A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities
Abstract
Let π be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space πβ. Let π : π β π·(π)β 2πβ be maximal monotone of type πͺπ π (i.e., there exist π β₯ 0 and a nondecreasing function π : [0,β) β [0,β) with π(0) = 0 such that β¨Vβ, π₯ β π¦β© β₯ βπβπ₯β β π(βπ¦β) for all π₯ β π·(π), Vβ β ππ₯, andπ¦ β π),πΏ : π β π·(πΏ) β πβ be linear, surjective, and closed such that πΏβ»ΒΉ : πβ β π is compact, and πΆ : π β πβ be a bounded demicontinuous operator. A new degree theory is developed for operators of the type πΏ+π+πΆ.The surjectivity of πΏ can be omitted provided that π
(πΏ) is closed, πΏ is densely defined and self-adjoint, and π = π», a real Hilbert space.The theory improves the degree theory of Berkovits and Mustonen for πΏ+πΆ, where πΆ is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when πΏ is monotone, a maximality result is included for πΏ and πΏ+π.The theory is applied to prove existence of weak solutions in π = πΏβ(0, π;π»ΒΉβ (Ξ©)) of the nonlinear equation given by ππ’/ππ‘βΞ£π π=1((π/ππ₯π)π΄π(π₯, π’, βπ’))+π»π(π₯, π’, βπ’) = π(π₯, π‘), (π₯, π‘) β Qπ; π’(π₯, π‘) = 0, (π₯, π‘) β πQπ; andπ’(π₯, 0) = π’(π₯, π), π₯ β Ξ©, whereπ > 0, ππ = Ξ©Γ(0,π), πππ = πΞ©Γ(0,π), π΄π(π₯, π’, βπ’) = (π/ππ₯π)π(π₯, π’, βπ’)+ππ(π₯, π’, βπ’) (π = 1, 2, . . . , π),π»π(π₯, π’, βπ’) = βπΞπ’ + π(π₯, π’, βπ’), Ξ© is a nonempty, bounded, and open subset of βπ with smooth boundary, and π, ππ, π : Ξ© Γ β Γ βπ β β satisfy suitable growth conditions. In addition, a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity.