On the solutions of the functional equation $x(t)+A(t)x(f(t))=F(t)$ when the function $F$ satisfies some special conditions

Abstract

The results of this paper are concerned with the soluiton $x(t)$ of the functional equation $x(t)+A(t)x(f(t))=F(t)$. Using regular summability methods $T$, we derive some necessary and also some sufficient conditions for the $T$-sum $x(t)$ of the series $\sum_{i=0}^{\bb}(-1)^iF(f^i(t))$ to be a solution of the above mentioned equation under the specific conditions for $F(t)$.