# Solvability for a system of generalized nonlinear ordered variational inclusions in ordered Banach spaces

## Main Article Content

## Abstract

In this paper, we deal a resolvent operator technique is applied to address a system

of generalized nonlinear ordered variational inclusions in real ordered Banach spaces and derived an algorithm for a solution of the considered system. Here, we prove an existence result for the solution of the system of generalized nonlinear ordered variational inclusions and discuss convergence of sequences suggested by the algorithms.

## Article Details

## References

[1] R. Ahmad, M. F. Khan and Salahuddin, Mann and Ishikawa type perturbed iterative algorithm for generalized nonlinear variational inclusions, Math. Comput. Appl. 6 (1) (2001), 47–52. Google Scholar

[2] M. K. Ahmad and Salahuddin, Resolvent equation technique for generalized non-linear variational inclusions, Adv. Nonlinear Var. Inequal. 5 (1) (2002), 91– 98. Google Scholar

[3] M. K. Ahmad and Salahuddin, Perturbed three step approximation process with errors for a general implicit nonlinear variational inequalities, Int. J. Math. Math. Sci. Article ID 43818, (2006), 1–14. Google Scholar

[4] X. P. Ding and H. R. Feng, The p-step iterative algorithm for a system of generalized mixed quasi variational inclusions with (A,η)-accretive operators in q-uniformly smooth Banach spaces, J. Comput. Appl. Math. 220 (2008), 163–174. Google Scholar

[5] X. P. Ding and Salahuddin, On a system of general nonlinear variational inclusions in Banach spaces, Appl. Math. Mech., 36(12)(2015), 1663-1672, DOI:10.1007/s10483-015-1972-6. Google Scholar

[6] Y. P, Du, Fixed points of increasing operators in ordered Banach spaces and applications, Appl. Anal. 38 (1990), 1–20. Google Scholar

[7] Y. P. Fang and N. J. Huang, H-accretive operator and resolvent operator technique for variational inclusions in Banach spaces, Appl. Math. Lett., 17(6)(2004), 647–653. Google Scholar

[8] Y. P. Fang and N. J. Huang, Approximate solutions for non-linear variational inclusions with (H,η)-monotone operator, Research report, Sichuan University (2003). Google Scholar

[9] Y. P. Fang and N. J. Huang, Iterative algorithm for a system of nonlinear variational inclusions involving H-accretive operators in Banach spaces, Acta Math. Hungar 108 (3) (2005), 183–195. Google Scholar

[10] Y. P. Fang, N. J. Huang and H. B. Thompson, A new system of variational inclusions with (H,η)-monotone operators in Hilbert spaces, Comput. Math. Appl. 49 (2005), 365–374. Google Scholar

[11] N. J. Huang and Y. P. Fang, Generalized m-accretive mappings in Banach spaces, J. Sichuan Univ. 38 (4) (2001), 591–592. Google Scholar

[12] S. Hussain, M. F. Khan and Salahuddin, Mann and Ishikawa type perturbed iterative algorithms for completely generalized nonlinear variational Inclusions, Int. J. Math. Anal. 3 (1) (2006), 51–62. Google Scholar

[13] M. F. Khan and Salahuddin, Mixed multivalued variational inclusions involving H-accretive operators, Adv. Nonlinear Var. Inequal. 9 (2) (2006), 29–47. Google Scholar

[14] M. F. Khan and Salahuddin, Generalized co-complementarity problems in p-uniformly smooth Banach spaces, JIPAM, J. Inequal. Pure Appl. Math. 7 (2) (2006), 1–11, Article ID 66. Google Scholar

[15] M. F. Khan and Salahuddin, Generalized multivalued nonlinear co-variational inequalities in Banach spaces, Funct. Diff. Equat. 14 (2-3-4) (2007), 299–313. Google Scholar

[16] S. H. Kim, B. S. Lee and Salahuddin, Fuzzy variational inclusions with (H, φ, ψ)- η-monotone mappings in Banach Spaces, J. Adv. Research Appl. Math. 4 (1) (2012), 10–22. Google Scholar

[17] B. S. Lee and Salahuddin, Fuzzy general nonlinear ordered random variational inequalities in ordered Banach spaces, East Asian Math. J. 32 (5) (2016), 685– 700. Google Scholar

[18] B. S. Lee, M. F. Khan and Salahuddin, Generalized nonlinear quasi variational inclusions in Banach spaces, Comput. Math. Appl. 56 (5) (2008), 1414–1422. Google Scholar

[19] B. S. Lee, M. F. Khan and Salahuddin, Hybrid-type set-valued variational-like inequalities in reflexive Banach spaces, J. Appl. Math. Inform. 27 (5-6) (2009), 1371–1379. Google Scholar

[20] H. G. Li, D. Qiu and M. M. Jin, GNM ordered variational inequality system with ordered Lipschitz continuous mappings in an ordered Banach space, J. Inequal. Appl. 2013 (2013), 514. Google Scholar

[21] H. G. Li, L. P. Li, J. M. Zheng and M. M. Jin, Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with (α, λ)-nodsm mappings in ordered Banach spaces, Fixed Point Theory Appl., 2014 (2014), 122. Google Scholar

[22] H. G. Li, X. B. Pan, Z. Y. Deng and C. Y. Wang, Solving GNOVI frameworks involving (γg , λ)-weak-GRD set-valued mappings in positive Hilbert spaces, Fixed Point Theory Appl. 2014 (2014), 146. Google Scholar

[23] H. G. Li, D. Qui and Y. Zou, