Boundary Value Problems for Generalized ODEs

Author(s)

Bonotto, E. M. ; Federson, M. ; Mesquita, C.

• Document type: Journal article
• Source: JOURNAL OF GEOMETRIC ANALYSIS; v. 33, n. 1, p. 37-pg., 2023-01-01.

Abstract

The aim of this paper is to investigate the existence and uniqueness of solutions of the following boundary value problem concerning generalized ODEs {dx/d tau = D[A(t)x + F(t)], integral(b)(a) d[K(s)] x(s) = r, for operators taking values in general Banach spaces. Up to now, this problemwas only tackled in the case of finite dimensional space-valued functions. We establish necessary and sufficient conditions not only for the existence of at least one solution, but also for the uniqueness of a solution. Another important result describes the solution in terms of a Green function and the fundamental operator of the corresponding homogeneous problem. We also explore the problem dx/d tau = D[A(t)x + F(t)] and we give necessary and sufficient conditions for the existence of a (theta, T)-periodic solution, where theta is an element of R, with. theta not equal 0, and T > 0, recalling that the notion of (theta, T)-periodic solution generalizes the notions of periodic, anti-periodic, almost periodic, and quasi-periodic solution. Examples are given to illustrate the main results. In addition, we apply the main theorems to abstract ODEs, as well as to a Volterra-Stieltjes-type integral equation, and we include examples of these as well. (AU)

• FAPESP's process: 20/14075-6 - Dynamical systems and their attractors under perturbations
• Grantee: Alexandre Nolasco de Carvalho
• Support Opportunities: Research Projects - Thematic Grants