a new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings abstract in this paper we introduce and study a new mapping generated by a finite family of nonexpansive mappings and finite real numbers and introduce a general iterative method concerning the new mappings for finding a common element of the set of solutions of an equilibrium problem and of the set of common fixed points of a finite family of nonexpansive mappings in a hilbert space then we prove a strong convergence theorem of the proposed iterative method for a finite family of nonexpansive mappings to the unique solution of variational inequality which is the optimality condition for a minimization problem our main result can be applied to obtain strong convergence of the general iterative methods which are modifications of those in g marino h k xu a general iterative method for nonexpansive mappings in hilbert spaces j math anal appl 318 1 2006 43 52 s plubtieng r punpaeng a general iterative method for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 336 1 2007 455 469 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 to a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping

an iterative algorithm for solving fixed point problems variational inequality problems and mixed equilibrium problems abstract in this paper we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of some variational inequality furthermore we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets our results extend and improve the corresponding results of zeng and yao l c zeng and j c yao a hybrid iterative scheme for mixed equilibrium problems and fixed point problems j comput appl math 214 2008 186 201 takahashi and takahashi s takahashi and w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2007 506 515 and many others

fitted reproducing kernel method for singularly perturbed delay initial value problems in this paper a fitted reproducing kernel is proposed for solving singularly perturbed initial value problems based on it an effective numerical method is proposed for a class of singularly perturbed delay initial value problem numerical results show that the present reproducing kernel is superior to the usual reproducing kernel for solving singularly perturbed initial value problems

asymptotic analysis of the exponential penalty trajectory in linear programming we consider the linear program min c x ax b and the associated exponential penalty functionfr x c x r exp aix bi r forr close to 0 the unconstrained minimizerx r offr admits an asymptotic expansion of the formx r x rd r wherex is a particular optimal solution of the linear program and the error term r has an exponentially fast decay using duality theory we exhibit an associated dual trajectory r which converges exponentially fast to a particular dual optimal solution these results are completed by an asymptotic analysis whenr tends to the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution

solutions of system of equilibrium and variational inequality problems on fixed points of infinite family of nonexpansive mappings in this paper we introduce a hybrid viscosity approximation scheme for finding a common element of the solution set for a system of equilibrium problems the solution set for a system of variational inequality problems and the common fixed point set for an infinite family of nonexpansive mappings in hilbert spaces then we prove the strong convergence of the proposed iterative scheme our results improve and extend the results announced by ceng and yao 2008 4

qualitative behaviour of numerical approximations to volterra integro differential equations in this paper we investigate the qualitative behaviour of numerical approximations to a nonlinear volterra integro differential equation with unbounded delay we consider the simple single species growth model d dt n t n t 1 c 1 t t k t s n s ds we apply the composite rule as a quadrature to discretize the equation we are particularly concerned with the way in which the long term qualitative properties of the analytical solution can be preserved in the numerical approximation using results in s n elaydi and s murakami j differ equations appl 2 1996 401 y song and c t h baker j differ equations appl 10 2004 379 for volterra difference equations we show that for a small bounded initial function and a small step size the corresponding numerical solutions display the same qualitative properties as found in the original problem in the final section of this paper we discuss how the analysis can be extended to a wider class of volterra integral equations and volterra integro differential equations with fading memory

a general iterative method for nonexpansive mappings in hilbert spaces let h be a real hilbert space consider on h a nonexpansive mapping t with a fixed point a contraction f with coefficient 0 0 let 0

a general iterative scheme for k strictly pseudo contractive mappings and optimization problems in this paper we introduce a new general iterative scheme for finding fixed points of a strictly pseudo contractive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a fixed point of the mapping which is a solution of a certain optimization problem related to a strongly positive bounded linear operator additional results of the main result are also obtained

