CONVERGENCE OF EXPONENTIALLY BOUNDED C SEMIGROUPS. Young S. Lee
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1 Kangweon-Kyungki Math. Jour. 9 (21, No. 2, pp CONVERGENCE OF EXPONENTIALLY BOUNDED C SEMIGROUPS Young S. Lee Abstract. In this paper, we establish the conditions that a mild C-existence family yields a solution to the abstract Cauchy problem. And we show the relation between mild C-existence family and C-regularized semigroup if the family of linear operators is exponentially bounded and C is a bounded injective linear operator. 1. Introduction Let X be a Banach space and let A be a linear operator from D(A X into X. The abstract Cauchy problem for A with initial data x X is to find a solution u(t to the following initial value problem du dt Au, t >, u( x. (ACP It is well known ([5] that when A is closed, generating a C semigroup {T (t : t } guarantees the abstract Cauchy problem to have a unique mild solution for all initial data x X, and the solution is given by u(t T (tx. If the abstract Cauchy problem does not have a mild solution for all x X, we may look for initial data in X that produce mild solutions. In [4], a family of linear operators was introduced that produces a solution of the abstract Cauchy problem for all initial data in the range of a bounded linear operator C. It is known [4] that for a bounded linear operator C mild C-existence family yields a mild solution for all initial data in the range of C. C-existence family is a generalization of the classical C semigroup such as integrated semigroups and regularized semigroups (see [1, 2, 3, 4, 6]. Received July 12, Mathematics Subject Classification: Primary 47D5, Secondary 47D1. Key words and phrases: Existence Family, C-semigroups, mild solution.
2 116 Young S. Lee In this paper, we establish the conditions that mild C-existence family for A produces a solution of (ACP. It is known that a mild C-existence family for A is a C-regularized semigroup generated by an extension of A if the family of operators and A commute. We will show the relation between C-regularized semigroup and mild C- existence family if the family of operators is exponentially bounded. If CA AC, mild C-existence family is a C-regularized semigroup generated by an extension of A. Throughout this paper, we denote by D(A the domain of the operator A on X and R(A the range of A. An operator T in X is said to commute with A if AT T A. It means that if x D(T, Ax D(T and T Ax AT x. By a solution u(t of (ACP, we mean a continuously differentiable function u : [, X such that u(t D(A for all t and satisfies (ACP. A mild solution of (ACP is a continuous function u : [, X such that v(t t u(sds D(A and d v(t Av(t + x, t. dt section2. Mild C-existence Families Let A be as in (ACP. Definition 2.1. The strongly continuous family {S(t : t } of bounded linear operators on X is called a mild C-existence family for A if for all x X, t >, t S(sxds D(A and A S(sxds S(tx Cx. Remark. If {S(t : t } is a mild C-existence family for A, then u(t S(tx is a mild solution of (ACP with u( Cx. Theorem 2.2. Let {S(t : t } be a mild C-existence family for A. Suppose that CA AC. Then there exits a solution u(t of (ACP with u( x C(D(A. Proof. Let x C(D(A. Then there exists y D(A such that x Cy. So Ax ACy CAy. Define u(t x + t S(sAyds.
3 Convergence of exponentially bounded C semigroups 117 Then du(t/dt S(tAy and Au(t Ax + A( t S(sAyds Ax + S(tAy CAy S(tAy, since {S(t : t } is a mild C-existence family for A.. So u(t is a solution of (ACP with u( x. Theorem 2.3. Let {S(t : t } be a mild C-existence family for A. If x D(A and Ax R(C, then there exists a solution u(t for (ACP with initial data x. Proof. Let y X such that Ax Cy. Define u(t x + t S(syds. Then du(t/dt S(ty and Au(t Ax+A t S(syds Ax+S(ty Cy S(ty, since {S(t : t } is a mild C-existence family for A. Thus u(t is a solution of (ACP with u( x. Next, we characterize an exponentially bounded mild C-existence family in terms of Laplace transforms. Theorem 2.4. Let {S(t : t } be a strongly continuous family of bounded linear operators on X such that S(t Me ωt, t, for some M, ω >. Suppose that A is closed and λ A is injective for λ > ω. Then the following are equivalent. (1 {S(t : t } is a mild C-existence family for A. (2 R(C R(λ A for λ > ω and (λ A 1 Cx e λt S(txdt, x X, λ > ω. Proof. Suppose that {S(t : t } is a mild C-existence family for A. Let x X. By integration by parts, we have e λt S(txdt λ e λt S(sxds dt.
