Finding the Fixed Points Inside Large Mapping Sets: Integral Equations

Open Access

Original research article

Author(s):

Theodore A. Burton^a, Ioannis K. Purnaras^b

^a Northwest Research Institute, 732 Caroline St., Port Angeles, WA, USA
^b Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece

Advances in the Theory of Nonlinear Analysis and its Applications 1(1), 41-47.

Received: August 17, 2017

Accepted: August 24, 2017

Published: August 25, 2017

Abstract

Let $x f(t,x) > 0$ for $x \neq 0$ and let $A(t-s)$ satisfy some classical properties yielding a nice resolvent. Using repeated application of a fixed point mapping and induction we develop an asymptotic formula showing that solutions of the Caputo equation
\[
{}^{c}D^{q}x(t) = -f(t, x(t)), \qquad 0 < q < 1, \quad x(0) \in \mathbb{R}, \quad x(0) \neq 0,
\]
and more generally of the integral equation
\[
x(t) = x(0) – \int_{0}^{t} A(t-s) f(s, x(s))\, ds, \quad x(0) \neq 0,
\]
all satisfy $x(t) \to 0$ as $t \to \infty$.

Keywords: Compact maps, repeated mappings, integral equations, fixed points, limit sets.

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APA Style

Burton, T. A., & Purnaras, I. K. (2017). Finding the Fixed Points Inside Large Mapping Sets: Integral Equations. Advances in the Theory of Nonlinear Analysis and its Application, 1(1), 41-47.