iterative convergence to cesaro means for continuous pseudocontractive mappings abstract let f be a fixed strongly pseudocontractive mapping and t be a continuous pseudocontractive mapping with f t 0 the sequence z m is iteratively defined as follows z m t m f z m 1 t m 1 m 1 j 0 m t j z m m 0 where t m 0 1 satisfies the condition lim m t m 0 we prove that z m converges strongly to the unique solution p to some variational inequality in f t our results develop and complement the corresponding ones by matsushita daishi kuroiwa j math anal appl 294 2004 206 214 and shioji takahashi arch math 72 1999 354 359 and moore nnoli j math anal appl 260 2001 269 278 and su li appl math comput 181 2006 332 341 and song chen appl math comput 186 2007 1120 1128 and wangkeeree appl math comput 201 2008 239 249

convergence analysis of fitted numerical method for a singularly perturbed nonlinear volterra integro differential equation with delay abstract in this paper the initial value problem for a quasilinear singularly perturbed delay volterra integro differential equation was considered by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form a fitted difference scheme is constructed and analysed it is shown that the method displays first order uniform convergence in perturbation parameter some numerical results are given to confirm the theoretical analysis

strong convergence theorems for a new iterative method of k strictly pseudo contractive mappings in hilbert spaces in this paper we introduce a new iterative method of a k strictly pseudo contractive mapping for some email protected k 1 and prove that the sequence x n converges strongly to a fixed point of t which solves a variational inequality related to the linear operator a our results have extended and improved the corresponding results of y j cho s m kang and x qin some results on k strictly pseudo contractive mappings in hilbert spaces nonlinear anal 70 2008 1956 1964 and many others

numerical modelling in biosciences using delay differential equations our principal purposes here are i to consider from the perspective of applied mathematics models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics ii to review the application of numerical techniques to investigate these models we show that there are prima facie reasons for using such models i they have a richer mathematical framework compared with ordinary differential equations for the analysis of biosystem dynamics ii they display better consistency with the nature of certain biological processes and predictive results we analyze both the qualitative and quantitative role that delays play in basic time lag models proposed in population dynamics epidemiology physiology immunology neural networks and cell kinetics we then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences comparing them with those implemented by the bio modellers

iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings abstract we introduce iterative algorithms for finding a common element of the set of solutions of a system of equilibrium problems and of the set of fixed points of a finite family and a left amenable semigroup of nonexpansive mappings in a hilbert space we prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality which is the optimality condition for a minimization problem our results extend for example the recent result of v colao g marino h k xu an iterative method for finding common solutions of equilibrium and fixed point problems j math anal appl 344 2008 340 352 to systems of equilibrium problems

regularized and inertial algorithms for common fixed points of nonlinear operators 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

convergence of composite iterative methods for finding zeros of accretive operators abstract strong convergence theorems on composite iterative schemes by the viscosity approximation methods for finding a zero of an accretive operator are established in banach spaces the main results generalize the recent corresponding results 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 banach spaces nonlinear anal 67 2007 2350 2360 ceng et al l c ceng a r khan q h ansari j c yao strong convergence of composite iterative schemes for zeros of m accretive operators in banach spaces nonlinear anal 70 2009 1830 1840 kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 and xu h k xu strong convergence of an iterative method for nonexpansive and accretive operators j math anal appl 314 2006 631 643 to viscosity methods in a strictly convex and reflexive banach space having a uniformly g teaux differentiable norm our results also improve the corresponding results of t d benavides g l acedo h k xu iterative solutions for zeros of accretive operators math nachr 248 249 2003 62 71 r chen z zhu viscosity approximation fixed points for nonexpansive and m accretive operators fixed point theory appl 2006 2006 1 10 s kamimura w takahashi approximation solutions of maximal monotone operators in hilbert spaces j approx theory 106 2000 226 240 p e mainge viscosity methods for zeroes of accretive operators j approx theory 140 2 2006 127 140 k nakajo strong convergence to zeros of accretive operators in banach spaces j nonlinear convex anal 7 2006 71 81

approximating fixed points of holomorphic mappings in the hilbert ball abstract we approximate fixed points of holomorphic and nonexpansive self mappings of the hilbert ball using both continuous and discrete schemes

