Abstract
We extend results of Favini, Nashed, and Zhao on singular differential equations using the -Drazin inverse and the order of a quasinilpotent operator in the sense of Miekka and Nevanlinna. Two classes of singularly perturbed differential equations are studied using the continuity properties of the -Drazin inverse obtained by Koliha and Rakočević.
1. Introduction
Let be a bounded or closed linear operator in a Banach space and let be a -valued function. The following initial value problemis central to the analysis of the abstract singular equationwhere and are closed linear operators from a Banach space to . Problem (2) and its variations were extensively studied in [1–3] and the references therein. In [4–6], Campbell studied (2) in matrix setting and applied his results in optimal control problems. More recently, related singular equations with delay are studied in [7–10]. Thus far problem (1) has been considered when is singular (noninvertible) but Drazin invertible in the classical sense. A bounded linear operator is Drazin invertible in the classical sense if is a pole of the resolvent of . In [11], Koliha generalized the concept of Drazin invertibility to the case where is only an isolated spectral point of the spectrum of . Drazin invertibility in the generalized sense for closed linear operators was studied in [12].
In this paper we study problem (1) for the case where the bounded linear operator is singular but -Drazin invertible. Even though the case of being closed can be dealt with using the -Drazin inverse for closed linear operators in [12], we focus on the bounded case since it has been pointed out in [1–3] that it is enough to consider problem (1) when is bounded. Following [11], a bounded linear operator is -Drazin invertible if is not an accumulated spectral point of . We write for the spectrum of . A bounded linear operator is called a -Drazin inverse of ifSuch an operator is unique, if it exists and is denoted by . It follows that if is -Drazin invertible, then can be decomposed to an invertible operator and a quasinilpotent operator. This fact plays a crucial role in our analysis. Recall that a bounded linear operator is quasinilpotent if the spectrum of is identical to and is nilpotent if there is a positive integer such that . The smallest such is the index of the nilpotency. The following result, which is due to Koliha [11], allows such decomposition of a -Drazin invertible operator.
Theorem 1 (see [11, Theorem 7.1]). If is a -Drazin invertible operator in a Banach space , then , , where is invertible, is quasinilpotent with respect to this direct sum, and Moreover, if is the spectral projection corresponding to , then .
We will show that, under certain condition on the rate of which the powers of the quasinilpotent part decay, the solution to problem (1) exists and is given by an explicit formula. A function is a solution to problem (1) if it is differentiable and satisfies the differential equation in and the initial condition .
In Section 3 we study two classes of the so-called “singular singularly perturbed initial value problem”:Problem (5) was extensively studied by Campbell [4, 6] in matrix setting. We will show that if the continuity of the -Drazin inverse is assumed, then the solution to (5) converges to the solution of the reduced system when converges to . We will also show that the solution to (6) converges to as , assuming the continuity of the -Drazin inverse and the appropriate location of the spectrum of . The operators under consideration are a family of bounded linear operators on a Banach space . For properties of the continuity of the classical Drazin inverse and the -Drazin inverse, see [13–15].
In the sequel we will use the following definition, which is attributed to to Miekkala and Nevanlinna [16].
Definition 2. A quasinilpotent operator is of finite order if the resolvent of is of finite order as an entire function in . The value of is a nonnegative number for which holds for with small enough but fails for .
Nilpotent operators are quasinilpotent of order zero but the converse is not true since a quasinilpotent is nilpotent of order if and only if the resolvent is a polynomial in of order . The following result in [16] is important for our analysis.
Theorem 3 (see [16, Proposition 3.5]). A quasinilpotent operator is of finite order if is finite, and then the order is equal to .
Using Theorem 1, we say that a -Drazin invertible operator is of order if the quasinilpotent part of is not and of order .
2. Singular Initial Value Problem
In this section we extend the results on singular differential equations in [1, Theorem 3.1] and [3, Theorem 4.1] for the case where is Drazin invertible in the classical sense to the case where is -Drazin invertible. The next theorem shows that when the function is analytic in , problem (1) can be solved when is quasinilpotent of order . This result extends [1, Lemma 3.1].
