Applications of Integral Inequalities to Fractional Hybrid Differential Equations

Abstract

The main focus of this thesis is applications of integral inequalities, in as many ways as possible, on hybrid differential equations of fractional order. For this purpose, generalizations of a certain type of integral inequalities are obtained. In addition to that, applications on a class of fractional hybrid differential equations using fixed point theory are established.

First, we present generalizations to some integral inequalities of Gronwall-Ballman type. This type of integral inequalities has many uses when it comes to differential equations. In that light, some applications to fractional hybrid differential equations with Hadamard derivative got included in this thesis.

Then, we present a different sense of applications of integral inequalities to a certain class of fractional hybrid differential equations. We study a boundary value problem which is a system of $n$-hybrid differential equations with Caputo derivative and nonlocal conditions. Accordingly, some results that address existence and uniqueness of the solution of the system are given. For the existence of at least one solution, two approaches are used: Shaefer fixed point theorem and another theorem developed by the mathematician Dhage. Illustrative examples will be presented as well to validate the results.

For stability of the system, we proceed through Ulam-Hyers stability as the main way to study it. We try to establish the necessary results that validate the stability of the system mentioned above.