hybrid iterative scheme for fixed point problem infinite systems of equilibrium and variational inequality problems the purpose of this paper is to construct a new iterative scheme by the hybrid method and prove a strong convergence theorem for the approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a common solution to infinite systems of equilibrium and variational inequality problems in a uniformly smooth and strictly convex real banach space with kadec klee property using the properties of a generalized f projection operator using this result we give some applications our results extend many known recent results in the literature

convergence theorems best approximation and best proximity for set valued dynamic systems of relatively quasi asymptotic contractions in cone uniform spaces abstract in cone uniform spaces x using the concept of the d family of cone pseudodistances the distance between two not necessarily convex or compact sets a and b in x is defined the concepts of cyclic and noncyclic set valued dynamic systems of d relatively quasi asymptotic contractions t a b 2 a b are introduced and the best approximation and best proximity point theorems for such contractions are proved also conditions are given which guarantee that for each starting point each generalized sequence of iterations of these contractions in particular each dynamic process converges and the limit is a best proximity point moreover d families are constructed characterized and compared the results are new for set valued and single valued dynamic systems in cone uniform cone locally convex and cone metric spaces various examples illustrating ideas methods definitions and results are constructed

best proximity points for asymptotic cyclic contraction mappings 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

two extragradient methods for generalized mixed equilibrium problems nonexpansive mappings and monotone mappings in this paper we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone lipschitz continuous mapping we obtain some weak convergence theorems for the sequences generated by these processes in hilbert spaces the results in this paper generalize extend and unify some well known weak convergence theorems in the literature

inertial extragradient algorithms for strongly pseudomonotone variational inequalities abstract the purpose of this paper is to study and analyze two different kinds of inertial type iterative methods for solving variational inequality problems involving strongly pseudomonotone and lipschitz continuous operators in hilbert spaces the projection method is used to design the algorithms which can be computed more easily the construction of solution approximations and the proof of convergence of the algorithms are performed without prior knowledge of the modulus of strong pseudomonotonicity and the lipschitz constant of cost operator instead of that the algorithms use variable stepsize sequences which are diminishing and non summable the numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with several previously known algorithms

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

on the achievable rate region of sequential decoding for a class of multiaccess channels the achievable rate region of sequential decoding for the class of pairwise reversible multiaccess channels is determined this result is obtained by finding tight lower bounds to the average list size for the same class of channels the average list size is defined as the expected number of incorrect messages that appear to a maximum likelihood decoder to be at least as likely as the correct message the average list size bounds developed may be of independent interest with possible applications to list decoding schemes

algorithms of common solutions for variational inclusions mixed equilibrium problems and fixed point problems in this paper we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of an infinite family of nonexpansive mappings and the set of a variational inclusion in a real hilbert space furthermore we prove that the proposed iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters

error estimates for approximation of coupled best proximity points for cyclic contractive maps we enrich the known results about coupled fixed and best proximity points of cyclic contraction ordered pair of maps the uniqueness of the coupled best proximity points for cyclic contraction ordered pair of maps in a uniformly convex banach space is proven we find a priori and a posteriori error estimates for the coupled best proximity points obtained by sequences of successive iterations when the underlying banach space has modulus of convexity of power type a looser conditions are presented for the existence and uniqueness of coupled fixed points of a cyclic contraction ordered pair of maps in a complete metric space and a priori a posteriori error estimates and the rate of convergence for the coupled fixed points are obtained for the sequences of successive iterations we apply these results for solving systems of integral equations systems of linear and nonlinear equations

