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The neutrosophic theory has been effectively used to address uncertainty and ambiguity. Neutrosophic Metric Space (NMS) was introduced by Krisci and Simsek in 2020. Following that, several kinds of compatible maps and their characteristics were investigated in the context of Intuitionistic fuzzy metric spaces and fuzzy metric spaces. In this paper, the author introduce the notion of compatible maps of type (α) and type (β) in neutrosophic metric space. For this purpose, four non-comparable mappings are used to prove the basic results. Furthermore, we prove several common fixed points results for compatible maps of type (α) and type (β) in neutrosophic metric space and provide a non-trivial examples.

The foremost target of the proposed work focuses on establishing fixed point theorems for the new concept named Intuitionistic fuzzy cone-b metric space (in short IFCbMS). Various definitions like compatible mappings, weakly-compatible maps (WC maps) and E.A. property are also used to prove new results for IFCbMS. The proofs of fixed points for FMS, IFMS, NFMS in the setting of E.A. property are taken as base for this research. Examples mentioned in the paper are increasing the importance of the research done. An application is also given to show the importance of the result proved. The application of this paper leads to the solution of an integral equation in context of implicit functions. [ABSTRACT FROM AUTHOR]

The neutrosophic theory has been effectively used to address uncertainty and ambiguity. Neutrosophic Metric Space (NMS) was introduced by Krisci and Simsek in 2020. Following that, several kinds of compatible maps and their characteristics were investigated in the context of Intuitionistic fuzzy metric spaces and fuzzy metric spaces. In this paper, the author introduce the notion of compatible maps of type (α) and type (β) in neutrosophic metric space. For this purpose, four non-comparable mappings are used to prove the basic results. Furthermore, we prove several common fixed points results for compatible maps of type (α) and type (β) in neutrosophic metric space and provide a non-trivial examples. [ABSTRACT FROM AUTHOR]

Fixed point theory is a crucial branch in mathematics with a colossal number of applications in countless subjects. It furnishes preeminent tools and resources for elucidating varied problems which at first glance do not look like a fixed point problem. Since and even before 1912 till now several authors launched the existence and uniqueness of common fixed points for pairs of single and set-valued maps satisfying certain compatibilities on different spaces. This paper proves existence and uniqueness of a common fixed point for pairs of occasionally weakly biased maps. This unique common fixed point is guaranteed under the concept of modified contractive modulus function. Our theorems improve some results existing in the fixed point literature. [ABSTRACT FROM AUTHOR]

A unique common fixed point is obtained for compatible and non-compatible self-maps on a b-metric space, through the notion of conditional reciprocal continuity, due to Pant and Bist. [ABSTRACT FROM AUTHOR]

In mathematical analysis, the Hausdorff derivatives or the fractal derivatives play an important role. Fixed-point theorems and metric fixed-point theory have varied applications in establishing a unique common solution to differential equations and integral equations. In the present work, some fixed-point theorems using the extension of Meir–Keeler contraction in the setting of bipolar metric spaces have been proved. The derived results have been supplemented with non-trivial examples. Our results extend and generalise the results established in the past. We have provided an application to find an analytical solution to an Integral Equation to supplement the derived result.

In this paper, we investigate a common fixed point problem for two pairs {f, S} and {g, T} of weakly compatible selfmaps of a complete b-metric (X, d; s), satisfying a contractive condition of Ćirić type. This contraction and some of its variants were used in the paper [29] published in 2016 by V. Ozturk and S. Radenovic, requiring the (E.A.)-property for the pairs {f, S} and {g, T}. The aim of this paper is to provide some improvements to the main result of [29]. Our main theorem will improve certain results published in 2015, by V. Ozturk and D. Turkoglu (see [30] and [31]). We improve also results from other related papers (see the references herin). Indeed, we remove the (E.A.)property and weaken certain assumptions imposed in these papers. So, our work aims to extend and unify, in one go, several common fixed point results known in a recent literature. We furnish two illustrative examples and we prove that the fixed point problem, considered here, for the pairs {f, S} and {g, T} is well-posed. We compare our main result with a recent result obtained in 2018 by N. Hussain, Z. D. Mitrović and S. Radenović in [19].

