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Cone expansion and cone compression fixed point theorems for sum of two operators and their applications

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Abstract

In this article, we first establish a series of user-friendly versions of fixed point theorems in cones for sum of two operators with one being either contractive or expansive and the other being compact, one of which covers the classical Krasnoselskii’s fixed point theorem concerning cone expansion and compression of norm type. Along the way, we also offer some sufficient conditions which slightly relax the compactness requirement on the one summand operator. Second, as applications to some of our main results, we consider the eigenvalue problems of Krasnoselskii type in critical case and the existence of one positive solution to one parameter operator equations. Finally, to illustrate the usefulness and the applicability of our fixed point results, we study the existence of one nontrivial positive solution to certain integral equations of Hammerstein type and of perturbed Volterra type.

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Acknowledgements

We greatly appreciate the four anonymous referees for giving positive and valuable comments from different perspectives, which further helped us to improve the exposition of this work. TX is funded by NSF of China (No. 11601516 and 11871226) and the Research Funds of Renmin University of China (No. 2018030199).

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Authors and Affiliations

  1. Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China

    Tian Xiang

  2. College of Business, Shanghai University of Finance and Economics, Shanghai, 200083, China

    Dong Zhu

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  1. Tian Xiang
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  2. Dong Zhu
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Correspondence to Tian Xiang.

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Xiang, T., Zhu, D. Cone expansion and cone compression fixed point theorems for sum of two operators and their applications. J. Fixed Point Theory Appl. 22, 49 (2020). https://doi.org/10.1007/s11784-020-00786-5

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  • DOI: https://doi.org/10.1007/s11784-020-00786-5

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