convergence and certain control conditions for hybrid viscosity approximation methods abstract very recently yao chen and yao 20 proposed a hybrid viscosity approximation method which combines the viscosity approximation method and the mann iteration method under the convergence of one parameter sequence to zero they derived a strong convergence theorem in a uniformly smooth banach space in this paper under the convergence of no parameter sequence to zero we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping which solves a variational inequality an appropriate example such that all conditions of this result are satisfied and their condition n 0 is not satisfied is provided furthermore we also give a weak convergence theorem for their method involving a nonexpansive mapping in a hilbert space

on the laplacian eigenvalues of a graph abstract in the note we present an upper bound for the spectral radius of laplacian matrix of a graph in terms of a 2 degree of a vertex

laplacian spectral characterization of 3 rose graphs abstract a 3 rose graph is a graph consisting of three cycles intersecting in a common vertex j wang et al showed all 3 rose graphs with at least one triangle are determined by their laplacian spectra in this paper we complete the above laplacian spectral characterization and prove that all 3 rose graphs are determined by their laplacian spectra

pointwise lipschitzian mappings in uniformly convex and uniformly smooth banach spaces abstract let x be a uniformly convex and uniformly smooth banach space let t c c be an asymptotic pointwise nonexpansive mapping where c is a bounded closed and convex subset of x in this paper we investigate conditions sufficient for the weak convergence of the generalized mann and ishikawa processes to a fixed point of t

graph zn and some graphs related to zn are determined by their spectrum abstract it is proved that graph zn is determined by its adjacency spectrum as well as its laplacian spectrum z n 1 z n 2 z n k is determined by its adjacency spectrum where n1 n2 nk are integers at least 2 wn is not determined by its adjacency spectrum but is determined by its laplacian spectrum kzn tn are determined by their laplacian spectrum respectively where k is a positive integer

strong convergence of modified mann iterations abstract the mann iterations for nonexpansive mappings have only weak convergence even in a hilbert space we propose two modifications of the mann iterations in a uniformly smooth banach space one for nonexpansive mappings and the other for the resolvent of accretive operators the two modified mann iterations are proved to have strong convergence

strong convergence of modified mann iterations for a countable family of nonexpansive mappings abstract in this paper we propose a new modified mann iteration for computing common fixed points of nonexpansive mappings in a banach space we give certain different control conditions for the modified mann iteration then we prove strong convergence theorems for a countable family of nonexpansive mappings in uniformly smooth banach spaces these results improve and extend results of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 yao et al y yao r chen and j yao strong convergence and certain control conditions for modified mann iteration nonlinear anal 68 2008 1687 1693 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 many others

strong convergence and certain control conditions for modified mann iteration abstract in this paper we propose a new modified mann iteration for computing fixed points of nonexpansive mappings in a banach space setting this new iterative scheme combines the modified mann iteration introduced by kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 and the viscosity approximation method introduced by moudafi a moudafi viscosity approximation methods for fixed points problems j math anal appl 241 2000 46 55 we give certain different control conditions for the modified mann iteration strong convergence in a uniformly smooth banach space is established

p r invex sets and functions abstract notions of invexity of a function and of a set are generalized the notion of an invex function with respect to can be further extended with the aid of p invex sets slight generalization of the notion of p invex sets with respect to leads to a new class of functions a family of real functions called in general p r pre invex functions with respect to without differentiability or p r invex functions with respect to in the differentiable case is introduced some geometric properties of these classes of functions are derived sufficient optimality conditions are obtained for a nonlinear programming problem involving p r invex functions with respect to

equivalence of saddle points and optima and duality for a class of non smooth non convex problems abstract the equivalence between saddle points and optima and duality theorems are established for a much larger class of non smooth non convex problems in which functions are locally lipschitz and are satisfying invex type conditions of hanson and craven

a class of b p r invex functions and mathematical programming abstract invexity of a function is generalized the new class of nonconvex functions called b p r invex functions with respect to and b being introduced includes many well known classes of generalized invex functions as its subclasses some properties of the introduced class of b p r invex functions with respect to and b are studied further mathematical programming problems involving b p r invex functions with respect to and b are considered the equivalence between saddle points and optima and different type duality theorems are established for this type of optimization problems

