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Existence of Solutions for Nonlinear Fractional Integro-Differential Equations with Three-Point Nonlocal Fractional Boundary Conditions
Advances in Difference Equations volume 2010, Article number: 691721 (2010)
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

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

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

provided the integral exists.
Definition 2.2.
The Riemann-Liouville fractional derivative of order for a function
is defined by

provided the right-hand side is pointwise defined on
Lemma 2.3 (see [16]).
For let
. Then

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

is given by

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

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

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

Let us set and choose

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

Now, for and for each
we obtain

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

Then there exists at least one solution of the boundary value problem (1.1) on
Proof.
Let us fix

and consider We define the operators
and
on
as

For we find that

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

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

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:

Here, With
we find that

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

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

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

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The authors are grateful to the referees for their careful review of the manuscript.
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Alsaedi, A., Ahmad, B. Existence of Solutions for Nonlinear Fractional Integro-Differential Equations with Three-Point Nonlocal Fractional Boundary Conditions. Adv Differ Equ 2010, 691721 (2010). https://doi.org/10.1155/2010/691721
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DOI: https://doi.org/10.1155/2010/691721