A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities
Let 𝑋 be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space 𝑋∗. Let 𝑇 : 𝑋 ⊇ 𝐷(𝑇)→ 2𝑋∗ be maximal monotone of type 𝚪𝜙 𝑑 (i.e., there exist 𝑑 ≥ 0 and a nondecreasing function 𝜙 : [0,∞) → [0,∞) with 𝜙(0) = 0 such that ⟨V∗, 𝑥 − 𝑦⟩ ≥ −𝑑‖𝑥‖ − 𝜙(‖𝑦‖) for all 𝑥 ∈ 𝐷(𝑇), V∗ ∈ 𝑇𝑥, and𝑦 ∈ 𝑋),𝐿 : 𝑋 ⊃ 𝐷(𝐿) → 𝑋∗ be linear, surjective, and closed such that 𝐿⁻¹ : 𝑋∗ → 𝑋 is compact, and 𝐶 : 𝑋 → 𝑋∗ be a bounded demicontinuous operator. A new degree theory is developed for operators of the type 𝐿+𝑇+𝐶.The surjectivity of 𝐿 can be omitted provided that 𝑅(𝐿) is closed, 𝐿 is densely defined and self-adjoint, and 𝑋 = 𝐻, a real Hilbert space.The theory improves the degree theory of Berkovits and Mustonen for 𝐿+𝐶, where 𝐶 is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when 𝐿 is monotone, a maximality result is included for 𝐿 and 𝐿+𝑇.The theory is applied to prove existence of weak solutions in 𝑋 = 𝐿₂(0, 𝑇;𝐻¹₀ (Ω)) of the nonlinear equation given by 𝜕𝑢/𝜕𝑡−Σ𝑁 𝑖=1((𝜕/𝜕𝑥𝑖)𝐴𝑖(𝑥, 𝑢, ∇𝑢))+𝐻𝜆(𝑥, 𝑢, ∇𝑢) = 𝑓(𝑥, 𝑡), (𝑥, 𝑡) ∈ Q𝑇; 𝑢(𝑥, 𝑡) = 0, (𝑥, 𝑡) ∈ 𝜕Q𝑇; and𝑢(𝑥, 0) = 𝑢(𝑥, 𝑇), 𝑥 ∈ Ω, where𝜆 > 0, 𝑄𝑇 = Ω×(0,𝑇), 𝜕𝑄𝑇 = 𝜕Ω×(0,𝑇), 𝐴𝑖(𝑥, 𝑢, ∇𝑢) = (𝜕/𝜕𝑥𝑖)𝜌(𝑥, 𝑢, ∇𝑢)+𝑎𝑖(𝑥, 𝑢, ∇𝑢) (𝑖 = 1, 2, . . . , 𝑁),𝐻𝜆(𝑥, 𝑢, ∇𝑢) = −𝜆Δ𝑢 + 𝑔(𝑥, 𝑢, ∇𝑢), Ω is a nonempty, bounded, and open subset of ℝ𝑁 with smooth boundary, and 𝜌, 𝑎𝑖, 𝑔 : Ω × ℝ × ℝ𝑁 → ℝ satisfy suitable growth conditions. In addition, a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity.