on the steepest descent approximation method for the zeros of generalized accretive operators abstract in this paper we present certain characteristic conditions for the convergence of the generalized steepest descent approximation process to a zero of a generalized strongly accretive operator defined on a uniformly smooth banach space our study is based on an important result of reich s reich an iterative procedure for constructing zeros of accretive sets in banach spaces nonlinear anal 2 1978 85 92 and given results extend and improve some of the earlier results which include the steepest descent approximation method

approximating common fixed points of asymptotically nonexpansive maps in uniformly convex banach spaces we introduce three step iterative schemes with errors for two and three nonexpansive maps and establish weak and strong convergence theorems for these schemes mann type and ishikawa type convergence results are included in the analysis of these new iteration schemes the results presented in this paper substantially improve and extend the results due to s h khan h fukhar ud din weak and strong convergence of a scheme with errors for two nonexpansive mappings nonlinear anal 8 2005 1295 1301 n shahzad approximating fixed points of non self nonexpansive mappings in banach spaces nonlinear anal 61 2005 1031 1039 w takahashi t tamura convergence theorems for a pair of nonexpansive mappings j convex anal 5 1995 45 58 k k tan h k xu approximating fixed points of nonexpansive mappings by the ishikawa iteration process j math anal appl 178 1993 301 308 and h f senter w g dotson approximating fixed points of nonexpansive mappings proc amer math soc 44 1974 375 380

inequalities in metric spaces with applications abstract analogues of the parallelogram identity and the cn inequality of bruhat and tits in uniformly convex metric spaces are established as an application of the new inequalities we prove two fixed point results for single valued and multi valued lipschitzian mappings

iterative solution of nonlinear equations involving set valued uniformly accretive operators abstract let e be a real normed linear space and let a e 2 e be a uniformly continuous and uniformly quasi accretive multivalued map with nonempty closed values such that the range of i a is bounded and the inclusion f ax has a solution x e it is proved that ishikawa and mann type iteration processes converge strongly to x further if t e 2 e is a uniformly continuous and uniformly hemicontractive set valued map with bounded range and a fixed point x e it is proved that both the mann and ishikawa type iteration processes converge strongly to x the strong convergence of these iteration processes with errors is also proved

a third order boundary value problem with jumping nonlinearities abstract a third order boundary value problem with jumping nonlinearities is studied some results obtained for second order boundary value problems with jumping nonlinearities are shown to hold also for the third order case

compactness and the fixed point property in l1 abstract in this paper we prove that compactness can be characterized by means of the existence of a fixed point for some classes of mappings defined on convex closed subsets of the space l 1 nominally our result involves nonexpansive mappings uniformly lipschitzian mappings and cascading nonexpansive mappings we also extend the results to some more general classes of banach spaces

on the peroidic bvp for the forced duffing equation with jumping nonlinearity on etudie le probleme suivant u cu f t u e t p p sur 0 t u t u 0 0 u t u 0 0 c est une fonction arbitraire dans l 1 0 t r c r et l application non lineaire f est a saut

strong convergence theorems for finitely many nonexpansive mappings and applications abstract let e be a uniformly convex banach space which satisfies opial s condition or whose norm is frechet differentiable recently takahashi and shimoji w takahashi k shimoji convergence theorems for nonexpansive mappings and feasibility problems math comput modelling 32 2000 1463 1471 introduced an iterative scheme given by finitely many nonexpansive mappings in e and proved weak convergence theorems which are connected with the problem of image recovery in this paper we introduce a new iterative scheme which includes their iterative scheme as a special case under the assumption that e is a reflexive banach space whose norm is uniformly g teaux differentiable and which has a weakly continuous duality mapping we prove strong convergence theorems which are connected with the problem of image recovery using the established results we consider the problem of finding a common fixed point of finitely many nonexpansive mappings

on uniformly lipschitzian multivalued mappings in banach and metric spaces abstract let x d be a metric space a mapping t x x is said to be uniformly lipschitzian if there exists a constant k such that d t n x t n y k d x y for all x y x and n 1 it is known that such mappings always have fixed points in certain metric spaces for k 1 provided k is sufficiently near 1 these spaces include uniformly convex metric and banach spaces as well as metric spaces having lifsic characteristic greater than 1 a uniformly lipschitzian concept for multivalued mappings is introduced in this paper and multivalued analogues of these results are obtained

on the asymptotic behavior of nonlinear semigroups and the range of accretive operators ii abstract it is known that certain problems in partial differential equations may be interpreted as initial value problems for ordinary differential equations in banach spaces when such an evolution equation is governed by an accretive operator then its solutions give rise to a nonlinear contraction semigroup in this paper we study certain aspects of the asymptotic behavior of nonlinear semigroups and or resolvents of accretive operators we also derive new results on their behavior at the origin it turns out that the behavior of a nonlinear semigroup resembles that of the resolvent of its generator both at infinity and at the origin author

fixed point theorems and convergence theorems for some generalized nonexpansive mappings we introduce some condition on mappings the condition is weaker than nonexpansiveness and stronger than quasinonexpansiveness we present fixed point theorems and convergence theorems for mappings satisfying the condition

strong convergence of modified ishikawa iteration for two asymptotically nonexpansive mappings and semigroups abstract in this paper we established strong convergence theorems for a common fixed point of two asymptotically nonexpansive mappings and for a common fixed point of two asymptotically nonexpansive semigroups by using the hybrid method in a hilbert space moreover we also proved a strong convergence theorem for a common fixed point of two nonexpansive mappings our results extend and improve the recent ones announced by kim and xu t w kim h w xu strong convergence of modified mann iteration for asymptotically nonexpansive mappings and semigroups nonlinear anal 64 2006 1140 1152 nakajo and takahashi k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 and many others

on uniformly convex functions abstract in this paper we study uniformly convex functions and uniformly convex functions at a point giving some properties and characterizations of them further we give some examples and applications of these types of functions

fixed point results for a generalized nonexpansive map in uniformly convex metric spaces abstract in this paper the existence of a unique fixed point of a map satisfying a very general contractive condition on a suitable subset of a uniformly convex metric space is proved this fixed point is approximated by averaging krasnosel skii iterations of a generalized nonexpansive map our results substantially improve and extend several known results existing in the literature

rings of invariants of modular p groups which are hypersurfaces abstract for l a finite non modular group whose invariants form a polynomial ring and h a subgroup of l containing the derived group of l nakajima found necessary and sufficient conditions on h for its invariant ring s h to be a hypersurface in a crucial step of his proof he showed that if s h is a hypersurface then between h and l there is a group g with polynomial invariant ring such that s h s g b for g a finite modular p group over f p with polynomial invariant ring and h a subgroup of g containing the derived group of g we find necessary and sufficient conditions on h to ensure that s h s g b

on some lp inequalities in best approximation theory h x jxp p ax p is strictly increasing in the interval p j o thus h x 0 has a unique solution x p note that x l 2 1 x p is strictly increasing in p and x 0 is the unique solution of the equation p 2 xp p l f2 1 in the interval 0 6 x 1 1 1 in 11 we proved the inequality

error bounds for approximation solutions to nonlinear equations of strongly accretive operators in uniformly smooth banach spaces abstract let x be a uniformly smooth banach space and t x x a strongly accretive operator in this paper we give the error bounds for the approximation solutions of the nonlinear equation tx f generated by both the mann and the ishikawa iteration process on the other hand let k be a nonempty convex subset of x and t k k a strictly pseudocontractive mapping the related results deal with the error bounds for the iterative approximation of the fixed point of t generated by these two processes

an analysis of a multi level projected steepest descent iteration for nonlinear inverse problems in banach spaces subject to stability constraints we consider nonlinear inverse problems described by operator equations in banach spaces assuming conditional stability of the inverse problem that is assuming that stability holds on a compact convex subset of the domain of the operator we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data we proceed with developing a multi level algorithm based on a nested family of compact convex subsets on which stability holds and the stability constants are ordered growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion after a finite number of steps

