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A unified existence theory for evolution equations and systems under nonlocal conditions

Tiziana Cardinali, Radu Precup, Paola Rubbioni

Abstract

We investigate the effect of nonlocal conditions expressed by linear continuous mappings over the hypotheses which guarantee the existence of global mild solutions for functional-differential equations in a Banach space. A progressive transition from the Volterra integral operator associated to the Cauchy problem, to Fredholm type operators appears when the support of the nonlocal condition increases from zero to the entire interval of the problem. The results are extended to systems of equations in a such way that the system nonlinearities behave independently as much as possible and the support of the nonlocal condition may differ from one variable to another.

A unified existence theory for evolution equations and systems under nonlocal conditions

Abstract

We investigate the effect of nonlocal conditions expressed by linear continuous mappings over the hypotheses which guarantee the existence of global mild solutions for functional-differential equations in a Banach space. A progressive transition from the Volterra integral operator associated to the Cauchy problem, to Fredholm type operators appears when the support of the nonlocal condition increases from zero to the entire interval of the problem. The results are extended to systems of equations in a such way that the system nonlinearities behave independently as much as possible and the support of the nonlocal condition may differ from one variable to another.

Paper Structure

This paper contains 3 sections, 9 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Existence and localization of solutions for evolution equations
  3. Existence and localization of solutions for evolution systems

Key Result

Theorem \oldthetheorem

Let $U$ be an open subset of a Banach space $X,$ and let $N:\overline{U}\rightarrow X$ be continuous. Assume that for some $u_{0}\in U$ the following conditions are satisfied: (a)$N\left( u\right) -u_{0}\neq \lambda \left( u-u_{0}\right)$ on $\partial U$ for all $\lambda >1;$ (b) if $C\subset \overl

Theorems & Definitions (17)

  • Theorem \oldthetheorem
  • Remark 2.1
  • Theorem \oldthetheorem
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3: the Kamke function of a nonlocal problem
  • Lemma 2.2
  • Theorem \oldthetheorem
  • Corollary 2.1: case of time-independent radius
  • ...and 7 more