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A fixed-point approach for decaying solutions of difference equations.

Zuzana Došlá1, Mauro Marini2, Serena Matucci2

  • 1Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|January 4, 2021
PubMed
Summary

This study proves the existence of intermediate solutions for a specific advanced argument difference equation. The research connects this to boundary value problems and fixed-point theory in applied mathematics.

Keywords:
boundary value problem on the half linedecaying solutionfixed-point theoremfunctional discrete equationsnonlinear difference equation

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Area of Science:

  • Applied Mathematics
  • Difference Equations
  • Nonlinear Analysis

Background:

  • Advanced argument difference equations present unique analytical challenges.
  • Intermediate solutions are a specific class of decaying solutions requiring focused study.
  • Fixed-point theory provides powerful tools for proving existence in differential and difference equations.

Purpose of the Study:

  • To investigate the existence of intermediate solutions for a specific advanced argument difference equation.
  • To reduce the problem to a boundary value problem involving a difference equation without deviating arguments.
  • To utilize fixed-point theorems adapted from continuous cases to the discrete domain.

Main Methods:

  • Formulation of a boundary value problem for the given advanced argument difference equation.
  • Transformation of the problem into an equivalent boundary value problem for a standard difference equation.
  • Application of a fixed-point theorem tailored for difference equations.

Main Results:

  • The existence of intermediate solutions for the studied difference equation is proven.
  • The method successfully reduces the complex problem to a more manageable one.
  • The approach demonstrates the applicability of fixed-point theory in this discrete setting.

Conclusions:

  • The study establishes the existence of intermediate solutions for a class of advanced argument difference equations.
  • The employed fixed-point approach offers a viable method for analyzing such equations.
  • This work contributes to the broader understanding of topological degree and fixed-point theories in difference equations.