Abstract
Calvert and Gupta’s results concerning the perturbations on the ranges of m-accretive mappings have been employed widely in the discussion of the existence of solutions of nonlinear elliptic differential equation with Neumann boundary. In this paper, we shall focus our attention on certain hyperbolic differential equation with mixed boundaries. By defining some suitable nonlinear mappings, we shall demonstrate that Calvert and Gupta’s results can be applied to hyperbolic equations, in addition to its wide usage in elliptic equations. Due to the differences between hyperbolic and elliptic equations, some new techniques have been developed in this paper, which can be regarded as the complement and extension of the previous work.
Highlights
Introduction and preliminaries1.1 Introduction Nonlinear boundary value problems involving the generalized p-Laplacian operator arise from many physical phenomena, such as reaction-diffusion problems, petroleum extraction, flow through porous media, and non-Newtonian fluids, just to name a few
We recall that Calvert and Gupta [ ] have used such a perturbation result to provide sufficient conditions so that some nonlinear boundary value problems with Neumann boundaries involving the Laplacian operator have solutions in Lp( )
Inspired by Calvert and Gupta’s perturbation result of Theorem . , the p-Laplacian boundary value problems and their general forms have been extensively studied in the work of [ – ]
Summary
≥. Similar to the proof of Lemma . For f ∈ L ( , T; L ( )) ⊂ Lp( , T; (W ,p( ))∗), there exists u ∈ Lp ( , T; W ,p( )) ⊂ L ( , T; L ( )) such that f = B( )u + F( )u + ∂ ( )u = A( )u + u, which implies that R(I + A( )) = L ( , T; L ( )) Let S be the same as that in Lemma. It can be seen from the definition of S that (c) is true Proof Let f ∈ L ( , T; L ( )) and satisfy
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