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The purpose of this paper is to study the solvability for a class of generalized strong vector variational-like inequalities in reflexive Banach spaces. Firstly, utilizing Brouwer's fixed point theorem, we prove the solvability for this class of generalized strong vector variational-like inequalities without monotonicity assumption under some quite mild conditions. Secondly, we introduce the new concept of pseudomonotonicity for vector multifunctions, and prove the solvability for this class of generalized strong vector variational-like inequalities for pseudomonotone vector multifunctions by using Fan's lemma and Nadler's theorem. Our results give an affirmative answer to an open problem proposed by Chen and Hou in 2000, and also extend and improve the corresponding results of Fang and Huang (2006).

In this paper, we derive several existence results for the variational and the generalized variational inequality problems where the mappings under consideration need not have any continuity properties. Uniqueness of the solution of the generalized variational inequality problem is also investigated. As a by-product, we obtain several existence and uniqueness results for the generalized complementarity problem. Applications to minimization problems with generalized convex functions are considered.

A new problem called the generalized quasi-variational inequality problem is introduced. This new problem extends all the existing variational inequality problems in finite dimensional spaces and enlarges the class of problems that can be approached by the variational inequality theory. Some existence results without convexity assumptions are presented. Applications to mathematical and equilibrium programming are given.

In this paper, we derive some new existence results for the generalized variational inequalities by introducing multivalued (S)1+ operators. Applications to nonlinear inequalities and the generalized complementarity problem are given.

We propose an approximate proximal algorithm for solving generalized variational inequalities in Hilbert space. Extension to Bregman-function-based approximate proximal algorithm is also discussed. Weak convergence of these two algorithms are established under the paramonotonicity and pseudomonotonicity assumptions of the operators.

In this paper we establish a generalized Minty’s lemma for generalized vector equilibrium problems. Some existence results for generalized vector equilibrium problems are derived by employing this lemma.

In this paper, we introduce a new class of generalized co-complementarity problems in Banach spaces. An iterative algorithm for finding approximate solutions of these problems is considered. Some convergence results for this iterative algorithm are derived and several existence results are obtained.

In this paper, we prove an existence result for the generalized vector equilibrium problem by using Fan-Browder type fixed-point theorem due to [1].

In this work, we study the F-implicit generalized variational inequalities in a real normed space setting. Weak solutions and strong solutions are introduced. Several existence results are derived. As an application, we study the F-implicit generalized complementarity problems and some existence results are obtained.

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient α∈(0,1), F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0, and A:H→H is a strongly positive bounded linear operator with coefficient γ̄∈(1,2). Let 0

In this paper we introduce general iterative methods for finding zeros of a maximal monotone operator in a Hilbert space which unify two previously studied iterative methods: relaxed proximal point algorithm [H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math Soc. 66 (2002) 240–256] and inexact hybrid extragradient proximal point algorithm [R.S. Burachik, S. Scheimberg, B.F. Svaiter, Robustness of the hybrid extragradient proximal-point algorithm, J. Optim. Theory Appl. 111 (2001) 117–136]. The paper establishes both weak convergence and strong convergence of the methods under suitable assumptions on the algorithm parameters.

In this paper, we derive several existence results for the vector variational inequality problem. The work of this paper is a continuation of the work in [1] where generalized monotonicity is assumed.

In this note, an extended version of the Zhou and Chen minimax inequality is obtained for reflexive Banach spaces. An application of the result to the existence of Nash equilibrium points without assuming that cost functions are convex is given.

In this paper, using the concept of w-distances, and we prove existence theorems for single-valued mappings and set-valued mappings in a complete metric space which generalize Takahashi, Wong, and Yao’s theorems.

In this paper, a new modified Ishikawa iterative algorithm with errors by a shrinking projection method for generalized mixed equilibrium problems and a countable family of uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings is introduced and investigated in the framework of a real Banach space. Strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions. These results are new and develop some recent results in this field.