strong convergence theorems for nonexpansive semigroup in banach spaces abstract let k be a nonempty closed convex subset of a reflexive and strictly convex banach space e with a uniformly g teaux differentiable norm and f t t t 0 a nonexpansive self mappings semigroup of k and f k k a fixed contractive mapping the strongly convergent theorems of the following implicit and explicit viscosity iterative schemes x n are proved x n n f x n 1 n t t n x n x n 1 n f x n 1 n t t n x n and the cluster point of x n is the unique solution to some co variational inequality

strong convergence theorems of modified ishikawa iterative process with errors for an infinite family of strict pseudo contractions abstract in this paper we study modified ishikawa iterative process with errors to have strong convergence for an infinite family of strict pseudo contractions in the framework of q uniformly smooth banach spaces our results improve and extend the recent ones announced by many others

a finite difference method for a singularly perturbed delay integro differential equation we consider the singularly perturbed initial value problem for a linear first order volterra integro differential equation with delay our purpose is to construct and analyse a numerical method with uniform convergence in the perturbation parameter the numerical solution of this problem is discretized using implicit difference rules for differential part and the composite numerical quadrature rules for integral part on a layer adapted mesh error estimations for the approximate solution are established numerical examples supporting the theory are presented

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

fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption let e be a real banach space with a uniformly gateaux differentiable norm and which possesses uniform normal structure k a nonempty bounded closed convex subset of e t i i 1 n a finite family of asymptotically nonexpansive self mappings on k with common sequence k n n 1 1 t n s n be two sequences in 0 1 such that s n t n 1 n 1 and f be a contraction on k under suitable conditions on the sequences s n t n we show the existence of a sequence x n satisfying the relation x n 1 1k n x n s nk nf x n t nk nt r n nx n where n l nn r n for some unique integers l n 0 and 1 r n n further we prove that x n converges strongly to a common fixed point of t i i 1 n which solves some variational inequality provided x n t ix n 0 as n for i 1 2 n as an application we prove that the iterative process defined by z 0 k z n 1 1 1k n z n s nk nf z n t nk nt r n nz n converges strongly to the same common fixed point of t i i 1 n

the viscosity approximation process for quasi nonexpansive mappings in hilbert spaces in this paper we discuss the strong convergence of the viscosity approximation method in hilbert spaces relatively to the computation of fixed points of operators in the wide class of quasi nonexpansive mappings our convergence results improve previously known ones obtained for the class of nonexpansive mappings

stability of linear multistep methods for delay integro differential equations this paper is concerned with the numerical solution of delay integro differential equations the adaptation of linear multistep methods is considered the emphasis is on the linear stability of numerical methods it is shown that every a stable strongly 0 stable linear multistep method of pouzet type can preserve the delay independent stability of the underlying linear systems in addition some delay dependent stability conditions for the stability of numerical methods are also given

viscosity approximation methods for nonexpansive mappings and monotone mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n s p c x n n a x n f is a contraction on c s is a nonexpansive self mapping of a closed convex subset c of a hilbert space h it is shown that x n converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping which solves some variational inequality

relaxed viscosity approximation methods for fixed point problems and variational inequality problems abstract let x be a real strictly convex and reflexive banach space with a uniformly g teaux differentiable norm and c be a nonempty closed convex subset of x let t n n 1 be a sequence of nonexpansive self mappings on c such that the common fixed point set f n 1 f t n 0 and f c c be a given contractive mapping and n be a sequence of nonnegative numbers in 0 1 consider the following relaxed viscosity approximation method x n 1 1 n n x n n f y n n w n y n y n 1 n x n n w n x n n 1 where w n is the w mapping generated by t n t n 1 t 1 and n n 1 1 for each n 1 it is proven that under very mild conditions on the parameters the sequence x n of approximate solutions generated by the proposed method converges strongly to some p f where p is the unique solution in f to the following variational inequality i f p j p x 0 x f

