implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps convergence theorems for approximation of common fixed points of strictly pseudocontractive mappings of browder petryshyn type are proved in banach spaces using an implicit iteration scheme recently introduced by xu and ori numer funct anal optim 22 2001 767 773

fixed point theorem for generalized φ pseudocontractive mappings abstract let e be a real banach space c be a nonempty closed convex subset of e and t c c be a continuous generalized pseudocontractive mapping it is proved that t has a unique fixed point in c

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

a hybrid algorithm for pseudo contractive mappings abstract in this paper we suggest and analyze a hybrid algorithm for pseudo contractive mappings in hilbert spaces further we prove the strong convergence of the proposed iterative algorithm for lipschitz pseudo contractive mappings in hilbert spaces

convergence theorems of common fixed points for a finite family of lipschitz pseudocontractions in banach spaces abstract in this paper we continue to study weak convergence problems for the implicit iteration process for a finite family of lipschitzian continuous pseudocontractions in general banach spaces the results presented in this paper improve and extend the corresponding ones of xu and ori h k xu r g ori an implicit iteration process for nonexpansive mappings numer funct anal optim 22 2001 767 773 osilike m o osilike implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps j math anal appl 294 2004 73 81 chen et al r chen y s song h y zhou convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings j math anal appl 314 2006 701 709 and others

weak convergence theorems for a finite family of strict pseudocontractions let e be a uniformly smooth real banach space which is also uniformly convex and k be a nonempty closed convex subset of e let t k k be a strict pseudocontraction for some 0 1 with x f t x k t x x 0 combining long solidus overlay for a fixed x0 k define a sequence xn by xn 1 1 n xn n t xn where n is a sequence in 0 1 satisfying the following conditions i n 0 n ii n 0 n2 then xn converges weakly to a fixed point of t furthermore weak convergence theorems are proved for a common fixed point for a finite family of strict pseudocontractions 2009 elsevier ltd all rights reserved

on some auxiliary mappings generated by nonexpansive and strictly pseudo contractive mappings abstract starting by a finite family of mappings we define the concept of procedure with lipschitzian dependence of the coefficients we give seven concrete examples of such procedures and prove the strong convergence of two viscosity methods

common fixed points of strict pseudocontractions by iterative algorithms in this paper we present iteration schemes to weakly and strongly approximate common fixed points of a finite family of a class of strict pseudocontractions

strong convergence of an explicit iterative algorithm for continuous pseudo contractions in banach spaces let c be a nonempty closed convex subset of a real banach space x whose norm is uniformly g teaux differentiable and t c c be a continuous pseudo contraction with a nonempty fixed point set f t for arbitrary given element u c and for t 0 1 let yt be the unique continuous path such that yt 1 t tyt tu assume that yt p f t as t 0 let n n and n be three real sequences in 0 1 satisfying the following conditions i n n n 1 ii limn n limn n 0 iii limn n1 n 0 or iii n 0 n1 n let n be a summable sequence of positive numbers for arbitrary initial datum x0 x00 c and a fixed n 0 construct elements xnm as follows xnm 1 nu nxn ntxnm m 0 1 2 suppose that there exists a least positive integer n n satisfying the following condition txnn n 1 txnn n n 1 1 n n define iteratively a sequence xn in an explicit manner as follows xn 1 xn 10 xnn n 1 nu nxn ntxnn n n 0 then xn converges strongly to a fixed point of t for all the continuous pseudo contractive mappings for which is possible to construct the sequence xn this result improves and extends a recent result of yao et al yonghong yao yeong cheng liou rudong chen strong convergence of an iterative algorithm for pseudocontractive mapping in banach spaces nonlinear anal 67 2007 3311 3317

recursive lower bounds on the nonlinearity profile of boolean functions and their applications the nonlinearity profile of a boolean function i e the sequence of its minimum hamming distances nlr f to all functions of degrees at most r for r ges 1 is a cryptographic criterion whose role against attacks on stream and block ciphers has been illustrated by many papers it plays also a role in coding theory since it is related to the covering radii of reed muller codes we introduce a method for lower bounding its values and we deduce bounds on the second order nonlinearity for several classes of cryptographic boolean functions including the welch and the multiplicative inverse functions used in the s boxes of the advanced encryption standard aes in the case of this last infinite class of functions we are able to bound the whole profile and we do it in an efficient way when the number of variables is not too small this allows showing the good behavior of this function with respect to this criterion as well

some distribution independent results about the asymptotic order of the average number of pivot steps of the simplex method this paper is concerned with the average number of pivot steps of the simplex method which are required to solve linear programming problems of the following kind mbox max v t x quad mbox subject to a t i x leq 1 quad mbox for i 1 ldots m mbox where a i v x in mathbb r n we investigate a certain variant of the simplex method which is particularly suited for theoretical considerations for every pair m nm n we introduce probability spaces whose random samples are the linear programming problems of the above mentioned type the vectors ai v are supposed to be distributed independently identically and symmetrically under rotations on a bounded subset of rn we deduce upper and lower bounds for the expectation values of the number of steps required in phase ii of the simplex method the evaluation of these bounds yields the following result for m and for fixed n 1 the expectation values become om1 n 1 2 there exists a distribution whose expectation values increase like m1 n 1 3 for any distribution which satisfies our conditions the expectation values tend to infinity 4 for every e 0 there are distributions whose expectation values become ome

on mann implicit iterations for strongly accretive and strongly pseudo contractive mappings abstract for a lipschitz strongly accretive map considered by chidume in c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for nonexpansive mappings and some of their generalizations nonlinear analysis and applications to v lakshmikantam on his 80th birthday vols 1 and 2 kluwer acad publ dordrecht 2003 pp 383 429 it is known that a classical picard type iteration process converges strongly to a zero of the operator he also proved that the rate of convergence in this case is at least as fast as a geometric progression in this paper we study the mann implicit iteration sequence for strongly accretive and strongly pseudo contractive mappings we showed that this implicit scheme gives better convergence rate estimate presented results improve the corresponding results of c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for nonexpansive mappings and some of their generalizations nonlinear analysis and applications to v lakshmikantam on his 80th birthday vols 1 and 2 kluwer acad publ dordrecht 2003 pp 383 429 l liu approximation of fixed points of a strictly pseudo contractive mapping proc am math soc 125 2 1997 1363 1366 w r sastry g v r babu approximation of fixed points of strictly pseudo contractive mappings on arbitrary closed convex sets in a banach space proc amer math soc 128 2000 2907 2909 y song r chen viscosity approximative methods to cesaro means for non expansive mappings appl math comput 186 2 2007 1120 1128

a family of simplex variants solving an m d linear program in expected number of pivot steps depending on d only we present a family of variants of the simplex method which are based on a constraint by constraint procedure the solution to a linear program is obtained by solving a sequence of subproblems with an increasing number of constraints we discuss several probabilistic models for generating linear programs in all of them the underlying distribution is assumed to be invariant under changing the signs of rows or columns in the problem data a weak regularity condition is also assumed under these models for linear programs with d variables and m d inequality constraints the expected number of pivots required by these algorithms is bounded by a function of minm d only in particular this means that for a fixed number of variables the expected number of pivots is bounded by a constant when the number of constraints tends to infinity since smale s original model smale s 1983 on the average speed of the simplex method of linear programming math programming27 satisfies our probabilistic assumptions the same results apply to his model although not to the particular algorithm he analyzes we also present some results for models generating only feasible linear programs and for bland s pivoting rule we conclude with a discussion of our probabilistic models and show why they are inadequate for obtaining meaningful results unless d and m are of the same order of magnitude

an approximation method for strictly pseudocontractive mappings let k be a closed convex subset of a q uniformly smooth separable banach space t k k a strictly pseudocontractive mapping and f k k an l lispschitzian strongly pseudocontractive mapping for any t 0 1 let xt be the unique fixed point of t f 1 t t we prove that if t has a fixed point then xt converges to a fixed point of t as t approaches to 0

strong convergence theorems of a general iterative process for a finite family of λi strict pseudo contractions in q uniformly smooth banach spaces in this paper we study a general iterative process to have strong convergence for a finite family of l i strict pseudo contractive non self mappings in the framework of q uniformly smooth banach spaces our results improve and extend the corresponding results announced by many others

on picard iterations for strongly accretive and strongly pseudo contractive lipschitz mappings abstract in this note we speed up the convergence of the picard sequence of iterations for strongly accretive and strongly pseudo contractive mappings our results improve the results of chidume c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for non expansive mappings and some of their generalizations in nonlinear analysis and applications to v lakshmikantham on his 80th birthday vol 1 2 kluwer acad publ dordrecht 2003 pp 383 429 and liu l liu approximation of fixed points of a strictly pseudo contractive mapping proc amer math soc 125 2 1997 1363 1366 and some other known results the technique of the proof presented in this paper is different from the technique used by chidume

viscosity approximation methods with weak contraction for l lipschitzian pseudocontractive self mapping abstract in this paper under the framework of real reflexive banach space which admits a weakly sequentially continuous duality mapping from e to e we study the strong convergence of an implicit and an explicit composite viscosity approximation algorithm i and ii respectively for a pseudocontractive mapping t by using the weakly contractive mapping f as follows i x t s t f x t s 1 t y t s y t s s x t s 1 s t x t s and ii x n 1 n f x n 1 n y n y n n x n 1 n t x n n 0 our results unify improve and complement several recent ones existing in the current literature

on the implicit iterative process for strictly pseudo contractive mappings in banach spaces in this paper we consider a composite implicit iterative process for an infinite family of strictly pseudo contractive mappings strong convergence theorems are established in an arbitrary real banach space the results presented in this paper improve and extend the recent ones announced by many others

weak and strong convergence theorems of an implicit iteration for a countable family of continuous pseudocontractive mappings to approximate a common fixed point of a countable family of continuous pseudocontractive mappings we introduce an implicit iteration sequence a necessary and sufficient condition for the convergence of a sequence of such iterates for countably many continuous pseudocontractive mappings is given we also prove the convergence theorems of an implicit iteration sequence for a countable family of strictly pseudocontractive mappings

convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings abstract in this paper a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real banach space using an implicit iteration scheme recently introduced by xu and ori h k xu r g ori an implicit iteration process for nonexpansive mappings numer fuct anal optim 22 2001 767 773 in condition n 0 1 and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of browder petryshyn type is obtained the results presented extend and improve the corresponding results of m o osilike m o osilike implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps j math anal appl 294 2004 73 81

iterative methods for common fixed points for a countable family of nonexpansive mappings in uniformly convex spaces abstract let e be a uniformly convex real banach space with a uniformly g teaux differentiable norm let k be a closed convex and nonempty subset of e let t i i 1 be a family of nonexpansive self mappings of k for arbitrary fixed 0 1 define a family of nonexpansive maps s i n 1 by s i 1 i t i where i is the identity map of k let f i 1 f t i 0 it is proved that an iterative sequence x n defined by x 0 k x n 1 n u i 1 i t n s i x n n 0 converges strongly to a common fixed point of the family t i i 1 where n and i t n are sequences in 0 1 satisfying appropriate conditions in each of the following cases a e l p 1 p and b at least one of the t i s is demicompact

convergence theorems for strict pseudo contractions in q uniformly smooth banach spaces abstract in this paper we discuss some properties of iterates generated by a strict pseudo contraction or a finite family of strict pseudo contractions in a real q uniformly smooth banach space we prove the weak convergence and strong convergence of pseudo contractive mappings the results presented in this paper are interesting extensions and improvements upon those known ones of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 349 acedo and xu g l acedo h k xu iterative methods for strict pseudo contractions in hilbert spaces nonlinear anal 67 2007 2258 2271 and zhou h zhou convergence theorems for strict pseudo contractions in 2 uniformly smooth banach spaces nonlinear anal 69 2008 3160 3173

convergence theorems for λ strict pseudo contractions in 2 uniformly smooth banach spaces abstract in this paper we discuss some properties of iterates generated by a strict pseudo contraction or a finite family of strict pseudo contractions in a real 2 uniformly smooth banach space the results presented in this paper are interesting extensions and improvements upon those known ones of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 324 2007 336 349 in order to get a strong convergence theorem we modify the normal mann s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in c this result extends a recent result of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 both from nonexpansive mappings to strict pseudo contractions and from hilbert spaces to 2 uniformly smooth banach spaces

polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems we show that a particular pivoting algorithm which we call the lexicographic lemke algorithm takes an expected number of steps that is bounded by a quadratic inn when applied to a random linear complementarity problem of dimensionn we present two probabilistic models both requiring some nondegeneracy and sign invariance properties the second distribution is concerned with linear complementarity problems that arise from linear programming in this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem and independent of the larger similar results have been obtained by adler and megiddo