Solvability and algorithms of generalized nonlinear variational-like inequalities in reflexive Banach spaces

Journal of inequalities and applications

Liangshi Zhao¹  Lili Wang²  Haiyan Gao³ 

[1] Center for Studies of Marine Economy and Sustainable Development, Liaoning Normal University, 116029, Dalian, Liaoning, People’s Republic of China;Department of Mathematics, Tonghua Teachers College, 134002, Tonghua, Jilin, People’s Republic of China;School of Mathematics, Dongbei University of Finance and Economics, 116025, Dalian, Liaoning, People’s Republic of China;Department of General Studies, Dalian University of Finance and Economics, 116000, Dalian, Liaoning, People’s Republic of China;

关键词: Generalized nonlinear variational-like inequality;    Banach’s fixed point theorem;    Schauder’s fixed point theorem;    FanKKM theorem;    Iterative algorithm;    Auxiliary principle technique;    Minimax inequality;    Reflexive Banach space;    47J20;   

DOI  :  10.1186/s13660-020-02490-x

来源: Springer

【 摘 要 】

This paper deals with solvability and algorithms for a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By employing the Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem, we obtain a sufficient condition which guarantees the existence of solutions for the generalized nonlinear variational-like inequality. We introduce also an auxiliary variational-like inequality and, by utilizing the minimax inequality, get the existence and uniqueness of solutions for the auxiliary variational-like inequality, which is used to suggest an iterative algorithm for solving the generalized nonlinear variational-like inequality. Under certain conditions, by means of the auxiliary principle technique, we both establish the existence and uniqueness of solutions for the generalized nonlinear variational-like inequality and discuss the convergence of iterative sequences generated by the iterative algorithm. Our results extend, improve, and unify several known results in the literature.

【 授权许可】

CC BY