Zero point problems of two accretive operators and fixed point problems of a nonexpansive mappings are investigated based on a Mann-like iterative algorithm. Weak convergence theorems are established in a Banach space.

The first purpose of this paper is to prove an existence and uniqueness result for the multivariate fixed point of a contraction type mapping in complete metric spaces. The proof is based on the new idea of introducing a convenient metric space and an appropriate mapping. This method leads to the changing of the non-self-mapping setting to the self-mapping one. Then the main result of the paper will be applied to an initial-value problem related to a class of differential equations of first order. The second aim of this paper is to prove strong and weak convergence theorems for the multivariate fixed point of a N-variables nonexpansive mapping. The results of this paper improve several important works published recently in the literature.

We introduce a Mann-type hybrid steepest-descent iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of solutions of finitely many variational inequalities, and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality.

The split common fixed point problem for two quasi-pseudo-contractive operators is studied. Some properties for quasi-pseudo-contractive operators are presented. An iterative algorithm for solving the split common fixed point problem for two quasi-pseudo-contractive operators is constructed. Strong convergence theorems are proved. A unified framework for the study of this class problem and class of operators is provided.

The split common fixed points problem for demicontractive operators has been studied in Hilbert spaces. An iterative algorithm is considered and the weak convergence result is given under some mild assumptions.

In this paper, we consider the split common null point problem in Banach spaces. Then using the hybrid method and the shrinking projection method in mathematical programming, we prove strong convergence theorems for finding a solution of the split common null point problem in Banach spaces.

This paper aims to deal with an iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings in the setting of real Hilbert spaces. We establish the strong convergence of the proposed method under some suitable conditions. Numerical examples are presented to illustrate the proposed method and convergence result. The algorithm and result presented in this paper extend and improve some well-known algorithm and results, respectively, in the literature.

The fixed point theory is a well-established subject in the area of nonlinear analysis. The iterative method for solving the fixed point problem was considered from the origin of this problem. Namely, Cauchy, Liouville, Lipschitz, Peano, Fredholm, Picard, Banach, Browder, Helpern, Mann, Ishikawa, etc., have given different kinds of iterative methods for solving fixed point problems. The mathematical formulation of the equilibrium problem (EP) is to find an element x of a set K such that f(x, y) ≥ 0 for all y I K, where f : K × K ® ℝ be a bifunction. It is most general problem and includes many known problems, namely, variational inequality problem, optimization problem, saddle point problem, fixed point problem, Nash EP, etc., as special cases. This general form of EP was first considered by Nikaido and Isoda in 1955 as an auxiliary problem to establish existence results for Nash equilibrium points in non-cooperative games. In the theory of EPs, the key contribution was made by Ky Fan, whose new existence results contained the original techniques which became a basis for most further existence theorems in topological spaces. This special issue contains 37 articles accepted on different topics in the area of fixed point theory and EPs. Ten articles are devoted to general fixed point theory, four are on iterative methods for solving variational inequalities, two on hierarchical variational inequalities, one is on some iterative methods for solving systems of variational inequalities, six are devoted to different kinds of iterative methods for solving EPs, seven are on iterative methods for systems of EP, and one is on the existence of solutions of system of generalized vector quasi-EPs. There are six more articles on different topics from nonlinear analysis. Finally, we are grateful to Professor Ravi P. Agarwal and the publishers of the journal Fixed Point Theory and Applications for giving us the opportunity to edit this special

In this paper, we introduce a new class of mappings called Bregman weak relatively nonexpansive mappings and propose new hybrid iterative algorithms for finding common fixed points of an infinite family of such mappings in Banach spaces. We prove strong convergence theorems for the sequences produced by the methods. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding common fixed points of finitely many Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. These algorithms take into account possible computational errors. We also apply our main results to solve equilibrium problems in reflexive Banach spaces. Finally, we study hybrid iterative schemes for finding common solutions of an equilibrium problem, fixed points of an infinite family of Bregman weak relatively nonexpansive mappings and null spaces of a γ -inverse strongly monotone mapping in 2-uniformly convex Banach spaces. Some application of our results to the solution of equations of Hammerstein-type is presented. Our results improve and generalize many known results in the current literature. MSC: 47H10; 37C25

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradient-like iterative algorithm. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings. 1991 MSC: 47H09; 47J20.

