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In this article, we introduce the fundamentals of the theory of demicontractive mappings in metric spaces and expose the key concepts and tools for building a constructive approach to approximating the fixed points of demicontractive mappings in this setting. By using an appropriate geodesic averaged perturbation technique, we obtained strong convergence and Δ-convergence theorems for a Krasnoselskij-Mann type iterative algorithm to approximate the fixed points of quasinonexpansive mappings within the framework of CAT(0) spaces. The main results obtained in this nonlinear setting are natural extensions of the classical results from linear settings (Hilbert and Banach spaces) for both quasi-nonexpansive mappings and demicontractive mappings. We applied our results to solving an equilibrium problem in CAT(0) spaces and showed how we can approximate the equilibrium points by using our fixed point results. In this context we also provided a numerical example in the case of a demicontractive mapping that is not a quasi-nonexpansive mapping and highlighted the convergence pattern of the algorithm in Table 1. It is important to note that the numerical example is set in non-Hilbert CAT(0) spaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish some fixed point results for fuzzy mappings in a complete dislocated b-metric space. Our results generalize and extend the results of Joseph et al. (SpringerPlus 5:Article ID 217, 2016). We also give examples to support our results, and applications relating the results to a fixed point for multivalued mappings and fuzzy mappings are studied. [ABSTRACT FROM AUTHOR]

In paper, we prove some common fixed point theorems for the pair of self-mappings with the g-quasi-condition in modular metric spaces. Also, we modify and prove some common fixed point theorems by using the (CLRg)-property along with the weakly compatible mapping. Finally, we give some applications on integral equations to illustrate our main results. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish some new fixed point theorems for generalized F-Suzuki-contraction mappings in complete b-metric spaces. [ABSTRACT FROM AUTHOR]

In this manuscript, common fixed point results for self-mappings satisfying generalized weak integral type contraction in the setting of G-metric space are established. Using the derived results, some applications to the systems of non-linear integral and fractional differential equations are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we state and prove Wardowski type fixed point theorems in metric space by using a modified generalized F-contraction maps. These theorems extend other well-known fundamental metrical fixed point theorems in the literature (Dung and Hang in Vietnam J. Math. 43:743-753, and Piri and Kumam in Fixed Point Theory Appl. 2014:210, , etc.). Examples are provided to support the usability of our results. [ABSTRACT FROM AUTHOR]

We introduce the notion of a modified α- ϕ-fuzzy contractive mapping and prove some results in fuzzy metric spaces for such kind of mappings. The theorems presented provide a generalization of some interesting results in the literature. Two examples and an application to integral equations are given to illustrate the usability of our theory. [ABSTRACT FROM AUTHOR]

In this paper, we prove common coupled fixed point theorems under a Geraghty-type condition using monotone instead of the often used mixed monotone property. Also we give some sufficient conditions for the uniqueness of a common coupled fixed point. An example illustrating our results is provided. [ABSTRACT FROM AUTHOR]

In this paper, we extend the concept of contraction mappings in b-metric spaces and utilize this concept to prove the existence and uniqueness of fixed point theorems for such mappings in such a space. We also prove the generalized Ulam-Hyers stability and well-posed results for a fixed point equation employing the concept of α-admissibility in b-metric spaces. We shall construct some examples to support our novel results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce viscosity approximation methods based on generalized contraction mappings for finding a set of common fixed points of a countable family of strict pseudo-contraction mappings, the common element of the set solutions of a general system of variational inequalities with Lipschitzian and relaxed cocoercive mappings and the set of solutions of a generalized mixed equilibrium problem. Furthermore, strong convergence theorems of the purposed iterative process are established in the framework of Banach spaces. The results presented in this paper improve and extend many recent important results. [Copyright &y& Elsevier]

In this paper, we introduce a new hybrid iterative scheme for finding a common element of the set of zeroes of a maximal monotone operator, the set of fixed points of a relatively quasi-nonexpansive mapping, the sets of solutions of an equilibrium problem and the variational inequality problem in Banach spaces. As applications, we apply our results to obtain strong convergence theorems for a maximal monotone operator and quasi-nonexpansive mappings in Hilbert spaces and we consider a problem of finding a minimizer of a convex function. [ABSTRACT FROM AUTHOR]

Recently, Samet et al. (2012) introduced the notion of α-ψ-contractive mappings and established some fixed point results in the setting of complete metric spaces. In this paper, we introduce the notion of weak α-ψ-contractive mappings and give fixed point results for this class of mappings in the setting of partial metric spaces. Also, we deduce fixed point results in ordered partial metric spaces. Our results extend and generalize the results of Samet et al. [ABSTRACT FROM AUTHOR]

Recently, Basha (2011) established the best proximity point theorems for proximal contractions of the first and second kinds which are extension of Banach's contraction principle in the case of non-self-mappings. The aim of this paper is to extend and generalize the notions of proximal contractions of the first and second kinds which aremore general than the notion of self-contractions, establish the existence of an optimal approximate solution theorems for these non-self-mappings, and also give examples to validate ourmain results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we extend a multi-valued contraction mapping to a cyclic multi-valued contraction mapping. We also establish the existence of common fixed point theorem for a cyclic multi-valued contraction mapping. Our results extend, generalize and unify Nadler’s multi-valued contraction mapping and many fixed point theorems for multi-valued mappings. [Copyright &y& Elsevier]

