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In this paper, we introduce an algorithm for solving the split common fixed point problem for asymptotically quasi-nonexpansive mapping in the intermediate sense in Hilbert spaces. The strong and weak convergence of the presented algorithm to split common fixed point are obtained. The results presented in this paper improve and extend some recent corresponding results announced by some authors. Mathematics Subject Classifications: 47H09, 47J25

In the recent developments of fixed point theorems is proving the existence of fixed points on partially ordered metric spaces [20, 23], a generalization of the Banach contraction principle for an integral-type inequality [2, 10, 21], and even some applications to matrix equations and ordinary differential equations. In this paper, we want to illustrate some new fixed point theorems for generalized distances with τ -function in the partially ordered metric space rather than w-distance [23]. In short, we would like to utilize τ -function to provide the existence of fixed points on partially ordered metric spaces which is our goal.

We present basic structures, definitions, normal forms, and a hierarchy of languages based on catenation, shuffle and their iterations, defined by algebraic closures or least fixed point solutions to systems of equations.

Over the past few decades, there have been a lot of activity about combining fixed point theory andanotherbranchesinmathematicssuchdifferentialequations, geometryandalgebraictopology. In2005, Echenique started combining fixed point theory and graph theory by giving a short constructive proof for the Tarski fixed point theorem by using graphs. In 2006, Espinola and Kirk started combining fixed point theory and graph theory. Recently, this field have been of great interest for fixed point theorists. In this paper, we give some fixed point results for generalized quasi-contractive multifunctions on graphs.

In this paper, we introduce a class Ψ of real functions defined on the set of non-negative real numbers, and obtain a new unique common fixed point theorem for four mappings satisfying Ψ-contractive condition on a non-complete 2-metric space and give the versions of the corresponding result for two and three mappings.

A distributed algorithm is described for finding a common fixed point of a family of m > 1 nonlinear maps M i : Rn → Rn assuming that each map is a paracontraction. The common fixed point is asynchronously computed in real time by m agents assuming each agent i knows only M i , the current estimates of the fixed point generated by its neighbors, and nothing more. Each agent recursively updates its estimate of the fixed point at its own event times, by utilizing estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph N(t) whose vertices correspond to agents and whose arcs depict neighbor relations.

In this article, we define the new concept of a fixed set for a set valued map with set valued domain in the setting of metric space endowed with a directed graph. This notion of fixed set is analogous to the notion of a fixed point for a multivalued map and not for a classical single-valued map. We also introduce the new concept of the start set of a graph whose vertices are closed and bounded subsets of a metric space. Characterizations for such a graph to have a start set are given. Further, the notion of a self-path set valued map is defined and its relation with the start set is established. Finally, the existence of fixed sets is established in this context.

In this paper, we investigate the fixed points of multi-valued contractions constructed by manageable functions in order to extend the theory of multi-valued contractions and introduce a new fixed point result in the set of multi-valued mappings which generalize Liu’s result as a simple corollary of our main result.

Existence of fixed point for C-class functions was first proved by Ansari in 2014. Then, many authors gave interesting results using C-class functions. In this paper, we prove the existence of strong coupled proximity point for generalised cyclic-coupled proximal maps. Our result generalises the results of Kadwin and Marudai.

In this article, a new multidirectional monotone hybrid iteration algorithm for finding a solution to the split common fixed point problem is presented for two countable families of quasi-nonexpansive mappings in Banach spaces. Strong convergence theorems are proved. The application of the result is to consider the split common null point problem of maximal monotone operators in Banach spaces. Strong convergence theorems for finding a solution of the split common null point problem are derived. This iteration algorithm can accelerate the convergence speed of iterative sequence. The results of this paper improve and extend the recent results of Takahashi and Yao (Fixed Point Theory Appl 2015:87, 2015) and many others .

