## $$L^{2}$$ 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

### Keywords:

Lebesgue integration; Sobolev spaces
Full Text:

### References:

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