fixed points of uniformly lipschitzian mappings in spaces with uniformly normal structure dans un espace de banach x on etudie quand reflexivite et structure normale sont suffisantes pour assurer pour de bon 1 la propriete de point fixe pour des auto applications uniformement lipschitziennes

on the structure of fixed point sets of asymptotically regular semigroups abstract we show that the set of fixed points of an asymptotically regular mapping acting on a convex and weakly compact subset of a banach space is in some cases a holder continuous retract of its domain our results qualitatively complement the corresponding fixed point existence theorems and extend a few recent results of gornicki 15 16 17 we also characterize bynum s coefficients and the opial modulus in terms of nets

strong convergence theorem for approximation of solutions of equations of hammerstein type abstract let h be a real hilbert space let k f h h be bounded continuous and monotone mappings suppose that u h is a solution to the hammerstein equation u k f u 0 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

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces abstract let m d be a complete 2 uniformly convex metric space let c be a nonempty bounded closed and convex subset of m and let t c c be an asymptotic pointwise nonexpansive mapping in this paper we prove that the modified mann iteration process defined by x n 1 t n t n x n 1 t n x n converges in a weaker sense to a fixed point of t

on a new system of generalized mixed quasi variational like inclusions involving a η m accretive operators with applications in this paper we introduce a new and interesting system of generalized mixed quasi variational like inclusions with a h m accretive operators and relaxed cocoercive mappings which contains variational inequalities variational inclusions systems of variational inequalities systems of variational like inequalities and systems of variational inclusions in the literature as special cases by using the resolvent technique for the a h m accretive operators we prove the existence of solutions and the convergence of a new p step iterative algorithm for this system of generalized mixed quasi variational like inclusions in real q uniformly smooth banach spaces the results in this paper unifies extends and improves some known results in the literature

strong convergence theorem for integral equations of hammerstein type in hilbert spaces let h be a real hilbert space let f h h be a bounded coercive and maximal monotone mapping let k h h be a bounded and maximal monotone mapping let k and f satisfy the range condition suppose that u h is a solution to hammerstein equation u kfu 0 we construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the hammerstein equation our iterative scheme in this paper seems far simpler than the iterative scheme used by chidume and ofoedu 2011 13 and their strong assumption is dispensed with we give some examples of our result so that it will find serious applications and be of much interest to our readers

on strong convergence by the hybrid method for families of mappings in hilbert spaces abstract let c be a nonempty closed convex subset of a real hilbert space and let t n be a family of mappings of c into itself such that the set of all common fixed points of t n is nonempty we consider a sequence x n generated by the hybrid method in mathematical programming we give the new conditions of t n under which x n converges strongly to a common fixed point of t n and generalize the unified result for families of nonexpansive mappings k nakajo k shimoji w takahashi strong convergence theorems by the hybrid method for families of nonexpansive mappings in hilbert spaces taiwanese j math 10 2006 339 360 and the results for asymptotically nonexpansive mappings and semigroups t h kim h k xu strong convergence of modified mann iterations for asymptotically nonexpansive mappings and semigroups nonlinear anal 64 2006 1140 1152 s plubtieng and k ungchittrakool strong convergence of modified ishikawa iteration for two asymptotically nonexpansive mappings and semigroups nonlinear anal 67 2007 2306 2315

fixed point theorems for uniformly lipschitzian semigroups in uniformly convex spaces for all x y in c and each n 1 2 goebel and kirk 2 first studied these mappings and they proved that such t has a fixed point if c is a closed bounded convex subset of a uniformly convex banach space x and k k k is the unique solution of the equation 1 a l k k 1 with 6 being the modulus of convexity of x lifschitz s showed that in a hilbert space i a uniformly k lipschitzian mapping t with k c 21j2 has a fixed point lim 6 7 and lim et al 8 extended goebel kirk s theorem in lp spaces recently downing and ray l and ishihara and takahashi 3 verified that lifschitz s theorem is valid for uniformly lipschitzian semigroups in hifbert spaces the purpose of this paper is to prove fixed point theorems for a uniformly k lipschitzian semigroup in uniformly convex banach spaces which is left reversible or for which the space of right uniformly continuous functions on the semigroup has a left invariant mean the results generalize those of downing and ray l lau 4 lim 6 7 and ishihara and takahashi 3

