Fuzzy resolvent equation with $H(cdot,cdot)$-$phi$-$eta$-accretive operator in Banach spaces

Document Type: Research Paper

Authors

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

Abstract

In this paper, we introduce and study fuzzy variational-like inclusion, fuzzy resolvent equation and $H(cdot,cdot)$-$phi$-$eta$-accretive operator in real  uniformly smooth Banach spaces. It is established that fuzzy variational-like inclusion is equivalent to a fixed point problem as well as to a fuzzy resolvent equation. This equivalence is used to define an iterative algorithm for solving fuzzy resolvent equation. Some examples are given.

Keywords


[1] Q. H. Ansari, Certain problems concerning variational inequalities, Ph.D Thesis, Aligarh
Muslim University, Aligarh, India, 1988.
[2] S. S. Chang, Fuzzy quasi-variational inclusions in Banach spaces, Appl. Math. Comput., 145
(2003), 805-819.
[3] S. S. Chang, Y. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems,
32 (1989), 359-367.
[4] H. X. Dai, Generalized mixed variational-like inequalities with fuzzy mappings, J. Comput.
Appl. Math., 224 (2009), 20-28.
[5] X. P. Ding and J. Y. Park, A new class of generalized nonlinear implicit quasivariational
inclusions with fuzzy mappings, J. Comp. Math. Appl., 138 (2002) 249-257.
[6] X. P. Ding, Algorithm of solutions for mixed implicit quasi-variational inequalities with fuzzy
mappings, Comput. Math. Appl., 38 (1999), 231-241.

[7] C. F. Hu, Solving variational inequalities in fuzzy environment, J. Math. Anal., 249 (2000),
527-538.
[8] P. Kumam and N. Petrol, Mixed variational-like inequality for fuzzy mappings in re
exive
Banach spaces, J. Inequal. Appl., 2009 (2009), 1-15.
[9] Z. Liu, L. Debnath, S. M. Kang and J. S. Ume, Generalized mixed quasi-variational inclusions
and generalized mixed resolvent equations for fuzzy mappings, Appl. Math. Comput., 149
(2004), 879-891.
[10] B. S. Lee, M. F. Khan, Salahuddin, Fuzzy nonlinear set-valued variational inclusions, Com-
put. Math. Appl., 60 (2010), 1768-1775.
[11] Jr. S. B. Nadler, Multivalued contraction mappings, Paci c. J. Math., 30 (1969), 475-488.
[12] M. A. Noor, Variational inequalities for fuzzy mappings (II), Fuzzy Sets and System, 110
(2000) 101-108.
[13] Z. Wu and J. Xu, Generalized convex fuzzy mappings and fuzzy variational-like inequalities,
Fuzzy Sets and Systems, 160(11) (2009), 1590-1619.
[14] H. K. Xu, Inequalities in Banach spaces and applications, Nonlinear Analysis, Theory Meth-
ods and Applications, 16(12) (1991), 1127-1138.
[15] L. A. Zadeh, Fuzzy Sets, Inform. Contr., 8 (1965), 338-353.
[16] Y. Z. Zou and N. J. Huang, H(; )-accretive operator with an application for solving varia-
tional inclusions in Banach spaces, Appl. Math. Comput., 204 (2008), 809-816.