Abstract
In studying the surjectivity of set-valued mappings, a modification of the “acute-angle lemma” (or the “equilibrium theorem”) is used. This allows one to weaken the coerciveness condition. Some applications to differential equations (inclusions) with Neumann boundary conditions are considered on Sobolev spaces W p 1(Ω) in which operators are used that are not coercive in the classical sense.
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REFERENCES
B. N. Pshenichnyi, Convex Analysis and Extremum Problems [in Russian], Nauka, Moscow (1980).
B. N. Pshenichnyi, “Theorems on implicit functions for set-valued mappings,” Kibernetika, No. 4, 36–43 (1986).
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, N.Y. (1993).
F. E. Browder and P. Hess, “Nonlinear mappings of monotone type in Banach spaces,” J. of Functional Analysis, 11, No. 2, 251–294 (1972).
J.-L Lions, Some Methods of Solution of Nonlinear Boundary-Value Problems [Russian translation], Mir, Moscow (1972).
V. S. Mel'nik and O. V. Solonukha, “Stationary variational inequalities with set-valued operators,” Kibern. Sist. Anal., No. 3, 74-89 (1997).
O. V. Solonukha, “On solvability of monotone type problems with non-coercive set-valued operators,” Methods of Functional Analysis and Topology, 6, No. 2, 66–72 (2000).
Ali M. Ben Cheikh and O. Guib'e, “Result d'existence et d'unicit ‘e pour une classe de probl ‘emes non lin ‘eares et non coercifs,” C.R. Acad. Sci. Paris., 329, S'erie I, 967–972 (1999).
M. Z. Zgurovskii and V. S. Mel'nik, “The Ki-Fan inequality and operator inclusions in Banach spaces,” Kibern. Sist. Anal., No. 2, 70–85 (2002).
O. V. Solonoukha, “On solvability of variational inequalities with “+”-- coercive multivalued mappings,” Nonlinear Boundary Value Problems, 9, 126–129 (1999).
O. V. Solonoukha, “On the stationary variational inequalities with the generally pseudomonotone operators,” Methods of Funct. Analysis and Topology, 3, No. 4, 81–95 (1997).
G. I. Laptev, “The first boundary-value problem for quasilinear elliptic equations of the second order with double degeneracy,” Differential Equations, 30(6), 1057–1068 (1994).
I. V. Skrypnik, Methods of Investigation of Nonlinear Elliptic Boundary Problems [in Russian], Nauka, Moscow (1990).
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).
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Solonukha, O.V. On Solvability of Operator Inclusions: Application to Elliptic Problems with the Neumann Boundary Condition. Cybernetics and Systems Analysis 40, 587–593 (2004). https://doi.org/10.1023/B:CASA.0000047879.58723.e2
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DOI: https://doi.org/10.1023/B:CASA.0000047879.58723.e2