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Abstract: Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant and for each there are cones with normal constant . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results. [Copyright &y& Elsevier]

Abstract: W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd–Wong fixed point theorem. Subsequently, I.D. Arand̄elović [I.D. Arand̄elović, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384–385] established somewhat more general version of Kirk''s result and he gave an elementary proof of it. However, our purpose is to show that there is an error in this proof and, moreover, Arand̄elović''s theorem is false. We also explain how to correct this result. [Copyright &y& Elsevier]

In a previous paper of ours we used the notion of porosity to show that most of the nonexpansive self-mappings of bounded, closed and convex subsets of a Banach space are contractive and possess a unique fixed point which is the uniform limit of all iterates. In this paper we extend this result to nonexpansive self-mappings of closed and convex sets in a Banach space which are not necessarily bounded. As a matter of fact, it turns out that our results are true for all complete hyperbolic metric spaces. [ABSTRACT FROM AUTHOR]

In 2008, T. Suzuki [33] defined one of most relevant extensions of the notion of nonexpansivity. This notion was extended in [15] defining the, so called, C λ -mappings. In the last years, some papers have appeared trying to extend the most important results about existence of fixed points for nonexpansive mappings to this wider class of mappings (see, for instance, [2,5,9,10,15] ). In this paper we continue this project proving that Banach spaces with extended unconditional bases satisfy the fixed point property for C λ -mappings if the unconditional constant of the basis is small enough. We also begin a new project on renorming with the fixed point property for C λ -mappings, proving that any separable Banach space can be renormed to satisfy the fixed point property for C λ -mappings in a stable sense, that is, the space with the new norm satisfies the fixed point property which, in addition, is inherited by those isomorphic spaces which are close to it in the sense of the Banach–Mazur distance. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with an operator equation on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane–Emden–Fowler equations. [Copyright &y& Elsevier]

In this paper, we define the class of ( α , β ) -nonexpansive mappings which is properly larger than the class of α -nonexpansive mappings and prove that every ( α , β ) -nonexpansive mapping T : C → C has an approximate fixed point sequence, where C is a nonempty bounded subset of a Banach space X , α > 0 and β ≥ 0 . This, in particular, gives an affirmative answer to the open question posed by Ariza-Ruiz and et al. concerning the existence of an approximate fixed point sequence for α -nonexpansive mappings, Ariza-Ruiz et al. (2016) [4] . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and exponential stability of mild solutions for a class of impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup in Hilbert spaces. Sufficient conditions for the existence of mild solutions are obtained using the Hausdorff measure of noncompactness and the Mönch fixed point theorem. Further, we establish a new impulsive-integral inequality to prove the exponential stability of mild solutions in the mean square moment. Finally, an example is presented to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the general solution of the following general radical quintic functional equation f ( a x 5 + b y 5 5 ) = r f ( x ) + s f ( y ) for f a mapping from the field of real numbers into a vector space, where a , b , r , s are fixed nonzero reals. Also, we prove the generalized hyperstability results for the general radical quintic functional equation by using the fixed point theorem (cf. Dung and Hang (2018) [15] , Theorem 2.1) in quasi- β -Banach spaces. Namely, we show, under some weak natural assumptions, functions satisfying the above equation approximately (in some sense) must be actually solutions to it. [ABSTRACT FROM AUTHOR]

The moduli R ( a , X ) and M ( X ) , introduced by Domínguez Benavides, play an important role in the fixed point theory for nonexpansive mappings. In the paper we show that if inf i ∈ I ⁡ M ( X i ) > 1 , then M ( ( ⨁ i ∈ I X i ) Z ) > 1 , where ( ⨁ i ∈ I X i ) Z is the direct sum of Banach spaces X i with respect to a Banach lattice Z , under some conditions for Z and I . Similar results are obtained for the modulus R ( a , X ) . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings. Furthermore, we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions. We also discuss some special cases. Results obtained in this paper improve the previously known results in this area. [Copyright &y& Elsevier]

