Search Results (39)

We develop a comprehensive dynamic Walrasian framework entirely in $L^{\infty }$ so that prices and allocations are essentially bounded, and market clearing holds pointwise almost everywhere. Utilities are allowed to be locally Lipschitz and quasi-concave; we employ Clarke subgradients to derive generalized quasi-variational inequalities (GQVIs). We endogenize inventories through a capital-accumulation constraint, leading to a differential QVI (dQVI). Existence is proved under either strong monotonicity or pseudo-monotonicity and coercivity. We establish Walras’ law, and the complementarity, stability, and sensitivity of the equilibrium correspondence in $L^2$-metrics, incorporate time-discounting and uncertainty into $\Omega \times [0,T]$, and present convergent numerical schemes (Rockafellar–Wets penalties and extragradient). Our results close the “in mean vs pointwise” gap noted in dynamic models and connect to modern decomposition approaches for QVIs. Full article

This research focuses on developing a novel approach to finding fixed points of quasi-nonexpansive mappings without relying on the demi-closedness condition, a common requirement in previous studies. The approach is based on the Subgradient Extragradient technique, which builds upon the foundational extragradient method introduced by G.M. Korpelevich. Korpelevich’s method is a widely recognized tool in the fields of optimization and variational inequalities. This study extends Korpelevich’s technique by adapting it to a broader class of operators while maintaining critical convergence properties. This research demonstrates the effectiveness and practical applicability of this new method through detailed computational examples, highlighting its potential to address complex mathematical problems across various domains. Full article

This paper considers a new nonlocal fractional differential quasi-variational inequality (NFDQVI) comprising a fractional differential equation with a nonlocal condition and a time-dependent quasi-variational inequality in Hilbert spaces. Qualitative properties of the solution for the time-dependent parameterized quasi-variational inequality are investigated, which improve some known results in the literature. Moreover, the unique existence of the solution and Hyers–Ulam stability are obtained for the novel NFDQVI under mild conditions. Finally, the obtained abstract results for NFDQVI are applied to analyze the unique solvability and stability, addressing a time-dependent multi-agent optimization problem and a time-dependent price control problem. Full article

This paper presents the convergence analysis of a newly proposed algorithm for approximating solutions to split equality variational inequality and fixed point problems in real Hilbert spaces. We establish that, under reasonably mild conditions, specifically when the involved mappings are quasimonotone, uniformly continuous, and quasi-nonexpansive, the sequences generated by the algorithm converge strongly to a solution of the problem. Furthermore, we provide several numerical experiments to demonstrate the practical effectiveness of the proposed method and compare its performance with that of existing algorithms. Full article

This paper investigates the structural properties of solutions of vector equilibrium systems and mixed variational inequalities in topological vector spaces. Based on Himmelberg-type fixed point theorem, combined with the analysis of set-valued mapping and quasi-monotone conditions, the existence criteria of solutions for two classes of generalized equilibrium problems with weak compactness constraints are constructed. This work introduces an innovative application of the measurable selection theorem of semi-continuous function space to eliminate the traditional compactness constraints, and provides a more universal theoretical framework for game theory and the economic equilibrium model. In the analysis of mixed variational problems, the topological stability of the solution set under the action of generalized monotone mappings is revealed by constructing a new KKM class of mappings and introducing the theory of pseudomonotone operators. In particular, by replacing the classical compactness assumption with pseudo-compactness, this study successfully extends the research boundary of scholars on variational inequalities, and its innovations are mainly reflected in the following aspects: (1) constructing a weak convergence analysis framework applicable to locally convex topological vector spaces, (2) optimizing the monotonicity constraint of mappings by introducing a semi-continuous asymmetric condition, and (3) in the proof of the nonemptiness of the solution set, the approximation technique based on the family of relatively nearest neighbor fields is developed. The results not only improve the theoretical system of variational analysis, but also provide a new mathematical tool for the non-compact parameter space analysis of economic equilibrium models and engineering optimization problems. This work presents a novel combination of measurable selection theory and pseudomonotone operator theory to handle non-compact constraints, advancing the theoretical framework for economic equilibrium analysis. Full article