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

on a cyclic jungck modified ts iterative procedure with application examples abstract this article investigates some convergence properties of quasi cyclic and cyclic jungck modified ts iterative schemes in complete metric spaces and banach spaces the uniqueness of the best proximity points is investigated it is basically assumed that one of the self mappings is asymptotically nonexpansive while the other is asymptotically contractive with several particular cases some application examples are also discussed

projected reflected gradient methods for monotone variational inequalities this paper is concerned with some new projection methods for solving variational inequality problems with monotone and lipschitz continuous mapping in hilbert space first we propose the projected reflected gradient algorithm with a constant stepsize it is similar to the projected gradient method namely the method requires only one projection onto the feasible set and only one value of the mapping per iteration this distinguishes our method from most other projection type methods for variational inequalities with monotone mapping also we prove that it has r linear rate of convergence under the strong monotonicity assumption the usual drawback of algorithms with constant stepsize is the requirement to know the lipschitz constant of the mapping to avoid this we modify our first algorithm so that the algorithm needs at most two projections per iteration in fact our computational experience shows that such cases with two projections are very rare this scheme at least theoretically seems to be ve

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

viscosity iteration method for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings abstract in this paper we introduce an iterative scheme for finding a common element of the set of solutions of generalized equilibrium problem and the set of common fixed point of family of nonexpansive mappings in a hilbert space we prove the strong convergence of viscosity iterative algorithm to common element of the set of generalized equilibrium problem and the set of common fixed point of family of nonexpansive mappings the result improve the recent ones of ceng and yao 2008 l c ceng j c yao hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings appl math comput 198 2008 729 741

existence of the solution to variational inequality optimization problem and elliptic boundary value problem through revisited best proximity point results abstract in this paper we establish some results regarding the existence of the solution to variational inequality optimization problem and elliptic boundary value problem in hilbert spaces our strategy consists in establishing new best proximity point results in the metric spaces by introducing the concept of cyclic orbital simulative contractions we also provide nontrivial examples to show that our results are proper generalization further we improve the recent best proximity results for mappings satisfying proximal simulative conditions due to abbas et al 2017 samet 2015 and tchier et al 2016 via new class of simulation functions our results unify extend and generalize various existing results

best proximity points for cyclic meir keeler contractions abstract we introduce a notion of cyclic meir keeler contractions and prove a theorem which assures the existence and uniqueness of a best proximity point for cyclic meir keeler contractions this theorem is a generalization of a recent result due to eldred and veeramani

best proximity point theorems an exploration of a common solution to approximation and optimization problems abstract given a non self mapping t a b in the setting of a metric space this work concentrates on the resolution of the non linear programming problem of globally minimizing the real valued function x d x tx thereby yielding an optimal approximate solution to the equation tx x an iterative algorithm is also presented to compute a solution of such problems as a sequel it is possible to compute an optimal approximate solution to some non linear equations

weak and strong convergence criteria of noor iterations for asymptotically nonexpansive mappings in this paper weak and strong convergence theorems are established for a three step iterative scheme for asymptotically nonexpansive mappings in banach spaces mann type and ishikawa type iterations are included by the new iterative scheme the results obtained in this paper extend and improve the recent ones announced by xu and noor glowinski and le tallec noor ishikawa and many others

best proximity points convergence and existence theorems for p cyclic mappings abstract we introduce a new class of mappings called p cyclic contractions which contains the p cyclic contraction mappings as a subclass then convergence and existence results of best proximity points for p cyclic contraction mappings are obtained moreover we prove results of the existence of best proximity points in a reflexive banach space these results are generalizations of the results of al thagafi and shahzad 2009 8

viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems abstract in this paper we first introduce a general iterative scheme by halpern approximation method for finding a common element of the set of solutions of an equilibrium problem the set of solutions of the variational inequality for an infinite family of i inverse strongly monotone mappings and the set of common fixed points for a family of infinitely nonexpansive mappings in a hilbert space then we extend the halpern iterative scheme to a general viscosity approximation scheme by a simple and different method as applications we utilize our results to investigate some convergence problems for strictly pseudocontractive mappings and maximal monotone operators the results obtained in this paper extend and improve the recent ones announced by many others

on modified iterative method for nonexpansive mappings and monotone mappings abstract in this paper we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for inverse strongly monotone mappings we show that the sequence converges strongly to a common element of the above two sets under some parameters controlling conditions

convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems abstract the purpose of this work is to introduce a hybrid projection method for finding a common element of the set of a generalized equilibrium problem the set of solutions to a variational inequality and the set of fixed points of a strict pseudo contraction in a real hilbert space

