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Triple Hierarchical Variational Inequalities

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Nonlinear Analysis

Part of the book series: Trends in Mathematics ((TM))

Abstract

In this chapter, we give a survey on hierarchical variational inequality problems and triple hierarchical variational inequality problems. By combining hybrid steepest descent method, Mann’s iteration method, and projection method, we present a hybrid iterative algorithm for computing a fixed point of a pseudo-contractive mapping and for finding a solution of a triple hierarchical variational inequality in the setting of real Hilbert space. We prove that the sequence generated by the proposed algorithm converges strongly to a fixed point which is also a solution of this triple hierarchical variational inequality problem. On the other hand, we also propose another hybrid iterative algorithm for solving a class of triple hierarchical variational inequality problems concerning a finite family of pseudo-contractive mappings in the setting of real Hilbert spaces. Under very appropriate conditions, we derive the strong convergence of the proposed algorithm to the unique solution of this class of problems.

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Acknowledgments

In this research, the first author was support by a research project no. SR/S4/MS:719/11 of the Department of Science and Technology, Govt. of India. While, the second author was supported by the National Science Foundation of China (11071169), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002).

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Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Qamrul Hasan Ansari & Himanshu Gupta

  2. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Lu-Chuan Ceng

  3. Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, China

    Lu-Chuan Ceng

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  1. Qamrul Hasan Ansari
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    Qamrul Hasan Ansari

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Ansari, Q.H., Ceng, LC., Gupta, H. (2014). Triple Hierarchical Variational Inequalities. In: Ansari, Q. (eds) Nonlinear Analysis. Trends in Mathematics. Birkhäuser, New Delhi. https://doi.org/10.1007/978-81-322-1883-8_8

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