Results of Semigroup of Linear Equation Generating a Wave Equation

Abstract

In this paper, we present results of $\omega$-order preserving partial contraction mapping generating a wave equation. We use the theory of semigroup to generate a wave equation by showing that the operator $ \begin{pmatrix} 0 & I\\ \Delta & 0 \end{pmatrix}, $ which is $A,$ is the infinitesimal generator of a $C_0$-semigroup of operators in some appropriately chosen Banach of functions. Furthermore we show that the operator $A$ is closed, unique and that operator $A$ is the infinitesimal generator of a wave equation.