Abstract
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.
Highlights
Consider the Φ-Laplacian type equation d dt Φ t, x f t, x0, t ∈ I : a, b, 1.1 where f ∈ C I × R, R is Lipschitz function with respect to x, Φ ∈ C I × R, R is Lipschitz and monotone function with respect to x, together with the boundary conditionsΦ a, x a 0, x b 0.This equation even in a greater generality was intensively studied in the last time 1–3 and references therein
If the original nonlinear problem allows for quasilinearization with respect to the linear parts with different types of nonresonance, this problem is expected to have multiple solutions
One says that a linear part LX t in 2.1 is i-nonresonant with respect to the boundary conditions 1.7 if the angular function φ t, defined by the initial condition φ a π/2, takes exactly i times i 0, 1, . . . values of the form πn in the interval a, b and φ b / πn, n ∈ N
Summary
0, t ∈ I : a, b , 1.1 where f ∈ C I × R, R is Lipschitz function with respect to x, Φ ∈ C I × R, R is Lipschitz and monotone function with respect to x , together with the boundary conditions. X −ky y kx is nonresonant with respect to the boundary conditions y a 0, x b 0. Notice that the type of a solution x t to the problem 1.1 , 1.2 is induced by oscillatory type of a solution x t , y t to the quasilinear problem 1.6 , 1.7 , which, in turn, is defined by oscillatory properties of the extracted nonresonant linear part LX t see below. If the original nonlinear problem allows for quasilinearization with respect to the linear parts with different types of nonresonance, this problem is expected to have multiple solutions.
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