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

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

[partial derivative]u(t,x)/[partial derivative]t = A [[partial derivative].sup.2]u(t,x)/[partial derivative][x.sup.2] u(0, x) = g(x) (1)

is studied in  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 A; = (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  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

[mathematical expression not reproducible] (2)

exists 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)x R for some T >0, and [lim.sub.t[right arrow]0+] u(0, x) = g(x) with g(x) being an analytic function satisfying the bounds [mathematical expression not reproducible], where a and b are some positive constants.

Secondly we consider the following delay differential equation

y' (z) = Ay (z-h) + f (z) (3)

in a Banach space X, which is studied by Gefter and Stulova in  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 [member of] > 0, there exists [C.sub.[member of] > 0 such that [parallel]f(z) [parallel] [less than or equal to] [mathematical expression not reproducible] for each z [member of] C. We generalize the results in  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 , a closed linear operator A is g-Drazin invertible if 0 is not an accumulated spectral point of A. By [sigma](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) [intersect] D(A), R(I - AB) c D(A), and

BA = AB, BAB = B, (4) [sigma](A(I-AB)) = {0}. (4)

Such an operator is unique, if it exists and is denoted by [A.sup.D]. From , we have the following decomposition result.

Theorem 1. If A is a g-Drazin invertible operator in a Banach space X, then X = R([A.sup.D]A) [direct sum] N([A.sup.D] A), A = [A.sub.1] [direct sum] [A.sub.2], where [A.sub.1] is closed and invertible, [A.sub.2] is bounded and quasinilpotent with respect to this direct sum, and

[A.sup.D] = [A.sup.-1.sub.1] [direct sum] 0. (5)

Moreover, if P is the spectral projection corresponding to 0, then P = I- [AA.sup.D].

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 [mathematical expression not reproducible] for some positive constants a and b. Then the system (2) has a unique solution given by the formula

[mathematical expression not reproducible], (6)

where [mathematical expression not reproducible], represents the [C.sub.0]-semigroup of linear bounded operators generated by - [A.sup.D], and [([A.sup.D]).sup.1/2] denotes a bounded operator B such that [B.sup.2] (I - P) = [A.sup.D](I - P).

Proof. Since A is g-Drazin invertible, by Theorem (1), X = R (I - P) [direct sum] N(I - P), A = [A.sub.1] [direct sum] [A.sub.2], where [A.sub.1] is closed invertible and [A.sub.2] 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

[mathematical expression not reproducible], (7)

[mathematical expression not reproducible]. (8)

Since the operator A is positive, it is self-adjoint. Therefore, [A.sub.2] is self-adjoint and the imaginary part of [A.sub.2] is zero. Applying [1, Theorem 2] to Problem (8),

[mathematical expression not reproducible] (9)

the unique solution of Problem (8). Next we will show that

[mathematical expression not reproducible] (10)

is the unique solution of Problem (7). The operator [([A.sub.1.sup.-1]).sup.1/2] denotes an operator B such that [A.sub.1.sup.-1] = [B.sup.2]. The existence of such an operator B is guaranteed by the positivity of [A.sub.1.sup.-1].

Since A is positive, [sigma](A) [subset] [0, [theta]), which implies [sigma](-[A.sub.1.sup.-1]) [subset] (0, [infinity]). Therefore, there exist constants [mu] > 0 and M > 0 such that

[mathematical expression not reproducible]. (11)

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

[partial derivative]u (t,x)/[partial derivative]t = 1/[mu] [[partial derivative].sup.2]u(t,x)/ [partial derivative][x.sup.2]

u(0,x) = g(x). (12)

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 [parallel]g(x) [parallel], and the fundamental solution to the heat equation, one can differentiate under the integrals and verify that the integrals for [u.sub.1], [[partial derivative]u.sub.1]/ [partial derivative]t and [[partial derivative].sup.2] [u.sub.1]/ [partial derivative][x.sup.2] all converge. Using the derivative of the [C.sub.0]-semigroup [mathematical expression not reproducible], it is straightforward to check that [u.sub.1] (t, x) satisfies the partial differential equation (7). Moreover, [u.sub.1] (t, x) is the only solution if (x, t) e [0, [mu]/4b) x R, and [lim.sub.t[right arrow]0+] [u.sub.1] (t, x) = (I - P)g(x).

