Periodic solutions of integro-differential equations in Banach space having Fourier type

TL;DR

This paper investigates the existence of periodic solutions for a class of integro-differential equations in Banach spaces with Fourier type, utilizing advanced operator theory and harmonic analysis tools.

Contribution

It introduces a novel approach combining M-boundedness, Fourier type, and Besov spaces to establish periodic solutions for integro-differential equations.

Findings

Established conditions for existence of periodic solutions.

Applied Fourier multiplier techniques in Banach spaces.

Extended analysis to equations with convolution terms.

Abstract

The aim of this work is to study the existence of a periodic solutions of integro-differential equations d dt [x(t)-- L(x t)] = A[x(t)-- L(x t)]+ G(x t)+ t --$\infty$ a(t-- s)x(s)ds+ f (t), (0 $\le$ t $\le$ 2$\pi$) with the periodic condition x(0) = x(2$\pi$), where a $\in$ L 1 (R +). Our approach is based on the M-boundedness of linear operators, Fourier type, B s p,q-multipliers and Besov spaces.