s), cDrx(s)

ds

(t s)q

T

(q) f s,

x(s), cDrx(s)

ds

(T s)q

+ (t)

T

(T s)qp

(q p) f s,

x(s), cDrx(s)

ds

+ (t)

T

(T s)qp

(q p )f s,

x(s), cDrx(s)

ds, (.)

where

(t) =

(p) + tT (p) (p))

Tp .

Observe that the problem (.)-(.) has a solution only if the operator G has a xed point.

Before proceeding further, let us introduce some notations:

N = max{N, N}, (.)

where

N = Tq

(q + ) +

( p)(T t) Tp ,

(t) =

( p)(T t (p)T

( p) (q p + ) +

( p) (q p)

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

,

N = Tqr

(q r + ) +

( p) ( r) (q p + )

+

( p) (q p)

( p) ( p) ( r)

( r)

.

Theorem . Assume that f : [, T]

R

R

R is a continuous function satisfying the

condition

f

(t, x, x) f (t, y, y)

L |x

y| + |x y| ,

t [, T], x, y, x, y

R

with L < N, where N is given by (.). Then the anti-periodic boundary value problem (.)-(.) has a unique solution.

Proof Let us set supt[,T] |f (t, , )| = M < and R MN( LN) to show that GBR BR, where BR = {x C : x R}. For x BR, we have

(

Gx)(t)

t

(t s)q

(q)

f s,

x(s), cDrx(s)

ds

+

T

(T s)q

(q)

f

s, x(s), cDrx(s)

ds

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+

(t)

T

(T s)qp

(q p)

f s,

x(s), cDrx(s)

ds

+

(t)

T

(T s)qp

(q p )

f s,

x(s), cDrx(s)

ds

t

(t s)q

(q)

f s,

x(s), cDrx(s)

f (s, , )

+

f

(s, , )

ds

+

T

(T s)q

(q)

f s,

x(s), cDrx(s)

f (s, , )

+

f

(s, , )

ds

+

(t)

T

(T s)qp

(q p)

f s,

x(s), cDrx(s)

f (s, , )

+

f

(s, , )

ds

+

(t)

T

(T s)qp

(q p )

f s,

x(s), cDrx(s)

f (s, , )

+

f

(s, , )

ds

(LR + M)

t

(t s)q

(q) ds +

T

(T s)q

(q) ds

+

(t)

T

(T s)qp

(q p) ds + (t)

T

(T s)qp

(q p ) ds

(LR + M)N (LR + M)N R.

Using the facts cDrb = (b is a constant), cDrt = t

r (r) , cDrt =

tr (r) , cDr+t =

tr (r) , for

< r < , we get

cDrGx (t) = t(t s)qr (q r) f s,x(s), cDrx(s)

ds

( p)tr

( r)Tp

T

(T s)qp

(q p) f s,

x(s), cDrx(s)

ds

+ ( p)Tp

Ttr ( p)

( p) ( r)

tr

( r)

T

(T s)qp

(q p )f s,

x(s), cDrx(s)

ds.

As in the previous step, it can be shown that

cDr

Gx (t)

(LR + M)N (LR + M)N R.

Thus we get Gx BR. Hence GBR BR. Next, for x, x C and for each t [, T], we obtain

(

Gx)(t) (Gx)(t)

t

(t s)q

(q)

f s,

x, cDrx

f

s, x, cDrx

ds

+

T

(T s)q

(q)

f s,

x, cDrx

f

s, x, cDrx

ds

+

(t)

T

(T s)qp

(q p)

f

s, x, cDrx

f

s, x, cDrx

ds

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+

(t)

T

(T s)qp

(q p )

f s,

x, cDrx

f

s, x, cDrx

ds

L x x

t

(t s)q

(q) ds +

T

(T s)q

(q) ds

+

(t)

T

(T s)qp

(q p) ds + (t)

T

(T s)qp

(q p ) ds

< LN x x LN x x .

In a similar manner, we nd that

cDr

Gx (t) cDrG

x

(t)

LN x x LN x x .

By the given assumption, L < /N, it follows that the operator G is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach xed point theorem).

3.2 Uniqueness result for the problem (1.3)-(1.2)

Here, we study the uniqueness of solutions for the problem of (.)-(.). For that, let

C = {x : x, cDrx, cDr+x(t) C([, T])} be a Banach space endowed with the norm x =

sup{|x(t)| + |cDrx(t)| + |cDr+x(t)|, t [, T]}, < r .Relative to the problem (.)-(.), we dene an operator

G :

C

C as

(

Gx)(t) =

t

(q) f s,

x(s), cDrx(s), cDr+x(s)

ds

(t s)q

T

(T s)q

(q) f s,

x(s), cDrx(s), cDr+x(s)

ds

+ (t)

T

(T s)qp

(q p) f s,

x(s), cDrx(s), cDr+x(s)

ds

+ (t)

T

(T s)qp

(q p )f s,

x(s), cDrx(s), cDr+x(s)

ds. (.)

In what follows, we set

N = max{N, N}, (.)

where N is given by (.) and

N = Tqr

(q r) +

( p) ( r) (q p)

.

