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 A be a bounded linear operator in a Hilbert space X and g be a holomorphic X-valued function. The following initial value problem(1)∂ut,x∂t=A∂2ut,x∂x2u0,x=gxis studied in [1] under the assumption that A is a Volterra operator and its imaginary part of A is of trace class. In particular, it has been proved that if A is quasinilpotent and its imaginary part AI≔1/2i(A-A∗) 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 A is a positive operator, and 0 is not an accumulated spectral point of A. Our results are extensions of [1] in the sense that the class of g-Drazin invertible operators A is more general than that of quasinilpotent operators.

We will show that if A is positive and g-Drazin invertible then the solution to the system(2)∂ut,x∂t=A∂2ut,x∂x2limt→0+ut,x=gxexists and is given by an explicit formula. We say a function u(t,x) is a solution to the above initial value problem if it satisfies the partial differential equation in [0,T)×R for some T>0, and limt→0+u(0,x)=g(x) with g(x) being an analytic function satisfying the bounds g(x)≤aebx2, where a and b are some positive constants.

Secondly we consider the following delay differential equation(3)y′z=Ayz-h+fzin a Banach space X, which is studied by Gefter and Stulova in [2] under the assumption that A is an invertible closed linear operator with a bounded inverse in X; the delay term h is a complex constant, and f is an X-valued holomorphic function of zero exponential type. Recall that an entire function f is of zero exponential type if, for every ϵ>0, there exists Cϵ>0 such that f(z)≤Cϵeϵ|z| for each z∈C. We generalize the results in [2] by replacing the invertible closed linear operator A with a g-Drazin invertible operator. We will show that if A is g-Drazin invertible and f 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 A is g-Drazin invertible if 0 is not an accumulated spectral point of A. By σ(A), R(A), D(A), and N we denote the spectrum, range, domain, and nullspace of A, respectively. A bounded linear operator B is called a g-Drazin inverse of A if R(B)⊂D(A), R(I-AB)⊂D(A), and(4)BA=AB,BAB=B,σAI-AB=0.Such an operator is unique, if it exists and is denoted by AD. From [3], we have the following decomposition result.

Theorem 1.
If A is a g-Drazin invertible operator in a Banach space X, then X=R(ADA)⊕N(ADA), A=A1⊕A2, where A1 is closed and invertible, A2 is bounded and quasinilpotent with respect to this direct sum, and(5)AD=A1-1⊕0.Moreover, if P is the spectral projection corresponding to 0, then P=I-AAD.

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 2] in the sense that the coefficient operator A is assumed to be g-Drazin invertible instead of quasinilpotent.

Theorem 2.
Let A be a closed positive operator which is g-Drazin invertible, and let g be an analytic function in R that satisfies the bound g(x)≤aebx2 for some positive constants a and b. Then the system (2) has a unique solution given by the formula (6)ut,x=Pgx+∑n=1∞tnn!AnPg2nx+AD1/24πt∫-∞∞e-ADx-y2/4tI-Pgydy,where P=I-AAD, e-ADs represents the C0-semigroup of linear bounded operators generated by -AD, and (AD)1/2 denotes a bounded operator B such that B2(I-P)=AD(I-P).

Proof.
Since A is g-Drazin invertible, by Theorem (1), X=R(I-P)⊕N(I-P), A=A1⊕A2, where A1 is closed invertible and A2 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 R(I-P) and R(P), respectively.(7)∂u1∂t=A1∂2u1∂x2limt→0+u10,x=I-Pgx,(8)∂u2∂t=A2∂2u2∂x2limt→0+u20,x=Pgx.Since the operator A is positive, it is self-adjoint. Therefore, A2 is self-adjoint and the imaginary part of A2 is zero. Applying [1, Theorem 2] to Problem (8),(9)u2t,x=Pgx+∑n=1∞tnn!A2nPg2nxis the unique solution of Problem (8). Next we will show that(10)u1t,x=A1-11/24πt∫-∞∞e-A1-1x-y2/4tI-Pgydyis the unique solution of Problem (7). The operator (A1-1)1/2 denotes an operator B such that A1-1=B2. The existence of such an operator B is guaranteed by the positivity of A1-1.

Since A is positive, σ(A)⊂[0,∞), which implies σ(-A1-1)⊂(-∞,0). Therefore, there exist constants μ>0 and M>0 such that(11)∫-∞∞e-A1-1x-y2/4tI-Pgydy≤∫-∞∞Me-μx-y2/4tI-Pgydy.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(12)∂ut,x∂t=1μ∂2ut,x∂x2u0,x=gx.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 g(x), and the fundamental solution to the heat equation, one can differentiate under the integrals and verify that the integrals for u1, ∂u1/∂t and ∂2u1/∂x2 all converge. Using the derivative of the C0-semigroup de-A1-1s/ds=-A1-1e-A1-1s, it is straightforward to check that u1(t,x) satisfies the partial differential equation (7). Moreover, u1(t,x) is the only solution if (x,t)∈[0,μ/4b)×R, and limt→0+u1(t,x)=(I-P)g(x).

Since AD(I-P)=A1-1(I-P) and AnP=A2nP, we obtain (13)ut,x=Pgx+∑n=1∞tnn!AnPg2nx+AD1/24πt∫-∞∞e-ADx-y2/4tI-Pgydy,limt→0+ut,x=limt→0+u1t,x+limt→0+u2t,x=I-Pgx+Pgx=gx.

An application of the above result can be illustrated by taking (14)A=-d2dx2,where (15)DA=w∈L2-π,π, Aw∈L2-π,π, w-π=wπ, w′-π=w′-π.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 2].

Theorem 3.
Let A be a closed linear operator which is g-Drazin invertible, and let f be an entire function of zero exponential type. Then (3) has a zero exponential type solution given by the formula (16)yz=-∑n=0∞ADn+1I-Pfnz+n+1h+∑n=0∞AnPFn+1z-nh,where P=I-AAD and F(n) is the n-th primitive of f; that is, dnF(n)(z)/dzn=f(z).

Proof.
Since A is g-Drazin invertible, X=R(I-P)⊕N(I-P), A=A1⊕A2, where A1 is closed and invertible and A2 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(I-P) and R(P), respectively.(17)y1′z=A1y1z-h+f1z,(18)y2′z=A2y2z-h+f2z.Since the operator A1 is closed and invertible, applying [2, Theorem 2] to (17), we have(19)y1z=-∑n=0∞A1-n+1f1nz+n+1hbeing the unique solution of Problem (17). Next we will show that (20)y2z=∑n=0∞A2nF2n+1z-nhis a zero exponential type solution of Problem (18). Following [2, Lemma 1], we first show that if f2(z) is of zero exponential type then so is F2(n)(z). Let f2(z)=∑n=0∞αmzm be of zero exponential type and ϵ>0. Since limm→∞(m!αm)1/m=0 for each m∈N, αm≤Mϵm/m! for some M>0. Letting m+n=k, we have (21)F2nz=∑m=0∞αmzm+nm+1m+2⋯m+n=∑k=n∞αk-nzkk-n!k!≤∑k=n∞Mϵk-nk!zk≤Mϵneϵz.Now, modifying the proof of [2, Theorem 1] with the n-th derivative replaced by the n-th primitive F2(n)(z), ϵn by ϵ-n and nh by -nh, we obtain the convergence of y2(z) 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 AD(I-P)=A1-1(I-P) and AnP=A2nP, we obtain (22)yz=-∑n=0∞ADn+1I-Pfnz+n+1h+∑n=0∞AnPFn+1z-nh.