Stig-Olof Londen

An abstract nonlinear Volterra integrodifferential equation

We study the nonlinear Volterra equation u′(t) + Bu(t) + ∫0t a(t − s) Au(s) ds ϵ F(t) (0 < t < ∞) (′ = ddt), u(0) = u0, (∗) as well as the corresponding problem with infinite delay u′(t) + Bu(t) + ∫−∞t a(t − s) Au(s) ds ϵ ƒ(t) (0 < t < ∞), u(t) = h(t) (−∞ < t ⩽ 0). (∗∗) Under various assumptions on the nonlinear operators A, B and on the given functions a, F, f, h existence theorems are obtained for (∗) and (∗∗, followed by results concerning boundedness and asymptotic behaviour of solutions on (0 ⩽ < ∞); two applications of the theory to problems of nonlinear heat flow with “infinite memory” are also discussed.

On an Integral Equation in a Hilbert Space

We consider the nonlinear Volterra equation \[(1.1)\qquad u(t) + \int_0^t {a(t - \tau )g(u(\tau ))d\tau \ni f(t)} ,\qquad t \geqq 0,\] where a, g, f are given and u is the unknown function taking values in a real Hilbert space H. The kernel

On the solutions of a nonlinear Volterra equation

a(t)

Some Nonoscillation Theorems for a Second Order Nonlinear Differential Equation

maps

On the Asymptotic Behavior of the Bounded Solutions of a Nonlinear Volterra Equation

R^ + \to R

Maximal regularity for stochastic integral equations

whereas f is a map of

On a Stochastic Parabolic Integral Equation

R^ + \to H

On a Fractional Partial Differential Equation with Dominating Linear Part

. The nonlinear function g has its domain and range contained in H.Making use of the theory of monotone operators we give at first an existence and uniqueness theorem on (1.1). This is followed by a result detailing the asymptotic behavior of solutions of (1.1). Finally we give some applications of our results. The results extend earlier results by Barbu.

On the Asymptotic Behavior of the Solution of a Nonlinear Integrodifferential Equation

Publisher Summary This chapter presents the solutions of a nonlinear Volterra equation. A real nonlinear Volterra equation is given by: where b(t), f(t), g(x) are given real functions. There exists a solution x(t) on 0 ≤ t ≤ ∞. Moreover, under this hypothesis any solution of il) on 0 ≤ t ≤ ∞ satisfies sup |x(t)| < ∞. These results generalize earlier results and partially overlap recent results obtained.

A note on Volterra equations in a Hilbert space

We investigate the equation \[\left[ {p(t)x^\prime (t)} \right]^\prime + q(t)g(x(t)) = f(t) \] and give sufficient hypotheses for the approach to zero of all nonoscillatory solutions. The conditions are related to earlier theorems of Bhatia and Hammett.