Existence and uniqueness of solutions of some nonlinear integral equations

This thesis is structured into six chapters. Chapter 1, titled "Introduction", provides basic information on integral equations, including their classification, applications in various scientific fields, and an overview of methods commonly employed to establish the existence and uniqueness of their solutions. Chapter 2, titled Literature Review, outlines previous research related to the existence and uniqueness of solutions for various forms of integral equations. The methodologies and significant findings from these studies are concisely summarized. Chapter 3 outlines the materials and methods used throughout the thesis. Chapter 4 introduces essential theoretical concepts relevant to the study, including definitions and theorems from fixed point theory. Key notions such as metric spaces, the Bielecki metric, Cauchy sequences, norm definitions (including Euclidean and Chebyshev norms), contraction mappings, Picard operators, and other fundamental results are discussed, supported by illustrative examples. Chapter 5 presents a series of results from the literature concerning the existence and uniqueness of solutions for certain classes of nonlinear integral equations. These results are rigorously proved using the Banach contraction principle and the progressive contraction technique introduced by T.A. Burton. A new example is also provided to demonstrate the applicability of one of the main results. Finally, Chapter 6 explores the existence and uniqueness of solutions for a functional Volterra-Fredholm integral equation in a Banach space. Furthermore, special cases of the proposed mathematical models are examined through newly developed illustrative examples. Keywords: Contraction mapping, Existence and uniqueness of solutions, Integral equations, Progressive contractions