layer adapted methods for quasilinear singularly perturbed delay differential problems abstract in this work we consider a class of initial value problems for quasilinear singularly perturbed first order delay differential equations to solve this class of problems numerically we consider two finite difference schemes the backward euler scheme and a high order hybrid scheme which is a blend of the trapezoidal scheme and the backward euler scheme we derive general convergence results for both the schemes based on which a number of layer adapted meshes can be constructed and analyzed as consequences of these results we establish uniform convergence of the schemes on certain layer adapted meshes numerical experiments confirm our theoretical findings

an iterative scheme for equilibrium problems and fixed point problems of strict pseudo contraction mappings in this paper we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo contraction mapping in the setting of real hilbert spaces we establish some weak and strong convergence theorems of the sequences generated by our proposed scheme our results combine the ideas of marino and xu s result g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 346 and takahashi and takahashi s result s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2007 506 515 in particular necessary and sufficient conditions for strong convergence of our iterative scheme are obtained

uniform numerical method for singularly perturbed delay differential equations this paper deals with the singularly perturbed initial value problem for a linear first order delay differential equation a numerical method is constructed for this problem which involves an appropriate piecewise uniform mesh on each time subinterval the difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter numerical results are presented

uniform difference method for singularly perturbed volterra integro differential equations singularly perturbed volterra integro differential equations is considered an exponentially fitted difference scheme is constructed in a uniform mesh which gives first order uniform convergence in the discrete maximum norm numerical experiments support the theoretical results

variational inequalities for φ pseudo contractive mappings abstract usually results on strongly accretive operators involve various assumptions on the function such as continuity being positive and or being strictly increasing however by redefining this function we are able to eliminate all these additional assumptions and leave only the positiveness condition t 0 for t 0 this fact allows us to extend a theorem of deimling and obtain a general result for the solvability of a variational inequality for strongly pseudo contractive mappings

a general iterative method for a finite family of nonexpansive mappings abstract we study the approximation of common fixed points of a finite family of nonexpansive mappings and suggest a modification of the iterative algorithm without the assumption of any type of commutativity also we show that the convergence of the proposed algorithm can be proved under some types of control conditions

iterative schemes for computing fixed points of nonexpansive mappings in banach spaces abstract let x be a real banach space with a normalized duality mapping uniformly norm to weak continuous on bounded sets or a reflexive banach space which admits a weakly continuous duality mapping j with gauge let f be an contraction and t n a sequence of nonexpansive mappings we study the strong convergence of explicit iterative schemes 1 x n 1 n f x n 1 n t n x n with a general theorem and then recover and improve some specific cases studied in the literature k aoyoma y kimura w takahashi m toyoda approximation of common fixed point of a countable family of nonexpansive mappings nonlinear anal 67 8 2007 2350 2360 g lopez v martin h k xu perturbation techniques for nonexpansive mappings with applications nonlinear anal real world appl in press available online 4 may 2008 h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 1 2004 279 291 t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 1 2 2005 51 60 y song r chen strong convergence theorems on an iterative method for a family of finite nonexpansive mappings appl math comput 180 2006 275 287 y song r chen viscosity approximation methods for nonexpansive nonself mappings j math anal appl 321 1 2006 316 326 j chen l zhang t fan viscosity approximation methods for nonexpansive mappings and monotone mappings j math anal appl 334 2 2007 1450 1461 y kimura w takahashi m toyoda convergence to common fixed points of a finite family of nonexpansive mappings arch math 84 2005 350 363

viscosity approximation methods for fixed points problems the aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed point of a given nonexpansive self mapping

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

stability analysis of volterra delay integro differential equations and their backward differentiation time discretization we study the stability of the analytical solutions of initial value problems of a general class of systems of volterra delay integro differential equations numerical methods based on backward differentiation formulae and repeated quadrature formulae are suggested nonlinear and linear stability conditions for the presented methods are derived

viscosity solutions of minimization problems viscosity methods for minimization problems are revisited from some modern perspectives in variational analysis variational convergences for sequences of functions epi convergence gamma convergence mosco convergence and for sequences of operators graph convergence provide a flexible tool for such questions it is proved in a rather large setting that the solutions of the approximate problems converge to a viscosity solution of the original problem that is a solution that is minimal among all the solutions with respect to some viscosity criteria various examples coming from mathematical programming calculus of variations semicoercive elliptic equations phase transition theory hamilton jacobi equations singular perturbations and optimal control theory are considered

