Fixed Point Theorems and Applications

ABSTRACT: Fixed point theory be one of the advanced topics in both pure and applied mathematics, it also has seen great interest since recent decades, because it is considered an essential tool for nonlinear analysis and many other branches of modern mathematics. In particular, when we deal with the solvability of a certain functional equation (differential equation, fractional differential equation, integral equation, matrix equation, etc), we are reformulating the problem in terms of investigating the existence and uniqueness of a fixed point of a mapping. In addition, this theory has several applications in many different fields such as biology, chemistry, economics, game theory, optimization theory, physics, etc. The basic purpose of this thesis is to present some recent advances in this theory with some applications that is an important for our life. For example, first and second order of ordinary differential equations in Banach space and fractional differential equations involving Riemmann-Liouville and Caputo differential operators. Keywords: Fixed points, Banach’s contraction theorem, Contraction, Schauder’s fixed point theorem, Brouwer’s fixed point theorem, Uniqueness, Existence, Fractional differential equations, Boundary value problems.