## Abstract

In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem *D*^{α}*u*(*t*)=*f*(*t*,*u*(*t*)) with a Riemann–Liouville fractional derivative via the different boundary-value problems *u*(0)=*u*(*T*), and the three-point boundary condition *u*(0)=*β*_{1}*u*(*η*) and *u*(*T*)=*β*_{2}*u*(*η*), where *T*>0, *t*∈*I*=[0,*T*], 0<*α*<1, 0<*η*<*T*, 0<*β*_{1}<*β*_{2}<1.

## 1. Introduction

The field of fractional differential equations has been subjected to an intensive development of theory and applications ([1–9] and references therein). For a new history of fractional calculus, see Machado *et al.* [10]. It should be noted that most papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations in terms of special functions. Recently, there have been some papers dealing with the existence of solutions of nonlinear initial-value problems of fractional differential equations by using techniques of nonlinear analysis such as fixed-point results, the Leray–Schauder theorem and stability [11–19]. In fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena and aerodynamics ([6,20–22] and references therein). The main advantage of using fractional nonlinear differential equations is related to the fact that we can describe the dynamics of complex non-local systems with memory. In this line of thought, the equations involving various fractional orders are important from both theoretical and applied view points. As is known, fixed-point theory has wide applications in several areas, e.g. economics, dynamic systems, the theory of differential and integral equations and so on [23,24]. In particular, fractional nonlinear differential equations is a still new arena where new fixed-point methods can be applied successfully.

The organization of the manuscript is as follows. In §2, we present briefly the basic definitions and results from the area of fractional differential equations used in the following chapters. Section 3 presents the main results. Finally, §4 depicts our conclusions.

## 2. Basic tools

First, let us recall some basic definitions of fractional calculus [4,5]. For a continuous function , the Caputo derivative of fractional order *α* is defined as
where [*α*] denotes the integer part of the real number *α*. In addition, the Riemann–Liouville fractional derivative of order *α* for a continuous function *f*(*t*) is defined by
provided the right-hand side is point-wise defined on [3].

Recently, the notions of *α*−*ψ*-contractive and *α*-admissible mappings were introduced in [7].

### Definition 2.1

Let (*X*,*d*) be a metric space and be a given mapping. We say that *T* is an *α*−*ψ*-contractive mapping whenever there exist two functions *ψ*∈*Ψ* and such that *α*(*x*,*y*) d(*Tx*,*Ty*)≤*ψ*(d(*x*,*y*)) for all *x*,*y*∈*X*. Also, we say that *T* is *α*-admissible whenever *α*(*x*,*y*)≥1 implies that *α*(*Tx*,*Ty*)≥1.

Here, *Ψ* is the family of non-decreasing functions . The following results, which we need in our results, have been proved in [7].

### Theorem 2.2

*Let (X,d) be a complete metric space,* *an α−ψ-contractive and α-admissible self-map on X such that α(x*_{0}*,Tx*_{0}*)≥1 for some x*_{0}*∈X. If x*_{n} *is a sequence in X such that α(x*_{n}*,x*_{n+1}*)≥1 for all n and x*_{n}*→x for some x∈X, then α(x*_{n}*,x)≥1 for all n. Then, T has a fixed point.*

## 3. Main results

In the following, we are going to state and prove our main results. We provide our results in three main parts.

*Case I*. Here, we discuss the nonlinear fractional differential equation
3.1via the integral boundary condition
where *D*^{α} denotes the Caputo fractional derivative of order *α* and is a continuous function. Here, (*X*,∥.∥) is a Banach space and *C*=*C*([0,1],*X*) denotes the Banach space of continuous functions from [0,1] into *X* endowed with uniform topology.

### Theorem 3.1

*Suppose that*

(i)

*there exists a function**and ψ∈Ψ such that**for all t∈I and**with ξ(a,b)≥0;*(ii)

*there exists x*_{0}*∈C(I) such that ξ(x*_{0}*(t),Fx*_{0}*(t))≥0 for all t∈I, where the operator**is defined by*(iii)

*for each t∈I and x,y∈C(I), ξ(x(t),y(t))≥0 implies ξ(Fx(t),Fy(t))≥0; and*(iv)

*if {x*_{n}*} is a sequence in C(I) such that**in C(I) and ξ(x*_{n}*,x*_{n+1}*)≥0 for all n, then ξ(x*_{n}*,x)≥0 for all n.*

*Then, the problem (3.1) has at least one solution.*

### Proof.

It is well known that *x*∈*C*(*I*) is a solution of (3.1) if and only if *x*∈*C*(*I*) is a solution of the integral equation
Then, the problem (3.1) is equivalent to finding *x**∈*C*(*I*), which is a fixed point of *F*. Now, let *x*,*y*∈*C*(*I*) such that *ξ*(*x*(*t*),*y*(*t*))≥0 for all *t*∈*I*. By using (i), we have
Thus, for each *x*,*y*∈*C*(*I*) with *ξ*(*x*(*t*),*y*(*t*))≥0 for all *t*∈*I*, we have
Now, define the function by
Hence, *α*(*x*,*y*) d(*Fx*,*Fy*)≤*ψ*(*d*(*x*,*y*)) for all *x*,*y*∈*C*(*I*). This implies that *F* is an *α*−*ψ*-contractive mapping. By using the condition (iii), we get
for all *x*,*y*∈*C*(*I*). Thus, *F* is *α*-admissible. From (ii), there exists *x*_{0}∈*C*(*I*) such that *α*(*x*_{0},*Fx*_{0})≥1. Finally, from (iv) and using theorem 2.2, we deduce the existence of *x**∈*C*(*I*) such that *x**=*Fx**. Hence, *x** is a solution of the problem. □