an upper bound on the cutoff rate of sequential decoding an upper bound is given on the cutoff rate of discrete memoryless channels this upper bound which coincides with a known lower bound determines the cutoff rate and settles a long standing open problem

viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove a strong convergence theorem which is connected with combettes and hirstoaga s result p l combettes s a hirstoaga equilibrium programming in hilbert spaces j nonlinear convex anal 6 2005 117 136 and wittmann s result r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 using this result we obtain two corollaries which improve and extend their results

viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications abstract in this paper we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems the set of common fixed points of a family of infinite non expansive mappings strong convergence theorems are established in a real hilbert space under suitable conditions as some applications we consider convex feasibility problems and equilibrium problems the results presented improve and extend the corresponding results of many others

on cyclic meir keeler contractions in metric spaces abstract cyclic meir keeler contractions are considered under the recently introduced wuc and hw properties on pairs of subsets of metric spaces we show that in contrast with previous results in the theory best proximity point theorems under these properties do not directly extend from cyclic contractions to cyclic meir keeler contractions we obtain however a positive result for cyclic meir keeler contractions under additional properties which is shown to be an extension of already existing results for cyclic contractions moreover we give examples supporting the necessity of our additional conditions

a viscosity approximation scheme for system of equilibrium problems nonexpansive mappings and monotone mappings abstract in this paper we introduce a new viscosity approximation scheme based on the extragradient method 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 of nonexpansive mappings and the set of solutions to the variational inequality for a monotone lipschitz continuous mapping several convergence results for the sequences generated by these processes in hilbert spaces were derived

equilibrium programming using proximal like algorithms we compute constrained equilibria satisfying an optimality condition important examples include convex programming saddle problems noncooperative games and variational inequalities under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization in the main algorithm each stage of computation requires two proximal steps possibly using bregman functions one step serves to predict the next point the other helps to correct the new prediction to enhance practical applicability we tolerate numerical errors

a hybrid extragradient method for approximating the common solutions of a variational inequality a system of variational inequalities a mixed equilibrium problem and a fixed point problem abstract in this paper we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem a system of variational inequality problems a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real hilbert space further we establish a strong convergence theorem based on this method the results presented in this paper improves and generalizes the results given in yao et al 36 and ceng et al 7 and some known corresponding results in the literature

existence and uniqueness of best proximity points in geodesic metric spaces abstract a mapping t a b a b such that t a b and t b a is called a cyclic mapping a best proximity point x for such a mapping t is a point such that d x t x dist a b in this work we provide different existence and uniqueness results of best proximity points in both banach and geodesic metric spaces we improve and extend some results on this recent theory and give an affirmative partial answer to a recently posed question by eldred and veeramani in a a eldred p veeramani existence and convergence of best proximity points j math anal appl 323 2 2006 1001 1006

a new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings in this paper we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for a inverse strongly monotone mapping in hilbert spaces we prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main result improve and extend plubtieng and punpaeng s corresponding result s plubtieng r punpaeng a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings applied mathematics and computation 197 2008 548 558 using this theorem we obtain three corollaries

some new algorithms for solving mixed equilibrium problems in this paper we suggest and analyze two projection methods one implicit and one explicit for finding a particular solution of a mixed equilibrium problem in a real hilbert space furthermore we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem

sequential decoding for multiple access channels the use of sequential decoding in multiple access channels is considered the fano metric which achieves all achievable rates in the one user case fails to do so in the multiuser case a new metric is introduced and an inner bound is given to it achievable rate region this inner bound region is large enough to encourage the use of sequential decoding in practice the new metric is optimal in the sense of achieving all achievable rates in the case of one user and pairwise reversible channels whether the metric is optimal for all multiple access channels remains an open problem it is noted that even in the one user case the new metric differs from the fano metric in a nontrivial way showing that the fano metric is not uniquely optimal for such channels a new and stricter criterion of achievability in sequential decoding is also introduced and examined

