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In this paper, we establish some point of φ-coincidence and common φ-fixed point results for two self-mappings defined on a metric space via extended CG-simulation functions. By giving an example we show that the obtained results are a proper extension of several well-known results in the existing literature. As applications of our results, we deduce some results in partial metric spaces besides proving an existence and uniqueness result on the solution of system of integral equations.

In this paper, we introduce and study a Lipschitz version of the Bishop–Phelps–Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map F and a pair of points at which F almost attains its norm by a Lipschitz map G and a pair of points such that G strongly attains its norm at the new pair of points. We first show that if M is a finite pointed metric space and Y is a finite-dimensional Banach space, then the pair (M , Y) has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if M is a uniformly Gromov concave pointed metric space (i.e. the molecules of M form a set of uniformly strongly exposed points), then (M , Y) has the Lip-BPB property for every Banach space Y. We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from (M , R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed. [ABSTRACT FROM AUTHOR]

We introduce a contractive condition involving new auxiliary function and prove a fixed point theorem for closed multivalued maps on complete metric spaces. An example and an application to integral equation are given in support of our findings.

We show that, unlike minima of superharmonic functions which are again superharmonic, the same property fails for Q -quasisuperminimizers. More precisely, if u i is a Q i -quasisuperminimizer, i = 1 , 2 , where 1 < Q 1 ≤ Q 2 , then u = min { u 1 , u 2 } is a Q -quasisuperminimizer, but there is an increase in the optimal quasisuperminimizing constant Q . We provide the first examples of this phenomenon, i.e. that Q > Q 2 . In addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to Kinnunen and Martio. Moreover, our lower and upper bounds turn out to be quite close. We also study a similar phenomenon in pasting lemmas for quasisuperminimizers, where Q = Q 1 Q 2 turns out to be optimal, and provide results on exact quasiminimizing constants of piecewise linear functions on the real line, which can serve as approximations of more general quasiminimizers. [ABSTRACT FROM AUTHOR]

We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on (ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we extend the coupled fixed point theorems for mixed monotone operators obtained by Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393] and Luong and Thuan [N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011) 983–992], by weakening the involved contractive condition. An example as well as an application to nonlinear Fredholm integral equations is also given in order to illustrate the effectiveness of our generalizations. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the coupled fixed point theorems for mixed monotone operators obtained in [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379–1393] by significantly weakening the contractive condition involved. Our technique of proof is essentially different and more natural. An example as well as an application to periodic BVP is also given in order to illustrate the effectiveness of our generalizations. [Copyright &y& Elsevier]

Abstract: In this note, we prove a simple fixed point theorem for a special class of complete metric spaces (namely, complete non-Archimedean metric spaces which are connected with some problems coming from quantum physics, -adic strings and superstrings). We also show that this theorem is a very efficient and convenient tool for proving the Hyers–Ulam stability of a quite wide class of functional equations in a single variable. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We introduce some notions of generalized nonlinear contractive maps and prove some fixed point results for such maps. Consequently, several known fixed point results are either improved or generalized including the corresponding recent fixed point results of Ciric [L.B. Ciric, Multivalued nonlinear contraction mappings, Nonlinear Anal. 71 (2009) 2716–2723], Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132–139], Feng and Liu [Y. Feng, S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112] and Mizoguchi and Takahashi [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188]. [Copyright &y& Elsevier]

Abstract: We establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane. As its application, we get that various contractive conditions using the so-called altering distance functions coincide with classical ones. We consider several classes of mappings both on metric spaces and ordered metric spaces. In particular, we show that unexpectedly, some very recent fixed point theorems for generalized contractions on ordered metric spaces obtained by Harjani and Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010) 1188–1197], and Amini-Harandi and Emami [A. Amini-Harandi, H. Emami A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010) 2238–2242] do follow from an earlier result of O’Regan and Petruşel [D. O’Regan and A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008) 1241–1252]. [Copyright &y& Elsevier]