Theorem 4. If the operator is quasinilpotent of order and is analytic in , then problem (1) has a unique solution if and only if , and the solution is given by
Proof. By direct verification it is clear that if converges uniformly on , then is a solution of (1) if and only if . Our proof of the existence of the solution is therefore reduced to showing that the infinite series converges uniformly on . Observe that converges pointwise in if it converges absolutely. Since each is continuous on , there exists such that for all . Hence by the Weierstrass -test, for uniform convergence it is sufficient to show that converges.
For each , Since the quasinilpotent is of order , by Theorem 3 there exist and such that which implies that . Since , for sufficiently large , which implies . Since , On the other hand, if a function is analytic in an open set (in the set of real numbers), for every compact set , there exists a constant such that for every and every nonnegative integer the following bound holds (see [17]): Using the above result and the condition that is analytic in , there exists a constant independent of and such that Dividing both sides of the above inequality by and raising it to the power of , we get . Therefore , which concludes that converges.
For the uniqueness of the solution, it is enough to show that is the only solution of (1) with and . Taking the Laplace transform of (1) with and , we obtainwhere denotes the Laplace transform of . Since the operator is invertible for every complex number , we can conclude that , which implies that is the only solution.
We are now in a position to show our main result.
Theorem 5. If is -Drazin invertible operator of order , then problem (1) has a unique solution if and only if , and the solution is given by where .
Proof. Since is -Drazin invertible of order , by Theorem 1, , , where is invertible and is quasinilpotent of order with respect to the direct sum. Therefore problem (1) has a unique solution if and only if each of the following two initial value problems has a unique solution on and , respectively:where and . Applying Theorem 4 to (19), is the unique solution of (19) if and only if .
Since is invertible, (18) has a unique solution given bySince and , we obtain
On modifying the proof of Theorem 5, we can extend [3, Theorem 4.1] for the case where is a closed linear operator. This can be done by replacing the -Drazin inverse for bounded linear operators by that of closed linear operators using Definition 2.1 in [12] and by replacing Theorem 1 by Theorem 2.3 in [12].
3. Singularly Perturbed Differential Equations
In this section we use the results in previous sections and the continuity of the -Drazin inverse to study two classes of singularly perturbed differential equations in the forms of (5) and (6).
We first show the stability of (6) under some Lyapunov-type conditions. Let and denote the open left half- and right half-plane of the complex plane, respectively. In the next two results, we write and for and , respectively.
Theorem 6. Let be a -Drazin invertible operator of order , let be the corresponding Drazin inverse, and let be analytic function in for each . Equation (6) has a unique solution for each if and only if , and the solution is given bywhere is the spectral projection of corresponding to . Furthermore, if , , , , and as , then
Proof. The fact that the solution of (6) exists and that is given by (23) follows from Theorem 5. Since , , and as , we can show that converges uniformly in a compact set of ; hence it converges to zero as . Since and , it follows that as (see [15, Theorem 2.4]). By [11, Theorem 4.4], , and . Since is an isolated spectral point, there are disjoint open sets and such that By the upper semicontinuity of the spectrum there is such that and there are a bounded open set such that and a Cauchy cycle with respect to . Since as , it follows that uniformly for . Therefore there exists a constant such that for all . Let . Then as and which implies . On the other hand, since as . Therefore the first two terms of (23) converge to zero as . We conclude as .
We can now easily show that the solution of (5) converges to the solution of the associated reduced equation as if the continuity of the -Drazin inverse is assumed.
Theorem 7. Let be -Drazin invertible operator of order and let be the corresponding Drazin inverse for each . Equation (5) has a unique solution for each if and only if , and the solution is given bywhere is the spectral projection of corresponding to . Furthermore, if , , , and as , then where is the solution of the associated reduced equation
Proof. The proof follows from Theorem 5 and [15, Theorem 2.4].
4. Conclusions
We have obtained some results on abstract singular differential equations on a Banach space using the generalized Drazin inverse. In particular, the associated singular operator is assumed to have a generalized Drazin inverse instead of a classical one. Furthermore, two classes of singularly perturbed system have been studied. Under the continuity conditions of the generalized Drazin inverses, we have shown that the solution to the singularly perturbed differential equation converges to the solution of the reduced equation.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.