Finding the Fixed Points Inside Large Mapping Sets: Integral Equations

Abstract

Let xf(t,x) > 0 for x 6= 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 cDqx(t) = −f(t,x(t)), 0 < q < 1, x(0) ∈<, x(0) 6= 0, and more generally of the integral equation x(t) = x(0)−Zt 0 A(t−s)f(s,x(s))ds,x(0) 6= 0, all satisfy x(t) → 0 as t →∞.

Keywords

• compact maps,repeated mappings,integral equations,fixed points,limit sets

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