Abstract: In this paper the concept of a contraction for multi-valued mappings in a metric space is introduced and the existence theorems for fixed points of such contractions in a complete metric space are proved. Presented results generalize and improve the recent results of Y. Feng, S. Liu [Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112], D. Klim, D. Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132–139] and several others. The method used in the proofs of our results is new and is simpler than methods used in the corresponding papers. Two examples are given to show that our results are genuine generalization of the results of Feng and Liu and Klim and Wardowski. [Copyright &y& Elsevier]

Abstract: The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995), 655–666]. [Copyright &y& Elsevier]

Abstract: As an important iteration, the Mann and Ishikawa iteration has extensive application in fixed point theory. In 1991, David Borwein and Jonathan Borwein proved the convergence of the Mann iteration on a closed bounded interval in their paper. In this paper, we will extend their result to an arbitrary interval and to the Ishikawa iteration, indicating the necessary and sufficient condition for the convergence of Ishikawa iteration of continuous functions on an arbitrary interval. [Copyright &y& Elsevier]

In this paper, by using a new fixed point alternative, we establish a new generalization of the Hyers-Ulam-Rassias stability concerning a Cauchy-Jensen functional equation in normed spaces which are not necessarily Banach spaces. Moreover, our paper consists of several non-trivial examples which signify the motivation of such investigations. [ABSTRACT FROM AUTHOR]

In this paper, we study the hyperstability for the general linear equation in the setting of quasi-Banach spaces. We first extend the fixed point result of Brzdek et al. [5, Theorem 1] in metric spaces to b -metric spaces, in particular to quasi-Banach spaces. Then we use this result to generalize the main results on the hyperstability for the general linear equation in Banach spaces to quasi-Banach spaces. We also show that we can not omit the assumption of completeness in [5, Theorem 1] . As a consequence, we conclude that we need more explanations to replace a normed space by its completion in the proofs of some results in the literature. [ABSTRACT FROM AUTHOR]

In this article, by introducing a new set and a new concept of φ − ( h , e ) -concave operators, and by using the cone theory and monotone iterative method, we present some new existence and uniqueness theorems of fixed points for increasing φ − ( h , e ) -concave operators without requiring the existence of upper and lower solutions. As an application, we establish the existence and uniqueness of a nontrivial solution for a new form of fractional differential equation with integral boundary conditions. The main results of this paper improve and extend some known results, and present a new method to study nonlinear equation problems. [ABSTRACT FROM AUTHOR]

In this paper we prove the admissibility of modular function spaces E ρ considered and defined by Kozłowski in [17] . As an application we get that any compact and continuous mapping T : E ρ → E ρ has a fixed point. Moreover, we prove that the same holds true for any retract of E ρ . [ABSTRACT FROM AUTHOR]

In this paper we prove that compactness can be characterized by means of the existence of a fixed point for some classes of mappings defined on convex closed subsets of the space ℓ 1 . Nominally, our result involves nonexpansive mappings, uniformly Lipschitzian mappings and cascading nonexpansive mappings. We also extend the results to some more general classes of Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, using the theory of fixed point index, we prove some results on the multiplicity of positive solutions for a class of nonlocal elliptic problem. [ABSTRACT FROM AUTHOR]

In this paper, we establish the demiclosedness principle for the class of mean nonexpansive mappings, introduced in 2007 by Goebel and Japón Pineda, defined on closed, convex subsets of Banach spaces satisfying Opial's condition. We also establish the demiclosedness principle for a subclass of the mean nonexpansive maps whose domains are closed, bounded, convex sets in uniformly convex spaces. These results extend known demiclosedness results for nonexpansive maps and lead to some fixed point results for mean nonexpansive maps which partially answer an open question posed by Goebel and Japón Pineda. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a coincidence point theorem for sequentially weakly continuous maps. Moreover, as a consequence, a critical point theorem for functionals possibly containing a nonsmooth part is obtained. Finally, as an application, existence results for nonlinear differential problems depending also on the derivative of the solution are established. [ABSTRACT FROM AUTHOR]

A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T : C → C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a closed convex subset C of a Hilbert space, assuming that there is a point c ∈ C with a bounded orbit and assuming that certain subspace of C b ( S ) has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups introduced by von Neumann in 1929 [28] and formalized by Day in 1957 [5] . In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi. [ABSTRACT FROM AUTHOR]