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

on some forests determined by their laplacian or signless laplacian spectrum we consider the class of graphs whose each component is either a proper subgraph of some smith graphs or belongs to a precized subset of smith graphs we classify the graphs from the considered class into those which are determined or not determined by laplacian or signless laplacian spectrum

strong convergence of the modified mann iterative method for strict pseudo contractions in this paper we introduce a modified mann iterative process for approximating a common fixed point of a finite family of strict pseudo contractions in hilbert spaces we establish the strong convergence theorem of the general iteration scheme under some mild conditions our results extend and improve the recent ones announced by many others

the complement of the path is determined by its spectrum abstract it is proved that a graph whose 0 1 adjacency matrix has the spectrum of p n the complement of the path on n vertices must be p n

the spectral characterization of graphs abstract a graph b r s is a graph consisting of two cycles c r 1 and c s 1 with just a vertex in common fan and luo in theorem 4 4 8 alleged that a graph is determined by its adjacency spectrum if it contains no cycle c 4 however according to theorem 6 3 16 we find that the result in theorem 4 4 8 is not completely correct in this paper we prove that b r s s r 7 is das if and only if s r 2 and b r s s r 7 has a unique cospectral mate if s r 2

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

strong convergence of an iterative method for nonexpansive mappings with new control conditions abstract the purpose of this paper is to study the convergence problem of an iterative method for nonexpansive mappings in banach spaces under some new control conditions on parameters

on p r invexity type nonlinear programming problems abstract mathematical programming problems involving p r invex functions with respect to are considered we introduce new classes of nonlinear programming problems called kt p r invex wd p r invex and hc p r invex problems where p r are some real numbers it is shown that for these types of problems kuhn tucker conditions are both necessary and sufficient for optimality furthermore these p r invexity type problems with r 0 are not sufficient for wolfe weak duality in this way it was shown that the optimization problems possessing some kind of invexity need not be equivalent to the class of optimization problems for which kuhn tucker necessary conditions for optimality are also sufficient and wolfe weak duality holds

graphs determined by their generalized characteristic polynomials abstract for a given graph g with 0 1 adjacency matrix a g the generalized characteristic polynomial of g is defined to be g g t det i a g td g where i is the identity matrix and d g is the diagonal degree matrix of g in this paper we are mainly concerned with the problem of characterizing a given graph g by its generalized characteristic polynomial g we show that graphs with the same generalized characteristic polynomials have the same degree sequence based on which a unified approach is proposed to show that some families of graphs are characterized by g we also provide a method for constructing graphs with the same generalized characteristic polynomial by using gm switching

a note on the spectral characterization of θ graphs we consider graphs that is graphs obtained by subdividing the edges of the multigraph consisting of 3 parallel edges it is shown that any graph g is determined by the spectrum the multiset of eigenvalues except possibly when it contains a unique 4 cycle

spectral characterizations of graphs with small spectral radius abstract a graph is said to have a small spectral radius if it does not exceed the corresponding hoffmann limit value in the case of signless laplacian matrix the hoffmann limit value is equal to 2 4 38 with being the real root of x 3 4 x 4 here the spectral characterization of connected graphs with small signless laplacian spectral radius is considered it is shown that all connected graphs with small laplacian spectral radius are determined by their laplacian spectra and all but one of connected graphs with small signless laplacian spectral radius are determined by their signless laplacian spectra

implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings in this paper we propose a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive mappings we establish some convergence theorems for this implicit iteration scheme in particular necessary and sufficient conditions for strong convergence of this implicit iteration scheme were obtained

generalized fractional minimax programming with b p r invexity optimality conditions are proved for a class of generalized fractional minimax programming problems involving b p r invexity functions subsequently these optimality conditions are utilized as a basis for constructing various duality models for this type of fractional programming problems and proving appropriate duality theorems

