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\(L^{2}\) spaces and boundary value problems on time-scales. (English) Zbl 1116.34021

This paper is concerned with extensions of the Sturm-Liouville theory for the boundary value operator \[ Lu:=-\left[pu^{\triangle}\right]^{\triangle}+qu^{\sigma} \] on a bounded time scale \(\mathbb{T}\). Following the introductory Section 1, Section 2 provides the fundamentals of mathematical analysis on time scales. Section 3 introduces the reader to the essentials of the Lebesgue measure and integration theory and outlines alternative ways of the Lebesgue measure construction. In addition, Section 3 provides convergence theorems for sequences of functions in the Lebesgue spaces \(L^p(\mathbb{T})\). Section 4 introduces the concept of the generalized derivative followed by definitions of Sobolev-type spaces \(H^n(\mathbb{T})\), \(n \geq 1\) and the statements of embedding theorems. Section 5 is devoted to the study of the operator \(L\) under the assumptions of \(p \in H^1(\mathbb{T})\) and \(q \in L^2(\mathbb{T})\). Such important properties of the operator \(L\) are studied as positivity, injectivity, invertibility, and compactness of the inverse. Finally, Section 5 provides a construction of a Green’s function for \(L\). This paper would be of interest to those interested in obtaining weak solutions for boundary value problems for dynamic equations on time scales.

MSC:

34B24 Sturm-Liouville theory
39A12 Discrete version of topics in analysis
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