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Abstract
In this paper, we deal with the regularity and a variation of constant formula for solutions of the nonlinear differential equation with delay nonlinear terms governed by the variational inequality in Hilbert spaces. Without the conditions of the uniform boundedness of the nonlinear terms and the compactness of the principal operators, we obtain the wellposedness and the norm estimate of the given equation by converting the problem into the contraction mapping principle on state space.MSC:35K85, 35F25.
Highlights
Let H and V be two complex Hilbert spaces
The theory of variational evolution inequalities is one of the most important domains of application of the ideas and techniques of differential equations associated with maximal monotone operators and semigroups of nonlinear contractions
In this paper we are primarily interested in the regular problem that arise as direct consequences of the general theory developed previously, and we consider to put in perspective those models of initial value problems which can be formulated as nonlinear differential equations of variational inequalities
Summary
Let H and V be two complex Hilbert spaces. Assume that V is dense subspace in H and the injection of V into H is continuous. The theory of variational evolution inequalities is one of the most important domains of application of the ideas and techniques of differential equations associated with maximal monotone operators and semigroups of nonlinear contractions. In this paper we are primarily interested in the regular problem that arise as direct consequences of the general theory developed previously, and we consider to put in perspective those models of initial value problems which can be formulated as nonlinear differential equations of variational inequalities. Without conditions of the uniform boundedness of the nonlinear terms and the compactness of the principal operators, we obtain the wellposedness of (NDE) by converting the problem into the contraction mapping principle and the norm estimate of a solution of the above nonlinear equation on L ( , T; V ) ∩ W , ( , T; V ∗) ∩ C([ , T]; H).
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