Results of Semigroup of Linear Equation Generating a Wave Equation

In this paper, we present results of ω -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


Introduction
Consider the initial value problem for the wave equation in R n , that is the initial value problem ∂ 2 u ∂t 2 = ∆u f or x ∈ R n , t > 0 u(0, x) = u 1 (x), ∂u ∂t (0, x) = u 2 (x), f or x ∈ R n . The problem is equivalent to the first order system: and Suppose X is a Banach space, X n ⊆ X is a finite set, ω − OCP n is the ω-order preserving partial contraction mapping, M m is a matrix, L(X) is a bounded linear operator on X, P n is a partial transformation semigroup, ρ(A) is a resolvent set, σ(A) is a spectrum of A and A is a generator of C 0 -semigroup. This paper consists of results of ω-order preserving partial contraction mapping generating a wave equation. Akinyele et al. [1], generated a continuous time Markov semigroup of linear operators and also in [2], Akinyele et al., obtained results of ω-order reversing partial contraction mapping generating a differential operator. Balakrishnan [3], introduced an operator calculus for infinitesimal generators of semigroup. Banach [4], established and introduced the concept of Banach spaces. Brezis and Gallouet [5], obtained nonlinear Schrodinger evolution equation. Chill and Tomilov [6], presented some resolvent approach to stability operator semigroup. Davies [7], obtained linear operators and their spectra. Engel and Nagel [8], deduced one-parameter semigroup for linear evolution equations. Omosowon et al. [9], established some analytic results of semigroup of linear operator with dynamic boundary conditions, and also in [10], Omosowon et al., obtained dual Properties of ω-order Reversing Partial Contraction Mapping in Semigroup of Linear Operator. Omosowon et al. [11], established a regular weak*-continuous semigroup of linear operators, and also in [12], Omosowon et al., generated a quasilinear equations of evolution on semigroup of linear operator. Pazy [13], presented asymptotic behavior of the solution of an abstract evolution and some applications and also in [14], obtained a class of semi-linear equations of evolution. Rauf and Akinyele [15], established ω-order preserving partial contraction mapping and obtained its properties, also in [16], Rauf et al., presented some results of stability and spectra properties on semigroup of linear operator.
Vrabie [17], proved some results of C 0 -semigroup and its applications. Yosida [18], established some results on differentiability and representation of one-parameter semigroup of linear operators.

Preliminaries
Definition 2.1 (C 0 -semigroup) [17] A C 0 -semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space.
A transformation α ∈ P n is called ω-order preserving partial contraction mapping if ∀x, y ∈ Domα : x ≤ y =⇒ αx ≤ αy and at least one of its transformation must satisfy αy = y such that T (t+s) = T (t)T (s) whenever t, s > 0 and otherwise for T (0) = I.

Definition 2.3 (Evolution Equation) [13]
An evolution equation is an equation that can be interpreted as the differential law of the development (evolution) in time of a system. The class of evolution equations includes, first of all, ordinary differential equations and systems of the form etc., in the case where u(t) can be regarded naturally as the solution of the Cauchy problem; these equations describe the evolution of systems with finitely many degrees of freedom.

Definition 2.4 (Mild Solution) [14]
A continuous solution u of the integral equation.
will be called a mild solution of the initial value problem if the solution is a Lipschitz continuous function.
The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. For each f ∈ X and each t, s ∈ R + , one may easily verify that {T (t); t ∈ R + } satisfies Examples 1 and 2.

Main Results
This section present results of semigroup of linear operator by using ω-OCP n to generates a wave equation: Since f ∈ H k (R n ), (1 + |ξ| 2 ) k/2f (ξ) ∈ L 2 (R n ) and therefore (1 + |ξ| 2 ) (k+2)/2 u(ξ) ∈ L 2 (R n ). If u is defined by then u ∈ H k+2 (R n ) and u is a solution of (3.1). The uniqueness of the solution u of (3.1) follows from the fact that if w ∈ H k+2 (R n ) satisfies w − v∆w = 0, then w = 0 and therefore w = 0. Hence the proof is complete.

Proof:
Let λ = 0 be real and let w 1 , w 2 be solutions of From Theorem 3.1 it is clear that such solutions exist and that w 1 ∈ H k (R n ) for every k ≥ 0. Set u 1 = w 1 + λw 2 , u 2 = w 2 + λ∆w 1 .
Denoting · o the scalar product in L 2 (R n ) we have Therefore if 0 < |λ| < 1 2 , we have and it shows that the range of operator I − λA contains C ∞ 0 (R n ) × C ∞ 0 (R n ) for all real λ satisfying 0 < |λ| < 1 2 . Since the operator A defined by Definition 2.5 is closed, the range of I −λA belongs to H = H (R n )×L 2 (R n ) for all A ∈ ω −OCP n . Then for every F ∈ H (R n ) × L 2 (R n ) and real λ satisfying 0 < |λ| < 1 2 , the equation has a unique solution u ∈ H 2 (R n ) × H (R n ) and Hence the proof is complete. (3.8)

Proof:
Let T (t) be the semigroup generated by A and set where A ∈ ω − OCP n and u 1 is the desired solution. Then we have that the operator A, defined in Definition 2.5 is the infinitesimal generator of a C 0 -semigroup on H = H (R n ) × L 2 (R n ) satisfying We have that the domain of A, H 2 (R n ) × H (R n ) is clearly dense in H. From (3.6) it follows that (µI − A) −1 exists for |µ| > 2 and satisfies for |µ| > 2 and A ∈ ω−OCP n . Then it follows that A is the infinitesimal generator of a C 0 -smigroup satisfying (3.9) and this achieved the proof.
Estimating D α v(x), by the Cauchy-Schwartz inequality, we find for every N > n 2 , where C 1 and C 2 are constants depending on N and |α|. Let u ∈ H k (R n ) and let u n ∈ C ∞ 0 (R n ) be such that u n → u in H k (R n ). Then from (3.12) it follows that D α u n → D α u uniformly in R n for all α satisfying |α| ≤ m < k − n/2 and therefore u ∈ C m (R n ) as desired. Hence the proof is complete.

Conclusion
In this paper, it has been established that ω-order preserving partial contraction mapping generates some results of a wave equation.