uniform convexity strong convexity and property uc abstract property uc introduced in 2009 plays a crucial role in several existence and convergence results for best proximity points and fixed points in this paper we present two approximation theoretic characterizations of uniform convexity and as consequences of these results we characterize the uniform convexity in terms of property uc characterizations of strong convexity in terms of property uc are also presented

convergence theorems of common elements for equilibrium problems and fixed point problems in banach spaces the purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi f nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of banach spaces our results improve and extend the corresponding results announced by many others

approximation theorems for solving common solution of a system of mixed equilibrium problems and variational inequality problems and fixed point problems for asymptotically strict pseudocontractions in the intermediate sense abstract in this paper we introduce a new iterative algorithm which is constructed by using the hybrid projection method for finding the common solution of a system of mixed equilibrium problems of bifunctions satisfying certain the conditions and the common solution of variational inequality problems of inverse strongly monotone mappings and the common solution of fixed point problems for a family of uniformly lipschitzian continuous and asymptotically i strict pseudocontractive mapping in the intermediate sense we prove the strong convergence theorem for a new iterative algorithm under some mild conditions in hilbert spaces finally we also give some numerical examples for support our results

strong convergence theorems for infinite family of relatively quasi nonexpansive mappings and systems of equilibrium problems abstract in this paper we construct a new iterative scheme by hybrid method for approximation of common element of set of common fixed points of countably infinite family of relatively quasi nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real banach space using the properties of generalized f projection operator then we prove strong convergence of the scheme to a common element of the two sets furthermore we apply our results to solve convex minimization problem our results extend important recent results

a new projection method for variational inequality problems we propose a new projection algorithm for solving the variational inequality problem where the underlying function is continuous and satisfies a certain generalized monotonicity assumption e g it can be pseudomonotone the method is simple and admits a nice geometric interpretation it consists of two steps first we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem this procedure requires a single projection onto the feasible set and employs an armijo type linesearch along a feasible direction then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set thus in contrast with most other projection type methods only two projection operations per iteration are needed the method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions preliminary computational experience is also reported

on the convergence of modified noor iterations with errors for asymptotically nonexpansive mappings abstract in this paper several weak and strong convergence theorems are established for a modified noor iterative scheme with errors for three asymptotically nonexpansive mappings in banach spaces mann type ishikawa type and noor type iterations are covered by the new iteration scheme our results extend and improve the recently announced ones b l xu m a noor fixed point iterations for asymptotically nonexpansive mappings in banach spaces j math anal appl 267 2002 444 453 y j cho h zhou g guo weak and strong convergence theorems for three step iterations with errors for asymptotically nonexpansive mappings comput math appl 47 2004 707 717 and many others

dynamical analysis of fractional order modified logistic model in this paper we study a fractional differential equation model of the single species multiplicative allee effect first we study the stability of equilibrium points further we give some sufficient conditions ensuring the existence and uniqueness of integral solution in the last section we perform several numerical simulations to validate our analytical findings

an outer approximation method for the variational inequality problem we study two outer approximation schemes applied to the variational inequality problem in reflexive banach spaces first we propose a generic outer approximation scheme and its convergence analysis unifies a wide class of outer approximation methods applied to the constrained optimization problem as is standard in this setting boundedness and optimality of weak limit points are proved to hold under two alternative conditions i boundedness of the feasible set or ii coerciveness of the operator to develop a convergence analysis where i and ii do not hold we consider a second scheme in which the approximated subproblems use a coercive approximation of the original operator under conditions alternative to both i and ii we obtain standard convergence results furthermore when the space is uniformly convex we establish full strong convergence of the second scheme to a solution

on linear convergence of iterative methods for the variational inequality problem abstract we present new linear convergence results for iterative methods for solving the variational inequality problem the methods include the extragradient method the proximal point method a matrix splitting method and a certain feasible descent method the proofs of the results are based on certain error bounds related to the algorithmic mappings moreover we show that all these error bounds hold if a projection type error bound holds

a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings abstract in this paper we introduce a new iterative scheme 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 inverse strongly monotone mappings we show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main theorem extends a recent result of yao and yao y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings applied mathematics and computation 186 2 2007 1551 1558

