University of TabrizComputational Methods for Differential Equations2345-3982Articles in Press20240501On Unique Solutions of Integral Equations by Progressive Contractions1786910.22034/cmde.2024.59214.2516ENOsmanTuncDepartment of Computer Programming, Baskale Vocational School, Van Yuzuncu Yil University, 65080, Campus, Van, Turkey.J. R.GraefDepartment of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.CemilTuncDepartment of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Van, Turkey.0000-0003-2909-8753Journal Article20231113The authors consider Hammerstein type integral equations for the purpose of obtaining new results on the uniqueness of solutions on an infinite interval. The approach used in the proofs is based on the technique called progressive contractions due to T. A. Burton. Here, the authors apply Burton's method to a general Hammerstein type integral equation that also yields existence of solutions. In much of the existing literature, investigators prove uniqueness of solutions of integral equations by applying some type of fixed point theorem. This can prove to be a difficult process that sometimes involves patching together solutions on short intervals and perhaps involving translations. In this paper, using progressive contractions in three simple short steps, each one being an elementary contraction mapping on a short interval, the authors improve the technique due to Burton by considering a general Hammerstein type integral equation, and they obtain the uniqueness of solutions on an infinite interval.https://cmde.tabrizu.ac.ir/article_17869_b91b2eed55aa41016212c6aadc804bae.pdf