In their usual form, homogenization methods produce first-order approximations of the exact solutions of problems for differential equations with rapidly oscillating coefficients which model the physical behavior of microstructured media. However, there is need of approximations containing higher-order terms when the usual first-order approximations, which are formed by superposing a macroscopic trend and a local perturbation, are not capable of reproducing the local details of the exact solutions. Here, two-scale asymptotic solutions with second-order terms are provided for a Dirichlet problem modeling the steady state of functionally-graded microperiodic nonlinear rods. The need of considering higherorder terms is illustrated through numerical examples for various power-law nonlinearities.
This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation Utt=Lu+f(x,t) in ΩT=Ω×(0,T), where L is a nonlinear operator and ϕ(x,t,⋅), f(x,t) and the exponents of the nonlinearities p(x,t) and μ(x,t) are given functions.
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We prove existence and uniqueness theorems for Dirichlet boundary value problems of the form u" + f(t,u) = 0, u(0) = uo, u(1) = ui in ordered finite dimensional Banach spaces, involving one-sided estimates and quasimonotonicity.
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In this paper we consider some systems of partial differential equations with variable boundary data. Some sufficient conditions under which solutions of these systems continuously depend on boundary data are given. The proofs of the main result of this work are based on some variational methods.
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