Iterative equations in Banach spaces

Let X be a Banach space, let

     

Semilinear evolution equations in Banach spaces

We study the evolution equation u′(t) = Au(t) + J(u(t)), t ? 0, where e^tA is a C₀ semi-group on a Banach space E, and J is a “singular” non-linear mapping defined on a subset of E. In Sections 1 and 2 of the paper we suppress the map J and instead consider maps K_t: E → E, t > 0, which heuristically are just e^tAJ. Under certain integrability conditions on the K_t we prove existence and uniqueness of local solutions to the integral equation u(t) = e^tAφ + ∝₀^tK_{t ? s}(u(s)) ds for all φ in E, and investigate the regularity of the solutions. Conditions which insure existence of global solutions are given. In Section 3 we recover the map J from the maps K_t, and show that the generator of the semi-flow on E induced by the integral equation has dense domain. Finally, we apply these results to a large class of examples which includes polynomial perturbations to elliptic operators on a domain in Rⁿ.     

Delay difference equations in Banach spaces

We derive explicit stability conditions for semilinear delay difference equations in a Banach space. It is assumed that the nonlinearities of the considered equations satisfy the local Lipschitz condition. By virtue of the new estimates for the norm of functions of quasi-Hermitian operators, explicit stability and boundedness conditions are given. Applications to infinite dimensional delay difference systems are discussed.     

Nonlinear evolution equations in Banach spaces

The evolution problem 0∈du/dt+A(t)u(t),u(s)=x, where theA(t) are nonlinear operators acting in a Banach space, is studied. Evolution operators are constructed from theA(t) under various assumptions. Basic properties of these evolution operators are established and their relationship to the evolution equation is determined. The results obtained extend several known existence theorems and provide generalized solutions of the evolution equation in more general cases. Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462. and in part by the Office of Naval Research under Contract N000-14-69-A-0200-4022. Reproduction in whole or in part is permitted for any purpose of the United States Government. This research was partially supported by the NSF Grant #GP-18127.     

Orientor field equations in Banach spaces

This paper focuses on the interplay between seminormality requirements and convergence hypotheses on trajectories in the lower closure theorems for orientor field equations. With the use of a weak form of seminormality, called the intermediate property (Q*), we obtain lower closure theorems (and thereby closure theorems) for orientor field equations in Banach spaces, under the assumption of strong convergence of some coordinates of the trajectories, while only weak convergence is assumed in the others. In the Euclidean case, this requirement of property (Q*) is further reduced to mere Kuratowski property (K), under the usual growth-type conditions. Finally, in the appendix, property (Q*) is investigated in detail.This research has been partially supported by Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan.     

Multivalued differential equations in separable Banach spaces

This paper is concerned with multivalued differential equations of the form F(t,x), whereF is a multivalued mapping taking as its values nonempty compact, but not necessarily convex, subsets in a separable Banach space. The main result is connected with the existence of solutions of these equations.     

Oscillation of differential equations in Banach spaces

Summary In this paper some properties of solutions of the differential equation Y″(t)++P(t) Y(t)=0 in Banach spaces are investigated. In particular, conditions are given for some solutions of such equations to possess an infinite number of zeros as t → ∞ while another condition ensures some solutions possess only a finite number of zeros, Some examples and a theorem show the concept of an oscillatory solution of a differential equation in a Banach space involves pathologies not found in the case of finite dimensional spaces. Upon specialization of the Banach spaces involved the results reduce to known theorems. Entrata in Redazione il 13 maggio 1969.     

Singularly perturbed evolution equations in Banach spaces

The singular perturbation method is applied to systems of linear evolution equations in Banach spaces with one of the time derivatives multiplied by a small parameter. An asymptotic solution of any given order is shown to consist of inner and outer terms obtained by a proper choice of initial conditions, and to converge uniformly to the exact solution.     

Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space-time Hölder regularity for solutions of the parabolic stochastic evolution equation

     

Causal functional differential equations in Banach spaces

The aim of this paper is to establish the existence of solutions and some properties of set solutions for a Cauchy problem with causal operator in a separable Banach space.     

On the Hamilton-Jacobi-Bellman equations in Banach spaces

This paper is concerned with a certain class of distributed parameter control problems. The value function of these problems is shown to be the unique viscosity solution of the corresponding Hamiltonian-Jacobi-Bellman equation. The main assumption is the existence of an increasing sequence of compact invariant subsets of the state space. In particular, this assumption is satisfied by a class of controlled delay equations. This research was partly supported by the Institute for Mathematics and Its Applications with funds provided by the National Science Foundation and the Office of Naval Research. The author is indebted to Professor P. L. Lions for stimulating discussions and helpful suggestions.