some new projection methods for variational inequalities in this paper we propose some new double projection methods for solving variational inequalities by using the wiener hopf equations technique it is shown that these methods converge linearly under mild conditions and include some existing projection methods as special cases some examples are given to illustrate the efficiency of the proposed methods

hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems in this paper we introduce a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real hilbert space then we prove strong convergence of the scheme to a common element of the three above described sets our result improves and extends the corresponding results announced by many others

a new type of fixed point theorem in metric spaces abstract we prove a generalization of edelstein s fixed point theorem though there are thousands of fixed point theorems in metric spaces our theorem is a new type of theorem

asymptotically non expansive self maps and global stability with ultimate boundedness of dynamic systems this paper investigates self maps t x x which satisfy a distance constraint in a metric space with mixed point dependent non expansive properties or in particular contractive ones and potentially expansive properties related to some distance threshold the above mentioned constraint is feasible in certain real world problems of usefulness for instance when discussing ultimate boundedness in dynamic systems which guarantees lyapunov stability this fact makes the proposed analysis to be potentially useful to investigate global stability properties in dynamic systems in the potential presence of some locally unstable equilibrium points the results can be applied to stability problems of dynamics systems and circuit theory as the given examples suggest

wiener hopf equation technique for variational inequalities and nonexpansive mappings in this paper we introduce and consider a new class of wiener hopf equations involving the nonlinear operator and nonexpansive operators essentially using the projection technique we establish the equivalence between the wiener hopf equations and variational inequalities using this alternative equivalent formulation we suggest and analyze an iterative method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of the variational inequalities we also study the convergence criteria of iterative methods under some mild conditions our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results

on best proximity pair theorems for relatively u continuous mappings abstract a new class of mappings called relatively u continuous is introduced and used to investigate the existence of best proximity points as an application of the existence theorem we obtain a generalized version of the markov kakutani theorem for best proximity points in the setting of a strictly convex banach space

best proximity point theorems generalizing the contraction principle abstract the primary objective of this article is to elicit some interesting extensions of the simple but powerful banach s contraction principle to the case of non self mappings in fact due to the fact that best proximity point theorems fit the bill to this end the proposed extensions are presented as best proximity point theorems for non self proximal contractions which are more general than the notion of self contractions

weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations abstract let h be a real hilbert space k a nonempty closed convex subset of h and t k k a lipschitz pseudocontraction with a nonempty fixed point set weak and strong convergence theorems for the iterative approximation of fixed points of t are proved furthermore if t h h is a lipschitz monotone operator and f r t where r t denotes the range of t weak and strong convergence theorems for the iterative approximation of solutions of the operator equation tx f are proved

approximate homotopy symmetry and infinite series solutions to the perturbed mkdv equation abstract we devise a homotopy model for the perturbed mkdv equation with weak fourth order dispersion and weak dissipation and demonstrate that the related approximate equations lead to similarity reduction equations and similarity reduction solutions of different orders which are formally coincident respectively series reduction solutions for the perturbed mkdv equation are thus derived painleve ii type equations or hyperbolic secant function solutions are obtained for zero order similarity reduction equations higher order similarity reduction equations are linear variable coefficients ordinary differential equations

best proximity points for cyclic and noncyclic set valued relatively quasi asymptotic contractions in uniform spaces abstract given a uniform space x and nonempty subsets a and b of x we introduce the concepts of some families v of generalized pseudodistances on x of set valued dynamic systems of relatively quasi asymptotic contractions t a b 2 a b with respect to v and best proximity points for t in a b and we describe the methods which we use to establish the conditions guaranteeing the existence of best proximity points for t when t is cyclic i e t a 2 b and t b 2 a or when t is noncyclic i e t a 2 a and t b 2 b moreover we establish conditions guaranteeing that for each starting point each generalized sequence of iterations of these contractions in particular each dynamic process converges and the limit is a best proximity point for t in a b these best proximity points for t are determined by unique endpoints in a b for a map t 2 when t is cyclic and for a map t when t is noncyclic the results and the methods are new for set valued and single valued dynamic systems in uniform locally convex metric and banach spaces various examples illustrating the ideas of our definitions and results and fundamental differences between our results and the well known ones are given

