A New Hybrid Algorithm for a Pair of Quasi-𝜙-Asymptotically Nonexpansive Mappings and Generalized Mixed Equilibrium Problems in Banach Spaces by Jinhua Zhu, Shih-Sen Chang

“…The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi-𝜙-asymptotically nonexpansive mappings. …”
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A Mixed Traffic Assignment Problem and Its Upper Bound of Efficiency Loss by Xiao-Jun Yu, Jiancheng Long

“…A variational inequality (VI) formulation of a mixed traffic assignment problem associated with user equilibrium (UE) player and altruistic players (UE-AP) is developed. …”
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Nonzero-Sum Stochastic Differential Game between Controller and Stopper for Jump Diffusions by Yan Wang, Aimin Song, Cheng-De Zheng, Enmin Feng

“…By a dynamic programming approach, we give a verification theorem in terms of variational inequality-Hamilton-Jacobi-Bellman (VIHJB) equations for the solutions of the game. …”
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Dynamic Combined-Mode Traffic Network Model considering Transfer Behaviors by ShuGuang Li, QingHua Zhou

“…We present an integrated variational inequality formulation to capture the complex traveler choice behaviors such as departure time choices, transfer point, and route choices. …”
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Hybrid Gradient-Projection Algorithm for Solving Constrained Convex Minimization Problems with Generalized Mixed Equilibrium Problems by Lu-Chuan Ceng, Ching-Feng Wen

“…It is proven that three sequences generated by this algorithm converge strongly to the unique solution of some variational inequality, which is also a common element of the set of solutions of a constrained convex minimization problem, the set of solutions of a generalized mixed equilibrium problem, and the set of fixed points of a strict pseudocontraction in a real Hilbert space.…”
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The Nonsteady Boussinesq System with Mixed Boundary Conditions including Conditions of Friction Type by Tujin Kim

“…The formulations consist of a variational inequality for velocity due to the boundary conditions of friction type and a variational equation for temperature. …”
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A Novel System and Iterative Schemes for Generalized Variational Inclusion Problems and its Approximation Solvability by Mohd Sarfaraz, Md. Kalimuddin Ahmad

“…The variational inclusion problems are the generalizations of the variational inequality problems. First, it was noted by Hasounni and Moudafi in 1994. …”
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Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces by Yali Li, Jianjun Liu, Lei Deng

“…We prove that the following implicit and modified implicit viscosity iterative schemes {xn} defined by xn=αnf(xn)+(1−αn)T(tn)xn and xn=αnyn+(1−αn)T(tn)xn, yn=βnf(xn−1)+(1−βn)xn−1 strongly converge to p∈F as n→∞ and p is the unique solution to the following variational inequality: 〈f(p)−p,j(y−p)〉≤0 for all y∈F.…”
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A New Modified Hybrid Steepest-Descent by Using a Viscosity Approximation Method with a Weakly Contractive Mapping for a System of Equilibrium Problems and Fixed Point Problems wit... by Uamporn Witthayarat, Thanyarat Jitpeera, Poom Kumam

“…The sequence is generated from an arbitrary initial point which converges in norm to the unique solution of the variational inequality under some suitable conditions in a real Hilbert space. …”
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Bounding the Inefficiency of the Multiclass, Multicriteria C-Logit Stochastic User Equilibrium in a Transportation Network by Lekai Yuan, Xi Zhang, Chaofeng Shi

“…To find efficiency loss upper bounds of the multiclass CL-SUE, two equivalent variational inequalities for the multiclass CL-SUE model, i.e., time-based variational inequality (VI) and monetary-based VI, are formulated. …”
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A Reliability-Based Network Equilibrium Model with Adaptive Risk-Averse Travelers by Lei Zhao, Hongzhi Guan, Junze Zhu, Yunfeng Wei

“…Moreover, its equivalent variational inequality (VI) problem is formulated and a route-based algorithm is used to solve the proposed model. …”
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The Role of Initial Credit Distribution Scheme in Managing Network Mobility and Maximizing Reserve Capacity: Considering Traveler’s Cognitive Illusion by Fei Han, Lin Cheng

“…To comprehensively reveal the impacts of ICDS on UE solution, four different types of ICDS are introduced and analyzed in a unified variational inequality (VI) modeling framework. The uniqueness of the UE link flow and market equilibrium (ME) credit price is also investigated. …”
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Multiple Equilibrium Behaviors considering Human Exposure to Vehicular Emissions by Zhanbo Sun, Yu Tan, Rui Ma, Xia Yang, Jie Zhang

“…The multiple equilibrium problems are further converted into variational inequality (VI) problems and they are solved using a method of successive average- (MSA-) based diagonalization method. …”
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A Trial-and-Error Method with Autonomous Vehicle-to-Infrastructure Traffic Counts for Cordon-Based Congestion Pricing by Zhiyuan Liu, Yong Zhang, Shuaian Wang, Zhibin Li

“…Then, the stochastic user equilibrium (SUE) with asymmetric link travel time functions is used to assess each feasible toll pattern. Based on a variational inequality (VI) model for the optimal toll pattern, this study proposes a theoretically convergent trial-and-error method for the addressed problem, where only traffic counts data are needed. …”
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Decision Optimization of Low-Carbon Dual-Channel Supply Chain of Auto Parts Based on Smart City Architecture by Zheng Liu, Bin Hu, Bangtong Huang, Lingling Lang, Hangxin Guo, Yuanjun Zhao

“…This paper analyzes the optimal decision-making behavior and interaction of suppliers, producers, physical retailers, online retailers, demand markets, and recyclers in the auto parts industry, constructs the economic and environmental objective functions of low-carbon supply chain management, applies variational inequality to analyze the optimal conditions of the whole low-carbon supply chain system, and finally carries out simulation calculation. …”
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Two General Algorithms for Computing Fixed Points of Nonexpansive Mappings in Banach Spaces by Shuang Wang

“…Moreover, the fixed points solve the same system of nonlinear variational inequalities.…”
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A New System of Generalized Mixed Quasivariational Inclusions with Relaxed Cocoercive Operators and Applications by Zhongping Wan, Jia-Wei Chen, Hai Sun, Liuyang Yuan

“…A new system of generalized mixed quasivariational inclusions (for short, SGMQVI) with relaxed cocoercive operators, which develop some preexisting variational inequalities, is introduced and investigated in Banach spaces. …”
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On Abstract Economies and Their Applications by Chien-Hao Huang, Liang-Ju Chu

“…As an application, we derive an existence theorem of generalized quasi-variational inequalities in the general setting of -spaces without any linear structure.…”
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On Mann’s Type Method for Nonexpansive and Strongly Quasinonexpansive Mappings in Hilbert Spaces by Nawab Hussain, Giuseppe Marino, Badriah A. S. Alamri

“…In the setting of Hilbert spaces, we study Mann’s type method to approximate strong solutions of variational inequalities. We show that these solutions are fixed points of a nonexpansive mapping and/or a strongly quasinonexpansive mapping, depending on the coefficients involved in the algorithm.…”
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L∞-error estimate for a system of elliptic quasivariational inequalities by M. Boulbrachene, M. Haiour, S. Saadi

“…Under W2,p(Ω)-regularity of the continuous solution, a quasi-optimal L∞-convergence of a piecewise linear finite element method is established, involving a monotone algorithm of Bensoussan-Lions type and standard uniform error estimates known for elliptic variational inequalities (VIs).…”
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