A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities
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Let ๐ be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space ๐โ. Let ๐ : ๐ โ ๐ท(๐)โ 2๐โ be maximal monotone of type ๐ช๐ ๐ (i.e., there exist ๐ โฅ 0 and a nondecreasing function ๐ : [0,โ) โ [0,โ) with ๐(0) = 0 such that โจVโ, ๐ฅ โ ๐ฆโฉ โฅ โ๐โ๐ฅโ โ ๐(โ๐ฆโ) for all ๐ฅ โ ๐ท(๐), Vโ โ ๐๐ฅ, and๐ฆ โ ๐),๐ฟ : ๐ โ ๐ท(๐ฟ) โ ๐โ be linear, surjective, and closed such that ๐ฟโปยน : ๐โ โ ๐ is compact, and ๐ถ : ๐ โ ๐โ be a bounded demicontinuous operator. A new degree theory is developed for operators of the type ๐ฟ+๐+๐ถ.The surjectivity of ๐ฟ can be omitted provided that ๐ (๐ฟ) is closed, ๐ฟ is densely defined and self-adjoint, and ๐ = ๐ป, a real Hilbert space.The theory improves the degree theory of Berkovits and Mustonen for ๐ฟ+๐ถ, where ๐ถ is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when ๐ฟ is monotone, a maximality result is included for ๐ฟ and ๐ฟ+๐.The theory is applied to prove existence of weak solutions in ๐ = ๐ฟโ(0, ๐;๐ปยนโ (ฮฉ)) of the nonlinear equation given by ๐๐ข/๐๐กโฮฃ๐ ๐=1((๐/๐๐ฅ๐)๐ด๐(๐ฅ, ๐ข, โ๐ข))+๐ป๐(๐ฅ, ๐ข, โ๐ข) = ๐(๐ฅ, ๐ก), (๐ฅ, ๐ก) โ Q๐; ๐ข(๐ฅ, ๐ก) = 0, (๐ฅ, ๐ก) โ ๐Q๐; and๐ข(๐ฅ, 0) = ๐ข(๐ฅ, ๐), ๐ฅ โ ฮฉ, where๐ > 0, ๐๐ = ฮฉร(0,๐), ๐๐๐ = ๐ฮฉร(0,๐), ๐ด๐(๐ฅ, ๐ข, โ๐ข) = (๐/๐๐ฅ๐)๐(๐ฅ, ๐ข, โ๐ข)+๐๐(๐ฅ, ๐ข, โ๐ข) (๐ = 1, 2, . . . , ๐),๐ป๐(๐ฅ, ๐ข, โ๐ข) = โ๐ฮ๐ข + ๐(๐ฅ, ๐ข, โ๐ข), ฮฉ is a nonempty, bounded, and open subset of โ๐ with smooth boundary, and ๐, ๐๐, ๐ : ฮฉ ร โ ร โ๐ 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.