Tabling is a powerful resolution mechanism for logic programs that captures their least fixed point semantics more faithfully than plain Prolog. In many tabling applications, we are not interested in the set of all answers to a goal, but only require an aggregation of those answers. Several works have studied efficient techniques, such as lattice-based answer subsumption and mode-directed tabling, to do so for various forms of aggregation. While much attention has been paid to expressivity and efficient implementation of the different approaches, soundness has not been considered. This paper shows that the different implementations indeed fail to produce least fixed points for some programs. As a remedy, we provide a formal framework that generalises the existing approaches and we establish a soundness criterion that explains for which programs the approach is sound. This article is under consideration for acceptance in TPLP., Paper presented at the 32nd International Conference on Logic Programming (ICLP 2016), New York City, USA, 16-21 October 2016, 15 pages, LaTeX, 0 PDF figures

In this paper, we introduce the concept of coupled fixed point in partially ordered multiplicative metric space by proving some theorems for the existence and uniqueness of coupled fixed point. Also, we discuss the application to the existence and uniqueness of solution for a periodic boundary value problem.

The fixed point theorems on partially ordered quasi-metric space has proved in previous work. In this manuscript, we forward to present a fixed point theorems into the same space by using a generalized altering distance function into its contractive mapping. This result expands other results from the references.

The existing artificial intelligence techniques for games of Go have made remarkable achievements. However, the strongest AI for Go does not have a deterministic search capability. To address this issue, we introduce the model checking technique to the field of Go algorithms. First, we use an Alternating Finite Automaton (AFA) to establish a formal model for Go, and we employee a formula of Alternating-time Temporal Logic (ATL) to formalize win conditions. On the basis of it, the Go problem is mapped to the model checking problem between the ATL formula and the AFA. By employing the Ordered Binary Decision Diagram (OBDD) technique, we obtain a method for computing the least fixed point to implicitly perform model checking, in order to avoid the huge state space of Go that cause insufficient memory. As shown in the simulation results, the symbolic model checking can efficiently conduct some deterministic searches in the state space of Go.

Stationany default extensions have recently been introduced by Przymusinska and Przymusinsky as an interesting alternative to classical extensions in Reiter′s default logic. An important property of this new approach is that the set of all stationary extensions has a unique minimal element. In this paper we investigate the computational complexity of the main reasoning tasks in propositional stationary default logic, namely, cautious and brave reasoning. We show that cautious reasoning is complete for Δ P 2 while brave reasoning is Σ P 2 -complete. We also show that for normal default theories, the least fixed point iteration used to compute the unique minimal stationary extension is noticeably simplified. Based on this observation we show that cautious reasoning with normal defaults is complete for P NP[log n] . This is the class of decision problems solvable in polynomial time with a logarithmic number of queries to an oracle in NP.

Concurrent systems are notoriously difficult to analyze, and technological advances such as weak memory architectures greatly compound this problem. This has renewed interest in partial order semantics as a theoretical foundation for formal verification techniques. Among these, symbolic techniques have been shown to be particularly effective at finding concurrency-related bugs because they can leverage highly optimized decision procedures such as SAT/SMT solvers. This paper gives new fundamental results on partial order semantics for SAT/SMT-based symbolic encodings of weak memory concurrency. In particular, we give the theoretical basis for a decision procedure that can handle a fragment of concurrent programs endowed with least fixed point operators. In addition, we show that a certain partial order semantics of relaxed sequential consistency is equivalent to the conjunction of three extensively studied weak memory axioms by Alglave et al. An important consequence of this equivalence is an asymptotically smaller symbolic encoding for bounded model checking which has only a quadratic number of partial order constraints compared to the state-of-the-art cubic-size encoding., 15 pages, 3 figures

Based on the concept of a $C^{*}$ -algebra-valued b-metric space, this paper establishes some coupled fixed point theorems for mapping satisfying different contractive conditions on such space. As applications, we obtain the existence and uniqueness of a solution for an integral equation.

We present Flix, a declarative programming language for specifying and solving least fixed point problems, particularly static program analyses. Flix is inspired by Datalog and extends it with lattices and monotone functions. Using Flix, implementors of static analyses can express a broader range of analyses than is currently possible in pure Datalog, while retaining its familiar rule-based syntax. We define a model-theoretic semantics of Flix as a natural extension of the Datalog semantics. This semantics captures the declarative meaning of Flix programs without imposing any specific evaluation strategy. An efficient strategy is semi-naive evaluation which we adapt for Flix. We have implemented a compiler and runtime for Flix, and used it to express several well-known static analyses, including the IFDS and IDE algorithms. The declarative nature of Flix clearly exposes the similarity between these two algorithms.