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

nonsmooth multiple objective optimization in separable hilbert spaces abstract this paper provides new necessary and sufficient optimality conditions for nonsmooth multiple objective optimization programs in separable hilbert spaces

symmetric duality with p r ρ η θ invexity abstract in this paper we introduce a new type of generalized invex function called p r invex function and study symmetric duality results under these assumptions in our study the nonnegative orthants for the constraints are replaced by closed convex cones and their polars we establish weak and strong duality theorems under p r invexity assumptions for the symmetric dual problems we also give many examples to justify our results

penalty function methods and a duality gap for invex optimization problems abstract in this paper the penalty function method is used to study duality in nonconvex mathematical programming problems in particular we prove the zero duality gap between optimization problems involving invex functions with respect to the same function and their lagrangian dual problems the results proved in the paper are illustrated by suitable examples of nonconvex optimization problems

convergence of a general iterative method for nonexpansive mappings in hilbert spaces in this paper we introduce a modified ishikawa iterative process for approximating a fixed point of nonexpansive mappings in hilbert spaces we establish some strong convergence theorems of the general iteration scheme under some mild conditions the results improve and extend the corresponding results of many others

necessary and sufficient conditions in constrained optimization additional conditions are attached to the kuhn tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization under appropriate constraint qualifications necessary and sufficient conditions are also given for optimality of the dual problem duality and converse duality are treated accordingly

strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings let c be a nonempty closed convex subset of a real strictly convex and reflexive banach space e which has a uniformly g teaux differentiable norm let f c c be a given contractive mapping and tn n 1 c c be an infinite family of nonexpansive mappings such that the common fixed point sets f n 1 f tn let n and n be two real sequences in 0 1 for given x0 c arbitrarily let the sequence xn be generated iteratively by xn 1 nf xn nxn 1 n n wnxn wherewn is the w mapping generated by the mappings tn tn 1 t1 and n n 1 1 suppose the iterative parameters n and n satisfy the following control conditions c1 limn n 0 c2 n 0 n b5 limsupn n 1 then the sequence xn converges strongly to p f where p is the unique solution in f to the following variational inequality i f p j p x 0for allx f

exact penalty functions method for mathematical programming problems involving invex functions in this paper some new results on the exact penalty function method are presented simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method the equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption furthermore the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved

strong convergence of a new iterative scheme for a finite family of strict pseudo contractions the purpose of this paper is to introduce a new iterative scheme for finding a common fixed point of a finite family of strict pseudo contractions the results obtained in this paper extend and improve the corresponding results of marino colao qin and kang g marino v colao x qin s m kang strong convergence of the modified mann iterative method for strict pseudo contractions comput math appl 57 2009 455 465

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 sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

nondifferentiable multiobjective programming under generalized di invexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce new concepts of d i invexity and generalized d i invexity in which each component of the objective and constraint functions is directionally differentiable in its own direction d i new fritz john type necessary and karush kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be weakly efficient efficient or properly efficient moreover we prove weak strong converse and strict duality results for a mond weir type dual under various types of generalized d i invexity assumptions

a unified hybrid iterative method for solving variational inequalities involving generalized pseudocontractive mappings we study in this paper the existence and the approximation of solutions of variational inequalities involving generalized pseudocontractive mappings in banach spaces the convergence analysis of a proposed hybrid iterative method for approximating common zeros or fixed points of a possibly infinitely countable or uncountable family of such operators will be conducted within the conceptual framework of the viscosity approximation technique in reflexive banach spaces with uniform g teaux differentiable norms this technique should make existing or new results in solving variational inequalities more applicable

strong convergence theorems for an infinite family of nonexpansive mappings in banach spaces in an infinite dimensional hilbert space the normal mann s iteration algorithm has only weak convergence in general even for nonexpansive mappings in order to get a strong convergence result we modify the normal mann s iterative process for an infinite family of nonexpansive mappings in the framework of banach spaces our results improve and extend the recent results announced by many others

note the t shape tree is determined by its laplacian spectrum a t shape is a tree with exactly one of its vertices having maximal degree 3 it is proved in this paper that the t shape tree is determined by its laplacian spectrum

