On the existence of solutions for nonlocal boundary value problems

Abstract

This paper deals with the existence of solutions for a system of equations ὔὔ = ( , , ὔ ), where : [0, 1] × ℝ × ℝ → ℝ is a vector function, subject to various nonlocal boundary conditions. The method is to impose an a priori bound condition on and apply the Leray-Schauder xed point theorem. Our results extend some results in the references and also provide the information on the existence of stationary solutions for the heat equation with nonlinear gradient source terms.

An inverse problem for an integro‐differential operator on a star‐shaped graph

“…Here, we illustrate the idea of generalizing such models to the case of the star-shaped graph. Firstly, we describe the model of a thermostat, following the ideas of [12,57]. Although in [12,57] nonlinear equations were studied, we consider the simplest linear heat equation…”

“…Firstly, we describe the model of a thermostat, following the ideas of [12,57]. Although in [12,57] nonlinear equations were studied, we consider the simplest linear heat equation…”

“…However, for modeling some physical processes, nonlocal BCs appear to be more adequate than local ones. Differential equations with integral BCs were used in the study of diffusion and heating processes, in the theory of elasticity, biotechnology, and other applications (see [5][6][7][8][9][10][11][12]). It is shown in Appendix B of this paper that various models with nonlocal BCs on intervals can be generalized to graph-like structures.…”

Generalized fractional delay functional equations with Riemann-Stieltjes and infinite point nonlocal conditions

“…So, these conditions allow us to get more accurate model of the phenomena being studied and therefore, better results can be derived. Recently, the study of existence of solutions for initial value problems subject to non-classical conditions such as the Riemann-Stieltjes and the infinite point nonlocal conditions becomes an issue of great importance, see [14,18]. As a continuation in that progress, El-Sayed et al studied in [10] the evolution of a physical model described by a nonlocal integro-differential equation on the form dz dt = g t, z(t), to demonstrate the applicability of their model.…”

A topological degree approach to a nonlocal Neumann problem for a system at resonance

“…Boundary value problems with nonlocal boundary conditions arise in a variety of different areas of applied mathematics and physics. They occur naturally in chemical engineering, thermo-elasticity, underground water flow, population dynamics or heat-flow problems (see, for example, [6,11,18,22,24,25] and the references therein).…”