- Research Article
- Open Access
Existence of Solutions for Nonlinear Fractional Integro-Differential Equations with Three-Point Nonlocal Fractional Boundary Conditions
© A. Alsaedi and B. Ahmad. 2010
- Received: 17 March 2010
- Accepted: 11 June 2010
- Published: 4 July 2010
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.
- Fractional Order
- Fractional Calculus
- Fractional Differential Equation
- Impedance Boundary Condition
- Surface Current Density
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.
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 . 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].
provided the integral exists.
provided the right-hand side is pointwise defined on
Lemma 2.3 (see ).
where ( is the smallest integer such that ).
Lemma 2.4 (see ).
Let . Then
Substituting the values of and in (2.6), we obtain (2.5). This completes the proof.
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 .
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  which is needed to prove the following result to prove the existence of at least one solution of (1.1).
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
Then there exists at least one solution of the boundary value problem (1.1) on
Thus, It follows from the assumption that is a contraction mapping for
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
Thus, by Theorem 3.1, the boundary value problem (3.14) has a unique solution on
The authors are grateful to the referees for their careful review of the manuscript.
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