spectral determination of graphs whose components are paths and cycles we consider the class of graphs each of whose components is either a path or a cycle we classify the graphs from the class considered into those which are determined and those which are not determined by the adjacency spectrum in addition we compare the result with the corresponding results for the laplacian and the signless laplacian spectra it turns out that the signless laplacian spectrum performs the best confirming some expectations from the literature

approximation of common fixed points of an infinite family of nonexpansive mappings in banach spaces let c be a closed convex subset of a real uniformly smooth and strictly convex banach space e consider the following iterative algorithm given by x 0 x carbitrarily chosen y n b nx n 1 b n w nx n x n 1 a nf x n 1 a n y n n 0 where f is a contraction on c and w n is a mapping generated by an infinite family of nonexpansive mappings t i i 1 assume that the set of common fixed points of this infinite family of nonexpansive mappings is not empty in this paper we prove that the sequence x n generated by the above iterative algorithm converges strongly to a common fixed point of t i i 1 which solves some variational inequality our results improve and extend the results announced by many others

laplacian spectrum characterization of extensions of vertices of wheel graphs and multi fan graphs the graph c n email protected k k is the product of a circuit c n 1 and a clique k k in this paper we will prove that it is determined by their laplacian spectrum except when n 1 6 if n 1 6 there are several counterexamples we also prove that the product of s vertex disjoint paths and a clique p n email protected p n email protected email protected p n s k k is also determined by the laplacian spectrum

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

starlike trees are determined by their laplacian spectrum in this paper we show that if g is a starlike tree then it is determined by its laplacian spectrum moreover we prove some facts about trees with the same adjacency spectrum as a starlike tree

constructing graphs with given spectrum and the spectral radius at most 2 abstract two graphs are cospectral if their spectra coincide the set of all graphs that are cospectral to a given graph including the graph by itself is the cospectral equivalence class of the graph we say that a graph is determined by its spectrum or that it is a ds graph if it is a unique graph having that spectrum given n reals belonging to the interval 2 2 we want to find all graphs on n vertices having these reals as the eigenvalues of the adjacency matrix such graphs are called smith graphs our search is based on solving a system of linear diophantine equations we present several results on spectral characterizations of smith graphs

strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in this paper we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in hilbert spaces by the hybrid method in the mathematical programming

strong convergence of composite iterative schemes for a countable family of nonexpansive mappings in banach spaces abstract in this paper we propose a new modified viscosity approximation method for approximating common fixed points for a countable family of nonexpansive mappings in a banach space we prove strong convergence theorems for a countable family nonexpansive mappings in a reflexive banach space with uniformly gateaux differentiable norm under some control conditions these results improve and extend the results of jong soo jung j s jung convergence on composite iterative schemes for nonexpansive mappings in banach spaces fixed point theory and appl 2008 2008 14 pp article id 167535 further we apply our result to the problem of finding a zero of an accretive operator and extend the results of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 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 space nonlinear anal 70 2009 1830 1840 and chen and zhu r chen z zhu viscosity approximation methods for accretive operator in banach space nonlinear anal 69 2008 1356 1363

strong convergence theorems of modified mann iterative process for strict pseudo contractions in hilbert spaces abstract in this paper we modify the normal mann s iterative process to have strong convergence for a k strictly pseudo contractive non self mapping in the framework of hilbert spaces our results improve and extend the corresponding results announced by many others

spectral characterization of graphs with index at most 2 5 abstract a graph is said to be determined by the adjacency spectrum ds for short if there is no other nonisomorphic graph with the same spectrum all connected graphs with index at most 2 5 are known in this paper we show that with few exceptions all of these graphs are ds

