Semicontinuity of Solution Sets to Parametric Quasivariational Inclusions with Applications to Traffic Networks I: Upper Semicontinuities

We propose some notions related to semicontinuity of a multivalued mapping and provide a clear insight for various semicontinuity-related definitions. We establish verifiable sufficient conditions for solution sets of general quasivariational inclusion problems to have these semicontinuity-related properties. Our results are proved to include and improve recent ones in the literature by corollaries and examples. Part I is devoted to upper semicontinuity properties of solution sets. Part II discusses lower semicontinuities of these sets and applications, where we discuss in details a traffic network problem as a sample for employing the main results in practical situations

[1]  Habtu Zegeye,et al.  Existence and convergence theorems for a class of multi-valued variational inclusions in Banach spaces , 2004 .

[2]  BORIS S. MORDUKHOVICH,et al.  Coderivative Analysis of Quasi-variational Inequalities with Applications to Stability and Optimization , 2007, SIAM J. Optim..

[3]  Kok Lay Teo,et al.  On the Stability of Generalized Vector Quasivariational Inequality Problems , 2002 .

[4]  Nan-Jing Huang,et al.  H-Accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces , 2004, Appl. Math. Lett..

[5]  R. U. Verma,et al.  Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A, eta)-resolvent operator technique , 2006, Appl. Math. Lett..

[6]  Phan Quoc Khanh,et al.  On the existence of solutions to quasivariational inclusion problems , 2009, J. Glob. Optim..

[7]  Kaleem Raza Kazmi,et al.  Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces , 2005, Appl. Math. Comput..

[8]  D. T. Luc,et al.  Existence conditions in variational inclusions with constraints , 2004 .

[9]  Variational inclusions for contingent derivative of set-valued maps , 2004 .

[10]  Sanyang Liu,et al.  Generalized set-valued variational inclusions in q-uniformly smooth Banach spaces , 2004 .

[11]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[12]  Monica Bianchi,et al.  A note on stability for parametric equilibrium problems , 2003, Oper. Res. Lett..

[13]  Xie Ping Ding,et al.  Sensitivity analysis for generalized nonlinear implicit quasi-variational inclusions , 2004, Appl. Math. Lett..

[14]  Phan Quoc Khanh,et al.  Lower Semicontinuity and Upper Semicontinuity of the Solution Sets and Approximate Solution Sets of Parametric Multivalued Quasivariational Inequalities , 2007 .

[15]  Phan Quoc Khanh,et al.  The solution existence of general variational inclusion problems , 2007 .

[16]  B. Mordukhovich.  Variational analysis and generalized differentiation , 2006 .

[17]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[18]  Jun Li,et al.  Stability for parametric implicit vector equilibrium problems , 2006, Math. Comput. Model..

[19]  A. H. Siddiqi,et al.  Existence results for generalized nonlinear variational inclusions , 2005, Appl. Math. Lett..

[20]  Phan Quoc Khanh,et al.  Systems of Set-Valued Quasivariational Inclusion Problems , 2007 .

[21]  Alexandru Kristály,et al.  Set-valued versions of Ky Fan's inequality with application to variational inclusion theory , 2003 .

[22]  B. Mordukhovich.  Variational Analysis and Generalized Differentiation II: Applications , 2006 .

[23]  Nguyen Xuan Tan,et al.  On the Existence of Solutions of Quasivariational Inclusion Problems , 2004 .

[24]  Shouchuan Hu,et al.  Handbook of multivalued analysis , 1997 .

[25]  Jian-Wen Peng,et al.  Set-valued variational inclusions with T-accretive operators in Banach spaces , 2006, Appl. Math. Lett..

[26]  Phan Quoc Khanh,et al.  Upper Semicontinuity of the Solution set to Parametric Vector Quasivariational Inequalities , 2005, J. Glob. Optim..

[27]  Lam Quoc Anh,et al.  On the Stability of the Solution Sets of General Multivalued Vector Quasiequilibrium Problems , 2007 .

[28]  Heng-you Lan.  (A, eta)-Accretive mappings and set-valued variational inclusions with relaxed cocoercive mappings in Banach spaces , 2007, Appl. Math. Lett..

[29]  Xie Ping Ding,et al.  A new class of completely generalized quasi-variational inclusions in Banach spaces , 2002 .

[30]  Zubair Khan,et al.  Proximal point algorithm for generalized multivalued nonlinear quasi-variational-like inclusions in Banach spaces , 2005, Appl. Math. Comput..

[31]  Salahuddin.  Parametric generalized set-valued variational inclusions and resolvent equations , 2004 .

[32]  Iterative processes with errors for nonlinear set-valued variational inclusions involving accretive type mappings☆ , 2004 .

[33]  Ram U. Verma,et al.  General nonlinear variational inclusion problems involving A-monotone mappings , 2006, Appl. Math. Lett..

[34]  Monica Bianchi,et al.  Sensitivity for parametric vector equilibria , 2006 .

[35]  L. Q. Anh,et al.  Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems , 2004 .

[36]  Qing-bang Zhang,et al.  Generalized implicit variational-like inclusion problems involving G-eta-monotone mappings , 2007, Appl. Math. Lett..

[37]  Lokenath Debnath,et al.  Sensitivity analysis for parametric completely generalized nonlinear implicit quasivariational inclusions , 2003 .