We establish a general common fixed point problem for two pairs {f, S} and {g, T} of weakly compatibles selfmaps of a complete b-metric (X, d; sj. These maps are satisfying a contractive condition defined by a class of implicit relations in five variables. This contraction unifies, in onego, several contractive conditions previously used in a set of recent papers dealing with fixed point or common fixed results for selfmaps of b-metric spaces. We provide an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we define the concept of (owc)-property for two single-valued mappings and two multi-valued mappings in Probabilistic metric spaces and give some new common fixed point results for these mappings. Also, we give some examples to illustrate the main results in this paper. [ABSTRACT FROM AUTHOR]

In the present paper, we pointed out that there is a gap in the proof of the main result of Rouzkard et al. (The Bulletin of the Belgian Mathematical Society 2012). Then after, utilizing the concept of (E.A.) property in convex metric space, we obtained an alternative and correct version of this result. Finally, it is clarified that in the theory of common fixed point, the notion of (E.A.) property in the set up of convex metric space develops some new dimensions in comparison to the hypothesis that a range set of one map is contained in the range set of another map.

In the present paper, we extend the concept of Weak Reciprocal Continuity for a hybrid pair of single-valued and multi-valued maps and introduce (T, f)-completeness of the space. Further, we establish some results on the existence of coincidence and fixed points for the hybrid pair of maps. Our results generalize several well known results available in the literature. [ABSTRACT FROM AUTHOR]

In the present paper, we extend the concept ofWeak Reciprocal Continuity for a hybrid pair of single-valued and multi-valued maps and introduce (T; f)-completeness of the space. Further, we establish some results on the existence of coincidence and fixed points for the hybrid pair of maps. Our results generalize several well known results available in the literature

In this paper, we prove coincidence and fixed point theorems for two hybrid pairs of single valued and multivalued mappings on noncomplete metric spaces. We prove our results without using conditions of compatibility and continuity of mappings. We also give an example to validate our result.

The main purpose of this paper is to prove a common fixed point theorem for two pairs of weakly compatible mappings in Menger space using implicit relation.

In the framework of a b-metric space, by using the compatible and weak compatible conditions of self-mapping pair, we discussed the existence and uniqueness of the common fixed point for a class of ϕ-type contraction mapping, some new common fixed point theorems are obtained. In the end of the paper, we give some illustrative examples in support of our new results. The results presented in this paper extend and improve some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a unique common fixed point theorem for two pairs of occasionally weakly compatible single and multi-valued maps in a metric space. This result improves the result of Türkoglu et al. (Math. Commun. 2002) and references therein.

Here, we extend the notion of (E.A.) property in a convex metric space defined by Kumar and Rathee (Fixed Point Theory Appl 1-14, 2014) by introducing a new class of self-maps which satisfies the common property (E.A.) in the context of convex metric space and ensure the existence of common fixed point for this newly introduced class of self-maps. Also, we guarantee the existence of common best proximity points for this class of maps satisfying generalized non-expansive type condition. We furnish an example in support of the proved results. [ABSTRACT FROM AUTHOR]

We prove that a continuous triangular map G of the n-dimensional cube In has only fixed points and no other periodic points if and only if G has a common fixed point with every continuous triangular map F that is nontrivially compatible with G. This is an analog of Jungck theorem for maps of a real compact interval. We also discuss possible extensions of Jungck theorem, Jachymski theorem and some related results to more general spaces. In particular, the spaces with the fixed point property and the complete invariance property are considered.

We demonstrate a usefulness of the notion of a connected graph for obtaining some common fixed point theorems. In particular, we establish two theorems of this type involving one, two and four sequences of maps. This generalizes among others the recent results of S. Chang [1], J. Jachymski [2], S. Sessa, R. N. Mukherjee and T. Som [3], and T. Taniguchi [4].

Following Gerald Jungck's formulation of compatibility of maps, several interesting fixed point theorems for such maps have been obtained, wherein at least one of the maps is continuous. The main purpose of this note is to avoid the continuity condition from some of the main results of Jungck et al.

The purpose of the present paper is to prove some new common fixed point theorems in intuitionistic fuzzy metric spaces for a pair of self-mappings satisfying minimal type contractive and continuity conditions. Our results substantially generalize and improve a multitude of relevant common fixed point theorems of the existing literature in metric as well as intuitionistic fuzzy metric spaces. We also provide an affirmative answer in the settings of intuitionistic fuzzy metric space to the open problem posed by Rhoades [28] regarding existence of a contractive definition which is strong enough to generate a fixed point, but which does not force the map to be continuous at the fixed point. [ABSTRACT FROM AUTHOR]

In the recent articles, Khan and Sumitra (Appl. Math. Sci., 5(29)(2011), 1421-1430) and Singh et al. (Int. J. Math. Anal., 5(27) (2011), 1301-1308) claim a fuzzy version of common fixed point theorem of Bouhadjera and Godet-Thobie [3]. The results of aforementioned articles contain flaws and are not correct in their present form. We provide some examples to demonstrate that this claim is false unless some additional conditions are imposed. Our note is desired to complete the interesting results in the quoted paper. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, using the setting of generalized metric spaces, existence of unique common fixed point of two pairs of power contraction mappings satisfying (E.A)-property in generalized metric spaces is established. We also provide example to support the results presented herein. [Copyright &y& Elsevier]

