The Generalized Solution Space And Its Applications To Integrated C-Semigroups

The abstract Cauchy problem and the fc-times integrated abstract Cauchy problem play important roles in many practical problems. Many equations and physics problems can be modeled as a abstract Cauchy problem or a fc-times integrated abstract Cauchy problem on a Banach space, and the theories of semigroups provide us a very useful tool to investigate them. In this paper, by using the theories of semigroups as basis and using the generalized solution space as a inportant tool, we investigate the abstract Cauchy problem and the fc-times integrated abstract Cauchy problem and their relationships between fc-times integrated C-semigroups. moreover, fc-times integrated C-semigroups provide a simple method of approximating the generalized solution space and its topology.For a given closed generator and arbitary initial value, we can not necessarily make the solution of the abstract Cauchy problem or the fc-times integrated abstract Cauchy problem unique. But, with the help of the theories of semigroups and the generalized solution space, we can always make the solution of the original abstract Cauchy problem and fc-times integrated abstract Cauchy problem unique. Moreover, the introduction of bounded injective operator C help us so much that we obtain important results in this paper.First, we introduce the bounded injective operator C, which commutes with A, and we have the following theorem:Theorem 2.1.1. If A is a closed operator, k N U {0}, then the following are equivalent:(a) A generates a nondegenerate fc-times integrated C-semigroup;(b) C-1 AC = A, and ACPk+1 has a unique solution for all x ImC, that is, ImC C In this case, the fc-times integrated C-semigroup is given byW(t) = Sk(t)C,where Sk(t) is defined as in Definition 1.3.4.Second, we proved that the introduction of bounded injective operator C provide a simple method of approximating the genaralized solution space and its topology and have the following result:Theorem 2.1.2. If A is a closed operator, k N U {0}, then the following are equivalent:(a) ACPk+1 has a unique solution for all x ImC;(b) [ImC] Zk+1;(c) A generates a nondegenerate fc-times integrated C-semigroup. In this case, the fc-times integrated C-semigroup is given byW(t) = Sk(t)C,where Sk(t) is defined as in Definition 1.3.4.Finally, when A generates a nondegenerate k-times integrated C-semigroup, we may useit to characterize Zk+1 and Zn(n N). It is interesting to know that the choice of C in thefollowing is irrelevant, that is we still obtain the same Zn. Therefore, We have the followingtheorem and corollary:Theorem 2.1.3. If closed operator A generates a nondegenerate fc-times integrated C-semigroup, then we have: is continuously n-times differentiable}with the seminorms:for all a, b Q+, j N. In particualar, C(D(Ak+1-n)) Zn for n = 0,1,..., k.Corollary 2.1.4. If A generates a nondegenerate k-times integrated C-semigroup, then we have: with the seminorms :for all...