Summary
Let H be a real Hilbert space, ϕ:H → [0, + ∞] a proper l.s.c., convex function with Lk:={u ε H; ∥u∥2 + ϕ(u) ≦ k} compact for every k > 0, let τ > 0 be a given constant and\(C_{\partial _\varphi } ([ - \tau ,0];H): = \{ v \in C([ - \tau ,0];H); v(t) \in D(\partial _\varphi ) a.e. for t \in ( - \tau ,0)\} \). We prove an existence result for strong solutions to a class of functional differential equations of the form
, where F: [0, T] × D(∂ϕ) ×\(C_{\partial _\varphi } \)([−τ, 0]; H) → H satisfies a certain demiclosedness condition, while v ε\(C_{\partial _\varphi } \)([−τ, 0]; H), v(0) ε D(ϕ) and\(\mathop \smallint \limits_{ - \tau }^0 ||\partial \varphi ^0 (v(s))||^2 ds< + \infty \).
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Research done while the second author was a C.N.R. visiting professor at the University of Trieste, Italy.
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Mitidieri, E., Vrabie, I.I. A class of strongly nonlinear functional differential equations. Annali di Matematica pura ed applicata 151, 125–147 (1988). https://doi.org/10.1007/BF01762791
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DOI: https://doi.org/10.1007/BF01762791