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Zbl 0993.34060
Jeong, Jin Mun; Jeong, Doo Hoan
Regularity for nonlinear variational evolution inequalities in Hilbert spaces.
(English)
[A] Cho, Yeol Je (ed.), Differential equations and applications. Proceedings of the international conference on mathematical analysis and applications, Chinju, China, August 3-4, 1998. Huntington, NY: Nova Science Publishers. 157-169 (2000). ISBN 1-56072-767-5/hbk

Let $H$ and $V$ be two real separable Hilbert spaces such that $V$ is a dense subspace of $H$. Let the single-valued operator $A$ be given which is hemicontinuous and coercive from $V$ to $V^{*}$. Here, $V^{*}$ stands for the dual space of $V$. Let $\varphi: V\to (-\infty,+\infty]$ be a lower semicontinuous proper convex function. The authors study the existence, uniqueness and a variation of solution to the following initial value problem $$\frac{d x(t)}{d t}+A x(t)+\partial \varphi (x(t))\notin f(t,x(t))+h(t), \quad t\in [0,T],$$ $x(0)=x_{0},$ where $f:\bbfR\times V\to H$ is Lipschitz continuous, $h:\bbfR\to H$ and $\partial \varphi: V\to V^{*}$ is the subdifferential multivalued operator of $\varphi$ defined by $$\partial\varphi (x)=\{x^{*}\in V^{*}: \varphi(x)\leq \varphi(y) + (x^{*},x-y), \quad y\in V \},$$ where $(.,.)$ denotes the duality pairing between $V^{*}$ and $V$.
[Mouffak Benchohra (Sidi Bel Abbes)]
MSC 2000:
*34G25 Evolution inclusions
49J24 Optimal control problems with differential inclusions (existence)
47J20 Inequalities involving nonlinear operators
49J40 Variational methods including variational inequalities

Keywords: nonlinear variational evolution inequality; maximal monotone operators; subdifferential operator; regularity

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