*Case II*. Now, we discuss the nonlinear fractional differential equation
3.2via the two-point boundary value condition *x*(0)=*x*(1)=0, where is a continuous function and *I*=[0,1]. Recall that the Green function associated with the problem (3.2) is given by

### Theorem 3.2

*Suppose that*

(i)

*there exists a function**and ψ∈Ψ such that |f(t,a)−f(t,b)|≤ψ(|a−b|) for all t∈I and**with ξ(a,b)≥0;*(ii)

*there exists x*_{0}*∈C(I) such that**for all t∈I;*(iii)

*for each t∈I and x,y∈C(I), ξ(x(t),y(t))≥0 implies*(iv)

*if {x*_{n}*} is a sequence in C(I) such that**in C(I) and ξ(x*_{n}*,x*_{n+1}*)≥0 for all n, then ξ(x*_{n}*,x)≥0 for all n.*

*Then, the problem (3.2) has at least one solution.*

### Proof.

It is well known that *x*∈*C*(*I*) is a solution of (3.2) if and only if is a solution of the integral equation for all *t*∈*I*. Define the operator by for all *t*∈*I*. Thus, for finding a solution of the problem (3.2), it is sufficient that we find a fixed point of *F*. Now, let *x*,*y*∈*C*(*I*) be such that *ξ*(*x*(*t*),*y*(*t*))≥0 for all *t*∈*I*. By using (i), we get
This implies that for each *x*,*y*∈*C*(*I*) with *ξ*(*x*(*t*),*y*(*t*))≥0 for all *t*∈*I*, we have
Let us define the function by
Therefore, for all *x*,*y*∈*C*(*I*), that is, *F* is an *α*−*ψ*-*contractive* mapping. Now by using (iii), we have
Hence, *F* is *α*-admissible. From (ii), there exists *x*_{0}∈*C*(*I*) such that *α*(*x*_{0},*Fx*_{0})≥1. Now using (iv) and theorem 2.2, there exists *x**∈*C*(*I*) such that *x**=*Fx**. □

*Case III*. We study now the nonlinear fractional differential equation
3.3via the two-point boundary-value condition *x*(0)=*x*(1)=0, where is a continuous function. Recall that the Green function associated with (3.3) is given by *G*(*t*)=*t*^{α−1}*E*_{α−β}(−*t*^{α−β}).

### Theorem 3.3

*Suppose that*

(i)

*there exists a function**and ψ∈Ψ such that**for all t∈I and**with ξ(a,b)≥0;*(ii)

*there exists x*_{0}*∈C(I) such that**for all t∈I;*(iii)

*for each t∈I and x,y∈C(I), ξ(x(t),y(t))≥0 implies*(iv)

*if {x*_{n}*} is a sequence in C(I) such that**in C(I) and ξ(x*_{n}*,x*_{n+1}*)≥0 for all n, then ξ(x*_{n}*,x)≥0 for all n.*

*Then, the problem (3.3) has at least one solution.*

### Proof.

It is well known that *x*∈*C*(*I*) is a solution of (3.3), if and only if is a solution of the integral equation for all *t*∈*I*.

Define the operator by for all *t*∈*I*. Thus, for finding a solution of the problem (3.3), it is sufficient we find a fixed point of *F*. Now, let *x*,*y*∈*C*(*I*) be such that *ξ*(*x*(*t*),*y*(*t*))≥0 for all *t*∈*I*. By using (i), we get
Note that *G*(*t*)=*t*^{α−1}*E*_{α−β,α}(−*t*^{α−β})≤*t*^{α−1}1/1+|−*t*^{α−β}|≤*t*^{α−1} for all *t*∈*I*. Thus, . Now, define the function by
Hence, for all *x*,*y*∈*C*(*I*), that is, *F* is an *α*−*ψ*-*contractive* mapping. Now by using (iii), we have
Hence, *F* is *α*-admissible. From (ii), there exists *x*_{0}∈*C*(*I*) such that *α*(*x*_{0},*Fx*_{0})≥1. Now by using (iv) and theorem 2.2, there exists *x**∈*C*(*I*) such that *Fx**=*x**. □

## 4. Concluding remarks

Fractional nonlinear differential equations and their applications represent a topic of high interest in the area of fractional calculus and its applications in various fields of science and engineering. As a result, new methods and techniques have been applied to this rapidly growing direction. In this article, based on the recently introduced notions of *α*−*ψ*-contractive and *α*-admissible mappings, we have proved three existence theorems for three nonlinear fractional differential equations for various boundary conditions.

## Acknowledgements

The research of S.R. and H.M. was supported by Azarbaijan University of Shahid Madani. The authors also express their gratitude to the referees for their helpful suggestions that improved the final version of this paper.

## Footnotes

One contribution of 14 to a Theme Issue ‘Fractional calculus and its applications’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.