The Solution Of Nonhomogen Abstract Cauchy Problem by Semigroup Theory of Linear Operator

In this article we will investigate how to solve nonhomogen degenerate Cauchy problem via theory of semigroup of linear operator. The problem is formulated in Hilbert space which can be written as direct sum of subset Ker M and Ran M*. By certain assumptions the problem can be reduced to nondegenerate Cauchy problem. And then by composition between invers of operator M and the nondegenerate problem we can transform it to canonic problem, which is easier to solve than the original problem. By taking assumption that the operator A is infinitesimal generator of semigroup, the canonic problem has a unique solution. This allow to define special operator which map the solution of canonic problem to original problem. ©2016 JNSMR UIN Walisongo. All rights reserved.


Introduction
Let us consider the homogen abstract Cauchy problem: For finite dimensional, the problem (1) homogen abstract degenerate Cauchy problem is discussed completely in the book [1], where we can tranform matrices M and A to a common normal form. We also can find many examples, applications to control theory, and references to the earlier literature in his book.
In the infinite dimensional case, it is mentioned in [2] that they treat the singular and degenerate Cauchy problem. In [3][4][5][6][7][8], the investigates degenerate Cauchy problem in Hilbert space. In his articles, the problem is treated also under the assumption that the Hilbert space of the system can be written as direct sum of the kernel of M (Ker M) and the range of adjoint M (Ran M*).
By certain asumptions [9,10] to discuss the Degenerate Cauchy problem in Banach are linear operators. In this section we are going to investigates how to solve nobhomogen abstract degenertae Cauchy problem, where M is not invertible. Problem (2) is called degenerate when M is not invertible. The solution of (2) is defined in the following. To solve the probelm we use several assumptions:  [15,16] problem: Operator M has close domain, so the problem (3) can be tranformed to problem: (4) Since the operator A -1 is a bounded operator, so we can define and then problem (4) can be written: and continuously differentiable then the solution of equation (5) is can be given by for all t  0.
Proof : Now we will prove the lemma by reductio ad absordum. We know that z(t) is solution of (1) and let ).
In order to solve nondegenerate Cauchy problem by factorization method, the operator where is located in the right handside must be generator of semigroup linear operator, so the solution of the problem is unique.
On the following lemma will be discused about differentiabelity of function

Lemma 4:
then we will define infinetimal genartor of semigroup.

Definition 5: The Infinitesimal generator
if only if the limit above exists. Every linear operator A in Hilbert space does not always be generator of Co-semigrup. By the Hille-Yosida theorema we know the characteristic the linear operator is a generator of Co-semigrup [18] Teorema 6: (Teorema Hille-Yosida) [

According
(6), the problem nondegenerate (5) can be tranformed to canonik form: Operator Moreover we must have the asumption to make the canonik problem has a unique solution. By assumption 8 dan Theorem 7, the solution problem (7) is

Example 9:
In this example, we have Hilbert space ) ( Next, we define an orthogonal projection operators: By the projection operators, we have operator: In this case, operator

Conclusion
After we discuss this topic, we can result how to solve degenerate abstract Cauchy problem by semigroup theory of linear operator. There are three stages to solve this problem. In the first stage, under certain assumptions we reduce the nonhomogen abstract degenertae Cauchy problem (2): to nondegenerate Cauchy problem (3): Operator M which is not invertible, can be reduced to invertible operator . r M The second stage the problem (3) can be tranformed to canonik form: . In order to use semigroup theory of linear operator , we assume that A1 is an infinitesimal generator of strongly continuous semigroup, so the solution is , ) (