Existence of Solutions for Nonlinear Fractional Integro-Differential Equations with Three-Point Nonlocal Fractional Boundary Conditions

Abstract

We prove the existence and uniqueness of solutions for nonlinear integro-differential equations of fractional order with three-point nonlocal fractional boundary conditions by applying some standard fixed point theorems.

1. Introduction

Fractional calculus (differentiation and integration of arbitrary order) is proved to be an important tool in the modelling of dynamical systems associated with phenomena such as fractal and chaos. In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, and fitting of experimental data [1–4].

Recently, differential equations of fractional order have been addressed by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. For some recent work on fractional differential equations, see [5–11] and the references therein.

In this paper, we study the following nonlinear fractional integro-differential equations with three-point nonlocal fractional boundary conditions

(1.1)

where is the standard Riemann-Liouville fractional derivative,  :  is continuous, for  : 

(1.2)

and satisfies the condition Here, is a Banach space and denotes the Banach space of all continuous functions from endowed with a topology of uniform convergence with the norm denoted by

We remark that fractional boundary conditions result in the existence of both electric and magnetic surface currents on the strip and are similar to the impedance boundary conditions with pure imaginary impedance, and in the physical optics approximation, the ratio of the surface currents is the same as for the impedance strip. For the comparison of the physical characteristics of the fractional and impedance strips such as radiation pattern, monostatic radar cross-section, and surface current densities, see [12]. The concept of nonlocal multipoint boundary conditions is quite important in various physical problems of applied nature when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the censors located at intermediate points. Some recent results on nonlocal fractional boundary value problems can be found in [13–15].

2. Preliminaries

Let us recall some basic definitions [1–3] on fractional calculus.

Definition 2.1.

The Riemann-Liouville fractional integral of order is defined as

(2.1)

provided the integral exists.

Definition 2.2.

The Riemann-Liouville fractional derivative of order for a function is defined by

(2.2)

provided the right-hand side is pointwise defined on

Lemma 2.3 (see [16]).

For let . Then

(2.3)

where ( is the smallest integer such that ).

Lemma 2.4 (see [2]).

Let . Then

  

   

Lemma 2.5.

For a given the unique solution of the boundary value problem

(2.4)

is given by

(2.5)

Proof.

In view of Lemma 2.3, the fractional differential equation in (2.4) is equivalent to the integral equation

(2.6)

where are arbitrary constants. Applying the boundary conditions for (2.4), we find that and

(2.7)

Substituting the values of and in (2.6), we obtain (2.5). This completes the proof.

3. Main Results

To establish the main results, we need the following assumptions.

• (A₁) There exist positive functions such that

  

  (3.1)

  Further,

  

  (3.2)

• (A₂) There exists a number such that , where

  

  (3.3)

• (A₃) for all

Theorem 3.1.

Assume that is a jointly continuous function and satisfies the assumption Then the boundary value problem (1.1) has a unique solution provided , where is given in the assumption .

Proof.

Define by

(3.4)

Let us set and choose

(3.5)

where is such that Now we show that where For we have

(3.6)

Now, for and for each we obtain

(3.7)

where we have used the assumption . As therefore is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.

Now, we state Krasnoselskii's fixed point theorem [17] which is needed to prove the following result to prove the existence of at least one solution of (1.1).

Theorem 3.2.

Let be a closed convex and nonempty subset of a Banach space Let be the operators such that (i) whenever ; (ii) is compact and continuous; (iii) is a contraction mapping. Then there exists such that

Theorem 3.3.

Let be jointly continuous, and the assumptions and hold with

(3.8)

Then there exists at least one solution of the boundary value problem (1.1) on

Proof.

Let us fix

(3.9)

and consider We define the operators and on as

(3.10)

For we find that

(3.11)

Thus, It follows from the assumption that is a contraction mapping for

In order to prove that is compact and continuous, we follow the approach used in [6, 7]. Continuity of implies that the operator is continuous. Also, is uniformly bounded on as

(3.12)

Now, we show that is equicontinuous. Since is bounded on the compact set , therefore, we define . Consequently, for , we have

(3.13)

which is independent of So, is relatively compact on . Hence, By Arzela-Ascoli's Theorem, is compact on . Thus all the assumptions of Theorem 3.2 are satisfied and the conclusion of Theorem 3.2 implies that the boundary value problem (1.1) has at least one solution on

Example 3.

Consider the following boundary value problem:

(3.14)

Here, With we find that

(3.15)

Thus, by Theorem 3.1, the boundary value problem (3.14) has a unique solution on

4. Conclusions

This paper studies the existence and uniqueness of solutions for nonlinear integro-differential equations of fractional order with three-point nonlocal fractional boundary conditions involving the fractional derivative . Our results are based on a generalized variant of Lipschitz condition given in , that is, there exist positive functions and such that

(4.1)

In case , and are constant functions, that is, , and ( and are positive real numbers), then Lipschitz-generalized variant reduces to the classical Lipschitz condition and in the assumption takes the form

(4.2)

In the limit , our results correspond to a second-order integro-differential equation with fractional boundary conditions:

(4.3)