Generalized cyclic contractions in partially ordered metric spaces

Abkar and Gabeleh in (J. Optim. Theory. Appl. doi:10.1007/s10957-011-9818-2) proved some theorems which ensure the existence and convergence of fixed points, as well as best proximity points for cyclic mappings in ordered metric spaces. In this paper we extend these results to generalized cyclic contractions and obtain some new results on the existence and convergence of fixed points for weakly contractive mappings, as well as on best proximity points for cyclic ??-contraction mappings in partially ordered metric spaces.     

具$w_0$-距离度量空间中$p$-逼近$\alpha$-$\eta$-$\beta$-拟压缩的最佳逼近点定理

本文提出了一类称为$p$-逼近$\alpha$-$\eta$-$\beta$-拟压缩的新的非自映射,并引进了关于$\eta$的$\alpha$-逼近可容许映射和关于$\eta$的$(\alpha,d)$正则映射的概念.基于这些新概念,在$w_0$-距离度量空间中研究了此类新压缩最佳逼近点的存在唯一性,并给出了一个新的定理,推广和补充了文[Ayari, M. I. et al. Fixed Point Theory Appl., 2017, 2017: 16]和[Ayari, M. I. et al. Fixed Point Theory Appl., 2019, 2019: 7]中的结果.给出了一个例子来说明主要结果的有效性.进一步地,作为推论得到关于两个映射的最佳逼近点和公共不动点定理.作为其中一个推论的应用,讨论了一类Volterra型积分方程组的求解问题.     

Equivalent conditions for generalized contractions on (ordered) metric spaces

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 Petru?el [D. O’Regan and A. Petru?el, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008) 1241-1252].     

Some new extensions of Edelstein-Suzuki-type fixed point theorem to G-metric and G-cone metric spaces

In this paper, we prove some fixed point theorems for generalized contractions in the setting of G-metric spaces. Our results extend a result of Edelstein [M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79] and a result of Suzuki [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317]. We prove, also, a fixed point theorem in the setting of G-cone metric spaces.     

Best proximity point theorems for single- and set-valued non-self mappings

We study the existence of best proximity points for single-valued non-self mappings. Also, we prove a best proximity point theorem for set-valued non-self mappings in metric spaces with an appropriate geometric property. Examples are given to support the usability of our results.     

Towards Lim

The paper contains an elegant extension of the Nadler fixed point theorem for multivalued contractions (see Theorem 21). It is based on a new idea of the α-step mappings (see Definition 17) being more efficient than α-contractions. In the present paper this theorem is a tool in proving some fixed point theorems for “nonexpansive” mappings in the bead spaces (metric spaces that, roughly speaking, are modelled after convex sets in uniformly convex spaces). More precisely the mappings are nonexpansive on a set with respect to only one point - the centre of this set (see condition (4)). The results are pretty general. At first we assume that the value of the mapping under consideration at this central point looks “sharp” (see Definition 6). This idea leads to a group of theorems (based on Theorem 7). Their proofs are compact and the theorems, in particular, are natural extensions of the classical results for (usual) nonexpansive mappings. In the second part we apply the idea of Lim to investigate the regular sequences and here the proofs are based on our extension of Nadler's Theorem. In consequence we obtain some fixed point theorems that generalise the classical Lim Theorem for multivalued nonexpansive mappings (see e.g. Theorem 26).     

Multivalued generalized nonlinear contractive maps and fixed points

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].     

Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces

First, we consider a strongly continuous semigroup of nonexpansive mappings defined on a closed convex subset of a complete CAT(0) space and prove a convergence of a Mann iteration to a common fixed point of the mappings. This result is motivated by a result of Kirk (2002) and of Suzuki (2002). Second, we obtain a result on limits of subsequences of Mann iterations of multivalued nonexpansive mappings on metric spaces of hyperbolic type, which leads to a convergence theorem for nonexpansive mappings on these spaces.     

BEST PROXIMITY POINT THEOREMS FOR SINGLE- AND SET-VALUED NON-SELF MAPPINGS

We study the existence of best proximity points for single-valued non-self map-pings. Also, we prove a best proximity point theorem for set-valued non-self mappings in metric spaces with an appropriate geometric property. Examples are given to support the usability of our results.     

Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces

In hyperconvex metric spaces we consider best approximation, invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive.     

Some common fixed-point results in generalized F-metric spaces.

In this paper, we establish a new common fixed-point theorem for multivalued mappings with the greatest lower bound property in generalized F-metric spaces. Also, we propose some new theorems via more general contractions.     

Fixed Points of Sequence of Ćirić Generalized Contractions of Perov Type

Perov used the concept of vector valued metric space and obtained a Banach type fixed point theorem on such a complete generalized metric space. In this article, we study fixed point results for the new extensions of sequence of ?iri? generalized contractions on cone metric space, and we give some generalized versions of the fixed point theorem of Perov. The theory is illustrated with some examples. It is worth mentioning that the main result in this paper could not be derived from ?iri?’s result by the scalarization method, and hence indeed improves many recent results in cone metric spaces.     

Coincidence and fixed point theorems for nonlinear hybrid generalized contractions

In this paper we first prove some coincidence and fixed point theorems for nonlinear hybrid generalized contractions on metric spaces. Secondly, using the concept of an asymptotically regular sequence, we give some fixed point theorems for Kannan type multi-valued mappings on metric spaces. Our main results improve and extend several known results proved by other authors.     

Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT(0) spaces

We establish △-convergence and strong convergence theorems for an iterative process for a finite family of generalized nonexpansive multivalued mappings in a CAT(0) space. Moreover, we present a fixed point theorem for a pair consisting of a finite family of generalized nonexpansive single valued mappings, and a generalized nonexpansive multivalued mapping in CAT(0) spaces.     

Fixed points, selections and common fixed points for nonexpansive-type mappings

We study the existence of fixed points in the context of uniformly convex geodesic metric spaces, hyperconvex spaces and Banach spaces for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity. Besides, we use the fixed point theorems proved here to give common fixed point results for commuting mappings.     

An Extension of Set-Valued Contraction Principle for Mappings on a Metric Space with a Graph and Application

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].     

On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces

We prove a strong convergence theorem for multivalued nonexpansive mappings which includes Kirk’s convergence theorem on CAT(0) spaces. The theorem properly contains a result of Jung for Hilbert spaces. We then apply the result to approximate a common fixed point of a countable family of single-valued nonexpansive mappings and a compact valued nonexpansive mapping.     

On a conjecture of S. Reich

Simeon Reich (1974) proved that the fixed point theorem for single-valued mappings proved by Boyd and Wong can be generalized to multivalued mappings which map points into compact sets. He then asked (1983) whether his theorem can be extended to multivalued mappings whose range consists of bounded closed sets. In this note, we provide an affirmative answer for a certain subclass of Boyd-Wong contractive mappings.

     

On some generalizations of Ekeland��s principle and inward contractions in gauge spaces

We give generalizations in complete gauge spaces of the following results: Bishop-Phelps’ theorem, Ekeland’s variational principle, Caristi’s fixed point theorem, the drop theorem and the flower petal theorem. We show that our generalizations are equivalent. We apply those results to obtain fixed point theorems for multivalued contractions defined on a closed subset of a complete gauge space and satisfying a generalized inwardness condition.     

KKM mappings in metric spaces

We introduce the class of KKM-type mappings on metric spaces and establish some fixed point theorems for this class. We also obtain a generalized Fan's matching theorem, a generalized Fan–Browder's type theorem, and a new version of Fan's best approximation theorem.