International audience; Well-founded fixed points have been used in several areas of knowledge representation and reasoning and in particular to give semantics to logic programs involving negation. They are an important ingredient of approximation fixed point theory. We study the logical properties of the (parametric) well-founded fixed point operation. We show that the operation satisfies several, but not all of the standard equational properties of fixed point operations described by the axioms of iteration theories.

International audience; Fixed points of endofunctors play a central role in program semantics (initial algebras as recursive specification of domains), in coalgebraic theory of systems (terminal coalgebras and coinduction) and in a number of other connections such as iterative theories (rational fixed point). In this survey we present some older and new results on the structure of the three fixed points we have mentioned.

In this paper, we introduce the concept of JS-quasi-contraction and prove some fixed point results for JS-quasi-contractions in complete metric spaces under the assumption that the involving function is nondecreasing and continuous. These fixed point results extend and improve many existing results since some assumptions made there are removed or weakened. In addition, we present some examples showing the usability of our results.

In this paper we give a fixed point theorem in the context of fuzzy metric spaces in the sense of George and Veeramani. As a consequence of our result we obtain a fixed point theorem due to D. Mihet and generalize a fixed point theorem due to D. Wardowski. Also, we answer in a positive way to a question posed by D. Wardowski, and solve partially an open question on Cauchyness and contractivity. (C) 2015 Elsevier B.V. All rights reserved., Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo of Generalitat Valenciana, Spain, by Programa Vali+d para investigadores en formacion under Grant ACIF/2012/040.

In this chapter, we discuss a new area that overlaps between metric fixed point theory and graph theory. This new area yields interesting generalizations of the Banach contraction principle in metric and modular spaces endowed with a graph. The bridge between both theories is motivated by the fact that they often arise in industrial fields such as image processing engineering, physics, computer science, economics, ladder networks, dynamic programming, control theory, stochastic filtering, statistics, telecommunications and many other applications.

Degree theory deals with equations of the form \(\varphi (u) = h\) on a space X (finite of infinite dimensional). It addresses the questions of existence, uniqueness, or multiplicity of solutions and their distribution in the space. Moreover, it examines how sensitive are these properties to variations of \(\varphi\) and h.

Altun, Ishak/0000-0002-7967-0554 WOS: 000385026900004 The main goal of this paper is to introduce a new type contraction, that is, multivalued F-G-contraction, on a metric space with a graph. In terms of this new contraction, we establish some fixed point results. At the end, we give an illustrative example, which shows the importance of graph on the contractive condition.

This chapter gives an overview how retractions are used to prove fixed point results in ordered sets. The primary focus is on comparative retractions, that is, retractions r so that, for each x, the image r(x) is comparable to the preimage x. We start with infinite ordered sets and the classical Abian-Brown Theorem, which establishes that, for every order-preserving self map f of a chain-complete ordered set, if there is an x ≤ f (x), then f has a fixed point. Subsequently, using comparative retractions, we prove that the unit ball in Lp has the (order-theoretical) fixed point property, that is, every order-preserving self map has a fixed point. On a finite ordered set, a comparative retraction is the composition of comparative retractions that each remove a single point. Such a point is called irreducible and the fixed point property is not affected by the presence or absence of irreducible points. An ordered set that can be reduced, by successive removal of irreducible points, to a singleton is called dismantlable by irreducibles. We exhibit the relation between ordered sets that are dismantlable by irreducibles and the application of constraint propagation methods to find fixed point free order-preserving self maps. Closely related to irreducible points are points that are removed by a, not necessarily comparative, retraction that removes a single point. These points are called retractable points. There is a fixed point theorem for retractable points that generalizes the one for irreducible points. However, connectedly collapsible ordered sets, a natural class of ordered sets that is defined based on this theorem, are computationally more challenging than ordered sets that are dismantlable by irreducibles. Whereas the definition of dismantlability can be directly verified in polynomial time, direct verification of the definition of connected collapsibility is worst-case exponential. For graphs, the natural analogue of the fixed point property for ordered sets is the fixed clique property. Although there is no analogue of the Abian-Brown Theorem for the fixed clique property, there are analogues of the fixed point theorems for irreducible and for retractable vertices. For simplicial complexes, the natural analogue of the fixed point property for ordered sets is the fixed simplex property. Although there is an analogue of the fixed point theorem for irreducible vertices, there is no full analogue of the corresponding theorem for retractable vertices. We conclude with the connection between the fixed point property for ordered sets and the iteration of the clique graph operator on the comparability graph. Specifically, it is shown that, for ordered sets that are dismantlable by irreducibles, iteration of the clique graph operator on the comparability graph leads to a graph with one vertex. It is also shown that, if iteration of the clique graph operator on the comparability graph leads to a graph with one vertex, then the ordered set has the fixed point property.

