www.ccsenet.org/jmr Journal of Mathematics Research Vol. 2, No. 4; November 2010 On Existence and Uniqueness of Generalized Solutions for a Mixed–type Differential Equation Tarig M. Elzaki Department of Mathematics, Sudan University of Science and Technology Western campus - Mogran, Khartoum, Sudan E-mail: Tarig.alzaki@gmail.com Adem Kılıc ¸man (Corresponding author) Department of Mathematics and Institute for Mathematical Research Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia E-mail: akilicman@putra.upm.edu.my Hassan Eltayeb Mathematics Department, College of Science, King Saud University P.O. Box 2455, Riyadh 11451, Kingdom of Saudi Arabia E-mail: hgadain@ksu.edu.sa Abstract In this paper, we study a boundary value problem for a mixed–type differential equation. The existence and uniqueness of generalized solution is proved. The proof is based on an energy inequality and the density of the range of the operator generated by this problem. Keywords: Energy inequality, Dual operator, Generalized solution 1. Introduction Partial differential equations are the bases of almost all physical theorems. In the theory of sound in gases, liquids, and solids, in the investigations of elasticity, in optics, everywhere partial differential equations formulate basic laws of nature which can be checked against the experiments. Many problems of physical interest are also described by ordinary or partial differential equations- with appropriate initial or boundary conditions. These problems are usually formulated as initial-boundary value problems that seem to be mathematically more rigorous and physically realistic in applied and engineering sciences, for example, see (Maazzucato, 2003) and (Bouziani and Benouar, 2002). The energy inequality and the density of the range are particularly useful for proving the existence and uniqueness of generalized solutions, see for example (Bougoffa, 1999), (Bougoffa and Moulay, 2003) and (Shi, 1993). In recent years, special equations of composite as well as mixed type have received many attention in several papers. Most of the papers were directed to parabolic-elliptic equations, and to hyperbolic-elliptic equations. Similarly, existence and uniqueness of generalized solutions for composite type was also discussed in (Tarig, 2009). In this study we prove a priori estimates and derive from the existence and uniqueness of generalized solutions for mixed type equations where proof is based on an energy inequality and density of the range. First of all we introduce appropriate Sobolev spaces and investigate the corresponding linear problem, see (Adam, 1975). Let Ω be a bounded domain in R n and x = ( x 1 ... x n ) with sufficiently smooth boundary Γ= ∂Ω and Q =Ω × (0, T ). Then we consider the following equation: lu =  ∂ 2 ∂t 2 − Δ  ∂ 2 u ∂t 2 +Δu  +Δ  ∂u ∂u  = f (t, x). (1.1) The initial conditions given by u(0, x) = ∂u ∂t (0, x) = ∂ 2 u ∂t 2 (0, x) = 0, ∀x ∈ Ω (1.2) the final condition: u(T , x) = 0, ∀x ∈ Ω (1.3) and the boundary conditions: u| S = ∂u ∂r = 0 (1.4) 88 ISSN 1916-9795 E-ISSN 1916-9809