On the existence of solutions of strongly damped nonlinear wave equations

We investigate the existence and uniqueness of solutions of the following equation of hyperbolic type with a strong dissipation: utt(t,x)−(α+β(∫Ω|∇u(t,y)|2dy)γ)Δu(t,x)                                −λΔut(t,x)+μ|u(t,x)|q−1u(t,x)=0,     x∈Ω,t≥0            u(0,x)=u0(x),          ut(0,x)=u1(x),      x∈Ω,  u|∂Ω=0, where q>1,λ>0,μ∈ℝ,α,β≥0,α+β>0, and Δ is the Laplacian in ℝN

Anti-periodic solutions to a parabolic hemivariational inequality

summary:In this paper we deal with the anti-periodic boundary value problems with nonlinearity of the form $b(u)$, where $b\in L^{\infty }_{{\rm loc}}({R}).$ Extending $b$ to be multivalued we obtain the existence of solutions to hemivariational inequality and variational-hemivariational inequality

Controllability of second-order neutral functional differential inclusions in Banach spaces

AbstractIn this paper, we prove the controllability of second-order neutral functional differential inclusions in Banach spaces. The result are obtained by using the theory of strongly continuous cosine families and a fixed point theorem for condensing maps due to Martelli

Nonlinear variational evolution inequalities in Hilbert spaces

The regular problem for solutions of the nonlinear functional differential equations with a nonlinear hemicontinuous and coercive operator A and a nonlinear term f(.,.):x′(t)+Ax(t)+∂ϕ(x(t))∋f(t,x(t))+h(t) is studied. The existence, uniqueness, and a variation of solutions of the equation are given

Asymptotic behavior of almost-orbits of reversible semigroups of non-Lipschitzian mappings in Banach spaces

Let C be a nonempty closed convex subset of a uniformly convex Banach space E with a Fréchet differentiable norm, G a right reversible semitopological semigroup, and ={S(t):t∈G} a continuous representation of G as mappings of asymptotically nonexpansive type of C into itself. The weak convergence of an almost-orbit {u(t):t∈G} of ={S(t):t∈G} on C is established. Furthermore, it is shown that if P is the metric projection of E onto set F(S) of all common fixed points of ={S(t):t∈G}, then the strong limit of the net {Pu(t):t∈G} exists

Existence and Asymptotic Stability of Solutions for Hyperbolic Differential Inclusions with a Source Term

We study the existence of global weak solutions for a hyperbolic differential inclusion with a source term, and then investigate the asymptotic stability of the solutions by using Nakao lemma