Constrained problem of fractional-order It\^o-integral equation subject to  a nonlocal fractional order stochastic-integral problem of It\^o-differential equation

maysa elgendy; Ahmed M. A.   El-Sayed; Hoda A.   Fouad

doi:10.5269/bspm.81731

Constrained problem of fractional-order It\^o-integral equation subject to a nonlocal fractional order stochastic-integral problem of It\^o-differential equation

maysa elgendy Department of Mathematics, College of Science, Qassim University, P. O. Box 6644, Burayda 51452, Saudi Arabia, Department of Mathematics and Computer Science, Faculty of Science, Damanhour University, Egypt
Ahmed M. A. El-Sayed Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Egypt
Hoda A. Fouad Department of Mathematics College of Science, Taibah University, Madinah, Saudi Arabia

Abstract

In this paper, an issue of fractional-order It\^o-integral equation constrained by a nonlocal fractional order stochastic-integral problem of It\^o-differential equation.\\
Firstly, the existence of solutions and the sufficient conditions for its uniqueness of the constraint will be proved. The dependency of the solution will be studied and the Hyers-Ulam stability of the constraint will be discussed. Secondly, we prove that for every solution of the constraint there exists a unique solution for the constrained problem of fractional-order It\^o-integral equation.
Also, for the unique solution that we get, we will discuss some dependency properties and the Hyers-Ulam stability for the constrained problem it-self will be proved. Finally some examples will be given.