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In ordered Banach spaces we prove exponential asymptotic stability estimates in Thompson’s metric for the semilinear equation u ′ = A u + f ( u ) , where A is the generator of a positive and exponentially stable C 0 -semigroup and f is a contraction with respect to Thompson’s metric. We apply our results to the study of the semilinear parabolic equation u ′ ( t ) = Δ u ( t ) + u ( t ) κ with homogeneous Dirichlet boundary conditions on bounded domains where ∣ κ ∣ < 1 . [ABSTRACT FROM AUTHOR]

Abstract: Consider in a real Hilbert space the differential equation (E), for a.a. , with the condition , where is a (possibly set-valued) maximal monotone operator, with (or, more generally, ); , with and either or . Recall that equation (E) in the case , subject to and , was investigated in the early 1970s by V. Barbu, who derived in particular from his results a definition for the square root of the nonlinear operator . Subsequently H. Brézis, N.H. Pavel, L. Véron and others have paid attention to equation (E). In this paper we prove the existence and uniqueness of the solution to equation (E) subject to in the weighted space , where , , under our weak assumptions on and (see above) and . For we prove the existence of a generalized solution. This is a classic solution if , . If , the solutions give rise to a nonlinear semigroup of contractions. If is linear its infinitesimal generator is given by . [Copyright &y& Elsevier]

Abstract: We prove a weak convergence theorem for generalized nonexpansive mappings in uniformly convex Banach spaces whose dual has the Kadec–Klee property. This theorem is connected with a famous convergence theorem for nonexpansive mappings proved by Reich in 1979. [Copyright &y& Elsevier]

Abstract: The notion of quasi-Fejér monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis. In this paper, we extend this notion in the context of variable metric algorithms, whereby the underlying norm is allowed to vary at each iteration. Applications to convex optimization and inverse problems are demonstrated. [Copyright &y& Elsevier]

Abstract: It is proved that every equivalent norm in is the starting point of a ray of norms which fail to have an asymptotically isometric copy of . In the case where is separable, it is proved that the set of all equivalent norms in which fail to have an asymptotically isometric copy of is residual. Similar results are obtained for the case of . In particular, we deduce that every separable Banach space has an equivalent norm for which it fails to contain an asymptotically isometric copy of and . [Copyright &y& Elsevier]

Abstract: Let be a 2-uniformly smooth real Banach space. For each , let be bounded and accretive mappings. Assume that the generalized Hammerstein equation has a solution in . We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein equation. Our result improves, generalizes and extends the results of Chidume and Ofoedu [C.E. Chidume, E.U. Ofoedu, Solution of nonlinear integral equations of Hammerstein type, Nonlinear Anal. 74 (2011) 4293–4299] and Chidume and Shehu [C.E. Chidume, Y. Shehu, Approximation of solutions of generalized equations of Hammerstein type, Comput. Math. Appl. 63 (2012) 966–974]. [Copyright &y& Elsevier]

Abstract: It is a known fact that the method of alternating projections introduced long ago by von Neumann fails to converge strongly for two arbitrary nonempty, closed and convex subsets of a real Hilbert space. In this paper, a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved. If the two operators are subdifferentials of indicator functions, this new algorithm coincides with the old method of alternating projections. Several other important algorithms, such as the contraction proximal point algorithm, occur as special cases of our algorithm. Hence our main results generalize and unify many results that occur in the literature. [Copyright &y& Elsevier]

Abstract: Let be a real Hilbert space. Let be bounded, continuous and monotone mappings. Suppose that is a solution to the Hammerstein equation . We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the Hammerstein equation. Furthermore, we give some examples to show that our result is interdisciplinary in nature, covers a large variety of areas and should be of much interest to a wide audience. [Copyright &y& Elsevier]

Abstract: We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that every noncommutative -space associated to a finite von Neumann algebra can be renormed to satisfy the fixed point property for nonexpansive affine mappings. Particular examples are , where is the hyperfinite factor and the function spaces and for any -finite measure space. This property does not hold for the usual norm. [Copyright &y& Elsevier]

Abstract: In this paper, the concept of eigenvalue is introduced to the monotone homogeneous mapping pairs , and its existence and uniqueness are established successfully under the boundedness of some orbits of in the Hilbert semi-norm. Also, the Collatz–Wielandt min–max type property is obtained for such a class of mapping pairs. In particular, the nonlinear Perron–Frobenius property for nonnegative tensor pairs is obtained without involving the calculation of the tensor inversion. By particularizing the mapping or , several results can emerge as corollaries. They lead to further existence results and open problems. [Copyright &y& Elsevier]

