Application of a noncompactness technique to an investigation of the existence of solutions to a nonlocal multivalued Darboux problem
type: Item
instance of: scholarly article
title: Application of a noncompactness technique to an investigation of the existence of solutions to a nonlocal multivalued Darboux problem
zbMATH Open document ID: 0936.35203
DOI: 10.1155/S1048953399000180
author: Ludwik Byszewski
author: Nikolaos S. Papageorgiou
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2000-03-14T00:00:00Z
full work available at URL: https://eudml.org/doc/48403
review text: For a given multifunction \(F:\Delta\times E\to P_{fc}(E)\) \((F:\Delta\times E\to P_f(E)\) respectively) and, given a \((\phi,\psi) \in C\), satisfying some assumptions, the authors study the existence of a solution to the following Darboux problem \[ \frac{\partial^2u(x,y)}{\partial x\partial y} \in F(x,y,u(x,y))\quad\text{a.e. in \(\Delta\)}, \] \[ u(x,0)+\sum_{j=1}^r h_j(x)u(x,b_j) = \phi(x),\quad x\in I, \] \[ u(0,y)+\sum_{i=1}^p k_i(y) u(a_i,y) = \psi(y),\quad y\in I, \] where \(a_i\) \((i=1,\dots,p)\) and \(b_j\) \((j=1,\dots,r)\) are given numbers such that \[ 0<a_1<\dots<a_p\leq c,\qquad 0<b_1<\dots<b_r\leq c. \] A compactness type condition involving the ball measures of noncompactness is applied.
Mathematics Subject Classification ID: 35R70
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 1359783
zbMATH Keywords: Darboux problem
zbMATH Keywords: existence
zbMATH Keywords: measures of noncompactness
reviewed by: A. A. Martynyuk
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1359783