Applications of the g-Drazin Inverse to the Heat Equation and a Delay Differential Equation

and Applied Analysis 3 Since AD(I − P) = A−1 1 (I − P) and AnP = A2P, we obtain u (t, x) = Pg (x) + ∞ ∑ n=1 tn n!A nPg(2n) (x) + (A D)1/2 √4πt ∫ ∞ −∞ e−AD(x−y)2/4t (I − P) g (y) dy, lim t→0 u (t, x) = lim t→0 u1 (t, x) + lim t→0 u2 (t, x) = (I − P) g (x) + Pg (x) = g (x) . (13) An application of the above result can be illustrated by taking A = − d 2 dx2 , (14)


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 problem is 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   fl (1/2)( −  * ) 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 0 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 system exists 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 [0, )×R for some  > 0, and lim →0 + (0, ) = () with () being an analytic function satisfying the bounds ‖()‖ ≤   2 , where  and  are some positive constants.
Secondly we consider the following delay differential equation in 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 ℎ 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  > 0, there exists   > 0 such that ‖()‖ ≤    || for each  ∈ C. 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 0 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 R() ⊂ D(), R( − ) ⊂ D(), and Such an operator is unique, if it exists and is denoted by   .From [3], we have the following decomposition result.
where  1 is closed and invertible,  2 is bounded and quasinilpotent with respect to this direct sum, and Moreover, if  is the spectral projection corresponding to 0, then  =  −   .
The above result is crucial to our analysis.

Solution for the Heat Equation with Positive Operator Coefficient
In this section we obtain an analytic solution for (2) that generalizes [1, Theorem 2] 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 R that satisfies the bound ‖()‖ ≤   2 for some positive constants  and .Then the system (2) has a unique solution given by the formula where  = −  ,  −   represents the  0 -semigroup of linear bounded operators generated by −  , and (  ) Since the operator  is positive, it is self-adjoint.Therefore,  2 is self-adjoint and the imaginary part of  2 is zero.
Applying [1,Theorem 2] to Problem (8), is the unique solution of Problem (8).Next we will show that is the unique solution of Problem (7).The operator ( −1 1 ) 1/2 denotes an operator  such that  −1 1 =  2 .The existence of such an operator  is guaranteed by the positivity of  −1 1 .Since  is positive, () ⊂ [0, ∞), which implies (− −1 1 ) ⊂ (−∞, 0).Therefore, there exist constants  > 0 and  > 0 such that Observe that the above inequality reduces the analysis of the heat equation with operator coefficient to that of the standard heat equation with scalar coefficient (0, ) =  () . ( This 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  1 ,  1 / and  2  1 / 2 all converge.Using the derivative of the  0semigroup , it is straightforward to check that  1 (, ) satisfies the partial differential equation (7).Moreover,  1 (, ) is the only solution if (, ) ∈ [0, /4) × R, and lim →0 +  1 (, ) = ( − )().
Since   ( − ) =  −1 1 ( − ) and    =   2 , 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].

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 2].
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,  = R(−)⊕(−),  =  1 ⊕  2 , where  1 is closed and invertible and  2 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 R( − ) and R(), respectively.
Since the operator  1 is closed and invertible, applying [2, Theorem 2] to (17), we have being the unique solution of Problem (17).Next we will show that

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 2] 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 2] in the sense that  is -Drazin invertible instead of invertible in the usual sense.