Fixed Point and Homotopy Methods in Cone A-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation

Arif, Anam; Nazam, Muhammad; Alsulami, Hamed; Hussain, Aftab; Mahmood, Hasan

doi:10.3390/sym14071328

Symmetry

2022

DOI: 10.3390/sym14071328

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Fixed Point and Homotopy Methods in Cone A-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation

Anam Arif

Muhammad Nazam

Hamed Alsulami

et al.

Abstract: The purpose of this article is to introduce an ordered implicit relation that can be used for the existence of fixed points of new contractions defined in cone A-metric spaces. We investigate a fixed-point method for proving the existence of Urysohn integral equation solutions. We prove an homotopy result by the application of obtained fixed-point theorem. The hypothesis is demonstrated with examples.

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2023

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Cited by 2 publications

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Generalized almost Contractions on Extended Quasi-Cone B-Metric Spaces

Liftaj,

Sila,

Selko

2023

Wseas Transactions on Mathematics

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Fixed Point Theory is among the most valued research topics nowadays. Over the years, it has been developed in three directions: by generalizing the metric space, by establishing new contractive conditions, and by applying its results to various fields such as Differential Equations, Integral Equations, Economics, etc. In this paper, we define a new class of cone metric spaces called the class of extended quasi-cone b-metric spaces. Extended quasi-cone b-metric spaces generalize cone metric spaces and quasi-cone b-metric spaces. We have studied topological issues, such as the right and left topologies, right (left) Cauchy, and convergent sequences. Furthermore, there are determined generalized τ-almost contractions, which extend the almost contractions. The highlight of this study is the investigation of the existence and uniqueness of a fixed point for some types of generalized τ-almost contractions in extended quasi-cone b-metric space. We prove some corollaries and theorems for known contractions in extended quasi-cone b-metric spaces. Our results generalize some known theorems given in literature due to the new cone metric spaces and contractions. Concrete examples illustrate theoretical outcomes. In addition, we show an application of the main results to Integral Equations, which provides the applicative side of them.

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Generalized almost Contractions on Extended Quasi-Cone B-Metric Spaces

Liftaj,

Sila,

Selko

2023

Wseas Transactions on Mathematics

View full text Add to dashboard Cite

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Fixed point equations for superlinear operators with strong upper or strong lower solutions and applications

Han

Zheng

2023

MATH

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<abstract><p>It is well known that sublinear operators and superlinear operators are two classes of important nonlinear operators in nonlinear analysis and dynamical systems. Since sublinear operators have only weak nonlinearity, this advantage makes it easy to deal with them. However, superlinear operators have strong nonlinearity, and there are only a few results about them. In this paper, the convergence of Picard iteration for the superlinear operator $ A $ is obtained based on the conditions that the fixed point equation $ Ax = x $ has a strong upper solution and a lower solution (or alternatively, an upper solution and a strong lower solution). Besides, the uniqueness of the fixed point of strongly increasing operators as well as the global attractivity of strongly monotone dynamical systems are also discussed. In addition, the main results are applied to monotone dynamics of superlinear operators and nonlinear integral equations. The method used in our work develops the traditional method of upper and lower solutions. Since a strong upper (upper) solution and a lower (strong lower) solution are easily checked, the obtained results are effective and practicable in the study of nonlinear equations and dynamical systems. The main novelty is that this paper provides new fixed point results for increasing superlinear operators and the obtained results are applied to strongly monotone systems to investigate their global attractivity.</p></abstract>

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Fixed Point and Homotopy Methods in Cone A-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation

Cited by 2 publications

References 37 publications

Generalized almost Contractions on Extended Quasi-Cone B-Metric Spaces

Generalized almost Contractions on Extended Quasi-Cone B-Metric Spaces

Fixed point equations for superlinear operators with strong upper or strong lower solutions and applications

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