Existence and Relaxation of Solutions for a Differential Inclusion with Maximal Monotone Operators and Perturbations

Pub Date : 2024-03-14 DOI:10.1134/S1064562423701399
A. A. Tolstonogov
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Abstract

A differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum with the identity operator multiplied by a positive square-integrable function is a monotone operator. The multivalued mapping is Lipschitz continuous with respect to the phase variable. We prove the existence of a solution and the density in the corresponding topology of the solution set of the initial inclusion in the solution set of the inclusion with a convexified multivalued mapping. For these purposes, new distances between maximal monotone operators are introduced.

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具有最大单调算子和扰动的微分包容的解的存在性和松弛性
在可分离的希尔伯特空间中,研究了具有依赖时间的最大单调算子和扰动的微分包容。扰动是一个随时间变化的单值算子与一个具有封闭非凸值的多值映射之和。单值算子的一个特点是,它与乘以正平方可积分函数的同值算子之和是一个单调算子。多值映射相对于相位变量是 Lipschitz 连续的。我们证明了初始包含的解集在具有凸化多值映射的包含解集的相应拓扑中的解存在性和密度。为此,我们引入了最大单调算子之间的新距离。
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