Abstract: By modifying von Neumann’s alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert space. [Copyright &y& Elsevier]

Abstract: We study the strong convergence of two viscosity iteration processes for pseudo-contractive semigroup and for -strongly pseudo-contractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. As special cases, we get strong convergence of two viscosity iteration processes for approximating common fixed points of nonexpansive semigroups in certain Banach spaces. The results presented in this paper extend and generalize previous results. [Copyright &y& Elsevier]

Abstract: The purpose of this article is to propose a new hybrid algorithm with variable coefficients and prove convergence theorems for asymptotically pseudocontractive mappings in the intermediate sense on unbounded domains. The results of the paper improve and extend the recent results of Qin et al. [X.L. Qin, S.Y. Cho, J.K. Kim, Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense, Fixed Point Theory Appl. 2010, doi:10.1155/2010/186874, Article ID 186874, 14 pages] and several others. The algorithm with variable coefficients introduced in this paper is of independent interest. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, it is proved that the sequence defined by Mann’s algorithm -converges to a fixed point in complete CAT() spaces. [Copyright &y& Elsevier]

Abstract: A new iterative method for approximating fixed points of bounded and continuous pseudocontractive mapping is proposed and a strong convergence theorem is obtained. As an application, we prove that a slight modification of our new scheme could be employed for approximating zeros of bounded and continuous accretive operators. Our theorems extend and unify most of the results that have been proved for this class of mappings. [Copyright &y& Elsevier]

Abstract: Let be a nonempty, closed and convex subset of a real Hilbert space . Let , be a finite family of Lipschitz pseudocontractive mappings. It is our purpose, in this paper, to prove strong convergence of Ishikawa’s method to a common fixed point of a finite family of Lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on or on . Moreover, computation of the closed convex set for each is not required. The results obtained in this paper improve on most of the results that have been proved for this class of nonlinear mappings. [Copyright &y& Elsevier]

Abstract: In this paper, we prove the existence and convergence of best proximity points for asymptotic cyclic contractions in metric spaces with the property UC, as well as for asymptotic proximal pointwise contractions in uniformly convex Banach spaces. Moreover, we consider a generalized cyclic contraction mapping and prove the existence of best proximity points for such a mapping in Banach spaces which do not necessarily satisfy the geometric property UC. [Copyright &y& Elsevier]

Abstract: The hybrid method in mathematical programming was introduced by Haugazeau (1968)  and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001)  proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009)  extended Bauschke and Combettes’s results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) . And we get strong convergence theorems for a family of asymptotically -strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings. [Copyright &y& Elsevier]

Abstract: Given a nonempty closed subset of a Hilbert space , we denote by the space of all bounded Lipschitz mappings from into , equipped with the supremum norm. We show that there is a continuous mapping such that for each , , , and . We also prove that the corresponding set-valued extension operator is lower semicontinuous. [Copyright &y& Elsevier]

Abstract: In this work, an implicit iterative process is considered for asymptotically pseudocontractive nonself-mappings. Weak and strong convergence theorems for common fixed points of a family of asymptotically pseudocontractive nonself-mappings are established in the framework of Hilbert spaces. [Copyright &y& Elsevier]

Abstract: Let be a nonempty closed convex subset of a real Hilbert space . Let be a non-expansive mapping and be an infinite family of non-expansive mappings. The purpose of this paper is to find the minimum norm solution of the following general hierarchical fixed point problem We introduce an explicit regularized algorithm with strong convergence for finding the minimum norm solution of the above hierarchical fixed point problem. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce hybrid pseudo-viscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of an infinite family and left amenable semigroup of non-expansive mappings in the frame work of Hilbert spaces. Our goal is to prove a result of strong convergence for hybrid pseudo-viscosity approximation schemes to approach a solution of systems of equilibrium problems which is also a common fixed point of an infinite family and left amenable semigroup of non-expansive mappings. The results presented in this paper can be treated as an extension and improvement of the corresponding results announced by Ceng et al. [L.C. Ceng, Q.H. Ansari, and J.C. Yao, Hybrid pseudo-viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many non-expansive mappings, Nonlinear Analysis 4 (2010) 743–754] and many others. [Copyright &y& Elsevier]

