Abstract
One of the most effective approaches to investigate nonlinear problems, represented by partial differential equations, inclusions and inequalities with boundary values, consists in the reduction of them into differential-operator inclusions, in infinite-dimensional spaces governed by nonlinear operators. In order to study these objects, the modern methods of nonlinear analysis have been used [7, 10, 11, 26]. Convergence of approximate solutions to an exact solution of the differential-operator equation or inclusion is frequently proved on the basis of the property of monotony or pseudomonotony of the corresponding operator.
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- 1.
That is, V is a real reflexive separable Banach space embedded into a real Hilbert space H continuously and densely, H is identified with its conjugated space H ∗ and V ∗ is a dual space to V. So, we have such chain of continuous and dense embeddings: \(V \subset H \equiv {H}^{{_\ast}}\subset {V }^{{_\ast}}\) (see, e.g., [49]).
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Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V. (2012). Properties of Resolving Operator for Nonautonomous Evolution Inclusions: Pullback Attractors. In: Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Advances in Mechanics and Mathematics, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28512-7_7
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DOI: https://doi.org/10.1007/978-3-642-28512-7_7
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