The main purpose of this paper is to prove a common fixed point theorem for two pairs of weakly compatible mappings in Menger space using implicit relation. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to generalize the notion of reciprocal continuity and obtain common fixed point theorems in diverse settings as an application of the new notion. The new notion unifies the approaches of three well known notions- reciprocal continuity, subsequential continuity and conditional commutativity. Our results generalize and extend several fixed point theorems. We also demonstrate that the new notion is a necessary condition for the existence of common fixed points. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, using the setting of generalized metric spaces, existence of unique common fixed point is established for mappings satisfying (E.A)-property under generalized strict contractive condition, as well as under some generalized contractions. Obtained results are proper extensions of the existing ones in metric space. We also provide example to support concepts and results presented herein. [Copyright &y& Elsevier]

The aim of this note is to point out a fallacy in the proof of Theorem 3.1 contained in the recent paper (Int. J. Contemp. Math. Sci. 5 (2010), 2699-2707) proved in intuitionistic fuzzy metric spaces employing the newly introduced notion of sub-compatible pair of mappings wherein our claim is also substantiated with the aid of an appropriate example. We also rectify the erratic theorem in two ways. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove common fixed point theorems for four mappings under the condition of weak compatible mappings in Quasi-gauge space. We point out that the continuity of any mapping for the existence of the fixed point is not required. [ABSTRACT FROM AUTHOR]

In this paper we prove a fixed point theorem for six mappings in Menger space using the notion of weakly compatible. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain coincidence and fixed point theorems for hybrid contractions consisting of single-valued and multivalued mappings in symmetric spaces. Some special cases and applications are discussed. [ABSTRACT FROM AUTHOR]

Abstract: In an interesting article, Bouhadjera and Godet-Thobie (2009)  introduced notions of subcompatibility and subsequential continuity, and utilized them to prove several common fixed point theorems. The results of the aforementioned article contain flaws, and they are not correct in their present form. But these results can be recovered in two ways by strengthening either of the newly introduced definitions. The results, in their corrected form, still bring about noted improvements over a multitude of relevant fixed point theorems of the existing literature, substantiating the utility of these (two) new notions. [Copyright &y& Elsevier]

The aim of the present paper is to introduce the notion of weak reciprocal continuity and obtain fixed point theorems by employing the new notion. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. Our results generalize several fixed point theorems. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to obtain a common fixed point theorem using a new type of contractive condition which is independent of the known contractive notions. In the following pages we prove a common fixed point theorem using an (ε,δ) contractive condition, without either imposing any additional condition on δ or assuming a π - contractive condition together with. Instead of a π - contractive condition we assume the Lipschitz type analogue of a plane contractive condition which is independent of the π -contractive condition. Our theorem improves and generalizes a number of Meir-Keeler type theorems including the results of Pant [ 17 , ( Theorem 1 ) ], Pant and Pant and Pant etal. [ABSTRACT FROM AUTHOR]

Some new common fixed point theorems for Gregus type contraction mappings have been obtained in convex metric spaces. As applications, invariant approximation results for these types of mappings are obtained. The proved results generalize, unify and extend some of the known results of M.A. Al-Thagafi (Int. J. Math. Sci. 18:613-616, ; J. Approx. Theory 85:318-323, ), M.A. Al-Thagafi and N. Shahzad (Nonlinear Anal. 64:2778-2786, ), L. Ćirić (Publ. Inst. Math. 49:174-178, ; Arch. Math. (BRNO) 29:145-152, ), M.L. Diviccaro, B. Fisher, S. Sessa (Publ. Math. (Debr.) 34:83-89, ), B. Fisher and S. Sessa (Int. J. Math. Math. 9:23-28, ), M. Gregus (Boll. Un. Mat. Ital. (5) 7-A:193-198, ), L. Habiniak (J. Approx. Theory 56:241-244, ), N. Hussain, B.E. Rhoades and G. Jungck (Numer. Func. Anal. Optim. 28:1139-1151, ), G. Jungck (Int. J. Math. Math. Sci. 13:497-500, ), G. Jungck and S. Sessa (Math. Jpn. 42:249-252, ), R.N. Mukherjee and V. Verma (Math. Jpn. 33:745-749, ), T.D. Narang and S. Chandok (Ukr. Math. J. 62:1367-1376, ), S.A. Sahab, M.S. Khan and S. Sessa (J. Approx. Theory 55:349-351, ), N. Shahzad (J. Math. Anal. Appl. 257:39-45, ; Rad. Math. 10:77-83, ; Int. J. Math. Game Theory Algebra 13:157-159, ), S.P. Singh (J. Approx. Theory 25:89-90, ), A. Smoluk (Mat. Stosow. 17:17-22, ), P.V. Subrahmanyam (J. Approx. Theory 20:165-172, ) and of few others. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we extend and generalize common fixed point theorems for six self-maps of Singh and Jain [B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439–448] from Menger and metric spaces to cone metric spaces. We also extend the notions of compatible and weakly compatible mappings from the setting of Menger and metric spaces to the setting of cone metric spaces. We do not impose the normality property on the cone, but suppose only that the cone , in the ordered Banach space , has a nonempty interior. Examples are given to illustrate the results. [Copyright &y& Elsevier]

