Self-reference (State-dependence) quadratic integral equation of fractional order

Document Type : Regular research papers

Author

Department of Mathematics, Faculty of Science, Alexandria University, Alexandria,Egypt.

Abstract

The theory of differential equations with deviating arguments is one of the important and significant branch of nonlinear analysis with numerous applications in most fields. Usually, equations of deviating arguments are with deviation depends only on the time, however, when the deviation of the arguments depend upon both the state variable $x$ and also the time $t $ is incredibly important theoretically and practically.
Differential equations with state-dependent delays attract interests of specialists since they widely arise from application models, such as two-body problem of classical Electrodynamics, also have may applications especially in the class of problems that have past memories.

In this paper, we study the existence of solutions of a self-reference (state dependence) quadratic integral equation of fractional order of the form
$$ x(t)~=~x_{0}~+I^{\alpha} f_{1}\big(s,x(x(\phi(s)))\big)ds~.~I^{\beta}f_{2}\big(s,x(x(\phi(s)))\big)
ds,~~~t\in[0,T],~~\alpha,~\beta \in (0,1).$$
The uniqueness of the solution will be studied also, moreover, the continuous dependence of the unique solution on the initial data and the functions $f_{1}$~and~ $f_{2}$ will be poved. Some examples are included.
The study establishes conditions for solution existence and uniqueness, according to Schauder fixed point Theorem

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