Pseudomonotone Mappings Research Articles

Optimal boundary control of nonlinear-viscous fluid flows

The optimal control problem for a stationary model of a nonlinear-viscous incompressible fluid flowing through a bounded domain is considered under the wall slip condition. As a control parameter, the dynamic pressure at the in-flow and out-flow parts of the boundary is used. Using methods of the theory of pseudomonotone mappings, the existence of a weak solution (a velocity–dynamic pressure pair) minimizing a given cost functional is proved. The behaviour of solutions and optimal values of the cost functional are studied when the set of admissible controls varies. In particular, it is shown that the marginal function of this control system is lower semicontinuous. Bibliography: 23 titles.

Nonlocal elliptic variational-hemivariational inequalities

We examine the existence of a weak solution to a class of variational-hemivariational inequalities governed by a fractional Laplace operator. Moreover, an existence result is proved for a bilateral obstacle problem for a nonlocal hemivariational inequality. Our approach is based on a surjectivity result for pseudomonotone mappings, properties of Clarke’s subgradient, and the Moreau–Yosida approximation.

Existence of solutions for double phase obstacle problems with multivalued convection term

The main goal of this paper is the study of an elliptic obstacle problem with a double phase phenomena and a multivalued reaction term which also depends on the gradient of the solution. Such term is called multivalued convection term. Under quite general assumptions on the data, we prove that the set of weak solutions to our problem is nonempty, bounded and closed. Our proof is based on a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping.

A Strongly Convergent Modified Halpern Subgradient Extragradient Method for Solving the Split Variational Inequality Problem

We propose a method for solving the split variational inequality problem (SVIP) involving Lipschitz continuous and pseudomonotone mappings. The proposed method is inspired by the Halpern subgradient extragradient method for solving the monotone variational inequality problem with a simple step size. A strong convergence theorem for an algorithm for solving such a SVIP is proved without the knowledge of the Lipschitz constants of the mappings. As a consequence, we get a strongly convergent algorithm for finding the solution of the split feasibility problem (SFP), which requires only two projections at each iteration step. A simple numerical example is given to illustrate the proposed algorithm.

Mixed invex equilibrium problems with generalized relaxed monotone and relaxed invariant pseudomonotone mappings

Mixed invex equilibrium problems with generalized relaxed monotone and relaxed invariant pseudomonotone mappings

ON THE STRONG CONVERGENCE OF A PROJECTION-BASED ALGORITHM IN HILBERT SPACES

In this paper, we introduce a new projection-based algorithm for solving variational inequality problems with a Lipschitz continuous pseudo-monotone mapping in Hilbert spaces. We prove a strong convergence of the generated sequences. The numerical behaviors of the proposed algorithm on test problems are illustrated and compared with previously known algorithms.

Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

In this paper, we introduce and investigate composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem (VIP) with a pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework infinite-dimensional Hilbert spaces. The proposed algorithms are based on an inertial subgradient–extragradient method with the line-search process, hybrid steepest-descent methods, viscosity approximation methods and Mann iteration methods. Under weak conditions, we prove strong convergence of the proposed algorithms to the element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Self-adaptive inertial subgradient extragradient algorithm for solving pseudomonotone variational inequalities

ABSTRACT In this paper, we introduce an inertial algorithm for solving classical variational inequalities with Lipschitz continuous and pseudomonotone mapping in real Hilbert space. The algorithm is inspired by subgradient extragradient method and the inertial method with a new step size. The convergence of algorithm is established without the knowledge of the Lipschitz constant of the mapping. Finally, some numerical experiments are presented to show the efficiency and advantage of the proposed algorithm.

Strong convergence of an extragradient-like algorithm involving pseudo-monotone mappings

The purpose of this paper is to introduce a new extragradient-like algorithm for solving a variational inequality problem with a pseudo-monotone and Lipschitz continuous mapping in a Hilbert space. The iterative algorithm combines inertial ideas and hybrid extragradient ideas with the Armijo-like step size rule. Strong convergence of the algorithm is obtained and numerical experiments are provided.

Iterative methods for fixed points and zero points of nonlinear mappings with applications

ABSTRACT The purpose of this article is to investigate a descent-type iterative algorithm for finding a common element of fixed-point sets of a finite family of nonexpansive mappings and zero-point sets of pseudomonotone mappings in Hilbert spaces. With suitable assumptions, we derive necessary and sufficient conditions for the strong convergence of the algorithm. Numerical examples and applications to signal processing problems are provided to support the main results.

On the existence of solutions of variational inequalities in nonreflexive Banach spaces

We are concerned in this article with an existence theorem for variational inequalities in nonreflexive Banach spaces with a general coercivity condition. The variational inequalities contain multivalued generalized pseudomonotone mappings and convex functionals, the nonreflexive Banach spaces form a complementary system, and the coercivity condition involves both the mapping and the functional. As an application, we study second-order elliptic variational inequalities with multivalued lower-order terms in general Orlicz–Sobolev spaces.

RETRACTED ARTICLE: On stability analysis for generalized Minty variational-hemivariational inequality in reflexive Banach spaces

The stability for a class of generalized Minty variational-hemivariational inequalities has been considered in reflexive Banach spaces. We demonstrate the equivalent characterizations of the generalized Minty variational-hemivariational inequality. A stability result is presented for the generalized Minty variational-hemivariational inequality with (f,J)-pseudomonotone mapping.

