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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 481853, 15 pages
Research Article

Forward Euler Solutions and Weakly Invariant Time-Delayed Systems

1Department of Mathematics and Applied Mathematics, VA Commonwealth University, Richmond, Virginia 23284, USA
2Departamento de Matemáticas, Facultad Experimental de Ciencias, Universidad del Zulia, Apartado 526, Maracaibo, Edo Zulia, Venezuela

Received 12 September 2012; Revised 10 December 2012; Accepted 11 December 2012

Academic Editor: Qiji J. Zhu

Copyright © 2012 Norma L. Ortiz-Robinson and Vinicio R. Ríos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents a necessary and sufficient condition for the weak invariance property of a time-delayed system parametrized by a differential inclusion. The aforementioned condition generalizes the well-known Hamilton-Jacobi inequality that characterizes weakly invariant systems in the nondelay setting. The forward Euler approximation scheme used in the theory of discontinuous differential equations is extended to the time-delayed context by incorporating the delay and tail functions featuring the dynamics. Accordingly, an existence theorem of weakly invariant trajectories is established under the extended forward Euler approach.