Existence theorems for weak solutions to \(n\)th-order differential equations in Banach spaces
type: Item
instance of: scholarly article
title: Existence theorems for weak solutions to \(n\)th-order differential equations in Banach spaces
zbMATH Open document ID: 0919.34054
author: Aldona Szukała
author: Stanisław Szufla
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1999-08-25T00:00:00Z
review text: The existence of weak solutions to the de La Vallee-Poussin boundary value problem \[ x^{(n)}=f(t,x,x' \dots, x^{(n-1)}), \quad x^{(i)}(a_k)=0, \tag{1} \] \[ k=1, \dots p, \;i=0,1, \dots i_k-1, \;a<a_1<a_2< \dots < a_p<b,\;1\leq p\leq n, \;\sum_{k=1}^pi_k=n, \] or to the equivalent integral equation \[ x(t)=\int_a^bG(s,x)f(s,x(s), \dots, x^{(n-1)}(s))ds \] is studied (where \(f:J\times E^n\to E\) is a weakly-weakly continuous function, \(J=[a,b], E\) is a sequentially weakly complete Banach space, and \(G(t,x)\) is the Green function of the problem \(x{(n)}=0, \;x^{(i)}(a_k)=0\)). Weak compactness type conditions are imposed in terms of the measure of weak noncompactness \(\beta\). It is proved that if \(p_1P_1+ \dots + p_nP_n<1,\) for \(P_i\) such that \[ \int_a^b| G^{(i-1)}(s,x)| ds\leq P_i \text{ and }\beta(f(J\times X_1 \times \dots \times X_n)) \leq p_1\beta(X_1) + \dots + p_n \beta(X_n) \] for any bounded subsets \(X_1, \dots X_n\) of \(E\), then there exists a weak solution to problem (1).
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B27
zbMATH DE Number: 1251495
zbMATH Keywords: initial and boundary value problem
zbMATH Keywords: measures of noncompactness
reviewed by: I. V. Melnikova
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1251495