%0 Journal Article
%T Quasilinearization Technique for -Laplacian Type Equations
%A Inara Yermachenko
%A Felix Sadyrbaev
%J International Journal of Mathematics and Mathematical Sciences
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/975760
%X An equation is considered together with the boundary conditions , . This problem under appropriate conditions can be reduced to quasilinear problem for two-dimensional differential system. The conditions for existence of multiple solutions to the original problem are obtained by multiply applying the quasilinearization technique. 1. Introduction Consider the -Laplacian type equation where is Lipschitz function with respect to , is Lipschitz and monotone function with respect to , together with the boundary conditions This equation (even in a greater generality) was intensively studied in the last time ([1¨C3] and references therein). If then it reduces to The equation (1.1) can also be interpreted as the Euler equation for the functional where and . Our aim is to obtain the multiplicity results. For this we denote and rewrite (1.1) as a two-dimensional differential system of the form and apply the quasilinearization process described in [4¨C7]. Namely, we reduce the system (1.5) to a quasilinear one of the form so that both systems (1.5) and (1.6) are equivalent in some domain and moreover the extracted linear part is nonresonant with respect to the boundary conditions If any solution of the quasilinear problem (1.6), (1.7) satisfies the inequalities , then we say that the original problem for -Laplacian type equation (1.1), (1.2) allows for quasilinearization. If a solution of the problem (1.6), (1.7) is located in , then this also solves the problem (1.5), (1.7) and therefore the respective solves the original problem (1.1), (1.2). Notice that the type of a solution to the problem (1.1), (1.2) is induced by oscillatory type of a solution to the quasilinear problem (1.6), (1.7), which, in turn, is defined by oscillatory properties of the extracted nonresonant linear part (see below). If the original nonlinear problem allows for quasilinearization with respect to the linear parts with different types of nonresonance, then this problem is expected to have multiple solutions. The paper is organized as follows. In Section 2 definitions are given. In Section 3 the main result is proved concerning the solvability of a quasilinear boundary value problem. Section 4 contains application of the main result and the quasilinearization technique for studying a nonlinear system; the numerical results are provided and a corresponding example was analyzed. 2. Definitions Consider the quasilinear system (1.6), where functions are continuous, bounded (i.e., there exists a positive constant such that and for all values of arguments) and satisfy the Lipschitz conditions in
%U http://www.hindawi.com/journals/ijmms/2012/975760/