Abstract and Applied Analysis

Relaxation Problems Involving Second-Order Differential Inclusions

Abstract

We present relaxation problems in control theory for the second-order differential inclusions, with four boundary conditions, $\stackrel{¨}{u}\left(t\right)\in F\left(t,u\left(t\right),\stackrel{˙}{u}\left(t\right)\right)$ a.e. on $\left[\mathrm{0,1}\right]$; $u\left(\mathrm{0}\right)=\mathrm{0}, u\left(\eta \right)=u\left(\theta \right)=u\left(\mathrm{1}\right)$ and, with $m\ge \mathrm{3}$ boundary conditions, $\stackrel{¨}{u}\left(t\right)\in F\left(t,u\left(t\right),\stackrel{˙}{u}\left(t\right)\right)$ a.e. on $\left[\mathrm{0,1}\right]; \stackrel{˙}{u}\left(\mathrm{0}\right)=\mathrm{0}, u\left(\mathrm{1}\right)={\sum }_{i=\mathrm{1}}^{m-\mathrm{2}}\mathrm{‍}{a}_{i}u\left({\xi }_{i}\right)$, where $\mathrm{0}<\eta <\theta <\mathrm{1}$, $\mathrm{0}<{\xi }_{\mathrm{1}}<{\xi }_{\mathrm{2}}<\cdots <{\xi }_{m-\mathrm{2}}<\mathrm{1}$ and $F$ is a multifunction from $\left[\mathrm{0,1}\right]×{ℝ}^{n}×{ℝ}^{n}$ to the nonempty compact convex subsets of ${ℝ}^{n}$. We have results that improve earlier theorems.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 792431, 9 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511840

Digital Object Identifier
doi:10.1155/2013/792431

Mathematical Reviews number (MathSciNet)
MR3044990

Zentralblatt MATH identifier
1272.49025

Citation

Gomaa, Adel Mahmoud. Relaxation Problems Involving Second-Order Differential Inclusions. Abstr. Appl. Anal. 2013 (2013), Article ID 792431, 9 pages. doi:10.1155/2013/792431. https://projecteuclid.org/euclid.aaa/1393511840