Implicit Linear q -Difference Equations in Banach Spaces
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Journal of Mathematical Sciences, Vol. 251, No. 6, December, 2020
IMPLICIT LINEAR q-DIFFERENCE EQUATIONS IN BANACH SPACES 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 517.962.22
We establish the existence and uniqueness of a holomorphic solution to an implicit linear q-difference equation in a Banach space. We consider holomorphic solutions in a neighborhood of zero if |q| < 1 and entire solutions if |q| > 1. The solutions are expressed an an explicit form. The case q = 0 is separately considered. Bibliography: 7 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, and q ∈ C, q = 1. For a holomorophic vector-valued function w(z) in some neighborhood of zero we introduce the operator of taking the q-derivative [1] Dq w(z) :=
w(qz) − w(z) . (q − 1)z
This paper continues the study of [2], where we considered holomorphic and entire solutions to the initial problem for the explicit linear q-difference equation in the space E Dq w(z) = Aw(z) + f (z),
(1.1)
where A is a closed linear operator in E. In this paper, we study the existence and uniqueness of holomorphic and entire solutions to the implicit q-difference equation BDq w(z) = w(z) − f (z) ∗
(1.2)
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 15-22. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2516-0787
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with an operator B ∈ L(E). For Equation (1.2) we study holomorphic solutions in some neighborhood of zero if |q| < 1 or entire solutions if |q| > 1. In Subsection 2.1, we establish the well-posedness of Equation (1.2) and an equation related to (1.1) with an invertible operator A in the space of holomorphic vector-valued functions (Theorem 2.1 and Corollaries 2.2, 2.3). This result can be regarded as a q-analog of the result of [3] concerning differential equations. In the case q = 0, we clarify (cf. Subsection 2.2) relationships between the existence of holomorphic solutions to Equation (1.2) and the solvability of the problem for implicit linear difference equations (Theorem 2.2). The case |q| > 1 is studied in Subsection 2.3, where for Equations (1.1) and (1.2) we find conditions guaranteeing the existence and uniqueness of an entire solution of q-exponential type (Theorem 2.3 and Corollary 2.5).
2
Holomorphic Solutions to q-Difference Equations
Properties of holomorphic solutions to Equation (1.2) depend on whether |q| < 1 or |q| > 1. Therefore, we consider these cases separately. 2.1. Case |q| < 1. We introduce the following q-analog of the number n! [1]: ⎧ 2 n ⎪ ⎨ (q − 1)(q − 1) . . . (q − 1) , n ∈ N, (q − 1)n [n]q ! = ⎪ ⎩ 1, n = 0. We denote by r(B) the spectral radius of an operator B ∈ L(E). Theorem 2.1. Assume that |q| < 1, B
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