A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities. (5th December 2016) - Asfaw, Teffera M.
Abstract : Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X ⁎ . Let T : X ⊇ D T → 2 X ⁎ be maximal monotone of type Γ d ϕ (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 ⁎ ∈ T x, and y ∈ 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 = L 2 0, T ; H 0 1 Ω of the nonlinear equation given by ∂ u / ∂ t - ∑ i = 1 N ( ∂ / ∂ x i ) A i x, u, ∇ u + H λ x, u, ∇ u = f x, t, x, t ∈ Q T ; u x, t = 0, x, t ∈ ∂ Q T ; and u x, 0 = u x, T, x ∈ Ω, where λ > 0, Q T = Ω × 0, T, ∂ Q T = ∂ Ω × 0, T, A i x, u, ∇ u = ∂ / ∂ x i ρ x, u, ∇ u + a i x, u, ∇ u ( i = 1, 2, …, N ), H λ x, u, ∇ u = - λ Δ u + g x, u, ∇ u, Ω is a nonempty, bounded, and open subset of R N with smooth boundary, and ρ, a i, g : Ω ¯ × R × R N → 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.