strong convergence of viscosity approximation methods for finding zeros of accretive operators in banach spaces abstract in this paper we introduce a composite iterative scheme by viscosity approximation method for finding a zero of an accretive operator in banach spaces then we establish strong convergence theorems for the composite iterative scheme the main theorems improve and generalize the recent corresponding results of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 qin and su x qin y su approximation of a zero point of accretive operator in banach spaces j math anal appl 329 2007 415 424 and xu h k xu strong convergence of an iterative method for nonexpansive and accretive operators j math anal appl 314 2006 631 643 as well as aoyama et al k aoyama y kimura w takahashi m toyoda approximation of common fixed points of a countable family of nonexpansive mappings in banach spaces nonlinear anal 67 2007 2350 2360 benavides et al t d benavides g l acedo h k xu iterative solutions for zeros of accretive operators math nachr 248 249 2003 62 71 chen and zhu r chen z zhu viscosity approximation fixed points for nonexpansive and m accretive operators fixed point theory and appl 2006 2006 1 10 and kamimura and takahashi s kamimura w takahashi approximation solutions of maximal monotone operators in hilberts spaces j approx theory 106 2000 226 240

dissipativity of θ methods for nonlinear volterra delay integro differential equations this paper deals with the analytic and numerical dissipativity of nonlinear volterra delay integro differential equations vdides a dissipativity criteria for vdides is given we investigate the dissipativity properties of one leg q methods and linear q methods for the underlying systems it is shown that for q 12 1 any one leg q method and linear q method can inherit the dissipativity of the underlying systems

dynamic responses of the in plane and out of plane vibrations for an axially moving membrane the dynamic responses of both the in plane and out of plane vibrations are investigated for an axially moving membrane using the extended hamilton principle equations of motion are derived for the moving membrane with no slip boundary conditions at rollers the equations of motion are discretized by using the galerkin method and then the generalized a time integration method is applied to compute the dynamic responses of the membrane based on the computed results the responses are compared between the in plane and out of plane vibrations in addition the effects of translating velocity acceleration critical speeds and velocity profiles on the dynamic behaviours for displacements and stresses are investigated and discussed r 2006 elsevier ltd all rights reserved

stability of runge kutta methods for delay integro differential equations we study stability of runge kutta rk methods for delay integro differential equations with a constant delay on the basis of the linear equation du dt lu t mu t k t t u d where l m k are constant complex matrices in particular we show that the same result as in the case k 0 koto bit 34 1994 262 267 holds for this test equation i e every a stable rk method preserves the delay independent stability of the exact solution whenever a step size of the form h m is used where m is a positive integer

on relaxed viscosity iterative methods for variational inequalities in banach spaces in this paper we suggest and analyze a relaxed viscosity iterative method for a commutative family of nonexpansive self mappings defined on a nonempty closed convex subset of a reflexive banach space we prove that the sequence of approximate solutions generated by the proposed iterative algorithm converges strongly to a solution of a variational inequality our relaxed viscosity iterative method is an extension and variant form of the original viscosity iterative method the results of this paper can be viewed as an improvement and generalization of the previously known results that have appeared in the literature

an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings abstract let h be a hilbert space consider on h a sequence of nonexpansive mappings t n with common fixed points a finite family of equilibrium functions g i i 1 k a contraction f with coefficient 0 1 and a strongly positive linear bounded operator a with coefficient 0 let 0 assuming there are common equilibrium points of the family g i i 1 k which are also fixed points for t n we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality a f x x x 0 for all the x in the intersection of the equilibrium points and the common fixed points of the sequence t n

some iterative methods for finding fixed points and for solving constrained convex minimization problems the present paper is divided into two parts in the first part we introduce implicit and explicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real hilbert space we establish results on the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping such a fixed point is also a solution of a variational inequality defined on the set of fixed points in the second part we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping the results of this paper extend and improve several results presented in the literature in the recent past