The aim of this paper is to construct a novel implicit iterative algorithm for the split common fixed point problem for the demicontractive operatorsU,T, and xn=αnfxn+1-αnUλxn-ρnA*I-TAxn,n≥0, whereUλ=(1-λ)I+λU, and we obtain the sequence which strongly converges to a solutionx^of this problem, and the solutionx^satisfies the variational inequality.〈x^-f(x^),x^-z〉≤0,∀z∈S, whereSdenotes the set of all solutions of the split common fixed point problem.

The aim of this paper is to prove generalized fixed point theorems on partially ordered cone metric space with c - distance by replacing the constants in contractive condition with functions. The result extends some well-known result in the literatures. After that, we provide an example that satisfies our theorems.

For a given partially ordered set (poset) and a given family of mappings of the poset into itself, we study the problem of the description of joint fixed points of this family. Well-known Tarski's theorem gives the structure of the set of joint fixed points of isotone automorphisms on a complete lattice. This theorem has several generalizations (see., e.g., Markowsky, Ronse) that weaken demands on the order structure and upgrade in an appropriate manner the assertion on the structure of the set of joint fixed points. However, there is a lack of the statements similar to Kantorovich or Kleene theorems, describing the set of joint fixed points in terms of convergent sequences of the operator degrees. The paper provids conditions on the poset and on the family; these conditions ensure that the iterative sequences of elements of this family approximate the set of joint fixed points. The result obtained develops in a constructive direction the mentioned theorems on joint fixed points., Comment: 6 pages

It is shown that a mixed monotone property in coupled fixed point results can be replaced by another property which is automatically satisfied in the case of a totally ordered space, the case which is the most important in applications. Hence, these results can be applied in a much wider class of problems.

In this paper, we prove existence results for common fixed points of two or three relatively asymptotically regular mappings satisfying the orbital continuity of one of the involved maps under implicit relation on ordered orbitally complete metric spaces. We furnish suitable examples to demonstrate the validity of the hypotheses of our results. At the end of the results, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented.

A behavioural pseudometric is often defined as the least fixed point of a monotone function F on a complete lattice of 1-bounded pseudometrics. According to Tarski@?s fixed point theorem, this least fixed point can be obtained by (possibly transfinite) iteration of F, starting from the least element @? of the lattice. The smallest ordinal @a such that F^@a(@?)=F^@a^+^1(@?) is known as the closure ordinal of F. We prove that if F is also continuous with respect to the sup-norm, then its closure ordinal is @w. We also show that our result gives rise to simpler and modular proofs that the closure ordinal is @w.

In this paper, we show that the mixed $$g$$ -monotone property in coupled coincidence point theorems can be replaced by generalized property. Hence, these results can be applied in a much wider class of problems. We also study the condition for the uniqueness of a common coupled fixed point and give some example of nonlinear contraction mappings where the existence of the common coupled fixed point cannot be obtained by the mixed monotone property, but it follows by our results. At the end of this paper, we give an open problems for further investigation.

In this paper, the study of necessary conditions for the existence of unique fixed point of \(T\)-Hardy–Rogers type contractions mappings in the setting of ordered generalized metric spaces is initiated. These results unify and extend various comparable results from literature. We also provide example in support of results presented herein.

The present paper deals with some common fixed point results on a metric space for compatible mappings satisfying a more general inequality condition. The results obtained generalize the fixed point results of Mukherjee (1981) and Som (1985) and many others in turn.

We prove quadruple fixed point theorems in partially ordered metric 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.

We define Sahlqvist fixed point equations and relativized fixed point Boolean algebras with operators (relativized fixed point BAOs). We show that every Sahlqvist fixed point equation is preserved under completions of conjugated relativized fixed point BAOs. This extends the result of Givant and Venema (1999) to the setting of relativized fixed point BAOs.

