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In this thesis, we prove some fixed point theorems in various spaces with applications. We discuss the uniqueness and existence of a fixed point in a dislocated metric and dislocated quasi metric spaces. Several results of fixed point under different kind of contractive assumptions are introduced that generalize and unify a number of related results in the existing literature. Some important examples are presented to support our main results. We also establish some fixed point theorems in ordered cone b–metric spaces. These new results generalize and unify numerous popular theorems in literature to ordered cone b–metric spaces. Several auxiliary examples are also provided. Some new theorems of fixed point are established in G–metric spaces by using integral type contraction with supporting examples and applications. We also analyze the Ulam type stability to some families of function equations via fixed point theory. In particular, of certain delay Volterra integro–differential equations are established by using a fixed point method in a generalized complete metric spaces. We provide some examples of our established results. After applying the idea of fixed point technique, we examine the Hyers–Ulam stability and the Hyers–Ulam–Rassias stability of impulsive Volterra integral equations in complete generalized metric spaces.
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