remarks and generalizations related to a recent multiplicity result of a lazer and p mckenna we will show here that this result has a natural generalization to the case that u 5 ai to state the result we consider the model problem lu au autel 2 where 0 0 and set t 0 we have a nonlinear eigenvalue problem lu yu au 4 the following proposition is known cf fueik 3 dancer 2 and also lo where the result appears in the formulation given below

proximal normal cone analysis on smooth banach spaces and applications unlike the frechet penot clarke and other order one notions of normal cones the notion of proximal normal cone describes variational behavior of order two and most of results on proximal normal cone are established in the hilbert space framework a possible reason is that in a general banach space such as in the classical banach spaces l p and l p with 1 leq p 2 the proximal normal cone of a smooth subset a at every point of a may consist of the zero element only nevertheless on the other hand we show in this paper that for a fairly big class of smooth banach spaces including the classical banach spaces l p and l p with 2 leq p infty the consideration of proximal normal cones provides a lot of useful information especially for leading to satisfactory treatment for constrained optimization problems some of our results are new even in the hilbert space case

hypersurfaces in modular invariant theory let the finite group g act linearly on the n dimensional vector space v over the field k of characteristic p 0 suppose h g is a normal subgroup of index l a prime number in this paper we shall study the relationship between the two invariant rings k v h and k v g as a corollary of our main result we get that if k v g is a polynomial algebra and k v h is factorial then k v h is a graded hypersurface algebra

on the number of solutions of two classes of sturm liouville boundary value problems abstract we employ global bifurcation theory of p h rabinowitz together with sturm comparison theorem to establish multiplicity results for some classes of sturm liouville boundary value problems

a quadratic rate of asymptotic regularity for cat 0 spaces abstract in this paper we obtain a quadratic bound on the rate of asymptotic regularity for the krasnoselski mann iterations of nonexpansive mappings in cat 0 spaces whereas previous results guarantee only exponential bounds the method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of groetsch s theorem obtained by kohlenbach using methods from mathematical logic so called proof mining

an lp inequality and its applications to fixed point theory and approximation theory on etablit une inegalite l p pour 1 p 2 et on donne des applications aux points fixes des applications uniformement lipschitziennes et aux meilleures approximations fortement uniques

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

resolvent operator technique for generalized implicit variational like inclusion in banach space a new class of g accretive mappings is introduced and studied in banach space by using the properties of g accretive mappings the concept of resolvent operators associated with the classical m accretive operators is extended and an iterative algorithm for a new class of generalized implicit variational like inclusion involving g accretive mappings and its convergence results are established in banach space

convergence of a general iterative scheme for a finite family of asymptotically quasi nonexpansive mappings in convex metric spaces and applications in this paper we introduce the iterative scheme due to khan domlo and fukhar ud din 2008 8 in convex metric spaces and establish strong convergence of this scheme to a unique common fixed point of a finite family of asymptotically quasi nonexpansive mappings as a consequence of our result we obtain some related convergence theorems our results generalize some recent results obtained in 8

a necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations abstract a necessary and sufficient condition is established which ensures the strong convergence of the steepest descent approximation to a solution of equations involving quasi accretive operators defined on a uniformly smooth banach space

nonautonomous boundary value problems with jumping nonlinearities soit f 0 1 r r une fonction continue continument differentiable par rapport au second argument sous l hypothese que les limites lim s f x s s x lim s f x s x x existent et sont uniformes sur 0 1 on etudie le probleme aux valeurs limites u f x u th u 0 u 1 0 ou t est un nombre reel et h une fonction continue pour f fixee on etudie le nombre de solutions

rings of invariants of certainp groups over the fieldfp abstract we use nakajima s j algebra85 1983 253 286 characterization ofp groups with polynomial rings of invariants over the fieldfpto prove that maximal proper subgroups of such groups have rings of invariants that are hypersurfaces we give an explicit construction for the ring of invariants of such a group as a module over the ring of invariants of the total group

on the existence of fixed points and ergodic retractions for lipschitzian semigroups in hilbert spaces and especially nonexpansive in the case of k 1 let s be a semitopological semigroup and let s c s e s be a continuous representation of s as lipschitzian mappings of c into itself which is said to be a lipschitzian semigroup on c a lipschitzian semigroup s t s e s with lipschitz constants k s e s is called a uniformly k lipschitzian semigroup and a nonexpansive semigroup when k k and k 1 for all s e s respectively goebel and kirk 7 first called a mapping t of c into itself a uniformly k lipschitzian mapping on c in the case when s t n e n is a lipschitzian semigroup with k k for all n e n lifschitz 12 obtained a fixed point theorem for uniformly k lipschitzian mappings on c with k 5 and this result was extended to uniformly k lipschitzian semigroups which are left reversible by downing and ray 6 lau l l showed that a nonexpansive semigroup has a common fixed point if there exists a left invariant mean on the space rug s of bounded right uniformly continuous functions on the semigroup furthermore ishihara and takahashi 9 extended lau s result to uniformly k lipschitzian semigroups and also gave a simpler proof of downing and ray s theorem on the other hand the first nonlinear ergodic theorem for nonexpansive mappings was established by baillon 11 let t be a nonexpansive mapping of c into itself and suppose that the set f t of fixed points of t is nonempty the the cesaro means

fixed point theorems in metric spaces abstract in this paper we establish two general theorems for equivalence between the meir keeler type contractive conditions and the contractive definitions involving gauge functions one of these theorems is an extension of a recent result of lim on characterization of meir keeler contractive maps nonlinear anal 46 2001 113 120 also we establish the following new fixed point theorem suppose r r is a contractive gauge function in the sense that for any e 0 there exists e such that e t implies t e and suppose t is a continuous and asymptotically regular selfmapping on a complete metric space x d satisfying the following i d tx ty d x y for all x y x and ii d tx ty d x y for all x y x with x y where d x y d x y d x tx d y ty with 0 then t has a unique fixed point and all of the picard iterates of t converge to this fixed point this result includes those of jachimski equivalent conditions and the meir keeler type theorems j math anal appl 194 1995 293 303 matkowski fixed point theorems for contractive mappings in metric spaces cas pest mat 105 1980 341 344 and others

strong convergence theorems for maximal monotone mappings in banach spaces let e be a uniformly convex and 2 uniformly smooth real banach space with dual e let a e e be a lipschitz continuous monotone mapping with a 1 0 for given u x1 e let xn be generated by the algorithm xn 1 nu 1 n xn najxn n 1 where j is the normalized duality mapping from e into e and n and n are real sequences in 0 1 satisfying certain conditions then it is proved that under some mild conditions xn converges strongly to x e where jx a 1 0 finally we apply our convergence theorems to the convex minimization problems

on some fixed point theorems on uniformly convex banach spaces abstract we introduce a notion of t regularity to generalize a well known fixed point theorem of browder we also give some related results in this direction

semigroup of nonexpansive mappings on a hilbert space soit s un semi groupe semi topologique soit h un espace de hilbert et s t s s s une representation continue de s comme applications non dilatation sur un sous ensemble convexe clos c de h dans c soit x c on etudie la convergence de t s x s s vers un point fixe commun z de s dans c

uniformly normal structure and uniformly lipschitzian semigroups abstract assume that x is a real banach space with uniformly normal structure and c is a nonempty closed convex subset of x we show that a uniformly lipschitzian semigroup of nonlinear self mappings of c 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

remarks on the structure of the fixed point sets of uniformly lipschitzian mappings in uniformly convex banach spaces abstract it is shown by asymptotic center techniques that the set of fixed points of any uniformly k lipschitzian mapping in a uniformly convex banach space is a retract of the domain when k is less than a constant bigger than the constant from the paper k goebel w a kirk a fixed point theorem for transformations whose iterates have uniform lipschitz constant studia math 47 1973 135 140 our result improves a recently result presented in e sedlak a wi nicki on the structure of fixed point sets of uniformly lipschitzian mappings topol methods nonlinear anal 30 2007 345 350