The fixed points of general quantum operations are considered. We characterize the set of fixed points of a general quantum operation and discuss some equivalent and sufficient conditions for this set to be nontrivial. Moreover, we also consider completely disturbed points of quantum operations.

In this paper the existence and unicity of fixed points for mappings in fuzzy metric spaces (in the sense of Kaleva and Seikkala) is discussed. Nonlinear contractions of the Boyd-Wong's type, Alber-Guerre Delabriere's type and Kannan-Reich's type are considered, and several new fixed point theorems for these contractions in complete fuzzy metric spaces are presented, respectively. Also, some error estimates are given for iterations to approximate fixed point. Previous work with respect to fixed point in fuzzy metric spaces is based on the t-conorm max. The presented work does away with this restriction, by proposing weaker conditions defining a generic class of suitable binary operations. As applications the corresponding fixed point theorems for Menger probabilistic metric spaces are obtained.

All “multiplied” fixed point results in ordered metric spaces (including the coupled, tripled and quadrupled ones) based on linear contractive conditions, are obtainable from the 1986 (standard) fixed point statement over such structures in Turinici (Dem Math 19:171–180, 1986).

The purpose of this article is to provide the existence of a unique best proximity point for non-self-mappings by using altering distance function in the setting of partially ordered set which is endowed with a metric. Further, our result provides an extension of a result due to Harjani and Sadarangani to the case of non-self-mappings.

The main purpose of this article is to address a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. Indeed, if A and B are non-void subsets of a partially ordered set that is equipped with a metric, and S is a non-self mapping from A to B, this paper scrutinizes the existence of an optimal approximate solution, called a best proximity point of the mapping S, to the operator equation Sx = x where S is a continuous, proximally monotone, ordered proximal contraction. Further, this paper manifests an iterative algorithm for discovering such an optimal approximate solution. As a special case of the result obtained in this article, an interesting fixed point theorem on partially ordered sets is deduced.

We establish some common fixed point theorems for Chatterjea fuzzy mappings on closed balls in a complete metric space. Our investigation is based on the fact that fuzzy fixed point results can be obtained simply from the fixed point theory of mappings on closed balls. In real-world problems, there are various mathematical models in which the mappings are contractive on the subsets of a space under consideration but not on the whole space itself. It seems that this technique of finding the fuzzy fixed points was ignored. Our results generalize several important results of the literature.

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P ( X ) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X . We prove that the subset of P ( X ) formed by the norms failing the fixed point property is dense in P ( X ) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of l 1 can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.

We define Sahlqvist fixed point formulas. By extending the technique of Sambin and Vaccaro we show that (1) for each Sahlqvist fixed point formula @f there exists an LFP-formula @g(@f), with no free first-order variable or predicate symbol, such that a descriptive @m-frame (an order-topological structure that admits topological interpretations of least fixed point operators as intersections of clopen pre-fixed points) validates @f iff @g(@f) is true in this structure, and (2) every modal fixed point logic axiomatized by a set @F of Sahlqvist fixed point formulas is sound and complete with respect to the class of descriptive @m-frames satisfying {@g(@f):@[email protected][email protected]}. We also give some concrete examples of Sahlqvist fixed point logics and classes of descriptive @m-frames for which these logics are sound and complete.

This study characterizes the convex sets whose complements in the unit square exhibit the fixed point property for mappings which are jointly continuous and order preserving. Hence, one can readily construct simple sets with this fixed point property, but which neither have the fixed point property individually for continuous mappings nor for order preserving mappings. This is the first characterization of any non-trivial set with this property.

Recently, Suzuki [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008) 1861–1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and characterizes the metric completeness. In this paper we prove an analogous fixed point result for a self-mapping on a partial metric space or on a partially ordered metric space. Our results on partially ordered metric spaces generalize and extend some recent results of Ran and Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435–1443], Nieto and Rodriguez-Lopez [J.J. Nieto, R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239]. We deduce, also, common fixed point results for two self-mappings. Moreover, using our results, we obtain a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends Suzukiʼs characterization of metric completeness.

In this paper, we prove some coupled fixed point theorems for nonlinear contractive mappings having the mixed monotone property in partially ordered G -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].