Using the theory of countable extension of t-norm we prove a common fixed point theorem for compatible mappings satisfying an implicit relation in fuzzy metric spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove some new common fixed point theorems for compatible and weakly compatible self-maps on Menger space under -contractive conditions. Our results improve and generalize the results of Singh and Jain [B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439–448], Razani and Shirdaryazdi [A. Razani, M. Shirdaryazdi, A common fixed point theorem of compatible maps in Menger space, Chaos Solitons Fractals 32 (2007) 26–34] and others. [Copyright &y& Elsevier]

Using Dotsonś convexity structure, the authors in [16, 17, 18] established some deterministic and random common fixed point results. In this note, we comment that the proofs of the results in [16, 17, 18] are incomplete and incorrect. [ABSTRACT FROM AUTHOR]

The existence of common fixed point results for a Banach operator pair under certain generalized φ-contractions is established. As applications, the corresponding invariant best approximation results are proved. Our results unify and generalize various known results to the more general class of non-commuting mappings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain common fixed point theorems for asymptotically commuting mappings in intuitionistic fuzzy metric spaces. This gives a generalization of the results by Mishra et al. [7]. Our results extend, generalize and fuzzify several fixed point theorems on metric spaces, Menger spaces and uniform spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the concept of compatibility for multivalued mappings and obtain a common coincidence point theorem in intuitionistic fuzzy metric spaces. Our results extend and generalize and fuzzify several fixed point theorems on metric spaces, Menger probabilistic metric spaces, uniform spaces and fuzzy metric spaces. [ABSTRACT FROM AUTHOR]

Abstract: The existence of common fixed point results for a Banach operator pair under certain generalized contractions is established. The invariant best approximation results are proved as applications and the existence of solutions of variational inequalities is obtained. We also study the solution of functional equations arising from dynamic programming. [Copyright &y& Elsevier]

In the present work, we introduce the concepts of compatible maps of type (P-1) and type (P-2), and prove common fixed point theorems for such maps without appeal to continuity in Menger spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the existence and approximation of a unique common fixed point of two families of weakly compatible self-maps on a complete metric space are investigated. An example is presented to show that our results for the mappings considered satisfying non-linear contractive type conditions are genuine generalizations of the recent result for metric spaces [B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439–448, Theorem 3.3] and many other known results. [Copyright &y& Elsevier]

The aim of this paper is to establish a unique common fixed point theorem for two pairs of occasionally weakly compatible single and multi-valued maps in a metric space. This result improves the result of Türkoğlu et al. [6] and references therein. [ABSTRACT FROM AUTHOR]

Abstract: The common fixed point theorems, similar to those of Ćirić [Lj.B. Ćirić, On a common fixed point theorem of a Gregus type, Publ. Inst. Math. (Beograd) (N.S.) 49 (1991) 174–178; Lj.B. Ćirić, On Diviccaro, Fisher and Sessa open questions, Arch. Math. (Brno) 29 (1993) 145–152; Lj.B. Ćirić, On a generalization of Gregus fixed point theorem, Czechoslovak Math. J. 50 (2000) 449–458], Fisher and Sessa [B. Fisher, S. Sessa, On a fixed point theorem of Gregus, Internat. J. Math. Math. Sci. 9 (1986) 23–28], Jungck [G. Jungck, On a fixed point theorem of Fisher and Sessa, Internat. J. Math. Math. Sci. 13 (1990) 497–500] and Mukherjee and Verma [R.N. Mukherjee, V. Verma, A note on fixed point theorem of Gregus, Math. Japon. 33 (1988) 745–749], are proved for a Banach operator pair. As applications, common fixed point and approximation results for Banach operator pair satisfying Ćirić type contractive conditions are obtained without the assumption of linearity or affinity of either T or I. Our results unify and generalize various known results to a more general class of noncommuting mappings. [Copyright &y& Elsevier]

The main purpose of this paper is to give some new common fixed point theorems in Menger space under strict contractive conditions for mappings satisfying the new property. [ABSTRACT FROM AUTHOR]