Abstract: Let be a complete metric space equipped with a doubling Borel measure supporting a -Poincaré inequality. We obtain various convergence results for solutions of double obstacle problems on open subsets of . In particular, we consider a sequence of double obstacle problems with converging obstacles and show that the corresponding solutions converge as well. We use the convergence properties to define and study a generalized solution of the double obstacle problem. [ABSTRACT FROM AUTHOR]

Abstract: We prove fixed point results for multimaps satisfying a generalized metric (or an -metric) inwardness condition. Our results extend, generalize or improve several known results. [Copyright &y& Elsevier]

Abstract: The notion of a modular is introduced as follows. A (metric) modular on a set is a function satisfying, for all , the following three properties: if and only if for all ; for all ; for all . We show that, given , the set is a metric space with metric , called a modular space. The modular is said to be convex if is also a modular on . In this case coincides with the set of all such that for some and is metrizable by . Moreover, if or , then ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces. [Copyright &y& Elsevier]

Abstract: Let be a partially ordered set and be a complete metric on . Let be two set-valued mappings on . We obtained sufficient conditions for the existence of common fixed point of and satisfying an implicit relation in partially ordered set . [Copyright &y& Elsevier]

Abstract: We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and -harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of hybrid fixed points involving two multivalued operators in a complete metric space. Our claim is also illustrated with applications to a class of integral inclusions for proving the existence results. [Copyright &y& Elsevier]

Abstract: We provide a positive answer to a question raised by Eldred and Veeramani [A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006) 1001–1006] about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. Moreover, we introduce a new class of maps, called cyclic -contractions, which contains the cyclic contraction maps as a subclass. Convergence and existence results of best proximity points for cyclic -contraction maps are also obtained. [Copyright &y& Elsevier]

Abstract: Given a uniform space and nonempty subsets and of , we introduce the concepts of some families of generalized pseudodistances on , of set-valued dynamic systems of relatively quasi-asymptotic contractions with respect to and best proximity points for in , and we describe the methods which we use to establish the conditions guaranteeing the existence of best proximity points for when is cyclic (i.e.  and ) or when is noncyclic (i.e.  and ). Moreover, we establish conditions guaranteeing that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and the limit is a best proximity point for in . These best proximity points for are determined by unique endpoints in for a map when is cyclic and for a map when is noncyclic. The results and the methods are new for set-valued and single-valued dynamic systems in uniform, locally convex, metric and Banach spaces. Various examples illustrating the ideas of our definitions and results, and fundamental differences between our results and the well-known ones are given. [Copyright &y& Elsevier]

Abstract: For set-valued dynamic systems in uniform spaces we introduce the concept of quasi-asymptotic contractions with respect to some generalized pseudodistances, describe a method which we use to establish general conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these contractions and exhibit conditions such that for each starting point each generalized sequence of iterations (in particular, each dynamic process) converges and the limit is an endpoint. The definition, result, ideas and techniques are new for set-valued dynamic systems in uniform, locally convex and metric spaces and even for single-valued maps. [Copyright &y& Elsevier]

Abstract: In this paper, the concept of the set-valued dynamical systems of contractions of Meir–Keeler type in uniform spaces is introduced and conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions are established. The definition and the result presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our result and the well-known ones. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones. [Copyright &y& Elsevier]

Abstract: The supremum metric D between fuzzy subsets of a metric space is the supremum of the Hausdorff distances of the corresponding level sets. In this paper some new criteria of compactness with respect to the distance D are given; they concern arbitrary fuzzy sets (see ), fuzzy sets having no proper local maximum points (see ) and, finally, fuzzy sets with convex sendograph (see ). In order to compare results with a previous characterization of compactness of Diamond–Kloeden, the criteria will be expressed by equi-(left/right)-continuity. In the proofs a first author''s purely topological criterion of D-compactness and a variational convergence (called -convergence) which was introduced by De Giorgi and Franzoni, are fundamental. [Copyright &y& Elsevier]

Abstract: We study discrete dynamic systems in a complete metric space (, ) defined by mappings which satisfy Prešić type contractive conditions. Their counterparts in an ordered Banach space are investigated and applied to solve the global asymptotic stability of the equilibriums of some nonlinear difference equations. [Copyright &y& Elsevier]