Giuseppe Anichini

Using solution sets for solving boundary value problems for ordinary differential equations

where f E C(I x R”, R”), S c C(Z, R”), consists of finding fixed points of a solution map E for a suitable linearized problem associated to (BV) (see e.g. [2, 5, 6, 71). In these problems the map X is either continuous and single-valued or upper semicontinuous set-valued with convex values. These are rather strong restrictions because the set of solutions of a general boundary value problem may well be neither a singleton nor a convex set. The case in which the solution map is upper semicontinuous but with nonnecessarily convex values is considered in this paper. More precisely we want to consider the case when the values of X form an acyclic set and the case when the values of C are sets with a finite number of elements. An application to the existence of periodic solutions for a LiCnard type equation is also given. Moreover an existence theorem for a boundary value problem of the following form: = f(l, x, x’, x”), teIcR, x E R”

Controllability and controllability with prescribed controls

In this paper, a controllability condition with prescribed controls (that is, a controllability condition for which the initial and final value of the control are givena priori), for classes of nonlinear control process with different controllability criteria, is obtained. The result is achieved by using the fixed-point argument as a crucial tool.

Multivalued differential equations in Banach space. An application to control theory

This paper is concerned with the study of existence theorems for multivalued differential systems in infinite-dimensional Banach space: the method used is based on techniques (extension theorem for linear operator, compactly convergent sequences) developed earlier by the authors for multivalued differential systems defined inn-dimensional vector spaces. As an application, the authors consider a distributed-parameter control problem arising in mathematical physics, more specifically, in the study of heat transfer in solids.

How to Make Use of the Solution Set to Solve Boundary Value Problems

The set of solutions of a differential problem can be conveniently used as a tool to get existence results for boundary value problems. In this paper a survey of results, methods, and applications concerning such a tool is given.

A Lipschitz boundary value problem for nonlinear ordinary differential equations on non compact intervals

(L(T) is called the Lipschitz constant of T.) Many authors have been interested in BVP for ordinary differential equations mainly when J is a real compact interval: Conti [3] deals with boundary conditions of the form x E Q c C(J, Rd) for a differential equation as x’ = A(t)x +f(t) (I) or, more generally, x’ = A(t)x +f(t, x). (If) Opial [7] considers systems of differential equations as x’ = F(t, x) for which Lx = y is a linear boundary condition. For many other cases, often for the two points problem for a second order differential equation, wider references may be found in the expository papers of Conti [3], Opial[7], Jackson [S] and in the book of Bernfeld and Lakshmikantham [2]. We want also to recall the papers of Furi, Martelli and Vignoli [4], Anichini [l], Tiba [lo], for the case of a nonlinear boundary condition, and the paper of Kartsatos [6] for the case in which .I is a noncompact real interval.