Linesearch methods for bilevel split pseudomonotone variational inequality problems

In this paper, we propose Linesearch methods for solving a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mapping in the upper-level problem and pseudomonotone mappings in the lower-level one. A strongly convergent algorithm for such a BSVIP is proposed and analyzed.

An improved projection method for solving generalized variational inequality problems

ABSTRACTIn this paper, we present a new algorithm for solving generalized variational inequality problems(GVIP for short) in finite-dimensional Euclidean space. In this method, our next iterate point is obtained by projecting the current iterate point onto a half-space. This half-space can separate strictly the current iterate point from the solution set of GVIP. Moreover, this method works without needing the current point belongs to the feasible set. Comparing with methods in Konnov [A combined relaxation method for variational inequalities with nonlinear constraints. Math Program. 1998;80(2):239–252] and Fang and Chen [Subgradient extragradient algorithm for solving multi-valued variational inequality. Appl Math Comput. 2014;229(3-4):123–130], our method can get rid of an auxiliary procedure in each iteration which is used to ensure the current iterate point belongs to feasible set. Consequently, our method is more simpler than those algorithms. The global convergence is proved under mild assumptions. Numerical results show that this method is much more efficient than the method in Li and He [An algorithm for generalized variational inequality with pseudomonotone mapping. J Comput Appl Math. 2009;228:212–218].

Existence results of the system of generalized variational inequalities with multi-valued mappings

In this article, we study a generalized variational inequality problem with multi-valued mappings over product sets and the system of generalized variational inequalities with multi-valued mappings which are equivalent problems. By developing the idea of generalized densely relatively pseudomonotone mappings, and by using well-known Fan-KKM theorem and fixed point theorem, we prove existence results of our problem. We construct an example of generalized Nash equilibrium problem in the context of our problem, and as an application of our results, we establish an existence of coincident point result.

Characterizations of weakly sharp solutions for a variational inequality with a pseudomonotone mapping

For a variational inequality problem with a pseudomonotone mapping F, we characterize the weak sharpness of its solution set C* without using gap functions. When the mapping F is also constant on C*, the characterizations of weak sharpness of C* become more succinct. As an example, a pseudomonotone+ mapping on C* is shown to be constant on C*. Consequently the weak sharpness of C* can further be described by a Gâteaux differentiable function which itself characterizes the pseudomonotonicity+ of F on C*. It turns out that several existing relevant results with differentiable gap functions can be obtained from ours.

Hemivariational inequalities modeling electro-elastic unilateral frictional contact problem

We study a class of abstract hemivariational inequalities in a reflexive Banach space. For this class, using the theory of multivalued pseudomonotone mappings and a fixed-point argument, we provide a result on the existence and uniqueness of the solution. Next, we investigate a static frictional contact problem with unilateral constraints between a piezoelastic body and a conductive foundation. The contact, friction and electrical conductivity condition on the contact surface are described with the Clarke generalized subgradient multivalued boundary relations. We derive the variational formulation of the contact problem which is a coupled system of two hemivariational inequalities. Finally, for such system we apply our abstract result and prove its unique weak solvability.

A Neurodynamic Model to Solve Nonlinear Pseudo-Monotone Projection Equation and Its Applications.

In this paper, a neurodynamic model is given to solve nonlinear pseudo-monotone projection equation. Under pseudo-monotonicity condition and Lipschitz continuous condition, the projection neurodynamic model is proved to be stable in the sense of Lyapunov, globally convergent, globally asymptotically stable, and globally exponentially stable. Also, we show that, our new neurodynamic model is effective to solve the nonconvex optimization problems. Moreover, since monotonicity is a special case of pseudo-monotonicity and also since a co-coercive mapping is Lipschitz continuous and monotone, and a strongly pseudo-monotone mapping is pseudo-monotone, the neurodynamic model can be applied to solve a broader classes of constrained optimization problems related to variational inequalities, pseudo-convex optimization problem, linear and nonlinear complementarity problems, and linear and convex quadratic programming problems. Finally, several illustrative examples are stated to demonstrate the effectiveness and efficiency of our new neurodynamic model.

On Bilevel Split Pseudomonotone Variational Inequality Problems with Applications

In this paper, we investigate a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mapping in the upper-level problem and pseudomonotone mappings in the lower-level one. A strongly convergent algorithm for such a BSVIP is proposed and analyzed. In particular, a problem of finding the minimum-norm solution of a split pseudomonotone variational inequality problem is also studied. As a consequence, we get a strongly convergent algorithm for finding the minimum-norm solution to the split feasibility problem, which requires only two projections at each iteration step. An application to discrete optimal control problems is considered.

On the convergence of solutions of inclusions containing maximal monotone and generalized pseudomonotone mappings

We are concerned in this paper with the existence, boundedness, and the convergence of solutions to a sequence of inclusions Ak(u)+Bk(u)∋Lk, where Ak is a maximal monotone mapping, Bk is a generalized pseudomonotone mapping defined on a reflexive Banach space X, and Lk∈X∗. We study appropriate kinds of convergence for Ak and Bk such that a limit of a sequence of solutions of these inclusions is also a solution of the limit inclusion.