Fixed point theorems without continuity condition in non-solid extended abstract metric spaces with applications

Yan Han; Jin Chen; Shaoyuan Xu; Xianhua Xie; Yan Han; Jin Chen; Shaoyuan Xu; Xianhua Xie

doi:10.3934/math.2026002

AIMS Mathematics

2026, Volume 11, Issue 1: 22-42. doi: 10.3934/math.2026002

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Fixed point theorems without continuity condition in non-solid extended abstract metric spaces with applications

1.
School of Mathematics and Statistics, Zhaotong University, Zhaotong, Yunnan, China
2.
School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi, China

Received: 11 September 2025 Revised: 26 November 2025 Accepted: 19 December 2025 Published: 04 January 2026
MSC : 47H10, 54H25

In this paper, several fixed point results for different contractions in extended abstract metric spaces with non-solid cones are established. The definitions of Cauchy sequences, convergent sequences, and completeness with respect to normality in the non-solid extended abstract metric spaces are reintroduced. We prove that the fixed point exists and is unique in these spaces by removing the orbital continuity assumption. Our results generalize and extend several significant theorems in classical abstract metric spaces and standard metric spaces. A consequence, we present several examples demonstrating that our main theorems serve as powerful tools for solving equations in different space frameworks, for both solid and non-solid cones.
- non-solid space,
- orbital continuity,
- fixed point,
- Reich type and Kannan type (almost) contractions
Citation: Yan Han, Jin Chen, Shaoyuan Xu, Xianhua Xie. Fixed point theorems without continuity condition in non-solid extended abstract metric spaces with applications[J]. AIMS Mathematics, 2026, 11(1): 22-42. doi: 10.3934/math.2026002

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Abstract

In this paper, several fixed point results for different contractions in extended abstract metric spaces with non-solid cones are established. The definitions of Cauchy sequences, convergent sequences, and completeness with respect to normality in the non-solid extended abstract metric spaces are reintroduced. We prove that the fixed point exists and is unique in these spaces by removing the orbital continuity assumption. Our results generalize and extend several significant theorems in classical abstract metric spaces and standard metric spaces. A consequence, we present several examples demonstrating that our main theorems serve as powerful tools for solving equations in different space frameworks, for both solid and non-solid cones.

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