Common Coupled Fixed Point Theorems of Single-Valued Mapping for c-Distance in Cone Metric Spaces

The existence and uniqueness of the common coupled fixed point in cone metric spaces have been studied by considering different types of contractive conditions. A new concept of the cdistance in cone metric space has been recently introduced in 2011. Then, coupled fixed point results for contraction-type mappings in ordered cone metric spaces and cone metric spaces have been considered. In this paper, some common coupled fixed point results on c-distance in cone metric spaces are obtained. Some supporting examples are given

Y -cone metric spaces and coupled common fixed point results with application to integral equation

This paper acquaints with a concept of Y -cone metric space and to study some topological properties of Y -cone metric space. We prove the coupled common fixed point results for mixed weakly monotone map in ordered Y -cone metric spaces. We give an example, which constitutes the main theorem.Publisher's Versio

On Some Common Fixed Point Theorems for Certain Contractive Mappings in Cone Metric Spaces

The aim of this paper is to establish the generalization of T-Reich and T-Rhoades type mappings on complete cone metric spaces. Huang and Zhang introduce the notion of cone metric spaces. He replaced real number system by ordered Banach space and gave the condition in the setting of cone metric spaces. These authors also described the convergence of sequences in the cone metric spaces and introduce the corresponding notion of completeness. The study of fixed point theorems in such spaces is followed by many researchers. In the present paper this study has been extended to analyse the existence and uniqueness of common fixed points of T-Reich contractive mappings defined on a complete cone metric space ( , ) as well as T-Rhoades contractive mappings. These results generalize and extend the existing fixed point theorems available in the literature

Common fixed-point theorems and c-distance in ordered cone metric spaces

We present a generalization of several fixed and common fixed point theorems on c -distance in ordered cone metric spaces. In this way, we improve and generalize various results existing in the literature.Наведено узагальнення деяких теорем про нерухому точку та спільну нерухому точку для с-відстані в упорядкованих конічних метричних просторах. Таким чином, покращено та узагальнено різноманітні результати, що наведені в літературі