This paper addresses the challenge of solving quasi-variational inequalities (QVIs) by developing and analyzing a forward–backward–forward algorithm from a continuous and iterative perspective. QVIs extend classical variational inequalities by allowing the constraint set to depend on the decision variable, a formulation that is particularly useful in modeling various problems. A critical computational challenge in these settings is the expensive nature of projection operations, especially when closed-form solutions are unavailable. To mitigate this, we consider the moving set case and propose a forward–backward–forward algorithm that requires only one projection per iteration. Under the assumption that the operator is strongly monotone, we establish that the continuous trajectories generated by the corresponding dynamical system converge exponentially to the unique solution of the QVI. We extend Tseng’s well-known forward–backward–forward algorithm for variational inequalities by adapting it to the more complex framework of QVIs. We prove that it converges when applied to strongly monotone QVIs and derive its convergence rate. We perform numerical implementations of our proposed algorithm and give numerical comparisons with other related gradient projection algorithms for quasi-variational inequalities in the literature. Full article

This study presents a novel category of nonconvex functions in Banach spaces, referred to as quasi-lower $C^2$ functions on nonempty closed sets. We establish the existence of solutions for nonconvex variational problems involving quasi-lower $C^2$ functions defined in Banach spaces. To illustrate the applicability of our findings, an example is provided in $L^p$ spaces. Full article

In this paper, we propose and study an averaged Halpern-type algorithm for approximating the solution of a common fixed-point problem for a couple of nonexpansive and demicontractive mappings with a variational inequality constraint in the setting of a Hilbert space. The strong convergence of the sequence generated by the algorithm is established under feasible assumptions on the parameters involved. In particular, we also obtain the common solution of the fixed point problem for nonexpansive or demicontractive mappings and of a variational inequality problem. Our results extend and generalize various important related results in the literature that were established for two pairs of mappings: (nonexpansive, nonspreading) and (nonexpansive, strongly quasi-nonexpansive). Numerical tests to illustrate the superiority of our algorithm over the ones existing in the literature are also reported. Full article

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results. Full article

In this study, we examine how the strategies of the players over multiple time scales impact the decision making, resulting payoffs and the costs in non-cooperative strategic games. We propose a dynamic generalized Nash equilibrium problem for non-cooperative strategic games which evolve in multidimensions. We also define an equivalent dynamic quasi-variational inequality problem. The existence of equilibria is established, and a spot electricity market problem is reformulated in terms of the proposed dynamic generalized Nash equilibrium problem. Employing the theory of projected dynamical systems, we illustrate our approach by applying it to a 39-bus network case, which is based on the New England system. Moreover, we illustrate a comparative study between multiple time scales and a single time scale by a simple numerical experiment. Full article

The numerical invariants of a variation of Hodge structure over a smooth quasi-projective variety are a measure of complexity for the global twisting of the limit-mixed Hodge structure when it degenerates. These invariants appear in formulas that may have correction terms called Arakelov inequalities. We investigate numerical Arakelov-type equalities for a family of surfaces fibered by curves. Our method uses Arakelov identities in weight-one and weight-two variations of Hodge structure in a commutative triangle of two-step fibrations. Our results also involve the Fujita decomposition of Hodge bundles in these fibrations. We prove various identities and relationships between Hodge numbers and degrees of the Hodge bundles in a two-step fibration of surfaces by curves. Full article

In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local $ϵ$-quasi solutions of the BLPP, under generalized approximate convexity assumptions, via limiting subdifferentials. Moreover, by employing the generalized Knaster–Kuratowski–Mazurkiewicz (KKM)-Fan’s lemma, we derive some existence results for the solutions of AMTVI and ASTVI. We have furnished suitable, non-trivial, illustrative examples to demonstrate the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that explores relationships between the approximate variational inequalities and BLPP under the assumptions of generalized approximate convexity by employing the powerful tool of limiting subdifferentials. Full article

We investigate a quasi-static-antiplane contact problem, examining a thermo-electro-visco-elastic material with a friction law dependent on the slip rate, assuming that the foundation is electrically conductive. The mechanical problem is represented by a system of partial differential equations, and establishing its solution involves several key steps. Initially, we obtain a variational formulation of the model, which comprises three systems: a hemivariational inequality, an elliptic equation, and a parabolic equation. Subsequently, we demonstrate the existence of a unique weak solution to the model. The proof relies on various arguments, including those related to evolutionary inequalities, techniques for decoupling unknowns, and certain results from differential equations. Full article

In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modified Mann method for solving the pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth real Banach spaces. Under some standard assumptions imposed on cost operators, we prove a strong convergence theorem for our proposed method. Finally, we perform numerical experiments to validate the efficiency of our proposed method. Full article