Stability of tripled fixed point iteration procedures for monotone mappings

The new concept of tripled fixed point introduced recently by Berinde and Borcut (Nonlinear Anal. 74:4889–4897, 2011) directed to several researches on this subject, in partially metric spaces and in cone metric spaces. In this paper, we introduce the notion of stability definition of tripled fixed point iteration procedures and establish stability results for monotone mappings which satisfy various contractive conditions. Our results extend and complete some existing results in the literature.     

Linear contractions in product ordered metric spaces

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).     

Meir-Keeler Type Contractions for Tripled Fixed Points

In 2011, Berinde and Borcut [6] introduced the notion of tripled fixed point in partially ordered metric spaces. In our paper, we give some new tripled fixed point theorems by using a generalization of Meir-Keeler contraction.     

Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces

In this paper we introduce the concept of a tripled coincidence point for a pair of nonlinear contractive mappings F : X³ → X and g : X → X. The obtained results extend recent coincidence theorems due to ?iri? and Lakshmikantham [V. Lakshmikantham, L. ?iri?, L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) 4341-4349].     

Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces

In this paper, we introduce the concept of tripled fixed point for nonlinear mappings in partially ordered complete metric spaces and obtain existence, and existence and uniqueness theorems for contractive type mappings. Our results generalize and extend recent coupled fixed point theorems established by Gnana Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379-1393]. Examples to support our new results are given.     

Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications

In this paper, we establish coincidence and common fixed point theorems for contractive mappings on a metric space endowed with an amorphous binary relation. The presented theorems extend the results of Samet and Turinici in [Commun. Math. Anal. 12 (2012), 82– 97] and generalize many existing results on metric and ordered metric spaces. We apply also our main results to derive coincidence and common fixed point theorems for cyclic contractive mappings.     

Applications of Fixed Point Theorems to the Vacuum Einstein Constraint Equations with Non-Constant Mean Curvature

In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer’s fixed point theorem (known methods use Schauder’s fixed point theorem), while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These methods allow to: unify some recent existence results, simplify many proofs (for instance, the one of the main theorems in Dahl et al., Duke Math J 161(14):2669–2697, 2012) and weaken the assumptions of many recent results.     

New fixed point results for mappings of contractive type with an application to nonlinear fractional differential equations

In this paper, the notion of α-ψ-contractive mappings in the setting of w-distance is introduced and some new fixed point theorems for such mappings are established. Presented fixed point theorems generalize recent results of Samet et al. [Nonlinear Anal. 75 (2012), 2154–2165] and others. Moreover, some examples and an application to nonlinear fractional differential equations are given to illustrate the usability of the new theory.     

Percolation and limit theory for the poisson lilypond model

The lilypond model on a point process in d ‐space is a growth‐maximal system of non‐overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decaying tail bounds for the size of the cluster at the origin. Finally, we consider the enhanced Poisson lilypond model where all the balls are enlarged by a fixed amount (the enhancement parameter), and show that for d > 1 the critical value of this parameter, above which the enhanced model percolates, is strictly positive. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

{\psi F}-Contractions: Not Necessarily Nonexpansive Picard Operators

The aim of this paper is to extend the result of Wardowski (Fixed Point Theory Appl 2012:94, 2012) by introducing a new class of Picard operators which strictly includes the family of F-contractions. For such operators some fixed point theorems are proved. It is showed that there exist Picard self-mappings on a complete metric space that are neither nonexpansive nor expansive.     

Fixed point theorems for contractions of Reich type on a metric space with a graph

In a metric space with a directed graph G, Jachymski (Proc Am Math Soc 1(136):1359–1373, 2008) introduced the concept of Banach G-contraction and proved two fixed point theorems for such mappings. Bojor (Nonlinear Anal 75:3895–3901, 2012) generalized this concept to Reich G-contraction and obtain a fixed point theorem. Note that Bojor’s theorem is established under the additional type of connectedness of G and it does not include Jachymski’s results as a special case. Moreover, there are some mistakes in several corollaries. Some examples and counterexamples are illustrated. It is our purpose to improve Bojor’s theorem and to present two fixed point theorems for Reich G-contractions. Our results are extensions of the two Jachymski’s theorems. Finally, we also discuss some priori error estimates.     

