Second-order functional differential equations with past, present and future dependence
type: Item
instance of: scholarly article
title: Second-order functional differential equations with past, present and future dependence
zbMATH Open document ID: 1242.34122
author: Rubén Figueroa
published in: Applied Mathematics and Computation
publication date: 2011-05-11T00:00:00Z
review text: Existence of extremal quasi-solutions to the problem \[ u''(t) = f(t, u(t), u'(t), u|_{I_{0}}, u ,u ) \;\text{for a.a.}\;\; t \in I_0 \] together with the boundary conditions \[ B_-(t, u(t), u|_{I_{0}}, u ,u) = 0 \;\text{for}\;\text{all} \;I_-, \;\; B_+(t, u(t), u|_{I_{0}}, u ,u) = 0 \;\text{for}\;\text{all} \;I_+ \] is established. Here \(I_0 = [a,b],\; I_-= [a-r_-,a],\; I_+ = [b, b+r_+], I = I_- \cup I_0 \cup I_+, r_-, r_+ \geq 0\). \(f\) is assumed to be Carathéodory with respect to its real variables and monotone with respect to its functional arguments. The boundary functionals are also assumed to satisfy monotonicity conditions with respect to their functional arguments. The space \(\chi = \{\gamma: I \rightarrow \mathbb{R}: \gamma_|{}_{I_{0}} \in C(I_{0}), \gamma|I_{\pm} \;\) is bounded\;\}, is endowed with the pointwise ordering and the supremum norm. The quasi-solutions of the above problem are elements of \(\chi\) satisfying \[ v''(t) = f(t, v(t), v'(t), v|_{I_{0}}, u ,w ) \;\text{for \;a.a.}\;\; t \in I_0, \] \[ B_{\pm}(t, v(t), v|_{I_{0}}, v ,w) = 0 \;\text{for \; all} \;I_{\pm} \] and \[ w''(t) = f(t, w(t), w'(t), w|_{I_{0}}, w ,v ) \;\text{for \;a.a.}\;\; t \in I_0,\; B_{\pm}(t, w(t), w|_{I_{0}}, w ,v) = 0 \;\text{for}\;\text{all} \;I_{\pm} \] Under some technical assumptions the existence of extremal quasi-solutions in \([\alpha, \beta]\), where \(\alpha\) and \(\beta\) are lower and upper solutions of the above problem, is proved. The above class of problems includes equations with delay, advance and both delay and advance. An example is discussed as an application of the result is also presented.
reviewed by: Gnana Bhaskar Tenali
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5887173
zbMATH Keywords: second order differential equations
zbMATH Keywords: functional differential equations
zbMATH Keywords: deviated arguments
MaRDI profile type: Publication
full work available at URL: j.amc.2011.02.041
OpenAlex ID: W2069735471
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DOI: 10.1016/J.AMC.2011.02.041
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description: scientific article; zbMATH DE number 5887173