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Lebesgue–Stieltjes combined \(\Diamond _\alpha \)-measure and integral on time scales

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we introduce the concepts of Lebesgue–Stieltjes \(\Diamond _\alpha \)-measure, \(\Diamond _\alpha \)-measurable function and \(\Diamond _\alpha \)-integral. Based on the theory of combined measurability on time scales, some basic properties including the relationships, the extensions and the composition theorems are established. Particularly, through the switch coefficient of combined theory on time scales, one can obtain the Lebesgue–Stieltjes \(\varDelta \)-measure and Lebesgue–Stieltjes \(\nabla \)-measure if by taking \(\alpha =1\) and \(\alpha =0\), respectively. In addition, several examples are provided to demonstrate the effectiveness of the obtained results in each section.

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Funding

This work is supported by National Natural Science Foundation of China (No. 11961077).

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  1. Department of Mathematics, Yunnan University, Kunming, 650091, Yunnan, China

    Guangzhou Qin & Chao Wang

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  1. Guangzhou Qin
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  2. Chao Wang
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Authors have equally contributed in preparing this manuscript. All authors read and approved the final manuscript.

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Correspondence to Chao Wang.

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Qin, G., Wang, C. Lebesgue–Stieltjes combined \(\Diamond _\alpha \)-measure and integral on time scales. RACSAM 115, 50 (2021). https://doi.org/10.1007/s13398-021-01000-y

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