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The study of the existence of fixed points is a lively and fascinating field of research and has many applications in engineering, physics, chemistry, biology, economics, computer science and etc. Formulating concrete problems abstractly in the framework of fixed point theory has the advantage of distilling the essentials and their relationships, of allowing a uniform treatment of differing practical problems and of enabling the use of deep and powerful mathematical methods, without which the problems could not be solved. In this monograph we aim to study hybrid of two classical theorems; Banach's fixed point theorem and Tarski's fixed point theorem. We focus on the existence of fixed point in partially ordered metric spaces for set valued mappings.
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