A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities

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2016-10-18
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Hindawi
Abstract

Let ๐‘‹ be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space ๐‘‹โˆ—. Let ๐‘‡ : ๐‘‹ โŠ‡ ๐ท(๐‘‡)โ†’ 2๐‘‹โˆ— be maximal monotone of type ๐šช๐œ™ ๐‘‘ (i.e., there exist ๐‘‘ โ‰ฅ 0 and a nondecreasing function ๐œ™ : [0,โˆž) โ†’ [0,โˆž) with ๐œ™(0) = 0 such that โŸจVโˆ—, ๐‘ฅ โˆ’ ๐‘ฆโŸฉ โ‰ฅ โˆ’๐‘‘โ€–๐‘ฅโ€– โˆ’ ๐œ™(โ€–๐‘ฆโ€–) for all ๐‘ฅ โˆˆ ๐ท(๐‘‡), Vโˆ— โˆˆ ๐‘‡๐‘ฅ, and๐‘ฆ โˆˆ ๐‘‹),๐ฟ : ๐‘‹ โŠƒ ๐ท(๐ฟ) โ†’ ๐‘‹โˆ— be linear, surjective, and closed such that ๐ฟโปยน : ๐‘‹โˆ— โ†’ ๐‘‹ is compact, and ๐ถ : ๐‘‹ โ†’ ๐‘‹โˆ— be a bounded demicontinuous operator. A new degree theory is developed for operators of the type ๐ฟ+๐‘‡+๐ถ.The surjectivity of ๐ฟ can be omitted provided that ๐‘…(๐ฟ) is closed, ๐ฟ is densely defined and self-adjoint, and ๐‘‹ = ๐ป, a real Hilbert space.The theory improves the degree theory of Berkovits and Mustonen for ๐ฟ+๐ถ, where ๐ถ is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when ๐ฟ is monotone, a maximality result is included for ๐ฟ and ๐ฟ+๐‘‡.The theory is applied to prove existence of weak solutions in ๐‘‹ = ๐ฟโ‚‚(0, ๐‘‡;๐ปยนโ‚€ (ฮฉ)) of the nonlinear equation given by ๐œ•๐‘ข/๐œ•๐‘กโˆ’ฮฃ๐‘ ๐‘–=1((๐œ•/๐œ•๐‘ฅ๐‘–)๐ด๐‘–(๐‘ฅ, ๐‘ข, โˆ‡๐‘ข))+๐ป๐œ†(๐‘ฅ, ๐‘ข, โˆ‡๐‘ข) = ๐‘“(๐‘ฅ, ๐‘ก), (๐‘ฅ, ๐‘ก) โˆˆ Q๐‘‡; ๐‘ข(๐‘ฅ, ๐‘ก) = 0, (๐‘ฅ, ๐‘ก) โˆˆ ๐œ•Q๐‘‡; and๐‘ข(๐‘ฅ, 0) = ๐‘ข(๐‘ฅ, ๐‘‡), ๐‘ฅ โˆˆ ฮฉ, where๐œ† > 0, ๐‘„๐‘‡ = ฮฉร—(0,๐‘‡), ๐œ•๐‘„๐‘‡ = ๐œ•ฮฉร—(0,๐‘‡), ๐ด๐‘–(๐‘ฅ, ๐‘ข, โˆ‡๐‘ข) = (๐œ•/๐œ•๐‘ฅ๐‘–)๐œŒ(๐‘ฅ, ๐‘ข, โˆ‡๐‘ข)+๐‘Ž๐‘–(๐‘ฅ, ๐‘ข, โˆ‡๐‘ข) (๐‘– = 1, 2, . . . , ๐‘),๐ป๐œ†(๐‘ฅ, ๐‘ข, โˆ‡๐‘ข) = โˆ’๐œ†ฮ”๐‘ข + ๐‘”(๐‘ฅ, ๐‘ข, โˆ‡๐‘ข), ฮฉ is a nonempty, bounded, and open subset of โ„๐‘ with smooth boundary, and ๐œŒ, ๐‘Ž๐‘–, ๐‘” : ฮฉ ร— โ„ ร— โ„๐‘ โ†’ โ„ satisfy suitable growth conditions. In addition, a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity.

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