Abstract: We present a sharp evaluation of the Lipschitz constant for every iterate for mean lipschitzian mappings, solving a problem posed by K. Goebel and M. A. Japón Pineda. [Copyright &y& Elsevier]

Abstract: The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35–61] has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. In this paper, we present several algorithms based on alternating resolvents of two maximal monotone operators, and , that can be used to approximate common zeros of and . In particular, we prove that the sequences generated by our algorithms converge strongly. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals. [Copyright &y& Elsevier]

Abstract: Let be a -uniformly real Banach space and be nonlinear-bounded accretive operators. Assume that the Hammerstein equation has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators and are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator and is not restricted to be an angle-bounded (necessarily linear) operator. [Copyright &y& Elsevier]

Abstract: We study the existence of monotonic solutions of a quadratic fractional Hammerstein–Volterra integral equation with linear modification of the argument. The quadratic integral equation studied below contains as a special case numerous integral equations encountered in the theory of radiative transfer and in the kinetic theory of gases. We show that the quadratic fractional Hammerstein–Volterra integral equation with linear modification of the argument has a monotonic solution in the Banach space of all real functions defined and continuous on a bounded and closed interval. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof. [Copyright &y& Elsevier]

Abstract: Let be a normed space and and be two nonexpansive self-maps on the closed convex subset . It is proved that if is a Banach operator pair and is compact then . A family of nonexpansive maps is also investigated; the well-known De Marr’s fixed point theorem is extended to the noncommuting case by introducing a notion of Banach operator family. As an application of the theorem, the problem for the common fixed point in invariant approximation for convex set is solved directly in a quite different way from the others before. [Copyright &y& Elsevier]

Abstract: Let be a finite von Neumann algebra. It is known that and every non-reflexive subspace of fail to have the fixed point property for non-expansive mappings (FPP). We prove a new fixed point theorem for this class of mappings in non-commutative Banach spaces which lets us obtain a sufficient condition such that a closed subspace of can be renormed to satisfy the FPP. As a consequence, we deduce that the predual of every atomic finite von Neumann algebra can be renormed with the FPP. [Copyright &y& Elsevier]

Abstract: Let be a complete CAT(0) space, be a generalized multivalued nonexpansive mapping, and be a single valued quasi-nonexpansive mapping. Under the assumption that and commute weakly, we shall prove the existence of a common fixed point for them. In this way, we extend and improve a number of recent results obtained by Shahzad (2009) , Shahzad and Markin (2008) , and Dhompongsa et al. (2005) . [ABSTRACT FROM AUTHOR]

Abstract: Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in Kurokawa and Takahashi [Y. Kurokawa, W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 1562–1568]. We explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of Halpern’s type for our class of mappings and hence resolved in the affirmative an open problem posed by Kurokawa and Takahashi in their final remark for the case where the mapping is averaged. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to prove the existence of a fixed point for a pointwise eventually nonexpansive mapping in a nearly uniformly convex Banach space. This provides an affirmative answer to a question given by Kirk and Xu [W.A. Kirk, Hong-Kun Xu, Asymptotic pointwise contraction, Nonlinear Anal. 69 (2008), 4706–4712]. [ABSTRACT FROM AUTHOR]

Abstract: Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm — one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) and later developed by Xu (2006) . The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature. [Copyright &y& Elsevier]

Abstract: We introduce an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems and common solutions of finite family of variational inequality problems for monotone mappings in Banach spaces. Our theorem extends and unifies most of the results that have been proved for this important class of nonlinear operators. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to extend the invariance of domain theorem to a large class of countably 1--contractive maps, by using homotopy theory and degree theory for countably 1--contractive maps. [ABSTRACT FROM AUTHOR]

Abstract: Assume that is a real Banach space with uniformly normal structure and is a nonempty closed convex subset of . We show that a -uniformly Lipschitzian semigroup of nonlinear self-mappings of admits a common fixed point if the semigroup has a bounded orbit and if is appropriately greater than one. This result applies, in particular, to the framework of uniformly convex Banach spaces. [ABSTRACT FROM AUTHOR]

Abstract: Let be a Hilbert space. Consider on a sequence of nonexpansive mappings with common fixed points, an equilibrium function , a contraction with coefficient and a strongly positive linear bounded operator with coefficient . Let . We define a suitable Mann type algorithm which strongly converges to the unique solution of the minimization problem , where is a potential function for and is the intersection of the equilibrium points and the common fixed points of the sequence . [Copyright &y& Elsevier]