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

the characteristic polynomial of the laplacian of graphs in a b linear classes in this work we deal with the characteristic polynomial of the laplacian of a graph we present some general results about the coefficients of this polynomial we present families of graphs for which the number of edges m is given by a linear function of the number of vertices n in some of these graphs we can find certain coefficients of the above named polynomial as functions just of n

spectral characterization of graphs whose second largest eigenvalue is less than 1 abstract graphs with second largest eigenvalue 2 1 are extensively studied however whether they are determined by their adjacency spectra or not is less considered in this paper we completely characterize all the connected bipartite graphs with 2 1 that are determined by their adjacency spectra in addition we prove that all the connected non bipartite graphs with girth no less than 4 and 2 1 are determined by their adjacency spectra

spectral characterizations of lollipop graphs abstract the lollipop graph denoted by h n p is obtained by appending a cycle c p to a pendant vertex of a path p n p we will show that no two non isomorphic lollipop graphs are cospectral with respect to the adjacency matrix it is proved that for p odd the lollipop graphs h n p and some related graphs h n p are determined by the adjacency spectrum and that all lollipop graphs are determined by its laplacian spectrum

spectral characterization of unicyclic graphs whose second largest eigenvalue does not exceed 1 abstract connected graphs with the same number of vertices and edges are called unicyclic graphs all unicyclic graphs whose second largest eigenvalue does not exceed 1 denoted by u 2 are determined by xu 10 let g t s denote the graph obtained by attaching t 1 pendent paths of length 2 and s 0 pendent edges to a vertex of a triangle in this paper we first prove that the graph g t s is determined by its adjacency spectrum then we prove that the graphs in u 2 are determined by their adjacency spectra except for four graphs

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

the cospectral equivalence classes of graphs having an isolated vertex let f g l be the characteristic polynomial of a graph g two graphs g and h are cospectral denoted by g h if f g l f h l by g f we denote the cospectral equivalence class determined by g under a graph g is said to be determined by its spectrum or simply g is a ds graph if h g whenever h g in this paper we determine the cospectral equivalence classes of three kinds of graphs having an isolated vertex find several ds graphs and identify the graph that has the fourth minimum index among all connected graphs with n vertices

viscosity approximation methods for countable families of nonexpansive mappings in banach spaces abstract let c be a nonempty closed convex subset of a banach space e and let s n be a family of nonexpansive mappings of c into itself such that the set of common fixed points of s n is nonempty we first introduce a sequence x n of c defined by x 1 x c and x n 1 n f x n 1 n s n x n for all n n where n 0 1 and f is a contraction of c into itself further we give the conditions of n and s n under which x n converges strongly to a common fixed point of s n this result generalizes the strong convergence theorem for nonexpansive mappings by suzuki t suzuki a sufficient and necessary condition for halpern type strong convergence to fixed points of nonexpansive mappings proc amer math soc 135 2007 99 106 and the strong convergence theorem for accretive operators by kamimura and takahashi s kamimura w takahashi weak and strong convergence of solutions to accretive operator inclusions and applications set valued anal 8 2000 361 374 simultaneously using this result we improve and extend the two above mentioned results

on a signless laplacian spectral characterization of t shape trees let m be an associated matrix of a graph g the adjacency laplacian and signless laplacian matrix two graphs are said to be cospectral with respect to m if they have the same m spectrum a graph is said to be determined by m spectrum if there is no other non isomorphic graph with the same spectrum with respect to m it is shown that t shape trees are determined by their laplacian spectra moreover among them those are determined by their adjacency spectra are characterized in this paper we identify graphs which are cospectral to a given t shape tree with respect to the signless laplacian matrix subsequently t shape trees which are determined by their signless laplacian spectra are identified

which wheel graphs are determined by their laplacian spectra the wheel graph denoted by w n 1 is the graph obtained from the circuit c n with n vertices by adding a new vertex and joining it to every vertex of c n in this paper the wheel graph w n 1 except for w 7 is proved to be determined by its laplacian spectrum and a graph cospectral with the wheel graph w 7 is given

the spectral characterization of graphs of index less than 2 with no path as a component a graph is said to be determined by the adjacency and laplacian spectrum or to be a ds graph for short if there is no other non isomorphic graph with the same adjacency and laplacian spectrum respectively it is known that connected graphs of index less than 2 are determined by their adjacency spectrum in this paper we focus on the problem of characterization of ds graphs of index less than 2 first we give various infinite families of cospectral graphs with respect to the adjacency matrix subsequently the results will be used to characterize all ds graphs with respect to the adjacency matrix of index less than 2 with no path as a component moreover we show that most of these graphs are ds with respect to the laplacian matrix