A new approach for fixed point theorems for $ C $-class functions in Hilbert $ C^{*} $-modules

Mi Zhou; Arsalan Hojjat Ansari; Choonkil Park; Snježana Maksimović; Zoran D. Mitrović; Mi Zhou; Arsalan Hojjat Ansari; Choonkil Park; Snježana Maksimović; Zoran D. Mitrović

doi:10.3934/math.20241400

AIMS Mathematics

2024, Volume 9, Issue 10: 28850-28869. doi: 10.3934/math.20241400

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A new approach for fixed point theorems for $ C $-class functions in Hilbert $ C^{*} $-modules

1.
Center for Mathematical Research, University of Sanya, Sanya 572022, China
2.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa-0204, South Africa
3.
Research Institite for Convergence of Basic Science, Hanyang University, Seoul 04763, Korea
4.
Faculty of Architecture, Civil Engineering and Geodesy, University of Banja Luka, Bosnia and Herzegovina
5.
Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina

Received: 11 August 2024 Revised: 26 September 2024 Accepted: 04 October 2024 Published: 12 October 2024
MSC : 47H10, 54H25

In this paper, we introduced a new contraction principle via altering distance and $ C $-class functions with rational forms which extends and generalizes the existing version provided by Hasan Ranjbar et al. [H. Ranjbar, A. Niknam, A fixed point theorem in Hilbert $ C^\ast $-modules, Korean J. Math., 30 (2022), 297–304]. Specifically, the rational forms involved in the contraction condition we presented involve the $ p $-th power of the displacements which can exceed the second power mentioned in Hasan Ranjbar et al.'s paper. Moreover, we also proved a fixed point theorem for this type of contraction in the Hilbert $ C^\ast $-module. Some adequate examples were provided to support our results. As an application, we applied our result to prove the existence of a unique solution to an integral equation and a second-order $ (p, q) $-difference equation with integral boundary value conditions.
- fixed point,
- Hilbert $ C^{\ast} $-modules,
- $ C $-class function
Citation: Mi Zhou, Arsalan Hojjat Ansari, Choonkil Park, Snježana Maksimović, Zoran D. Mitrović. A new approach for fixed point theorems for $ C $-class functions in Hilbert $ C^{*} $-modules[J]. AIMS Mathematics, 2024, 9(10): 28850-28869. doi: 10.3934/math.20241400

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Abstract

In this paper, we introduced a new contraction principle via altering distance and $ C $-class functions with rational forms which extends and generalizes the existing version provided by Hasan Ranjbar et al. [H. Ranjbar, A. Niknam, A fixed point theorem in Hilbert $ C^\ast $-modules, Korean J. Math., 30 (2022), 297–304]. Specifically, the rational forms involved in the contraction condition we presented involve the $ p $-th power of the displacements which can exceed the second power mentioned in Hasan Ranjbar et al.'s paper. Moreover, we also proved a fixed point theorem for this type of contraction in the Hilbert $ C^\ast $-module. Some adequate examples were provided to support our results. As an application, we applied our result to prove the existence of a unique solution to an integral equation and a second-order $ (p, q) $-difference equation with integral boundary value conditions.

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