R be a continuous function and there exists a positive number L < / N such that

f

(t, x, x, x) f (t, y, y, y)

Theorem . Let f : [, T]

R

R

L |x

y| + |x y| + |x y| ,

t [, T], x, y, x, y, x, y

R.

Then the anti-periodic boundary value problem (.)-(.) has a unique solution on [, T].

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Proof We dene BR = {x

C : x R}, R

M N

L N , M = supt[,T] |f (t, , , )| < and

show that

GBR BR. In view of the given assumption, we have

f s,

x(s), cDrx(s), cDr+x(t)

f s,

x(s), cDrx(s), cDr+x(t)

f (t, , , )

+

f

(t, , , )

L R + M, x BR. (.)

Thus

(

Gx)(t)

t

(t s)q

(q)

f s,

x(s), cDrx(s), cDr+x(t)

ds

+

T

(T s)q

(q)

f s,

x(s), cDrx(s), cDr+x(t)

ds

+

(t)

T

(T s)qp

(q p)

f s,

x(s), cDrx(s), cDr+x(t)

ds

+

(t)

T

(T s)qp

(q p )

f s,

x(s), cDrx(s), cDr+x(t)

ds

(L R + M)

t

(t s)q

(q) ds +

T

(T s)q

(q) ds

+

(t)

T

(T s)qp

(q p) ds + (t)

T

(T s)qp

(q p ) ds

(L R + M)N

(L R + M) N R.

Further, it can be shown in a similar way that

cDr

Gx (t)

(LR + M)N (L R + M) N R,

cDr+

Gx (t)

(LR + M)N (LR + M) N R.

Next, for x, x

C and for each t [, T], we obtain

(

Gx)(t) (

Gx)(t)

t

(t s)q

(q)

f s,

x, cDrx, cDr+x

f

s, x, cDrx, cDr+x

ds

+

T

(T s)q

(q)

f s,

x, cDrx, cDr+x

f

s, x, cDrx, cDr+x

ds

+

(t)

T

(T s)qp

(q p)

f s,

x, cDrx, cDr+x

f

s, x, cDrx, cDr+x

ds

+

(t)

T

(T s)qp

(q p )

f s,

x, cDrx, cDr+x

f

s, x, cDrx, cDr+x

ds

L x x

t

(t s)q

(q) ds +

T

(T s)q

(q) ds

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+

(t)

T

(T s)qp

(q p) ds + (t)

T

(T s)qp

(q p ) ds

< LN x x L N x x .

Also, we have

cDr

Gx (t)

LN x x L N x x ,

cDr+

Gx (t) cDr+

Gx (t)

LN x x L N x x .

Since L < / N, therefore, the operator

G is a contraction. Thus, it follows by the contraction mapping principle that the problem (.)-(.) has a unique solution on [, T].

4 Examples

(a) Consider the anti-periodic fractional boundary value problem given by

cD

x(t) = L

|

x(t)| + |x(t)|

+

+ t

|cD

x(t)| + |cD

+ +

sin t, L > , t [, ],

x(t)|

x() = x(), cD

x() = cD

x(), cD

x() = cD

x(), (.)

where q = /, p = /, r = /, T = , and

f (t, x, x) = L |

x| + |x|

+

+ t

|x| + |x|

+ + sin t, x = cD x(t).

Clearly

f

(t, x, x) f (t, y, y)

L

x + x

y + y

+

x + x

y

+ y

L |x

y| + |x y| ,

N =

+

, N =

.

With L < /N (N = N > N), all the assumptions of Theorem . hold. Therefore, the problem (.) has a unique solution on [, ].(b) Consider the following anti-periodic fractional boundary value problem:

cD

x(t) = L |

x(t)| + |x(t)|

+ tan

cD

x(t)

+ cos t

|c

D x(t)|

+ |cD

x(t)|

+ et, L > , t [, ],

x() = x(), cD

x() = cD

x(), cD

x() = cD

x(), (.)

where q = /, p = /, r = /, T = . With x = cD

x(t), x = cD

x(t), we can write

f (t, x, x, x) = L |

x| + |x|

+ tan

(

x)

+ cos t

|

x| + |x|

+ et.

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Furthermore, we have

f

(t, x, x, x) f (t, y, y, y)

N =

L |x

y| + |x y| + |x y ,

(/) (/), N = +

(/)

(/).

Clearly all the assumptions of Theorem . are satised with L < / N ( N = N > N > N). Hence, the problem (.) has a unique solution on [, ].

Competing interests

The authors declare that they have no competing interests.

Authors contributions

Each of the authors, AA, BA, NM, and SKN, contributed to each part of this work equally and read and approved the nal version of the manuscript.

Author details

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.

2Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece.

Authors information

Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

Acknowledgements

This research was partially supported by the Deanship of Scientic Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

Received: 20 February 2014 Accepted: 2 April 2014 Published: 7 May 2014

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doi:10.1186/1687-1847-2014-136Cite this article as: Alsaedi et al.: Uniqueness results for fully anti-periodic fractional boundary value problems with nonlinearity depending on lower-order derivatives. Advances in Dierence Equations 2014 2014:136.