Holomorphic Solutions to Linear q -Difference Equations in a Banach Space
- PDF / 198,679 Bytes
- 13 Pages / 594 x 792 pts Page_size
- 48 Downloads / 217 Views
Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
HOLOMORPHIC SOLUTIONS TO LINEAR q-DIFFERENCE EQUATIONS IN A BANACH SPACE S. L. Gefter ∗ Karazin Kharkiv National University 4, pl. Svobody, Kharkiv 61000, Ukraine [email protected]
A. L. Piven’ Karazin Kharkiv National University 4, pl. Svobody, Kharkiv 61000, Ukraine [email protected]
UDC 517.983
We prove the existence and uniqueness of a holomorphic solution to a linear q-difference equation for a closed operator with not necessarily dense domain in a Banach space. We consider holomorphic solutions in a neighborhood of zero in the case |q| < 1 and integer-valued solutions in the case |q| > 1. We obtain the solutions in an explicit form. Bibliography: 12 titles.
1
Introduction
Assume that E is a complex Banach space, L(E) is the space of bounded linear operators in E, f (z) is a holomorphic vector-valued function in some neighborhood of zero with the values in E, A is a closed linear operator in E with the domain D(A) not necessarily dense in E, and q ∈ C, q = 1. We consider the q-difference equation w(qz) − w(z) = Aw(z) + f (z) (q − 1)z
(1.1)
and study holomorphic solutions in some neighborhood of zero in the case |q| < 1 and integervalued solutions in the case |q| > 1. Different aspects connected with Equation (1.1) were studied in many works (cf., for example, [1] and the references therein). Equation (1.1) with a bounded operator A was studied on the real axis in [2, 3]. We consider Equation (1.1) together with the initial condition w(0) = w0 ∈ E. ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 43-53. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0602
602
(1.2)
In Subsection 3.1, in the case |q| < 1, we obtain an existence and uniqueness criterion for holomorphic solutions to the initial problem for the homogeneous q-difference equation with a closed linear operator. We also obtain an explicit form for the solution (Theorem 3.1). This result is a counterpart of the theorem on holomorphic solutions to linear differential–operator equations in a Banach space (cf. [4]). For bounded operators we additionally establish the existence and uniqueness of holomorphic solutions to problems with inhomogeneous equations (Theorem 3.2). The case |q| > 1 is treated in Subsection 3.2, where we find conditions for the existence and uniqueness of an entire solution to the homogeneous q-difference equation (Theorem 3.3 and Corollary 3.3). This result for a bounded operator is generalized to the case of inhomogeneous equations (Theorem 3.4). Section 4 contains examples. In particular, for some q-difference analogue of the heat equation we show that, in the case |q| < 1, the Dirichlet problem has a holomorphic solution with respect to z if and only if the initial condition is a sine polynomial (Example 4.3).
2
Preliminaries
We study particular properties of vectors in connection with the domain of a closed operator. Definition 2.1 ([5, Chapter 2, Section 14]). The l
Data Loading...
