Singular Differential Equations and g-Drazin Invertible Operators

where M and N are closed linear operators from a Banach space Y to X. 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 A is singular (noninvertible) but Drazin invertible in the classical sense. A bounded linear operator is Drazin invertible in the classical sense if 0 is a pole of the resolvent of A. In [11], Koliha generalized the concept of Drazin invertibility to the case where 0 is only an isolated spectral point of the spectrum of A. 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 A is singular but g-Drazin invertible. Even though the case ofA being closed can be dealt with using the g-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 A is bounded. Following [11], a bounded linear operator A is g-Drazin invertible if 0 is not an accumulated spectral point of A. We write σ(A) for the spectrum of A. A bounded linear operator B is called a g-Drazin inverse of A if


Introduction
Let  be a bounded or closed linear operator in a Banach space  and let  be a -valued function.The following initial value problem   ()  +  () =  ()  (0) =  0 ,  ∈ [0, ] is central to the analysis of the abstract singular equation where  and  are closed linear operators from a Banach space  to .Problem (2) and its variations were extensively studied in [1][2][3] and the references therein.In [4][5][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][8][9][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 0 is a pole of the resolvent of .In [11], Koliha generalized the concept of Drazin invertibility to the case where 0 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][2][3] that it is enough to consider problem (1) when  is bounded.Following [11], a bounded linear operator  is -Drazin invertible if 0 is not an accumulated spectral point of .We write () for the spectrum of .A bounded linear operator  is called a -Drazin inverse of  if (3) Such 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 0 and  is nilpotent if there is a positive integer  such that   = 0.The smallest such  is the index of the nilpotency.The following result, which is 2 International Journal of Analysis due to Koliha [11], allows such decomposition of a -Drazin invertible operator.
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 [0, ] and the initial condition (0) =  0 .
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 0 + .We will also show that the solution to (6) converges to 0 as  → 0 + , assuming the continuity of the -Drazin inverse and the appropriate location of the spectrum of (0).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][14][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 1/.The value of  is a nonnegative number for which holds for  > 0 with small enough || but fails for  < 0.
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 1/ of order .The following result in [16] is important for our analysis.

Theorem 3 (see [16, Proposition 3.5]). A quasinilpotent operator 𝐴 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 0 and of order .
We are now in a position to show our main result.

Theorem 5. If 𝐴 is 𝑔-Drazin invertible operator of order 𝜔 < 1, 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  < 1, by Theorem 1,  = ( − ) ⊕ ( − ),  =  1 ⊕  2 , where  1 is invertible and  2 is quasinilpotent of order  < 1 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  1 () = ( − )() and  2 () = ().Applying Theorem 4 to (19), is the unique solution of (19) if and only if  0 = ∑ ∞ =0 (−1)    2  () 2 (0).Since  1 is invertible, (18) has a unique solution given by ( 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].

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 can now easily show that the solution of (5) converges to the solution of the associated reduced equation as  → 0 + if the continuity of the -Drazin inverse is assumed.Theorem 7. Let () be -Drazin invertible operator of order  < 1 and let   () be the corresponding Drazin inverse for each  ∈ [0,  0 ).Equation ( 5

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.