4 118 Young S. Lee Since A is closed, A e λt S(txdt λa λ λ λ e λt e λt A S(sxds dt S(sxds dt e λt (S(tx Cxdt e λt S(txdt Cx. So Cx (λ A e λt S(txdt for all x X, and the result follows. Suppose that (2 is satisfied. Let x X. By Post-Widder inversion theorem of Laplace transform and the closedness of A, we have λ 1 A(λ A 1 Cx λ 1 A Since Cx (λ A(λ A 1 Cx, e λt A e λt S(txdt S(sxds dt. λ 1 A(λ A 1 Cx (λ A 1 Cx λ 1 Cx e λt (S(tx Cxdt. By the uniqueness of the Laplace transform, we have A S(sxds S(tx Cx. The following shows the relation between mild C-existence family and regularized semigroup. Theorem 2.5. Let {S(t : t } be an exponentially bounded mild C-existence family for A such that S(t Me ωt for t, some M and ω >, and let C be a bounded injective linear operator. Suppose that A is closed and has no eigenvalue in (ω, with CA AC. Then {S(t : t } is a C-regularized semigroup generated by an extension of A.
5 Convergence of exponentially bounded C semigroups 119 Proof. Let x X. Since {S(t : t } is a mild C-existence family and CA AC, CS(tx C 2 x CA S(sxds A CS(sxds. So CS(tx is a mild solution of (ACP with u( C 2 x. Clearly, S(tCx is a mild solution of (ACP with u( C 2 x. By the uniqueness of mild solution (Proposition in [4], CS(t S(tC. By Theorem 2.4, for x X, C(λ A 1 Cx C e λt S(txdt e λt S(tCxdt (λ A 1 C 2 x. So we have (λ A 1 C(µ A 1 Cx 1 λ A(λ A 1 C(µ A 1 Cx + 1 λ C(µ A 1 Cx 1 λ (λ A 1 CA(µ A 1 Cx + 1 λ (µ A 1 C 2 x 1 λ (λ A 1 C(µ(µ A 1 Cx Cx + 1 λ (µ A 1 C 2 x µ λ (λ A 1 C(µ A 1 Cx 1 λ (λ A 1 C 2 x + 1 λ (µ A 1 C 2 x Thus (λ µ(λ A 1 C(µ A 1 Cx (µ A 1 C 2 x (λ A 1 C 2 x.
6 12 Young S. Lee By Theorem 2.4 and integration by parts, 1 λ µ ((µ A 1 C 2 x (λ A 1 C 2 x e (λ µt (µ A 1 C 2 xdt 1 λ µ e (λ µt (µ A 1 C 2 xdt ( e (λ µt e µs S(sCxds dt e (λ µt e (λ µt ( e λt ( t e λt ( On the other hand, we have (λ A 1 C(µ A 1 Cx e µs S(sCxds dt e µs S(sCxds dt t e µ(s t S(sCxds dt e µw S(t + wcxdw dt e λt S(tCxdt 1 λ µ e (λ µt e µt S(tCxdt e λt S(t(µ A 1 Cxdt ( e λt S(t e µs S(sxds dt ( e λt e µs S(tS(sxds dt. By the uniqueness of the Laplace transform, we have S(t + sc S(tS(s. Therefore {S(t : t } is a C-regularized semigroup generated by an extension of A.
7 Convergence of exponentially bounded C semigroups 121 References [1] W. Arendt, Vector valued Laplace transform and Cauchy problems, Israel J. Math. 59 (1987, [2] E.B. Davies and M.M.H. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc. 55 (1987, [3] R. delaubenfels, C-semigroups and the Cauchy problem, J. Funct. Anal. 111 (1993, [4] R. delaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Math., vol. 157, Springer Verlag, [5] J. A. Goldstein, Semigoups of operators and applications, Oxford, New York, [6] M. Li and F. Huang, Characterization of contraction C-semigroups, Proc. Amer. Math. Soc. 126 (1998, Department of Mathematics Seoul Women s University Seoul, , Korea younglee@swift2.swu.ac.kr
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