Let C be a closed, bounded and convex subset of a uniformly convex and uniformly smooth Banach space. Let { T t } t ≥ 0 be a strongly-continuous nonexpansive semigroup on C . Consider the iterative process defined by the sequence of equations x k + 1 = c k T t k + 1 ( x k + 1 ) + ( 1 − c k ) x k . We prove that, under certain conditions, the sequence { x k } converges weakly to a common fixed point of the semigroup { T t } t ≥ 0 . There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property. However, many important spaces like L p for 1 ≤ p ≠ 2 do not possess the Opial property. In this paper, we do not assume the Opial property. We do assume instead that X is uniformly convex and uniformly smooth. L p for p > 1 are prime examples of such spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we extend Mizoguchi–Takahashi's fixed point theorem for multi-valued mappings on a metric space endowed with a graph. As an application, we establish a fixed point theorem on an ε-chainable metric space for mappings satisfying Mizoguchi–Takahashi contractive condition uniformly locally. Also, we establish a result on the convergence of successive approximations for certain operators (not necessarily linear) on a Banach space as another application. Consequently, this result yields the Kelisky–Rivlin theorem on iterates of the Bernstein operators on the space and also enables us study the asymptotic behaviour of iterates of some nonlinear Bernstein type operators on . [Copyright &y& Elsevier]

Abstract: Let be a metric space with a K-valued non-Archimedean metric p. In this paper, we prove the existence and approximation of a fixed point for operators satisfying the contractive condition in the form , where is an increasing operator. Then, we study the generalized Ulam–Hyers stability of fixed point equations. We next obtain an extension of the Krasnoselskii fixed point theorem for the sum of two operators. Finally, an application to functional equations is given. [Copyright &y& Elsevier]

Abstract: In this paper, we provide several extensions of the Abian–Brown Fixed Point Theorem from single-valued mappings to set-valued mappings on chain-complete posets. Then we examine some non-monetized, non-cooperative games where both the collections of the strategies and the ranges of the utilities for the players are posets. By applying the extensions of the Abian–Brown Fixed Point Theorem and by applying the order-preserving property of mappings, we prove some existence theorems of extended and generalized Nash equilibria for non-monetized, non-cooperative games on chain-complete posets. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the notion of generalized hybrid sequences, extending the notion of nonexpansive sequences introduced and studied in our previous work Djafari Rouhani (1981, 1990, 1990, 1997, 2002, 2004, 2002) [2–8], and prove ergodic and convergence theorems for such sequences in a Hilbert space . Subsequently, we apply our results to prove new fixed point theorems for generalized hybrid mappings, first introduced in Kocourek et al. (2010) [14], Takahashi and Takeuchi (2011) [20], defined on arbitrary nonempty subsets of . [Copyright &y& Elsevier]

Abstract: In this paper, we mainly consider the properties of differences of meromorphic solutions for the difference equation concerning a Gamma function, where and are nonzero polynomials. By these properties, we deduce that a Gamma function satisfies that for every , and have same zeros, at most except exceptional zeros, where denotes the order of growth of a meromorphic function , and and denote the exponents of convergence of fixed points and zeros of respectively. [Copyright &y& Elsevier]

Abstract: In this paper, we study the problem of solving multivalued mixed variational inequalities. By using some sequential approximation techniques of fixed point theory, we solve the multivalued mixed variational inequalities involving locally Lipschitzian or locally cocoercive multivalued mappings. We establish that the convexity of the multivalued mapping values is not needed and construct by using the Banach contraction principle converging sequences to the solutions. Also, we show how to choose regularization parameters to compute these solutions. [Copyright &y& Elsevier]

Abstract: Let be a complete 2-uniformly convex metric space. Let be a nonempty, bounded, closed, and convex subset of , and let be an asymptotic pointwise nonexpansive mapping. In this paper, we prove that the modified Mann iteration process defined by converges in a weaker sense to a fixed point of . [Copyright &y& Elsevier]

Abstract: A remarkable fundamental theorem established by Mehta plays an important role in proving existence of fixed points, maximal elements, and equilibria in abstract economies. In this paper, we extend Himmelbergʼs measure of precompactness to the general setting of -spaces and obtain a generalization of Mehtaʼs theorem. As an application, we develop some new fixed point theorems involving a kind of condensing mappings. [Copyright &y& Elsevier]

