Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Generalized ρ, θ -η Invariant Monotonicity and Generalized ρ, θ -η Invexity of Nondifferentiable Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 393940 16 pages doi 2009 393940 Research Article Generalized p ỡ n Invariant Monotonicity and Generalized p ỡ n Invexity of Nondifferentiable Functions Caiping Liu1 2 and Xinmin Yang2 1 College of Mathematics Science Inner Mongolia University Hohhot 010021 China 2 College of Mathematics and Computer Science Chongqing Normal University Chongqing 400047 China Correspondence should be addressed to Caiping Liu caipingliu99@ Received 11 December 2008 Accepted 9 February 2009 Recommended by Charles E. Chidume New concepts of generalized p ff -n invex functions for non-differentiable functions and generalized p 0 -y invariant monotone operators for set-valued mappings are introduced. The relationships between generalized p 0 -y invexity of functions and generalized p 0 -y invariant monotonicity of the corresponding Clarke s subdifferentials are studied. Some of our results are extension and improvement of some results obtained in Jabarootion and Zafarani 2006 Behera etal. 2008 . Copyright 2009 C. Liu and X. Yang. 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. 1. Introduction Convexity plays a central role in mathematical economics engineering management sciences and optimization. In recent years several extensions and generalizations have been developed for classical convexity. An important generalization of convex functions is invex functions introduced by Hanson 1981 1 . He has shown that the Kuhn-Tuker conditions are sufficient for optimality of nonlinear programming problems under invexity conditions. Kaul and Kaur 1985 2 presented the conpects of pseudoinvex and quasi-invex functions and investigated their applications in nonlinear programming. A concept closely related to the convexity of function
