Open Access

New classes of generalized invex monotonicity

Journal of Inequalities and Applications20062006:57071

DOI: 10.1155/JIA/2006/57071

Received: 26 December 2004

Accepted: 16 August 2005

Published: 4 May 2006


This paper introduces new classes of generalized invex monotone mappings and invex cocoercive mappings. Their differential property and role to analyze and solve variational-like inequality problem are presented.


Authors’ Affiliations

School of Management, Fudan University
Department of Management Science and Engineering, Nanchang University


  1. Baiocchi C, Capelo A: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. John Wiley & Sons, New York; 1984:ix+452.MATHGoogle Scholar
  2. Crouzeix J-P, Marcotte P, Zhu DL: Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities. Mathematical Programming. Series A 2000,88(3):521–539. 10.1007/PL00011384MATHMathSciNetView ArticleGoogle Scholar
  3. Fang YP, Huang NJ: Variational-like inequalities with generalized monotone mappings in Banach spaces. Journal of Optimization Theory and Applications 2003,118(2):327–338. 10.1023/A:1025499305742MATHMathSciNetView ArticleGoogle Scholar
  4. Harker PT, Pang J-S: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Mathematical Programming. Series B 1990,48(2):161–220.MATHMathSciNetView ArticleGoogle Scholar
  5. Karamardian S, Schaible S: Seven kinds of monotone maps. Journal of Optimization Theory and Applications 1990,66(1):37–46. 10.1007/BF00940531MATHMathSciNetView ArticleGoogle Scholar
  6. Luo HZ, Xu ZK: On characterizations of prequasi-invex functions. Journal of Optimization Theory and Applications 2004,120(2):429–439.MATHMathSciNetView ArticleGoogle Scholar
  7. Mohan SR, Neogy SK: On invex sets and preinvex functions. Journal of Mathematical Analysis and Applications 1995,189(3):901–908. 10.1006/jmaa.1995.1057MATHMathSciNetView ArticleGoogle Scholar
  8. Osuna-Gómez R, Rufián-Lizana A, Ruíz-Canales P: Invex functions and generalized convexity in multiobjective programming. Journal of Optimization Theory and Applications 1998,98(3):651–661. 10.1023/A:1022628130448MATHMathSciNetView ArticleGoogle Scholar
  9. Parida J, Sahoo M, Kumar A: A variational-like inequality problem. Bulletin of the Australian Mathematical Society 1989,39(2):225–231. 10.1017/S0004972700002690MATHMathSciNetView ArticleGoogle Scholar
  10. Ruiz-Garzón G, Osuna-Gómez R, Rufián-Lizana A: Generalized invex monotonicity. European Journal of Operational Research 2003,144(3):501–512. 10.1016/S0377-2217(01)00393-9MATHMathSciNetView ArticleGoogle Scholar
  11. Yang XQ: On the gap functions of prevariational inequalities. Journal of Optimization Theory and Applications 2003,116(2):437–452. 10.1023/A:1022422407705MATHMathSciNetView ArticleGoogle Scholar
  12. Yang XM, Yang XQ, Teo KL: Characterizations and applications of prequasi-invex functions. Journal of Optimization Theory and Applications 2001,110(3):645–668. 10.1023/A:1017544513305MATHMathSciNetView ArticleGoogle Scholar
  13. Yang XM, Yang XQ, Teo KL: Generalized invexity and generalized invariant monotonicity. Journal of Optimization Theory and Applications 2003,117(3):607–625. 10.1023/A:1023953823177MATHMathSciNetView ArticleGoogle Scholar
  14. Zhu DL, Marcotte P: New classes of generalized monotonicity. Journal of Optimization Theory and Applications 1995,87(2):457–471. 10.1007/BF02192574MATHMathSciNetView ArticleGoogle Scholar
  15. Zhu DL, Marcotte P: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM Journal on Optimization 1996,6(3):714–726. 10.1137/S1052623494250415MATHMathSciNetView ArticleGoogle Scholar


© B. Xu and D.L. Zhu 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.