Abstract: In this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing Henstock–Kurzweil integrable functions. [Copyright &y& Elsevier]

Abstract: Let X be a uniformly convex Banach space with the Opial property. Let be an asymptotic pointwise nonexpansive mapping, where C is bounded, closed and convex subset of X. In this paper, we prove that the generalized Mann and Ishikawa processes converge weakly to a fixed point of T. In addition, we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex Banach spaces, both processes converge strongly to a fixed point. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce two new classes of generalized nonexpansive mapping and we study both the existence of fixed points and their asymptotic behavior. [Copyright &y& Elsevier]

Abstract: This paper considers existence of multiple positive fixed points for some nonlinear operators, a particular case of the operators is sum of an e-concave operator and an e-convex operator. Then we apply the results to nonlinear integral equations. [Copyright &y& Elsevier]

Abstract: This paper investigates the existence of solutions for a class of weighted -Laplacian ordinary system boundary value problems, the sufficient conditions for the existence of solutions have been given via Leray–Schauder''s degree, and the existence of multiple solutions has been discussed via critical point theory. As an application, we discussed the existence of the radial solutions for the -Laplacian partial differential system boundary value problems. [Copyright &y& Elsevier]

Abstract: It is shown, by asymptotic center techniques, that the set of fixed points of any uniformly k-lipschitzian mapping in a uniformly convex Banach space is a retract of the domain when k is less than a constant bigger than the constant from the paper [K. Goebel, W.A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973) 135–140]. Our result improves a recently result presented in [E. Sędłak, A. Wiśnicki, On the structure of fixed-point sets of uniformly lipschitzian mappings, Topol. Methods Nonlinear Anal. 30 (2007) 345–350]. [Copyright &y& Elsevier]

Abstract: As a powerful mechanism, fixed point theorems have many applications in mathematical and economic analysis. In this paper, the well-known Brouwer fixed point theorem and Kakutani fixed point theorem are generalized to a class of nonconvex sets and a globally convergent homotopy method is developed for computing fixed points on this class of nonconvex sets. [Copyright &y& Elsevier]

Abstract: This paper deals with a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real Hilbert space. The proposed method combines two strategies: viscosity approximations (regularization) and inertial type extrapolation. The first strategy is known to ensure the strong convergence of some successive approximation methods, while the second one is intended to speed up the convergence process. Under classical conditions on the operators and the parameters, we prove that the sequence of iterates generated by our scheme converges strongly to the element of minimal norm in the solution set. This algorithm works, for instance, for approximating common fixed points of infinite families of demicontractive mappings, including the classes of quasi-nonexpansive operators and strictly pseudocontractive ones. [Copyright &y& Elsevier]

Abstract: In this paper, we study the fixed point set of a strongly continuous non-expansive semigroup of a semi-topological semigroup S for which is n-extremely left amenable. Also, we study the fixed point set of a strongly continuous semigroup of mappings when S is a semigroup which is a sub-semigroup of a locally convex topological vector space with addition. Some applications to harmonic analysis are also provided. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to present some fixed point results for self-generalized contractions in ordered metric spaces. Our results generalize and extend some recent results of A.C.M. Ran, M.C. 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], J.J. Nieto, R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205–2212], J.J. Nieto, R.L. Pouso, R. Rodríguez-López [J.J. Nieto, R.L. Pouso, R. Rodríguez-López, Fixed point theorem theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505–2517], A. Petruşel, I.A. Rus [A. Petruşel, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411–418] and R.P. Agarwal, M.A. El-Gebeily, D. O''Regan [R.P. Agarwal, M.A. El-Gebeily, D. O''Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., in press]. As applications, existence and uniqueness results for Fredholm and Volterra type integral equations are given. [Copyright &y& Elsevier]

Abstract: In this paper, we study the uniqueness and existence of fixed point of mixed monotone operators in the partially ordered Banach space. The results extend and improve recent related results. [Copyright &y& Elsevier]