a best proximity point theorem for weakly contractive non self mappings abstract let us consider a map t a b where a and b are two nonempty subsets of a metric space x the aim of this article is to provide sufficient conditions for the existence of a unique point x in a called the best proximity point which satisfies d x t x dist a b inf d a b a a b b our result generalizes a result due to rhoades b e rhoades some theorems on weakly contractive maps nonlinear analysis tma 47 2001 2683 2693 and hence it provides an extension of banach s contraction principle to the case of non self mappings

numerical approach to monotone variational inequalities by a one step projected reflected gradient method with line search procedure abstract in this paper we revisit a recent approach to classical monotone variational inequalities by means of a projected reflected gradient type method in r n a line search procedure is incorporated for possible varying step sizes and no lipschitz continuity condition on the operator is required a main feature of the proposed method is that it requires only one projection step onto the feasible set at each iteration

an iterative algorithm for approximating convex minimization problem we suggest and analyze an iterative algorithm without the assumption of any type of commutativity on an infinite family of nonexpansive mappings we show that the proposed iterative algorithm converges to the unique minimizer of some quadratic function over the common fixed point sets of an infinite family of nonexpansive mappings our result extend and improve many results announced by many authors

the existence of best proximity points in metric spaces with the property uc abstract eldred and veeramani in a a eldred p veeramani existence and convergence of best proximity points j math anal appl 323 2006 1001 1006 mr2260159 proved a theorem which ensures the existence of a best proximity point of cyclic contractions in the framework of uniformly convex banach spaces in this paper we introduce a notion of the property uc and extend the eldred and veeramani theorem to metric spaces with the property uc

some results on generalized equilibrium problems involving a family of nonexpansive mappings abstract in this paper we introduce an iterative method for finding a common element in the solution set of generalized equilibrium problems in the solution set of variational inequalities and in the common fixed point set of a family of nonexpansive mappings strong convergence theorems are established in the framework of hilbert spaces

convergence and existence results for best proximity points we provide a positive answer to a question raised by eldred and veeramani a a eldred p veeramani existence and convergence of best proximity points j math anal appl 323 2006 1001 1006 about the existence of a best proximity point for a cyclic contraction map in a reflexive banach space moreover we introduce a new class of maps called cyclic contractions which contains the cyclic contraction maps as a subclass convergence and existence results of best proximity points for cyclic contraction maps are also obtained 2008 elsevier ltd all rights reserved

a new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization abstract in this paper we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for inverse strongly monotone mappings in a hilbert space under suitable conditions some strong convergence theorems for approximating a common element of the above three sets are obtained as applications at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings the results presented in the paper extend and improve some recent results of yao and yao y y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput 186 2 2007 1551 1558 plubtieng and punpaeng s plubtieng r punpaeng a new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings appl math comput 2007 doi 10 1016 j amc 2007 07 075 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 2006 506 515 su shang and qin y f su m j shang x l qin an iterative method of solution for equilibrium and optimization problems nonlinear anal 2007 doi 10 1016 j na 2007 08 045 and chang cho and kim s s chang y j cho j k kim approximation methods of solutions for equilibrium problem in hilbert spaces dynam systems appl in print

existence and convergence of best proximity points abstract consider a self map t defined on the union of two subsets a and b of a metric space and satisfying t a b and t b a we give some contraction type existence results for a best proximity point that is a point x such that d x t x dist a b we also give an algorithm to find a best proximity point for the map t in the setting of a uniformly convex banach space

best proximity point theorems for cyclic orbital meir keeler contraction maps abstract we introduce a notion of cyclic orbital meir keeler contraction and give sufficient conditions for the existence of fixed points and best proximity points of such a map our main result is a generalization of a best proximity point result due to di bari et al c di bari t suzuki c vetro best proximity points for cyclic meir keeler contractions nonlinear anal 69 2008 3790 3794