Electronic Journal of Differential Equations, Vol. 2006(2006), No. 23, pp. 1-11. Title: Existence theory for perturbed hyperbolic differential inclusions Authors: Abdelkader Belarbi (Univ. de Sidi Bel Abbes, Algerie) Mouffak Benchohra (Univ. de Sidi Bel Abbes, Algerie) Abstract: In this paper, the existence of solutions and extremal solutions for a perturbed hyperbolic differential inclusion is proved under the mixed generalized Lipschitz and Caratheodory's conditions. Submitted March 25, 2005. Published February 23, 2006. Math Subject Classifications: 35L70, 35L20, 35R70. Key Words: Hyperbolic differential inclusion; fixed point; extremal solutions.