Abstract: In this paper, we prove a strong convergence theorem by the hybrid method for a family of nonexpansive mappings which generalizes Nakajo and Takahashi''s theorems [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379], simultaneously. Furthermore, we obtain another strong convergence theorem for the family of nonexpansive mappings by a hybrid method which is different from Nakajo and Takahashi. Using this theorem, we get some new results for a single nonexpansive mapping or a family of nonexpansive mappings in a Hilbert space. [Copyright &y& Elsevier]

Abstract: -convexity was introduced in [W. Briec, C. Horvath, -convexity, Optimization 53 (2004) 103–127]. Separation and Hahn–Banach like theorems can be found in [G. Adilov, A.M. Rubinov, -convex sets and functions, Numer. Funct. Anal. Optim. 27 (2006) 237–257] and [W. Briec, C.D. Horvath, A. Rubinov, Separation in -convexity, Pacific J. Optim. 1 (2005) 13–30]. We show here that all the basic results related to fixed point theorems are available in -convexity. Ky Fan inequality, existence of Nash equilibria and existence of equilibria for abstract economies are established in the framework of -convexity. Monotone analysis, or analysis on Maslov semimodules [V.N. Kolokoltsov, V.P. Maslov, Idempotent Analysis and Its Applications, Math. Appl., vol. 401, Kluwer Academic, 1997; V.P. Litvinov, V.P. Maslov, G.B. Shpitz, Idempotent functional analysis: An algebraic approach, Math. Notes 69 (2001) 696–729; V.P. Maslov, S.N. Samborski (Eds.), Idempotent Analysis, Advances in Soviet Mathematics, Amer. Math. Soc., Providence, RI, 1992], is the natural framework for these results. From this point of view Max-Plus convexity and -convexity are isomorphic Maslov semimodules structures over isomorphic semirings. Therefore all the results of this paper hold in the context of Max-Plus convexity. [Copyright &y& Elsevier]

Abstract: In this paper we propose a nonlinear elasticity model of macromolecular conformational change (deformation) induced by electrostatic forces generated by an implicit solvation model. The Poisson–Boltzmann equation for the electrostatic potential is analyzed in a domain varying with the elastic deformation of molecules, and a new continuous model of the electrostatic forces is developed to ensure solvability of the nonlinear elasticity equations. We derive the estimates of electrostatic forces corresponding to four types of perturbations to an electrostatic potential field, and establish the existence of an equilibrium configuration using a fixed-point argument, under the assumption that the change in the ionic strength and charges due to the additional molecules causing the deformation are sufficiently small. The results are valid for elastic models with arbitrarily complex dielectric interfaces and cavities, and can be generalized to large elastic deformation caused by high ionic strength, large charges, and strong external fields by using continuation methods. [Copyright &y& Elsevier]

Abstract: In this paper, ϕ convex −ψ concave mixed monotone operators are introduced and some new existence and uniqueness theorems of fixed points for mixed monotone operators with such convexity concavity are obtained. As an application, we give one example to illustrate our results. [Copyright &y& Elsevier]

Abstract: In this paper, for a commuting pair consisting of a point-valued nonexpansive self-mapping t and a set-valued nonexpansive self-mapping T of a hyperconvex metric space (or a CAT(0) space) X, we look for a solution to the problem of existence of such that [Copyright &y& Elsevier]

Abstract: This paper is concerned with α-convex operators on ordered Banach spaces. A surjection theorem for 1-convex operators in order intervals is established by means of the properties of cone and monotone iterative technique. It is assumed that 1-convex operator A is increasing and satisfies for , where θ denotes the zero element and is a constant. Moreover, we prove a fixed point theorem for -convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (1) (2007) 506–515], and many others. [Copyright &y& Elsevier]

Abstract: In this paper, we present the definitions of generalized e-concave operators and generalized e-convex operators, which are the generalizations of e-concave operators and e-convex operators, respectively. Without compactness or continuity assumption of generalized e-concave operators and generalized e-convex operators, we have proved the existence, uniqueness and monotone iterative techniques of their fixed points. Our results are even new to e-concave operators and e-convex operators. Finally, we apply the results to the singular boundary value problems for second order differential equations. [Copyright &y& Elsevier]