Approximation of common fixed points in 2-Banach spaces with applications

[EN] The purpose of this paper is to establish the existence and uniqueness of common fixed points of a family of self-mappings satisfying generalized rational contractive condition in 2-Banach spaces. An example is included to justify our results. We approximate the common fixed point by Mann and Picard type iteration schemes. Further, an application to well-posedness of the common fixed point problem is given. The presented results generalize many known results on 2-Banach spaces.The authors thank the reviewers for valuable comments. The first author D. Ramesh Kumar would like to thank the University Grants Commission, New Delhi, India for providing the financial support in preparation of this manuscript.Kumar, DR.; Pitchaimani, M. (2019). Approximation of common fixed points in 2-Banach spaces with applications. Applied General Topology. 20(1):43-55. https://doi.org/10.4995/agt.2019.9168SWORD4355201M. Abbas, B. E. Rhoades and T. Nazir, Common fixed points for four maps in cone metric spaces, Applied Mathematics and Computation 216 (2010), 80-86. https://doi.org/10.1016/j.amc.2010.01.003M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), 416-420. https://doi.org/10.1016/j.jmaa.2007.09.070M. Arshad, E. Karapinar and J. Ahmad, Some unique fixed point theorems for rational contractions in partially ordered metric spaces, J. Inequal. Appl. 2013(1) (2013), 1-16. https://doi.org/10.1186/1029-242x-2013-248A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numerical Functional Analysis and optimization 32, no. 3 (2011), 243-253. https://doi.org/10.1080/01630563.2011.533046S. Banach, Sur les opérations dans les ensembles abstraits et leur application auxéquations intégrales, Fund. Math. 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181I. Beg and A. R. Butt, Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal. 71, no.9 (2009), 3699-3704. https://doi.org/10.1016/j.na.2009.02.027K. Cieplinski, Approximate multi-additive mappings in 2-Banach spaces, Bull. Iranian Math. Soc. 41, no. 3 (2015), 785-792.B. K. Dass and S. Gupta, An extension of Banach's contraction principle through rational expression, Indian J. Pure appl. Math. 6, no. 4 (1975), 1445-1458.F. S. De Blasi and J. Myjak, Sur la porosité de l'ensemble des contractions sans point fixe, C. R. Acad. Sci. Paris 308 (1989), 51-54.S. Gähler, 2-metrische Räume and ihre topologische strucktur, Math. Nachr. 26 (1963), 115-148. https://doi.org/10.1002/mana.19630260109S. Gähler, Uber die unifromisieberkeit 2-metrischer Räume, Math. Nachr. 28 (1965), 235-244. https://doi.org/10.1002/mana.19640280309S. Gähler, Über 2-Banach-Räume, Math. Nachr. 42 (1969), 335-347. https://doi.org/10.1002/mana.19690420414L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332, no. 2 (2007), 1468-1476. https://doi.org/10.1016/j.jmaa.2005.03.087W. A. Kirk and N. Shahzad, Some fixed point results in ultrametric spaces, Topology and its Applications 159 (2012), 3327-3334. https://doi.org/10.1016/j.topol.2012.07.016K. Iseki, Fixed point theorems in 2-metric space, Math.Seminar. Notes, Kobe Univ. 3(1975), 133-136.E. Matouskova, S. Reich and A. J. Zaslavski, Genericity in nonexpansive mapping theory, Advanced Courses of Mathematical Analysis I, World Scientific Hackensack (2004), 81-98. https://doi.org/10.1142/9789812702371_0004S. B. Nadler, Sequence of contraction and fixed points, Pacific J.Math. 27 (1968), 579-585.H. K. Nashinea, M. Imdadb and M. Hasan, Common fixed point theorems under rational contractions in complex valued metric spaces, J. Nonlinear Sci. Appl. 7 (2014), 42-50. https://doi.org/10.22436/jnsa.007.01.05B. G. Pachpatte, Common fixed point theorems for mappings satisfying rational inequalities, Indian J. Pure appl. Math. 10, no. 11 (1979), 1362-1368.A.-D. Filip and A. Petrusel, Fixed point theorems for operators in generalized Kasahara spaces, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 109, no. 1 (2015), 15-26. https://doi.org/10.1007/s13398-014-0163-9M. Pitchaimani and D. Ramesh Kumar, Some common fixed point theorems using implicit relation in 2-Banach spaces, Surv. Math. Appl. 10 (2015), 159-168.M. Pitchaimani and D. Ramesh Kumar, Common and coincidence fixed point theorems for asymptotically regular mappings in 2-Banach Spaces, Nonlinear Funct. Anal. Appl.21, no. 1 (2016), 131-144.M. Pitchaimani and D. Ramesh Kumar, On construction of fixed point theory under implicit relation in Hilbert spaces, Nonlinear Funct. Anal. Appl. 21, no. 3 (2016), 513-522.M. Pitchaimani and D. Ramesh Kumar, On Nadler type results in ultrametric spaces with application to well-posedness, Asian-European Journal of Mathematics 10, no. 4(2017), 1750073(1-15). https://doi.org/10.1142/s1793557117500735M. Pitchaimani and D. Ramesh Kumar, Generalized Nadler type results in ultrametric spaces with application to well-posedness, Afr. Mat. 28 (2017), 957-970. https://doi.org/10.1007/s13370-017-0496-6V. Popa, Well-Posedness of fixed problem in compact metric space, Bull. Univ. Petrol-Gaze, Ploicsti, sec. Mat Inform. Fiz. 60, no. 1 (2008), 1-4.D. Ramesh Kumar and M. Pitchaimani, Set-valued contraction mappings of Presic-Reichtype in ultrametric spaces, Asian-European Journal of Mathematics 10, no. 4 (2017), 1750065 (1-15). https://doi.org/10.1142/s1793557117500656D. Ramesh Kumar and M. Pitchaimani, A generalization ofset-valued Presic-Reich type contractions in ultrametric spaces with applications, J. 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A unified theory of cone metric spaces and its applications to the fixed point theory

In this paper we develop a unified theory for cone metric spaces over a solid vector space. As an application of the new theory we present full statements of the iterated contraction principle and the Banach contraction principle in cone metric spaces over a solid vector space.Comment: 51 page

TVS-cone metric spaces as a special case of metric spaces

There have been a number of generalizations of fixed point results to the so called TVS-cone metric spaces, based on a distance function that takes values in some cone with nonempty interior (solid cone) in some topological vector space. In this paper we prove that the TVS-cone metric space can be equipped with a family of mutually equivalent (usual) metrics such that the convergence (resp. property of being Cauchy sequence, contractivity condition) in TVS sense is equivalent to convergence (resp. property of being Cauchy sequence, contractivity condition) in all of these metrics. As a consequence, we prove that if a topological vector space $E$ and a solid cone $P$ are given, then the category of TVS-cone metric spaces is a proper subcategory of metric spaces with a family of mutually equivalent metrics (Corollary 3.9). Hence, generalization of a result from metric spaces to TVS-cone metric spaces is meaningless. This, also, leads to a formal deriving of fixed point results from metric spaces to TVS-cone metric spaces and makes some earlier results vague. We also give a new common fixed point result in (usual) metric spaces context, and show that it can be reformulated to TVS-cone metric spaces context very easy, despite of the fact that formal (syntactic) generalization is impossible. Apart of main results, we prove that the existence of a solid cone ensures that the initial topology is Hausdorff, as well as it admits a plenty of convex open sets. In fact such topology is stronger then some norm topology.Comment: 14 page