On strongly continuous ρh-semigroup

In this paper, we introduce a semi group which it constructs the solution of the partial differential equation as the form: ρ(t)ρ′(t)∂u(t,x)∂t=h(t)h′(t)∂u(t,x)∂x,h(0)=1,ρ(0)=1 First, we introduce the operator theory and the fundamental theorems of the semigroup and certain notions of strongly continuous operators. These concepts are particular types of operator semigroups of functional analytic Using functional analytic tools and methods from ergodic theory, we describe various features of the On strongly continuous ρh -semigroup

In this paper , we introduce a semi group which it constructs the solution of the partial differential equation as the form:

Mathematics subject classification . 47Dxx
The

1.Introduction.
Many Scientists ( [1], [2], [3]) introduce several generations of analysts working in the area of operator semigroups. In particular, the progress has been made in the asymptotic theory of strongly continuous semigroups. One of the major results in this direction a strongly continuous semigroup on a Banach space with the norm of the resolvent of its generator A is uniformly bounded in the right half-plane.
We consider the following equations.
We introduce a new type of semigroup namely (Multipilicative Canonical semigroup) and its define by.
And also we introduce strongly continuous generalized canonical semigroup defined by.

Definition
The function f(t) is called continuous at a point t0 if ‖ ( ) − ( 0 )‖ → 0 , → 0 , continuous on the interval [a,b], if it is continuous at each point of this segment. [5].

Definition
The function f(t) is called differentiable in point t0, if there is an element ′ such that.
The element f' is called the derivative of the function f (t) at point t0 and denoted by.

Definition [7].
We will say that the operator function A(t) is continuous in norm at point t0 [a,b] if [8].

Definition
The operator-function A(t) is strongly continuous in a point t0 [a,b] if at any fixed x  E1

Definition
We say that an operator A is closed if for every xnD (A), then ‖x − x 0 ‖ → 0 and Ax0 = y0. [8].

Definition
A family of bounded operators T (t) (t >0), define on the Banach space E, is called strongly continuous semigroup of operators if T(t) strongly continuous and satisfies the condition T (t)T (s) = T(t+s) (t, s > 0).

Definition [6].
It is said that T (t) is a semigroup of class C0 if it is strongly continuous and the following condition for any x E.

Theorem [4].
The linear operator A is a generating operator (generator) of a semigroup T (t) of class C0 iff its closed with a dense in E.

Definition[6].
A family of bounded operators T (t) (t >0), define on the Banach space E, is called strongly continuous multiplicative semigroup of operators if T(t) strongly continuous and satisfies the conditions. Consider the differential equation.

Definition.
It is easy to see that the general solution of this equation is.
Where is an arbitrary differentiable function. we can assign the one-parameter equation (1) to a oneparameter family of operators.
under the assumption that φ belongs to the space of continuous and bounded functions C(a,b) with the norm.

Definition.
We define a binary operation ⨀ by.
We will prove that ℎ ( ) defined by (3) is a semigroup of linear and bounded in C(a, b)

Lemma.
The operational family ℎ ( ) defined by (3) is a semigroup of linear and bounded in C(a, b) of operators with the binary operation in (4).

Proof.
We note that.

Remark.
The function ( ) which given in the semigroup ℎ ( ) is invariant relative to the functions h(x) on h(x)+c , where c-constant. semigroup ℎ ( ) is called ℎ −semigroup and equation (1) is its generating equation. We note that the function ( ) it is possible to select such that the equation (1) generates a family ℎ-semigroup.
In the following proposition we show that ℎ −semigroup has a fixed point.

Definition.
If ( ) = ℎ( ) then the semigroup ℎ −semigroup can be written by the form.
In the following lemma we will prove that the family ℎℎ −semigroup has one symmetric semigroup. the family of semigroups produced by symmetric differential equation, contains only one symmetric semigroup.
Therefore ℎ is not symmetric for c1 .
There exists a special cases of partial differential equation can be solved by another method and we take some of these cases in the following examples.  We note that ℎ (0) ( ) with the binary operation ⨀ = . is called Arithmetic semigroup.

Definition.
Let f(t) be a vector function , define on 3.14 Remark.

Definition .
The function φ C(a,b) is called uniformly continuous if its −1 -deformation = ( ( )) is bounded and uniformly continuous function.
we note that .

Proof.
We note that. Now we can get the form of A generator operator of the semigroup ℎ (0) ( ) as the following theorem.

Theorem.
A generator operator of the semigroup ℎ (0) ( ) given by the differential expression .

Proof.
We have . In the following theorem we get the estimate of the operator ℎ ( ).

Theorem.
The family of operators ℎ ( ) is strongly continuous generalized canonical semigroup defines on the space , ,ℎ and the following estimation holds.