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Abstract This paper continues the study of the Axiom of Infinite Choice on transversal ordered spring spaces in terms of fixed point and increasing inductive sets. These principles unify a number of diverse results (about three thousand papers) in fixed point theory, especially recently published. Applications in partially ordered spaces and fixed point theory are also considered.

New fixed point and coupled fixed point theorems in partially ordered ν-generalized metric spaces are presented. Since the product of two ν-generalized metric spaces is not in general a ν-generalized metric space, a different approach is needed than in the case of standard metric spaces. [ABSTRACT FROM AUTHOR]

New generalized forms of Banach and Kannan type contractions have been described and some coupled coincidence and coupled common fixed point results have been established in the setting of metric space involving a graph. Our theorems extend and generalize many existing results in the literature. Furthermore, an application to integral equations is presented to affirm the importance. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show the existence of best proximity points for multivalued Suzuki-Edelstein-Wardowski type α-proximal contractions in the setting of complete metric spaces and partially ordered metric spaces. We give examples to illustrate the main results. Our results improve and extend the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

It is known since the late 1960's that the dual of the category of compact Hausdorff spaces and continuous maps is a variety - not finitary, but bounded by N1. In this note we show that the dual of the category of partially ordered compact spaces and monotone continuous maps is an N1-ary quasivariety, and describe partially its algebraic theory. Based on this description, we extend these results to categories of Vietoris coalgebras and homomorphisms on ordered compact spaces. We also characterise the N1-copresentable partially ordered compact spaces. [ABSTRACT FROM AUTHOR]

The notion of coincidence point and common fixed point were extended in generalized partially ordered fuzzy metric spaces. Under some conditions, some coincidence point and common fixed point theorems were established in generalized partially fuzzy metric spaces using weakly compatible mappings. These results improve some theorems in corresponding literature. [ABSTRACT FROM AUTHOR]

In this paper, we use the concept of Ab-metric space which is obtained by generalizing the definitions of A-metric space and b-metrc space. Using this concept, we prove some coupled fixed point theorems in partially ordered Ab-metric space. Examples are also presented to verify the obtained results. [ABSTRACT FROM AUTHOR]

In the paper the concept of covering (regularity) for set-valued mappings in partially ordered spaces is introduced. The coincidence points problem for set-valued mappings in partially ordered spaces is considered. Sufficient conditions for the existence of coincidence points of isotone and orderly covering set-valued mappings are obtained. It is shown that the known theorems on coincidence points of covering and Lipschitz mappings in metric spaces can be deduced from the obtained results. [ABSTRACT FROM AUTHOR]

The aim of the paper is to introduce new Suzuki and convex type contractions and prove new best proximity results for these contractions in the setting of a metric space. As applications, we deduce similar results for such type of contractions in partially ordered metric spaces and derive new Suzuki type fixed point results. An illustrative example is provided here to highlight our findings. [ABSTRACT FROM AUTHOR]

Under some weaker conditions, some coincidence point and common fixed point theorems are established in partially ordered fuzzy metric spaces using weakly compatible mappings. By using the theorems, we obtain some coupled and multidimensional fixed point results, which are generalization and improvement of very recent theorems in the corresponding literature. In order to illustrate our main results, we give three examples. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for the existence and uniqueness of best proximity points for a new class of non-self mappings involving simulation functions in a metric space endowed with a partial order. Some interesting consequences including fixed point results via simulation functions are presented. [ABSTRACT FROM AUTHOR]

In this paper, we establish some fixed point theorems with T-contractions and w-distances in partially ordered metric spaces. The main tool used in our proof is a generalized altering distance function. Our results can be applied directly to study multidimensional fixed point which covers the concepts of coupled, tripled, quadruple fixed point etc. Moreover, a Fredholm integral equation and an initial-value problem for partial differential equation of parabolic type are also discussed. [ABSTRACT FROM AUTHOR]

In this paper we establish some best proximity point results using generalized weak contractions with discontinuous control functions. The theorems are established in metric spaces with a partial order. We view the main problem in the paper as a problem of finding an optimal approximate solution of a fixed point equation. We also discuss several corollaries and give an illustrative example. We apply our result to obtain some coupled best proximity point results. [ABSTRACT FROM AUTHOR]

The aim of this article is to present, improve and generalize some recent coincidence and coupled coincidence point results from several papers in the framework of partially ordered b-metric spaces. Two examples are also provided to support the superiority of the obtained results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish random coincidence point results for weakly increasing random operators in the setting of ordered metric spaces by using generalized altering distance functions. Our results present random versions and extensions of some well-known results in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of cyclic weakly (ψ, ...)-contractive mappings and cyclic weakly (C, ψ, φ)-contractive mappings, and prove some fixed point theorems for such two types of mappings in complete partially ordered Menger PM-spaces. Some new results are obtained, which extend and generalize some fixed point results in metric and probabilistic metric spaces. Some examples are given to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of coincidence points for nonlinear mappings of any number of arguments under a weak (ψ,φ) -contractive condition. Our results generalize, extend and unify several classical and very recent related results in the literature (see Aydi et al. (2011), Berinde (2010), Gnana-Bhaskar and Lakshmikantham (2006), Berzig and Samet (2012), Borcut and Berinde (2012), Choudhury et al. (2011), Karapınar and Luong (2012), Lakshmikantham and Ćirić (2009), Luong and Thuan (2011), Roldán (2012)). [ABSTRACT FROM AUTHOR]

