Abstract

A Dirichlet problem for a functional-differential equation the operator of which is represented by the product of a quasilinear differential operator and a linear shift operator is considered. The nonlinear operator has differentiable coefficients. A sufficient condition for the strong ellipticity of the differential-difference operator is proposed. For a Dirichlet problem with an operator satisfying the strong ellipticity condition, the existence and uniqueness of a generalized solution is proved. The situation is considered in which the differential-difference operator belongs to the class of pseudomonotone \({{(S)}_{ + }}\) operators; in this case, a generalized solution of the Dirichlet problem exists. As an example, a nonlocal problem with a Bitsadze–Samarskii boundary condition is considered.

中文翻译：

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拟线性椭圆型微分方程的广义解

摘要

考虑泛函微分方程的Dirichlet问题，该函数的算子由拟线性微分算子和线性移位算子的乘积表示。非线性算子具有可微的系数。提出了微分算子强椭圆性的充分条件。对于一个满足强椭圆性条件的算子Dirichlet问题，证明了广义解的存在性和唯一性。考虑这样一种情况，其中微分差分算符属于伪单调\（{{（（S）} _ {{+}} \\）}类；在这种情况下，存在Dirichlet问题的广义解。例如，考虑一个具有Bitsadze–Samarskii边界条件的非局部问题。