Best proximity points: global minimization of multivalued non-self mappings

In this article, we give a best proximity point theorem for generalized contractions in metric spaces with appropriate geometric property. We also, give an example which ensures that our result cannot be obtained from a similar result due to Amini-Harandi (Best proximity points for proximal generalized contractions in metric spaces. Optim Lett, 2012). Moreover, we prove a best proximity point theorem for multivalued non-self mappings which generalizes the Mizoguchi and Takahashi’s fixed point theorem for multivalued mappings.     

A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert space and a Banach space

The main purpose of this paper is to introduce a viscosity-type iterative algorithm for approximating a common solution of a split variational inclusion problem and a fixed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating a common solution of a split variational inclusion problem and a fixed point problem for a multivalued quasi-nonexpansive mapping between a Hilbert space and a Banach space. Furthermore, we applied our results to study a split convex minimization problem. Also, a numerical example of our result is given. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13, 759–775, 2012), Moudafi (J. Optim. Theory Appl. 150, 275–283, 2011), Takahashi and Yao (Fixed Point Theory Appl. 2015, 87, 2015), and a host of other important results in this direction.     

Multidimensional fixed point theorems in partially ordered complete metric spaces

In this paper we propose a notion of coincidence point between mappings in any number of variables and we prove some existence and uniqueness fixed point theorems for nonlinear mappings verifying different kinds of contractive conditions and defined on partially ordered metric spaces. These theorems extend and clarify very recent results that can be found in [T. Gnana-Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7)(2006) 1379–1393], [V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 4889–4897] and [M. Berzig, B. Samet, An extension of coupled fixed point’s concept in higher dimension and applications, Comput. Math. Appl. 63 (8) (2012) 1319–1334].     

Amenability and nonlinear flows in dual Banach spaces

This paper is devoted to the study of nonlinear flows with weak* compact convex phase spaces sitting in conjugate Banach spaces. Nonlinear fixed point properties for the classes of amenable affine and convex semitopological semigroups are established; giving partial answers to question 1 in Lau and Zhang (J Funct Anal 263:2949–2977, 2012); we also prove related results for amenable semitopological semigroups. Furthermore, by almost periodicity techniques, we derive a non-commutative version of Bruck’s result (Pac J Math 53:59–70, 1974) on the existence of a nonexpansive retract; we also provide a characterization of left amenability property for the space of almost periodic functions for semitopological semigroups, and derive a fixed point property in \(\ell ^1\).     

Positive definite solution of a nonlinear matrix equation

Using fixed point theory, we present a sufficient condition for the existence of a positive definite solution of the nonlinear matrix equation \({X = Q \pm \sum^{m}_{i=1}{A_{i}}^*F(X)A_{i}}\), where Q is a positive definite matrix, A _i ’s are arbitrary n × n matrices and F is a monotone map from the set of positive definite matrices to itself. We show that the presented condition is weaker than that presented by Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443]. In order to do so, we establish some fixed point theorems for mappings satisfying (\({\psi, \phi}\))-weak contractivity conditions in partially ordered G-metric spaces, which generalize some existing results related to (\({\psi, \phi}\))-weak contractions in partially ordered metric spaces as well as in G-metric spaces for a given function f. We conclude, by presenting an example, that our fixed point theorem cannot be obtained from any existing fixed point theorem using the process of Jleli and Samet [Fixed Point Theory Appl. 2012 (2012), Article ID 210].     

Fixed point theorems for $$F_\mathfrak {R}$$-contractions with applications to solution of nonlinear matrix equations

In (Fixed Point Theory Appl 94:6, 2012), the author introduced a new kind of contractions, called F-contractions, that extended the Banach contractions in a newfangled way. In this work, we introduce the notion of \(F_\mathfrak {R}\)-contraction where \(\mathfrak {R}\) is a binary relation on its domain that has not to be neither transitive nor a partial order. Consequently, we establish some fixed point results for such contractions in complete metric spaces that improve the Wardowski’s original idea and we also give illustrative examples. Furthermore, we show some results to guarantee existence and uniqueness of fixed point of N-order. As an application, we apply our main result to study a class of nonlinear matrix equation.     

The 2D‐directed spanning forest is almost surely a tree

We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi‐infinite paths in the DSF. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Complementary Lidstone interpolation on scattered data sets

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691–1707, 2012) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell’Accio, Appl Numer Math 52:339–361, 2005). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77–90, 2005) and drawn on by Agarwal et al. (2009) and Agarwal and Wong (J Comput Appl Math 234:2543–2561, 2010). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided.