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

convergence criteria of iterative methods based on landweber iteration for solving nonlinear problems abstract landweber iteration xk 1 xk f xk f xk y for the solution of a nonlinear operator equation f x0 y0 can be viewed as a fixed point iteration with fixed point operator x f x f x y especially for nonlinear ill posed problems it seems impossible to verify that this fixed point operator is of contractive type which is a typical assumption for proving weak convergence of fixed point iteration schemes however for specific examples of nonlinear ill posed problems it is possible to verify conditions of quasi contractive type weak convergence of landweber iteration can be proven by application of general results for fixed point iterations based on quasi contractive type conditions in a recent paper by hanke et al a condition on the operator f has been investigated which guarantees convergence of the landweber s method a geometrical interpretation of this condition is given and is compared with well known conditions in the theory of fixed point iterations

thermoacoustic tomography for an integro differential wave equation modeling attenuation abstract in this article we study the inverse problem of thermoacoustic tomography tat on a medium with attenuation represented by a time convolution or memory term and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters under the assumption of being able to measure data on the whole boundary we prove uniqueness and stability and propose a convergent reconstruction method for a class of smooth variable sound speeds by a suitable modification of the time reversal technique we obtain a neumann series reconstruction formula

inversion of spherical means and the wave equation in even dimensions we establish inversion formulas of the so called filtered back projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball we also find several formulas to recover initial data of the form f 0 or 0 g for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball provided that the initial data has support in the ball

a general iterative method for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem these results extended and improved the corresponding results of marino and xu g marino h k xu a general iterative method for nonexpansive mapping in hilbert spaces j math anal appl 318 2006 43 52 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 1 2007 506 515 and many others

weak and strong convergence theorems for strict pseudo contractions in hilbert spaces abstract let c be a closed convex subset of a real hilbert space h and assume that t is a strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the mann s algorithm generates a sequence x n by the formula x n 1 n x n 1 n t x n n 0 it is proved that if the control sequence n is chosen so that n 1 and n 0 n 1 n then x n converges weakly to a fixed point of t however this convergence is in general not strong we then modify mann s algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence this result extends a recent result of nakajo and takahashi k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 from nonexpansive mappings to strict pseudo contractions

algorithms for approximating minimization problems in hilbert spaces in this paper we study the following minimization problem minx fix s w m2 12 x 2 h x where b is a bounded linear operator m 0 is some constant h is a potential function for c f fix t is the set of fixed points of nonexpansive mapping s and w is the solution set of an equilibrium problem this paper introduces two new algorithms one implicit and one explicit that can be used to find the solution of the above minimization problem

recovery of a source term or a speed with one measurement and applications we study the problem of recovery the source a t x f x in the wave equation in anisotropic medium with a known so that a 0 x 6d 0 with a single measurement we use carleman estimates combined with geometric arguments and give sharp conditions for uniqueness we also study the non linear problem of recovery the sound speed in the equation utt c x u d 0with one measurement we give sharp conditions for stability as well an application to thermoacoustic tomography is also presented

on iterative methods for equilibrium problems abstract in this paper we introduce a hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of finitely many nonexpansive mappings we prove that the approximate solution converges strongly to a solution of a class of variational inequalities under some mild conditions which is the optimality condition for some minimization problem we also give some comments on the results of plubtieng and punpaeng s plubtieng r punpaeng a general iterative method for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 336 2007 455 469 results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this area

approximating common fixed points of nonexpansive mappings and nonspreading mappings in a hilbert space abstract in this paper we obtain some fundamental properties for nonspreading mappings in a hilbert space further we study the approximation of common fixed points of nonexpansive mappings and nonspreading mappings in a hilbert space

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

an explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings abstract let h be a hilbert space and c be a nonempty closed convex subset of h t i i n be a family of nonexpansive mappings from c into h g i c c r be a finite family of equilibrium functions i 1 2 k a be a strongly positive bounded linear operator with a coefficient and b lipschitzian relaxed cocoercive map of c into h moreover let r k n k 1 k n satisfy appropriate conditions and f k 1 k e p g k v i c b n n f i x t n 0 we introduce an explicit scheme which defines a suitable sequence as follows x n 1 n f x n 1 n a w n p c i s n b s r 1 n 1 s r 2 n 2 s r k n k x n n n and x n strongly converges to x f which satisfies the variational inequality a f x x x 0 for all x f the results presented in this paper mainly extend and improve a recent result of colao v colao an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings nonlinear analysis 2009 doi 10 1016 j na 2009 01 115 and qin x qin m shang y su a general iterative method for equilibrium problems and fixed point problems in hilbert spaces nonlinear analysis 69 2008 3897 3909

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

convergence of mann s type iteration method for generalized asymptotically nonexpansive mappings let c be a nonempty closed and convex subset of a real hilbert space h let t i c h i 1 2 n be a finite family of generalized asymptotically nonexpansive mappings it is our purpose in this paper to prove strong convergence of mann s type method to a common fixed point of t i i 1 2 n provided that the interior of common fixed points is nonempty no compactness assumption is imposed either on t or on c as a consequence it is proved that mann s method converges for a fixed point of nonexpansive mapping provided that interior of f t 0 the results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings

a new general iterative scheme for split variational inclusion and fixed point problems of k strict pseudo contraction mappings with convergence analysis in this paper we modify the general iterative method to approximate a common element of the set of solutions of split variational inclusion problem and the set of common fixed points of a finite family of k strictly pseudo contractive nonself mappings strong convergence theorem is established under some suitable conditions in a real hilbert space which also solves some variational inequality problems results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors finally some examples to study the rate of convergence and some illustrative numerical examples are presented

modified szabo s wave equation models for lossy media obeying frequency power law szabo s models of acoustic attenuation szabo j acoust soc am 96 1 491 500 1994 comply well with the empirical frequency power law involving noninteger and odd integer exponent coefficients while guaranteeing causality but nevertheless encounter the troublesome issues of hypersingular improper integral and obscurity in implementing initial conditions the purpose of this paper is to ease or remove these drawbacks of the szabo s models via the caputo fractional derivative concept the positive time fractional derivative is also introduced to include the positivity of the attenuation processes

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

a new algorithm for two finite families of demicontractive mappings and equilibrium problems in this paper we introduce a new algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two finite families of demicontractive mappings the strong convergence theorem of the proposed algorithm is established under some suitable control conditions in a real hilbert space our result generalizes several recent results in the current literature

an iterative method for finding common solutions of equilibrium and fixed point problems we introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family 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

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

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

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

a unified hybrid iterative method for hierarchical minimization problems in this paper we introduce and analyze a new unified hybrid iterative method to compute the approximate solution of the general optimization problem defined over the set d fix t email protected gmep f j f where fix t is the set of common fixed points of a family t t t email protected t of nonexpansive self mappings on a hilbert space h and w gmep f j f is the set of solutions of the generalized mixed equilibrium problem in short gmep such type of minimization problem is called the hierarchical minimization problem we establish the strong convergence of the sequences generated by the proposed algorithm our strong convergence theorem extends improves and unifies the previously known results in the literature we also give a numerical example to illustrate our algorithm and results

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

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 general iterative method for equilibrium problems and strict pseudo contractions in hilbert spaces abstract in this paper we introduce two iterative schemes by the general iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a k strictly pseudo contractive non self mapping in the setting of real hilbert spaces our results improve and extend the corresponding results given by many others

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

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

time domain wave equations for lossy media obeying a frequency power law for attenuation described by a slowly varying power law function of frequency 0 y classical lossy time domain wave equations exist only for the restricted cases where y 0 or y 2 for the frequently occurring practical situation in which attenuation is much smaller than the wave number a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence in order to obtain the corresponding time domain lossy wave equation time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions three forms of lossy wave equations are found depending on whether y is an even or odd integer or a noninteger a time domain expression of causality analogous in function to the kramers kronig relations in the frequency domain is used to derive the causal wave equations final causal versions of the time domain wave equations are obtained even for the cases where y 1 which according to the paley wiener th

mixed equilibrium problems and optimization problems in this paper we introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings furthermore we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions we also discuss some special cases results obtained in this paper improve the previously known results in this area

strong convergence of a new composite iterative method for equilibrium problems and fixed point problems in this paper we propose a new composite iterative method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings in a hilbert space it is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings our results improve and extend the corresponding ones announced by many others

on perfect isometries and isotypies in alternating groups perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the brauer correspondents of these blocks these perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended broue conjecture holds for the principal block of an almost simple group with an abelian sylow psubgroup and a generalized fitting subgroup isomorphic to an alternating group let g be a finite group and let 0 be a complete discrete valuation ring with field of quotients k of characteristic 0 and residue class field k of characteristic p 0 we suppose that k contains a primitive igi th root of unity in 4 6 1 michel broue posed the following isotypy conjecture let e be a block of og with abelian defect group d and let f be the brauer correspondent of e in 9ng d then e and f are isotypic blocks if g has an abelian sylow p subgroup then the conjecture can be posed for the principal block of 0g in this case the authors have shown that the conjecture holds provided an extended conjecture holds for the principal block of almost simple groups with an abelian sylow p subgroup see 10 5e in this paper the isotypy conjecture for an arbitrary block with abelian defect group is proved for alternating groups in addition the extended conjecture is proved for the principal p block of almost simple groups with abelian sylow p subgroups and generalized fitting subgroup isomorphic to an alternating group we recall the basic definitions let 0 k be the canonical quotient mapping and let og kg be the induced 0 algebra homomorphism of the group algebras in particular e e induces a bijection between central idempotents of og and kg if e is a block idempotent of og let kgemod be the category of left kge modules of finite type and let zk g e be the grothendieck group of kgemod let gv be the set of irreducible characters of g over k we identify r k g e with the free abelian group on g e v x e gv x ge x g for all g e g let cf g k be the k space of k valued class functions on g and let cf g e k be the k subspace of class functions ae in cf g k such that ao ge a g the received by the editors march 26 1996 1991 mathematics subject classification primary 20c15 20c20 secondary 20c30 the first author was supported in part by nsf grant dms 9100310 the second author was supported in part by nsa grant mda 904 92 h 3027 1997 american mathematical society

modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian the efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics previous wave equations which use time domain fractional operators require the storage of the complete pressure field at previous time steps such operators are convolution based this makes them unsuitable for many three dimensional problems of interest here a wave equation that utilizes two lossy derivative operators based on the fractional laplacian is derived these operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into fourier based pseudospectral and k space methods without the increase in memory required by their time domain fractional counterparts a framework for encoding the developed wave equation using three coupled first order constitutive equations is discussed and the model is demonstrated through several one two and three dimensional simulations

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

strong convergence of the cq method for fixed point iteration processes strong convergence theorems are obtained for the cq method for an ishikawa iteration process a contractive type iteration process for nonexpansive mappings and the proximal point algorithm for maximal monotone operators in hilbert spaces

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

new refinements of the mckay conjecture for arbitrary finite groups let g be an arbitrary finite group and fix a prime number p the mckay conjecture asserts that g and the normalizer in g of a sylow p subgroup have equal numbers of irreducible characters with degrees not divisible by p the alperin mckay conjecture is version of this as applied to individual brauer p blocks of g we offer evidence that perhaps much stronger forms of both of these conjectures are true

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

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

weak and strong convergence theorems for nonspreading type mappings in hilbert spaces abstract weak and strong convergence theorems are proved in real hilbert spaces for a new class of nonspreading type mappings more general than the class studied recently in kurokawa and takahashi y kurokawa w takahashi weak and strong convergence theorems for nonspreading mappings in hilbert spaces nonlinear anal 73 2010 1562 1568 we explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of halpern s type for our class of mappings and hence resolved in the affirmative an open problem posed by kurokawa and takahashi in their final remark for the case where the mapping t is averaged

iterative methods for strict pseudo contractions in hilbert spaces abstract let t i i 1 n be n strict pseudo contractions defined on a closed convex subset c of a real hilbert space h consider the problem of finding a common fixed point of these mappings and consider the parallel and cyclic algorithms for solving this problem we will prove the weak convergence of these algorithms moreover by applying additional projections we further prove that these algorithms can be modified to have strong convergence

the isaacs navarro conjecture for the alternating groups abstract a recent refinement of the mckay conjecture is verified for the case of the alternating groups the argument builds upon the verification of the conjecture for the symmetric groups p fong the isaacs navarro conjecture for symmetric groups j algebra 250 1 2003 154 161

an explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings let h be a hilbert space t i i n a family of nonexpansive mappings from h into itself g i cxc r a finite family of equilibrium functions email protected 1 2 k a a strongly positive bounded linear operator with coefficient email protected and f an a contraction on h let w n be the mapping generated by t i and l n as in 1 5 let s r k n k be the resolvent generated by g k and r k n as in lemma 2 4 moreover let r k n k 1 k e n and l n satisfy appropriate conditions and email protected k 1 ksep g k n nfix t n email protected we introduce an explicit scheme which defines a suitable sequence as follows z n email protected email protected z n email protected na w ns r 1 n 1s r 2 n 2 s r k n kz email protected email protected n and z n strongly converges to x f which satisfies the variational inequality 0 for all email protected f the results presented in this paper mainly extend and improve some recent results in vittorio colao et al an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings nonlinear anal 71 2009 2708 2715 s plubtieng r punpaeng a general iterative method for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 336 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 2007 506 515

boundary rigidity with partial data we study the boundary rigidity problem with partial data consisting of determining locally the riemannian metric of a riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary we show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information in particular this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary the local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces

existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces a class of nonlinear operators in banach spaces is proposed we call each operator in this class a firmly nonexpansive type mapping this class contains the classes of firmly nonexpansive mappings in hilbert spaces and resolvents of maximal monotone operators in banach spaces we study the existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces

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