Abstract
We consider applications of the -Drazin inverse to some classes of abstract Cauchy problems, namely, the heat equation with operator coefficient and delay differential equations in Banach space.
1. Introduction
In this paper we utilize the generalized Drazin inverse for closed linear operators to obtain explicit solutions to two types of abstract Cauchy problem. The first type is the heat equation with operator coefficient. The second type is a delay differential equation.
Firstly let us consider the heat equation with operator coefficient. Let be a bounded linear operator in a Hilbert space and be a holomorphic -valued function. The following initial value problemis studied in [1] under the assumption that is a Volterra operator and its imaginary part of is of trace class. In particular, it has been proved that if is quasinilpotent and its imaginary part is of trace class, then the Cauchy problem has a unique holomorphic solution in a neighborhood of zero.
We study the above Cauchy problem for the case where is a positive operator, and is not an accumulated spectral point of . Our results are extensions of [1] in the sense that the class of -Drazin invertible operators is more general than that of quasinilpotent operators.
We will show that if is positive and -Drazin invertible then the solution to the systemexists and is given by an explicit formula. We say a function is a solution to the above initial value problem if it satisfies the partial differential equation in for some , and with being an analytic function satisfying the bounds , where and are some positive constants.
Secondly we consider the following delay differential equationin a Banach space , which is studied by Gefter and Stulova in [2] under the assumption that is an invertible closed linear operator with a bounded inverse in ; the delay term is a complex constant, and is an -valued holomorphic function of zero exponential type. Recall that an entire function is of zero exponential type if, for every , there exists such that for each . We generalize the results in [2] by replacing the invertible closed linear operator with a -Drazin invertible operator. We will show that if is -Drazin invertible and is an entire function of zero exponential type, then the delay equation (3) has an entire solution of zero exponential type and it is expressed by an explicit formula.
Following [3], a closed linear operator is -Drazin invertible if is not an accumulated spectral point of . By , , , and we denote the spectrum, range, domain, and nullspace of , respectively. A bounded linear operator is called a -Drazin inverse of if , , andSuch an operator is unique, if it exists and is denoted by . From [3], we have the following decomposition result.
Theorem 1. If is a -Drazin invertible operator in a Banach space , then , , where is closed and invertible, is bounded and quasinilpotent with respect to this direct sum, andMoreover, if is the spectral projection corresponding to , then .
The above result is crucial to our analysis.
2. Solution for the Heat Equation with Positive Operator Coefficient
In this section we obtain an analytic solution for (2) that generalizes [1, Theorem ] in the sense that the coefficient operator is assumed to be -Drazin invertible instead of quasinilpotent.
Theorem 2. Let be a closed positive operator which is -Drazin invertible, and let be an analytic function in that satisfies the bound for some positive constants and . Then the system (2) has a unique solution given by the formula where , represents the -semigroup of linear bounded operators generated by , and denotes a bounded operator such that .
Proof. Since is -Drazin invertible, by Theorem (1), , , where is closed invertible and is bounded quasinilpotent with respect to the direct sum. Therefore Problem (2) has a unique solution if and only if each of the following two initial value problems has a unique solution on and , respectively.Since the operator is positive, it is self-adjoint. Therefore, is self-adjoint and the imaginary part of is zero. Applying [1, Theorem ] to Problem (8),is the unique solution of Problem (8). Next we will show thatis the unique solution of Problem (7). The operator denotes an operator such that . The existence of such an operator is guaranteed by the positivity of .
Since is positive, , which implies . Therefore, there exist constants and such thatObserve that the above inequality reduces the analysis of the heat equation with operator coefficient to that of the standard heat equation with scalar coefficientThis allows us to apply standard results of the heat equation with scalar coefficient to Problem (7). In particular, using the last inequality, the bounds on , and the fundamental solution to the heat equation, one can differentiate under the integrals and verify that the integrals for , and all converge. Using the derivative of the -semigroup , it is straightforward to check that satisfies the partial differential equation (7). Moreover, is the only solution if , and .
Since and , we obtain
An application of the above result can be illustrated by taking where For more details about this operator we refer the reader to [4, page 389].
3. Solution to the Delay Differential Equation
In this section we obtain a holomorphic solution to the delay differential equation (3). The result generalizes [2, Theorem ].
Theorem 3. Let be a closed linear operator which is -Drazin invertible, and let be an entire function of zero exponential type. Then (3) has a zero exponential type solution given by the formula where and is the -th primitive of ; that is, .
Proof. Since is -Drazin invertible, , , where is closed and invertible and is bounded and quasinilpotent with respect to the direct sum. Therefore (3) has a solution if and only if each of the following two initial value problems has a solution on and , respectively.Since the operator is closed and invertible, applying [2, Theorem ] to (17), we havebeing the unique solution of Problem (17). Next we will show that is a zero exponential type solution of Problem (18). Following [2, Lemma ], we first show that if is of zero exponential type then so is . Let be of zero exponential type and . Since for each , for some . Letting , we have Now, modifying the proof of [2, Theorem ] with the -th derivative replaced by the -th primitive , by and by , we obtain the convergence of and its sum is an entire function of zero exponential type. It is straightforward to check that the infinite sum is a solution of (18). Since and , we obtain
4. Conclusion
In Section 2 we have obtained the unique solution for the heat equation with operator coefficient , which is assumed to be self-adjoint and positive in a Hilbert space. Our result extends [1, Theorem ] in the sense that is -Drazin invertible instead of quasinilpotent. In Section 3 we have obtained an explicit solution for the delay differential equation with singular operator coefficient. Our result extends [2, Theorem ] in the sense that is -Drazin invertible instead of invertible in the usual sense.
Conflicts of Interest
The authors declare that they have no conflicts of interest.