The concept of covering (regularity) for mappings in partially ordered spaces is introduced. Sufficient conditions for the existence of coincidence points and minimal coincidence points of isotone and orderly covering mappings are obtained. These results generalize classical fixed point theorems for isotone mappings. Moreover, the known theorems on coincidence points of covering and Lipschitz mappings in metric spaces are deduced from the obtained results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish some coupled coincidence point theorems for mappings having a mixed g- monotone property in partially ordered metric spaces. Also, we present a result on the existence and uniqueness of coupled common fixed points. The results presented in the paper generalize and extend several well-known results in the literature. [ABSTRACT FROM AUTHOR]

The concept of ordered g-contraction is introduced, and some fixed and common fixed point theorems for g-nondecreasing ordered g-contraction mapping in partially ordered metric spaces are proved. We also show the uniqueness of the common fixed point in the case of an ordered g-contraction mapping. The theorems presented are generalizations of very recent fixed point theorems due to Golubović et al. (Fixed Point Theory Appl. 2012:20, 2012). [ABSTRACT FROM AUTHOR]

Very recently, Abkar and Gabeleh (2013) observed that some best proximity point results under the P-property can be obtained fromthe same results in fixed-point theory. In this paper, motivated by thismentioned work, we show that themost best proximity point results on a metric space endowed with a partial order (under the P-property) can be deduced from existing fixed-point theorems in the literature. We present various model examples to illustrate this point of view. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the new notion of g-monotone mapping and prove some fixed point theorems for multi-valued and single-valued g-increasing mappings in partially ordered metric spaces. The mappings considered in this paper are assumed to satisfy certain metric inequalities which are established by an altering distance function. The presented results extend and improve the main results of Choudhury and Metiya [B.S. Choudhury, N. Metiya, Multi-valued and single-valued fixed point results in partially ordered metric spaces, Arab J. Math. Sci. 17 (2011) 135–151]. [Copyright &y& Elsevier]

Abstract: In this paper, we prove some coupled fixed point theorems for nonlinear contractive mappings having the mixed monotone property in partially ordered -metric spaces. The results on fixed point theorems are generalizations of the recent results of Choudhury and Maity [B.S. Choudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Modelling 54 (2011) 73–79]. [Copyright &y& Elsevier]

Two concepts--one of the coupled fixed point and the other of the generalized metric space-- play a very active role in recent research on the fixed point theory. The definition of coupled fixed point was introduced by Bhaskar and Lakshmikantham (2006) while the generalized metric space was introduced by Mustafa and Sims (2006). In this work, we determine some coupled fixed point theorems for mixedmonotone mapping satisfying nonlinear contraction in the framework of generalized metric space endowed with partial order. We also prove the uniqueness of the coupled fixed point for such mappings in this setup. [ABSTRACT FROM AUTHOR]

We generalize a fixed point theorem in partially ordered complete metric spaces in the study of A. Amini-Harandi and H. Emami (2010).We also give an application on the existence and uniqueness of the positive solution of a multipoint boundary value problem with fractional derivatives. [ABSTRACT FROM AUTHOR]

We investigate the existence and uniqueness of positive solutions for the following singular fractional three-point boundary value problem D0+αu(t) + f(t,u(t)) = 0,0 < t < 1, u(0) = u'(0) = u'(0) = 0,u''(1) = βu''(η), where 3 < α ≤ 4, D0+α is the standard Riemann-Liouville derivative and f : (0,1] x [0, ∞) → [0, ∞) with lim t→0+ f(t, .) = ∞ (i.e., f is singular at t = 0). Our analysis relies on a fixed point theorem in partially ordered metric spaces. [ABSTRACT FROM AUTHOR]

We prove a coupled coincidence point theorem for mappings F : XxX → X and g : X → X, where F has the mixed g-monotone property, in partially ordered metric spaces via implicit relations. Our result extends and improves several results in the literature. Examples are also given to illustrate our work. [ABSTRACT FROM AUTHOR]

We prove coupled coincidence point and coupled fixed point results of F : XxX → X and g : X → X involving Meir-Keeler type contractions on the class of partially ordered metric spaces. Our results generalize some recent results in the literature. Also, we give some illustrative examples and application. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present some coincidence and common fixed point results for generalized weakly G-contractive mappings in the setup of partially ordered G-metric space. We also provide an example to illustrate the results presented herein. As an application of our results, periodic points of weakly G-contractive mappings are obtained. [ABSTRACT FROM AUTHOR]

