Ahmad, R., Dilshad, M. (2015). Fuzzy resolvent equation with $H(cdot,cdot)$-$phi$-$eta$-accretive operator in Banach spaces. Iranian Journal of Fuzzy Systems, 12(2), 95-106. doi: 10.22111/ijfs.2015.1985

Rais Ahmad; Mohd Dilshad. "Fuzzy resolvent equation with $H(cdot,cdot)$-$phi$-$eta$-accretive operator in Banach spaces". Iranian Journal of Fuzzy Systems, 12, 2, 2015, 95-106. doi: 10.22111/ijfs.2015.1985

Ahmad, R., Dilshad, M. (2015). 'Fuzzy resolvent equation with $H(cdot,cdot)$-$phi$-$eta$-accretive operator in Banach spaces', Iranian Journal of Fuzzy Systems, 12(2), pp. 95-106. doi: 10.22111/ijfs.2015.1985

Ahmad, R., Dilshad, M. Fuzzy resolvent equation with $H(cdot,cdot)$-$phi$-$eta$-accretive operator in Banach spaces. Iranian Journal of Fuzzy Systems, 2015; 12(2): 95-106. doi: 10.22111/ijfs.2015.1985

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

^{}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.

[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, Pacic. 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.