On Unique Solutions of Integral Equations by Progressive Contractions

Document Type : Research Paper

Authors

1 Department of Computer Programming, Baskale Vocational School, Van Yuzuncu Yil University, 65080, Campus, Van, Turkey.

2 Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.

3 Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Van, Turkey.

Abstract

The 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.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 01 May 2024

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History

  • Receive Date: 13 November 2023
  • Revise Date: 01 February 2024
  • Accept Date: 26 February 2024

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