Abstract: We prove strong convergence theorems for a sequence which is generated by Halpern’s iteration. We also apply our result for finding zeros of an accretive operator. Our result improves the recent result of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350–2360] by removing some assumptions on the parameters. Finally we discuss the new sufficient condition studied by Song [Y. Song, A new sufficient condition for the strong convergence of Halpern type iterations. Appl. Math. Comput. 198 (2) (2008) 721–728; Y. Song, New strong convergence theorems for nonexpansive nonself-mappings without boundary conditions. Comput. Math. Appl. 56 (6) (2008) 1473–1478] and correct the main result of Song and Chai [Y. Song, X. Chai, Halpern iteration for firmly type nonexpansive mappings, Nonlinear Anal. 71 (10) (2009) 4500–4506]. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to prove by using a new hybrid method a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem, the set of solutions for a variational inequality problem and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space. As applications, we utilize our results to obtain some new results for finding a solution of an equilibrium problem, a fixed point problem and a common zero-point problem for maximal monotone mappings in Banach spaces. [Copyright &y& Elsevier]

Abstract: Very recently, Yao, Chen and Yao  proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new general iterative method for finding a common element of the set of solutions of a mixed equilibrium problem (MEP), the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of variational inequalities for a -inverse-strongly monotone mapping in Hilbert spaces. Furthermore, we establish the strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under some suitable conditions, which solves some optimization problems. Our results extend and improve the recent results of Yao et al. [Y. Yao, M.A. Noor, S. Zainab, Y.C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl. 354 (2009) 319–329; Y. Yao, M. A. Noor, Y.C. Liou, On iterative methods for equilibrium problems, Nonlinear Anal. 70 (1) (2009) 479–509] and many others. [Copyright &y& Elsevier]

Abstract: In this work, we give certain control conditions for a modified Ishikawa iteration to compute common fixed points of a kind of nonexpansive semigroup in Banach spaces. These results improve and extend those in Somyot and Rattanaporn (2008) and Song and Xu (2008) . [Copyright &y& Elsevier]

Abstract: Let be a real Hilbert space. Suppose that is a nonexpansive mapping on with a fixed point, is a contraction on with coefficient , and is a -Lipschitzian and -strongly monotone operator with . Let . We proved that the sequence generated by the iterative method converges strongly to a fixed point , which solves the variational inequality , for . [Copyright &y& Elsevier]

Abstract: In this paper we propose a new modified viscosity approximation method for approximating common fixed points for a countable family of nonexpansive mappings in a Banach space. We prove strong convergence theorems for a countable family nonexpansive mappings in a reflexive Banach space with uniformly Gateaux differentiable norm under some control conditions. These results improve and extend the results of Jong Soo Jung [J.S. Jung, Convergence on composite iterative schemes for nonexpansive mappings in Banach spaces, Fixed Point Theory and Appl. 2008 (2008) 14 pp., Article ID 167535]. Further, we apply our result to the problem of finding a zero of an accretive operator and extend the results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], Ceng, et al. [L.-C. Ceng, A.R. Khan, Q.H. Ansari, J.-C, Yao, Strong convergence of composite iterative schemes for zeros of -accretive operators in Banach space, Nonlinear Anal. 70 (2009)1830–1840] and Chen and Zhu [R. Chen, Z. Zhu, Viscosity approximation methods for accretive operator in Banach space, Nonlinear Anal. 69 (2008) 1356–1363]. [Copyright &y& Elsevier]

Abstract: We study the convergence of two iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in reflexive Banach spaces. Both algorithms take into account possible computational errors. We establish two strong convergence theorems and then apply them to the solution of convex feasibility, variational inequality and equilibrium problems. [Copyright &y& Elsevier]

Abstract: Consider a finite family of nonexpansive mappings which are defined on a closed convex subset of a Hilbert space . Suppose the set of common fixed points of this family is nonempty. We then address the problem of finding the minimum-norm element of this common fixed point set. We introduce both cyclic and parallel iteration methods to find this minimal-norm element. [Copyright &y& Elsevier]

Abstract: Consider the variational inequality (VI) of finding a point such that where are nonexpansive self-mappings of a closed convex subset of a Hilbert space, and is the set of fixed points of . Assume that the solution set of this VI is nonempty. This paper introduces two schemes, one implicit and one explicit, that can be used to find the minimum-norm solution of VI (∗); namely, the unique solution to the quadratic minimization problem: . [Copyright &y& Elsevier]