Since [A.sup.D](I - P) = [A.sub.1.sup.-1] (I - P) and [A.sup.n]P = [A.sup.n.sum.2]P, we obtain u (t, x)

[mathematical expression not reproducible], (13)

An application of the above result can be illustrated by taking

A = -[d.sup.2]/[dx.sup.2], (14)

where

[mathematical expression not reproducible]. (15)

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

[mathematical expression not reproducible], (16)

where P = I - [AA.sup.D] and [F.sup.(n)] is the n-th primitive of f; that is, [d.sup.n] [F.sup.(n)] (z)/[dz.sup.n] = f(z).

Proof. Since A is g-Drazin invertible, X = R(I-P)[direct sum] N(I-P), A = [A.sub.1] [direct sum] [A.sub.2], where [A.sub.1] is closed and invertible and [A.sub.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(I - P) and R(P), respectively

[y'.sub.1] (z) = [A.sub.1], [y.sub.1] (z-h) + [f.sub.1] (z), (17)

[y'.sub.2] (z) = [A.sub.2] [y.sub.2] (z-h) + [f.sub.2] (z). (18)

Since the operator [A.sub.1] is closed and invertible, applying [2, Theorem 2] to (17), we have

[y.sub.1] (z) = -[[infinity].summation over (n=0)] [A.sup.n.sub.2] [F.sub.1.sup.(n)] (z + (n + 1) h) (19)

being the unique solution of Problem (17). Next we will show that

[y.sub.2] (z) = -[[infinity].summation over (n=0)] [A.sup.n.sub.2] [F.sub.2.sup.(n+1)] (z + (n + 1) h) (20)

is a zero exponential type solution of Problem (18). Following [2, Lemma 1 ], we first show that if [f.sub.2] (z) is of zero exponential type then so is [F.sub.2.sup.(n)] (z). Let [f.sub.2] (z) = [[SIGMA].sup.[infinity].sub.n=0] [a.sub.m] [z.sup.m] be of zero exponential type and [member of] >0. Since [lim.sub.m[right arrow][infinity]] [(m![parallel][a.sub.m] [parallel]).sup.1/m] = 0 for each m [member of] N, [parallel][[alpha].sub.m][parallel] [less than or equal to] M([e.sup.m]/m!) for some M > 0. Letting m + n = k, we have

[mathematical expression not reproducible]. (21)

Now, modifying the proof of [2, Theorem 1] with the n-th derivative replaced by the n-th primitive [F.sub.2.sup.(n)] (z), [e.sup.n] by [e.sup.-n] and nh by -nh, we obtain the convergence of [y.sub.2](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 [A.sup.D] P(I-P) = [A.sub.1.sup.-1] (I-P) and [A.sup.n] P = [A.sup.n.sub.2]P, we obtain

[mathematical expression not reproducible], (22)

4. Conclusion

In Section 2 we have obtained the unique solution for the heat equation with operator coefficient A, which is assumed to be self-adjoint and positive in a Hilbert space. Our result extends [1, Theorem 2] in the sense that A is g-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 A is g-Drazin invertible instead of invertible in the usual sense.

https://doi.org/10.1155/2017/4248304

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

 S. Gefter and A. Vershynina, "On holomorphic solutions of the heat equation with a Volterra operator coefficient," Methods of Functional Analysis and Topology, vol. 13, no. 4, pp. 329-332, 2007.

 S. Gefter and T. Stulova, "On solutions of zero exponential type for some inhomogeneous differential-difference equations in a Banach space," in Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics and Statistics, vol. 54 of Springer Proc. Math. Stat., pp. 253-263, Springer, Berlin, Germany, 2013.

 J. J. Koliha and T. D. Tran, "The Drazin inverse for closed linear operators and the asymptotic convergence of C0-semigroups," The Journal of Operator Theory, vol. 46, no. 2, pp. 323-336, 2001.

 T. Kato, Perturbation Theory for Linear Operators, vol. 132 of Classics in Mathematics, Springer, Berlin, Germany, 1995.

Alrazi Abdeljabbar and Trung Dinh Tran

Department of Mathematics, College of Arts and Sciences, Khalifa University for Science and Technology-The Petroleum Institute, P O. Box 2533, Abu Dhabi, UAE

Correspondence should be addressed to Alrazi Abdeljabbar; aabdeljabbar@pi.ac.ae

Received 11 June 2017; Accepted 11 October 2017; Published 1 November 2017

Academic Editor: Carlos Lizama
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