An implicit iterative scheme for monotone variational inequalities and fixed point problems

In this paper we introduce an implicit iterative scheme for finding a common element of the set of common fixed points of N$N$ nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The implicit iterative scheme is based on two well-known methods: extragradient and approximate proximal. We obtain a weak convergence theorem for three sequences generated by this implicit iterative scheme. On the basis of this theorem, we also construct an implicit iterative process for finding a common fixed point of N+1$N+1$ mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other N$N$ mappings are nonexpansive.     

Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.     

Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators

In this paper, we introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators. We establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique. This equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces

A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved.     

Existence of solutions for generalized implicit vector variational-like inequalities

In this paper, we introduce a new class of generalized implicit vector variational-like inequalities in Hausdorff topological vector spaces and Banach spaces which contain implicit vector equilibrium problems, implicit vector variational inequalities and implicit vector complementarity problems as special cases. We derive some new results by using the KKM–Fan theorem, under compact and noncompact assumptions on underlying convex sets.     

A Fuzzy Extension of Generalized Implicit Vector Variational-like Inequalities

In this paper, we study the existence of solutions to the generalized implicit vector variational like inequality problems with fuzzy mappings in Hausdorff topological vector spaces using KKM-Fan Theorem.     

A continuity method for sweeping processes

We show that the existence and uniqueness of BV continuous sweeping processes can be easily reduced to the Lipschitz continuous case by means of a suitable reparametrization of the associated moving convex set. Moreover we put this approach in the wider framework of rate independent operators acting on curves in metric spaces and we prove an extension theorem for such operators. This abstract theorem is then applied in order to infer continuous dependence of the sweeping process on the data.     

Convergence of an implicit iterative process for asymptotically pseudocontractive nonself-mappings

In this work, an implicit iterative process is considered for asymptotically pseudocontractive nonself-mappings. Weak and strong convergence theorems for common fixed points of a family of asymptotically pseudocontractive nonself-mappings are established in the framework of Hilbert spaces.     

Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces

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.     

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.     

A generalization of Browder’s fixed point theorem with applications

In this paper, the concept of c$c$-compact mapping is introduced. A generalization of Browder’s fixed point theorem and some equivalence forms are given. As applications, the existence of solutions for some variational inequalities and monotone operator equations is discussed.     

Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities

Some new classes of extended general nonconvex set-valued variational inequalities and the extended general Wiener-Hopf inclusions are introduced. By the projection technique, equivalence between the extended general nonconvex set-valued variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener-Hopf inclusions is proved. Then by using this equivalent formulation, we discuss the existence of solutions of the extended general nonconvex set-valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set-valued variational inequalities. We also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set-valued variational inequalities. The results presented in this paper extend and improve some known results from the literature.     

Absolute Equivalence and Dirac Operators of Commuting Tuples of Operators

In this paper we define an equivalence relation of operators on Hilbert spaces which we call absolute equivalence. Two operators are called absolutely equivalent if both the absolute value of the operators and their adjoints are unitarily equivalent. We then use the properties of this equivalence relation to study the Koszul complex of a commuting tuple of operators through the Dirac operator of the tuple.     

Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings

In this paper, we prove strong convergence theorems to a zero of monotone mapping and a fixed point of relatively weak nonexpansive mapping. Moreover, strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions.     

Strong convergence studied by a hybrid type method for monotone operators in a Banach space

In this paper, we study a strong convergence for monotone operators. We first introduce the hybrid type algorithm for monotone operators. Next, we obtain a strong convergence theorem (Theorem 3.3) for finding a zero point of an inverse-strongly monotone operator in a Banach space. Finally, we apply our convergence theorem to the problem of finding a minimizer of a convex function.     

Optimal control of a nonconvex perturbed sweeping process

The paper concerns optimal control of discontinuous differential inclusions of the normal cone type governed by a generalized version of the Moreau sweeping process with control functions acting in both nonconvex moving sets and additive perturbations. This is a new class of optimal control problems in comparison with previously considered counterparts where the controlled sweeping sets are described by convex polyhedra. Besides a theoretical interest, a major motivation for our study of such challenging optimal control problems with intrinsic state constraints comes from the application to the crowd motion model in a practically adequate planar setting with nonconvex but prox-regular sweeping sets. Based on a constructive discrete approximation approach and advanced tools of first-order and second-order variational analysis and generalized differentiation, we establish the strong convergence of discrete optimal solutions and derive a complete set of necessary optimality conditions for discrete-time and continuous-time sweeping control systems that are expressed entirely via the problem data.     

Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces

In this paper, we first obtain a weak mean convergence theorem of Baillon’s type for nonspreading mappings in a Hilbert space. Further, using an idea of mean convergence, we prove a strong convergence theorem for nonspreading mappings in a Hilbert space.     

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C^1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.     

Generalized vector implicit quasi complementarity problems

In this paper, we introduce and study a generalized class of vector implicit quasi complementarity problem and the corresponding vector implicit quasi variational inequality problem. By using Fan-KKM theorem, we derive existence of solutions of generalized vector implicit quasi variational inequalities without any monotonicity assumption and establish the equivalence between those problems in Banach spaces.     

Hyperbolic mean curvature flow

In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations is strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with dimension larger than 4. We derive nonlinear wave equations satisfied by some geometric quantities related to the hyperbolic mean curvature flow. Moreover, we also discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the Minkowski space-time.