Row-summable matrices with application to generalization of Schröder’s and Abel’s functional equations

In this paper, the author discusses a generalization of Schroder’s and Abel’s functional equation of the form $$\sum _{n}\alpha _n(x)f\big (u_n(x)\big ) = g(x).$$ In fact, using the theory of operators and infinite matrices, we show under certain conditions this equation has only a unique bounded solution $$f:X\rightarrow {\mathbb {R}}.$$