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On Results of Fixed Point Theory of Monotone Operators and Applications

Mathew Aibinu, PhD; Department of Mathematics, University of Iowa

The existence or construction of solutions of differential and integral equations is often reduced to the problem of finding a fixed point for an operator which is defined on a space of the functions. Differential, integrals, and other forms of equations such as optimization and variational problems, can be modelled by x = Tx (1), where T is an operator defined on a space. Solutions of (1) are known as fixed points of T. Fixed Point Theory (FPT) is one of the most powerful tools of modern Mathematics. FPT includes theorems concerning the existence and properties of fixed points. Also, it blends analysis, topology, and geometry. It has numerous applications, and it has been applied in several fields, such as game theory, engineering, Physics, Economics, Biology, Chemistry, etc. FPT has also been applied to determine the existence of periodic solutions for functional differential equations. In addition to the deep involvement in the theory of differential equations, FPT has been found to be inevitable in solving problems such as finding zeros of nonlinear equations and proving surjectivity theorems. Some results on FPT of monotone operator will be presented and reference will be made to their applications.