We prove quadruple fixed point theorems in partially orderedmetric spaces depending on another function. Also, we state some examples showing that our results are real generalization of known ones in quadruple fixed point theory. [ABSTRACT FROM AUTHOR]

We prove a number of quadruple fixed point theorems under f-contractive conditions for a mapping F : X4 → X in ordered metric spaces. Also, we introduce an example to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we have generalized the weak contraction principle to coincidence point and common fixed point results in partially ordered metric spaces. Our results extend some existing results. Two examples illustrating our results are given. [Copyright &y& Elsevier]

Abstract: Abelʼs partial summation formula has been used classically to obtain convergence tests for certain types of series of real or complex numbers. We generalize the formula and convergence tests to the setting of directed partially ordered topological groups, where the notion of a convergent series is replaced by that of a Cauchy multipliable sequence. [Copyright &y& Elsevier]

Abstract: We prove coupled coincidence and coupled common fixed point theorems for a mixed-monotone mapping satisfying nonlinear contractions in partially ordered -metric spaces. Presented theorems are generalizations of the very recent results of Choudhury and Maity [B.S. Choudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Modelling 54 (1–2) (2011) 73–79]. [Copyright &y& Elsevier]

Abstract: Recently, Heman Kumar Nashine and Bessem Samet [H.K. Nashine, B. Samet, Fixed point results for mappings satisfying -weakly contractive condition in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 2201–2209] studied some coincidence fixed point and common fixed point theorems for two mappings satisfying -weakly contractive condition in an ordered complete metric space. In the present paper, we study some coincidence fixed point and common fixed point theorems for three mappings and satisfying -weakly contractive condition in an ordered complete metric space, where the mappings and are assumed to be weakly increasing with respect to . Our results generalize several well-known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, common fixed points of four mappings satisfying a generalized weak contractive condition in the framework of partially ordered metric space are obtained. We also provide examples of new concepts introduced herein. [Copyright &y& Elsevier]

Abstract: Through a simple extension of Brézis–Browder principle to partially ordered spaces, a very general strong minimal point existence theorem on quasi ordered spaces, is proved. This theorem together with a generic quasi order and a new notion of strong approximate solution allow us to obtain two strong solution existence theorems, and three general Ekeland variational principles in optimization problems where the objective space is quasi ordered. Then, they are applied to prove strong minimal point existence results, generalizations of Bishop–Phelps lemma in linear spaces, and Ekeland variational principles in set-valued optimization problems through a set solution criterion. [Copyright &y& Elsevier]

Abstract: Recent results of Đorić [D. Đorić, Common fixed point for generalized -weak contractions, Appl. Math. Lett. 22 (2009) 1896–1900] on generalized weakly contractive mappings are extended to the setting of partially ordered metric spaces. Thus, generalization of fixed point results of Harjani and Sadarangani [J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71 (2009) 3403–3410] is obtained. Some applications are presented. Examples are given to show that our results are proper generalizations of the existing ones. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the concept of a -distance on a complete partially ordered -metric space and prove some fixed point theorems. [Copyright &y& Elsevier]

Abstract: An abstract monotone iterative method is developed for operators between partially ordered Banach spaces for the nonlinear problem and the nonlinear time dependent problem . Under appropriate assumptions on and we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems. The results are illustrated by means of concrete examples. [Copyright &y& Elsevier]

Abstract: In this paper, a concept of monotone generalized contraction in partially ordered probabilistic metric spaces is introduced and some fixed and common fixed point theorems are proved. Presented theorems extend the results in partially ordered metric spaces of Nieto and Rodriguez-Lopez [Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239; Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205–2212], Ran and Reurings [A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435–1443] to a more general class of contractive type mappings in partially ordered probabilistic metric spaces and include several recent developments. [Copyright &y& Elsevier]

Abstract: It is well known that a sum (coproduct) of a family of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces , indexed by the ultrafilters u on the index set I. The nature of those subspaces indexed by the free ultrafilters u is not yet fully understood. In this article we study a certain dense subset satisfying exactly those sentences in the first-order theory of partial orders which are satisfied by almost all of the ''s. As an application we present a complete analysis of the coproduct of an increasing family of finite chains, in a sense the first non-trivial case which is not a Čech–Stone compactification of the disjoint union . In this case, all the ''s with u free turn out to be isomorphic under the Continuum Hypothesis. [Copyright &y& Elsevier]

Abstract: Let be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space , Oxtoby''s pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”. [Copyright &y& Elsevier]

We prove fixed point theorems for mixed-monotone mappings in partially ordered complete metric spaces which satisfy a weaker contraction condition than the classical Banach contraction condition for all points that are related by given ordering. We also give a global attractivity result for all solutions of the difference equation zn+1 = F(zn, zn-1), n = 2,3, ..., where F satisfies mixed-monotone conditions with respect to the given ordering. [ABSTRACT FROM AUTHOR]