Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: EXISTENCE OF SOLUTIONS FOR EQUATIONS INVOLVING ITERATED FUNCTIONAL SERIES | EXISTENCE OF SOLUTIONS FOR EQUATIONS INVOLVING ITERATED FUNCTIONAL SERIES V. MURUGAN AND P. V. SUBRAHMANYAM Received 26 August 2004 and in revised form 8 October 2004 Theorems on the existence and uniqueness of differentiable solutions for a class of iterated functional series equations are obtained. These extend earlier results due to Zhang. 1. Introduction The study of iterated functional equations dates back to the classical works of Abel Babbage and others. This paper offers new theorems on the existence and uniqueness of solutions to the iterated functional series equation 00 X Wffxf F x i 1 where Àị s are nonnegative numbers and f0 x x fk x f fk-1 x k e N. In the functions F Hi and constants Ảị i e N are given and the unknown function f is to be found. The above equation is more general than those considered by Dhombres 2 Mukherjea and Ratti 3 Nabeya 4 and Zhang 5 . 2. Preliminaries This section collects the standard terminology and results used in the sequel see 5 . Let I a b be an interval of real numbers. C1 I I the set of all continuously differentiable functions from I into I is a closed subset of the Banach Space C1 I R of all continuously differentiable functions from I into R with the norm II c1 defined by 0 c1 11011c0 0 c 0 e C1 I R where 11011c0 maxxeI 10 x I and 0 is the derivative of 0. Following Zhang 5 for given constants M 0 M 0 and 8 0 we define the families of functions 1 I M M 0 e C1 I I 0 a a 0 b b 0 0 x M Vx e I . 0 xj - 0 xf I M x1 - x2 Vx1 x2 e I and 1 I M M 0 e 1 I M M 8 0 x M for allx e I . Copyright 2005 Hindawi Publishing Corporation Fixed Point Theoryand Applications 2005 2 2005 219-232 DOI 220 Equations involving series of iterates In this context it is useful to note the following proposition. Proposition . Let 8 0 M 0 andM 0. Then i for M 1 1 I M M is empty and for M 1 1 I M M contains only the identity function ii for 8 1 8 I M M is empty and for 8 1 8 I M M contains only the identity .
