Weakly Nonlinear Boundary-Value Problems for Operator Equations in Banach Spaces

Abstract

We consider weakly nonlinear boundary-value problems for operator equations with generalized invertible operator in the linear part of the boundary-value problem in the critical case. We establish necessary and sufficient conditions for the existence of at least one and unique solution of this boundary-value problem and propose convergent iterative procedures for its construction.The investigations of weakly nonlinear boundary-value problems .Lz/.t / D f .t / C "Z.z; t; "/; `z.�; "/ D ˛C "J.z; "/ (1) for operator equations with generalized invertible operators in the linear part given in Banach spaces continues the development of the methods of perturbation theory and Lyapunov-Poincaré methods of small parameter [1] created by the Kyiv school of nonlinear mechanics.These boundary-value problems for systems of ordinary differential equations in the periodic case were considered in [2]. For systems of ordinary differential equations and functional-differential equations with impulsive action in the general Noetherian case, these problems were studied in [3][4][5].These problems were generalized to the case of Banach spaces in which a finite-dimensional Euclidean space of values of a function is replaced by a Banach space. In the case where L is an ordinary differential operator acting in a Banach space, weakly nonlinear boundary-value problems (1) were investigated in [6]. As an important specific feature of these problems, we can mention the fact that the equation Lz D f of the linear generating boundary-value problem is solvable for any right-hand side, i.e., the operator L is everywhere solvable [7].In the case where L is an integral Fredholm operator with degenerate kernel, which is not everywhere solvable, the boundary-value problem (1) was considered in [8].At present, there is no general approach to the investigation of weakly nonlinear boundary-value problems (1) whose linear part contains an operator, which is not everywhere solvable. The difficulties encountered in the investigation of these problems in Banach spaces are connected with the problem of generalized inversion of operators and operator matrices in Banach spaces. Statement of the ProblemLet l 1 .I; B 1 / be a Banach space of bounded vector functions z.t / given on a finite interval I with values in a Banach space B 1 ; i.e., z.�/W I ! B 1 ; with the norm jjjzjjj D sup t 2I kz.t /k B 1 ;