LINEAR APPROXIMATION AND ASYMPTOTIC EXPANSION ASSOCIATED WITH THE SYSTEM OF NONLINEAR FUNCTIONAL EQUATIONS

THIS PAPER IS DEVOTED TO THE STUDY OF THE FOLLOWING PERTURBED SYSTEM OF NONLINEAR FUNCTIONAL EQUATIONS EQUATION PRESENTED WHERE IS A SMALL PARAMETER, AIJK, BIJK ARE THE GIVEN REAL CONSTANTS, R IJK, SIJK, XIJK: , G I : R, : R2 R ARE THE GIVEN CONTINUOUS FUNCTIONS AND FI : R ARE UNKNOWN FUNCTIONS. FIRST, BY USING THE BANACH FIXED POINT THEOREM, WE FIND SUFFICIENT CONDITIONS FOR THE UNIQUE EXISTENCE AND STABILITY OF A SOLUTION OF (E). NEXT, IN THE CASE OF C2(R 2;R), WE INVESTIGATE THE QUADRATIC CONVERGENCE OF (E). FINALLY, IN THE CASE OF CN(R2,R) AND SUFFICIENTLY SMALL, WE ESTABLISH AN ASYMPTOTIC EXPANSION OF THE SOLUTION OF (E) UP TO ORDER N + 1 IN . IN ORDER TO ILLUSTRATE THE RESULTS OBTAINED, SOME EXAMPLES ARE ALSO GIVEN. COPYRIGHT BY FACULTY OF MATHEMATICS AND INFORMATION SCIENCE, WARSAW UNIVERSITY OF TECHNOLOGY.