The purpose of this paper is to introduce a new iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solutions of variational inequality problems, the zero point of maximal monotone operators, and the set of two countable families of quasi-f-nonexpansive mappings in Banach spaces. Moreover, the strong convergence theorems of this method are established under the suitable conditions of the parameter imposed on the algorithm. Finally, we apply our results to finding a zero point of inverse-strongly monotone operators and complementarity problems. Our results presented in this paper improve and extend the recently results by many others. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the notion of a hybrid generalized multi-valued contraction mapping and establish the common fixed point theorem for this mapping. Our results generalize, unify, extend and complement several common fixed point theorems of many authors in the literature. [Copyright &y& Elsevier]

We generalized the notion of proximal contractions of the first and the second kinds and established the best proximity point theorems for these classes. Our results improve and extend recent result of Sadiq Basha (2011) and some authors. [ABSTRACT FROM AUTHOR]

We introduce an implicit-relation-type cyclic contractive condition for a map in a metric space and derive existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. At the end of the paper, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented. [ABSTRACT FROM AUTHOR]

We extend the recent results of the coupled fixed point theorems of Cho et al. (2012) by weakening the concept of the mixed monotone property. We also give some examples of a nonlinear contraction mapping, which is not applied to the existence of the coupled fixed point by the results of Cho et al. but can be applied to our results. The main results extend and unify the results of Cho et al. and many results of the coupled fixed point theorems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider a modified hybrid steepest-descent method by using a viscosity approximation method with a weakly contractive mapping for finding the common element of the set of a common fixed point for an infinite family of nonexpansive mappings and the set of solutions of a system of an equilibrium problem. The sequence is generated from an arbitrary initial point which converges in norm to the unique solution of the variational inequality under some suitable conditions in a real Hilbert space. The results presented in this paper generalize and improve the results of Moudafi (2000), Marino and Xu (2006), Tian (2010), Saeidi (2010), and some others. Finally, we give an application to minimization problems and a numerical example which support our main theorem in the last part. [ABSTRACT FROM AUTHOR]

In this article, we define a tangential property which can be used not only for single-valued mappings but also for multi-valued mappings, and used it in the prove for the existence of a common fixed point theorems of Gregus type for four mappings satisfying a strict general contractive condition of integral type in metric spaces. Our theorems generalize and unify main results of Pathak and Shahzad (Bull. Belg. Math. Soc. Simon Stevin 16, 277-288, 2009) and several known fixed point results. [ABSTRACT FROM AUTHOR]

Abstract: Kirk and Xu introduced the concept of asymptotic pointwise contractions. In this paper, we investigate these kinds of mappings in modular spaces. Moreover, the fixed point theorem for asymptotic pointwise nonexpansive mappings in modular spaces is also studied. The results of this paper improve and extend the results of Razani et al. [A. Razani, E. Nabizadeh, M. Beyg Mohamadi, S. Homaei Pour, Fixed points of nonlinear and asymptotic contractions in the modular spaces, Abstract and Applied Analysis, 2007 (2007)], and Kirk and Xu [W. A. Kirk, Hong-Kun Xu, Asymptotic pointwise contractions, Nonlinear Analysis 69 (2008) 4706–4712] to modular spaces. [Copyright &y& Elsevier]

Abstract: In 2007, Kamran [T. Kamran, Multivalued -weakly Picard mappings, Nonlinear Anal. 67 (2007) 2289–2296] extended the notion of multi-valued mapping from weak contraction and generalized -weak contraction to -weak contraction and generalized multi-valued -weak contraction. He also obtained some common fixed point theorems with the notion of -weakly commuting at a coincidence point of a hybrid pair. In this paper, we can drop the condition of -weakly commuting in Theorems 2.9 and 3.5 in . We further extend the notion of generalized multi-valued -weak contraction and introduce the notion of generalized multi-valued -weak contraction. We also establish some coincidence and common fixed point theorems with generalized multi-valued -weak contraction mappings. Our results extend and generalize several common fixed point theorems of many authors. [Copyright &y& Elsevier]

We prove new fixed point and common fixed point theorems for generalized weak contractive mappings of integral type in modular spaces. Our results extend and generalize the results of A. Razani and R. Moradi (2009) and M. Beygmohammadi and A. Razani (2010). [ABSTRACT FROM AUTHOR]

We introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions of the variational inclusion for a β-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (2006), Su et al. (2008), Klin-eam and Suantai (2009), Tan and Chang (2011), and some other authors. [ABSTRACT FROM AUTHOR]

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others. [ABSTRACT FROM AUTHOR]

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors. [ABSTRACT FROM AUTHOR]

