A mixed problem with only integral boundary conditions for a
hyperbolic equation

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International Journal of Mathematics and Mathematical Sciences 
Volume 2004 (2004), Issue 24, Pages 1279-1291
doi:10.1155/S0161171204204240

A mixed problem with only integral boundary conditions for a hyperbolic equation

Abdelfatah Bouziani

Département de Mathématiques, Centre Universitaire Larbi Ben M'hidi, Oum El Bouagui 04000, Algeria

Received 29 April 2002

Abstract

We investigate an initial boundary value problem for a second-order hyperbolic equation with only integral conditions. We show the existence, uniqueness, and continuous dependence of a strongly generalized solution. The proof is based on an energy inequality established in a nonclassical function space, and on the density of the range of the operator associated to the abstract formulation of the studied problem by introducing special smoothing operators.