Up to now, a large number of practical problems such as signal processing and network resource allocation have been formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms for solving these problems have been proposed. The purpose of this article is to investigate a monotone variational inequality with variational inequality constraint over the fixed point set of one or finitely many nonexpansive mappings, which is called the triple-hierarchical constrained optimization. Two relaxed hybrid steepest-descent algorithms for solving the triple-hierarchical constrained optimization are proposed. Strong convergence for them is proven. Applications of these results to constrained generalized pseudoinverse are included. AMS Subject Classifications: 49J40; 65K05; 47H09.

We present a new and interesting extension theorem for concave operators as follows. Let be a real linear space, and let be a real order complete PL space. Let the set be convex. Let be a real linear proper subspace of , with , where for some . Let be a concave operator such that whenever and . Then there exists a concave operator such that (i) is an extension of , that is, for all , and (ii) whenever .

In the present paper, we study Newton’s method on Lie groups (independent of affine connections) for finding zeros of a mapping f from a Lie group to its Lie algebra. Under a generalized L-average Lipschitz condition of the differential of f, we establish a unified convergence criterion of Newton’s method. As applications, we get the convergence criteria under the Kantorovich’s condition and the γ-condition, respectively. Moreover, applications to optimization problems are also provided.

In this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, let Fix ( T ) be the fixed point set of a nonexpansive mapping, let MEP ( Θ , φ ) be the solution set of a mixed equilibrium problem (MEP), and let Γ be the solution set of a minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solution x ∗ ∈ Fix ( T ) ∩ MEP ( Θ , φ ) ∩ Γ of a variational inequality with a variational inequality constraint over the intersection of Fix ( T ) , MEP ( Θ , φ ) , and Γ. We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm. MSC:49J40, 47J20, 47H10, 65K05, 47H09.

In this paper, considering the problem of solving a system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in Banach spaces, we propose a two-step relaxed extragradient method which is based on Korpelevich’s extragradient method and viscosity approximation method. Strong convergence results are established. MSC:49J30, 47H09, 47J20.

Let be a nonempty closed convex subset of a Banach space with the dual , let be a continuous mapping, and let be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator we study the variational inequality (for short, VI( ): find such that for all , where is a given element. By combining the approximate proximal point scheme both with the modified Ishikawa iteration and with the modified Halpern iteration for relatively nonexpansive mappings, respectively, we propose two modified versions of the approximate proximal point scheme L. C. Ceng and J. C. Yao (2008) for finding approximate solutions of the VI( ). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of the VI( ), which is also a fixed point of .

The hybrid steepest-descent method introduced by Yamada (2001) is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi (1996) introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamada's hybrid steepest-descent and Lehdili and Moudafi's algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions. Our strong convergence theorems extend and improve certain corresponding results in the recent literature.

In this paper, we consider a hierarchical variational inequality problem (HVIP) defined over a common set of solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, a general system of variational inequalities, and the fixed point problem of a strictly pseudocontractive mapping. By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method and Mann’s iteration method, we introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of our HVIP. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a solution of a general system of variational inequalities defined over a common set of solutions of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to a unique solution of our HVIP. The results obtained in this paper improve and extend the corresponding results announced by many others. MSC:49J30, 47H09, 47J20.

In this paper, we suggest a hybrid method for finding a common element of the set of solution of a monotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method. Mathematics Subject Classification (2000): 47H05; 47H09; 47H10; 47J05; 47J25.

Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek et al. in Taiwan. J. Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo et al. (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J. Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J. Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive non-self mappings and the Kocourek et al. fixed point theorem (Taiwan. J. Math. 14:2497-2511, 2010) for super hybrid non-self mappings. In particular, we solve a fixed point problem. MSC:Primary 47H10; secondary 47H05.