the iterative solution of the equation f x tx for a monotone operator t in lp spaces suppose t is a multivalued monotone operator not necessarily continuous with open domain d t in lp 2 p f r i t and the equation f x tx has a solution q d t then there exists a neighbourhood b d t of q and a real number r1 0 such that for any r r1 for any initial guess x1 b and any singlevalued section t0 of t the sequence xn n 1 generated from x1 by xn 1 1 cn xn cn f t0xn remains in d t and converges strongly to q with xn q 0 n 1 2 furthermore for x lp e e

ball algorithms for constructing solutions of nonlinear operator equations abstract in the most general setting we extend the previous theory of the region contraction algorithm developed recently by several authors for solving nonlinear operator equations it is demonstrated that the extended algorithm still possesses properties such as generating error bounds automatically providing a computational test for the existence of solutions and being globally convergent under nominal conditions as example of applications we specialize the extended theory in particular to nonlinear operator equations of monotone type

convergence rates for inverse problems with impulsive noise we study inverse problems f f g with perturbed right hand side g rm obs corrupted by so called impulsive noise i e noise which is concentrated on a small subset of the domain of definition of g it is well known that tikhonov type regularization with an mathbf l 1 data fidelity term yields significantly more accurate results than tikhonov regularization with classical mathbf l 2 data fidelity terms for this type of noise the purpose of this paper is to provide a convergence analysis explaining this remarkable difference in accuracy our error estimates significantly improve previous error estimates for tikhonov regularization with mathbf l 1 fidelity term in the case of impulsive noise we present numerical results which are in good agreement with the predictions of our analysis

strong convergence of modified mann iterations for asymptotically nonexpansive mappings and semigroups abstract the mann iterations for nonexpansive mappings have in general only weak convergence in a hilbert space we modify an iterative method of mann s type introduced by nakajo and takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 for nonexpansive mappings and prove strong convergence of our modified mann s iteration processes for asymptotically nonexpansive mappings and semigroups

second order sufficient conditions for error bounds in banach spaces recently huang and ng presented second order sufficient conditions for error bounds of continuous and ga teaux differentiable functions in banach spaces wu and ye dropped the assumption of huang and ng on ga teaux differentiability but required the space to be a hilbert space we carry on this research in two directions first we extend wu and ye s result to some non hilbert spaces second same as huang and ng we work on banach spaces but provide different second order sufficient conditions that may allow the function to be non ga teaux differentiable

on an inequality of bynum and drew an extension of the bynum drew inequality in the abstract lp space 1 p 2 is presented the extended inequality is applied to show the strong uniqueness of best approximations and existence of fixed points for uniformly lipschitzian mappings and to prove an estimate for the self jung constants

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

characteristic conditions for convergence of generalized steepest descent approximation to multivalued accretive operator equations abstract in the present paper sufficient and necessary conditions for convergence of generalized steepest descent approximation to multivalued accretive operator equations are established for the sufficiency part and a specific error estimation is also given

on the rate of convergence of iterated bregman projections and of the alternating algorithm abstract we study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth banach spaces for banach spaces which are convex and smooth of power type we exhibit a condition which implies linear convergence of this method we show these convergence results for iterates of bregman projections onto closed linear subspaces using an intimate connection between the metric projection onto a closed linear subspace and the bregman projection onto its annihilator we deduce the convergence rate results for the alternating algorithm from the corresponding results for the iterated bregman projection method

iterative construction for common fixed points of finitely many nonexpansive mappings abstract many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a family of nonexpansive mappings in a banach space in this paper under the hypotheses that e is a reflexive and strictly convex banach spaces with a uniformly g eaux differentiable norm the strong convergence of a iteration algorithm is proved for finding a common fixed point of a finite family of nonexpansive self mappings defined on a closed convex subset k of e

asymptotic behavior of the solutions of non autonomous systems in banach spaces abstract in this paper we study the weak and strong convergence of the integral solution of the following nonlinear evolution inclusion u t a t u t f t t 0 u 0 x 0 where a t t 0 is a family of m accretive operators and f is a multi function which extends some results in this area