We introduce an iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions mapping, the set of common solutions of a system of two mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse strongly monotone mappings. Strong convergence theorems are established in the framework of Hilbert spaces. Finally, we apply our results for solving convex feasibility problems in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently. [ABSTRACT FROM AUTHOR]

We introduce a new modified Ishikawa iterative process and a new W-mapping for computing fixed points of an infinite family of strict pseudo contractions mapping in the framework of q-uniformly smooth Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions. The results obtained in this paper extend and improve the recent results of Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011 and many others in the literature. [ABSTRACT FROM AUTHOR]

We introduce new iterative algorithms by hybrid method for finding a common element of the set of solutions of fixed points of infinite family of nonexpansive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequality with inverse-strongly monotone mappings in a real Hilbert space. We prove the strong convergence of the proposed iterative method under some suitable conditions. Finally, we apply our results to complementarity problems and optimization problems. Our results improve and extend the results announced by many others. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce an iterative scheme by using the hybrid projection methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a -strict pseudo-contraction mapping in Hilbert spaces. We obtain two strong convergence theorems under mild assumptions on parameters for the sequences generated by these processes. The results presented in the paper extend and improve some recent results of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–346], Tada and Takahashi [A. Tada, W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl. 133 (2007) 359–370] and Ceng et al. [L.C. Ceng, S. Al-Homidan, Q.H. Ansari, J.C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009) 967–974] and many others. [Copyright &y& Elsevier]

Abstract: In this work, we establish new coincidence and common fixed point theorems for hybrid strict contraction maps by dropping the assumption “ is -weakly commuting” for a hybrid pair of multivalued maps in Theorem 3.10 of T. Kamran (2004) . As an application, an invariant approximation result is derived. [Copyright &y& Elsevier]

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for monotone, Lipschitz-continuous mappings. The iterative process is based on the so-called extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parametric controlling conditions. This main theorem extends a recent result of Yao, Liou and Yao [Y. Yao, Y. C. Liou and J.-C. Yao, "An Extragradient Method for Fixed Point Problems and Variational Inequality Problems," Journal of Inequalities and Applications Volume 2007, Article ID 38752, 12 pages doi:10.1155/2007/38752] and many others. [ABSTRACT FROM AUTHOR]

We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problems for β-inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality, which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area. [ABSTRACT FROM AUTHOR]

We prove some strong convergence theorems for fixed points of modified Ishikawa and Halpern iterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space by using the hybrid projection methods. Moreover,we also apply our results to a class of relatively nonexpansive mappings, and hence, we immediately obtain the results announced by Qin and Su's result (2007), Nilsrakoo and Saejung's result (2008), Su et al.'s result (2008), and some known corresponding results in the literatures. [ABSTRACT FROM AUTHOR]

This article is written due to a small gap in our published paper. In this erratum, we point out and fix the problem to set our existed results at the best of their perfection. [ABSTRACT FROM AUTHOR]

Some well known results from the existing literature are extended and generalized via new contractive type mappings in fuzzy metric spaces. A non trivial supporting example is also provided to demonstrate the validity of the obtained results. [ABSTRACT FROM AUTHOR]

The main purpose in this paper is to define the modification form of random α -admissible and random α - ψ -contractive maps. We establish new random fixed point theorems in complete separable metric spaces. The interpretation of our results provide the main theorems of Tchier and Vetro (2017) as directed corollaries. In addition, some applications to second order random differential equations are presenred here to interpret the usability of the obtained results. [ABSTRACT FROM AUTHOR]

The purpose of this article is to resolve a global optimization problem for quasi-noncyclic relatively nonexpansive mappings by giving an algorithm that determines an optimal approximate solution of the following minimization problem, min x ∈ A d (x , T x) , min y ∈ B d (y , T y) and min (x , y) ∈ A × B d (x , y) ; also, we provide some illustrative examples to support our results. As an application, the existence of a solution of the analytic complex function is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of random α -proximal admissible and random α - Z -contraction. Then we establish random best proximity point theorems for such mapping in complete separable metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we present fixed point theorems for a generalized Roger Hardy type F-contraction in metric-like spaces and also give some examples to illustrate the main results in this paper. Moreover the applications of second-order differential equations and fractional differential equations are shown. The existing results improve and extend the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish some fixed point results for fuzzy mapping in a complete b-metric space. Our results unify, extend and generalize several results in the existing literature. Example is also given to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we modify the general iterative method to approximate a common element of the set of solutions of split variational inclusion problem and the set of common fixed points of a finite family of k -strictly pseudo-contractive nonself mappings. Strong convergence theorem is established under some suitable conditions in a real Hilbert space, which also solves some variational inequality problems. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors. Finally, some examples to study the rate of convergence and some illustrative numerical examples are presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of A-cone metric spaces over Banach algebra as a generalization of A-metric spaces and cone metric spaces over Banach algebra. We also defined generalized Lipschitz and expansive maps in such maps and establish some fixed point theorems for such maps in the setting of the new space. As an application, we prove a theorem for integral equation. We provide illustrative example to verify our results. Our results generalize and unify some well-known results in the literature. [ABSTRACT FROM AUTHOR]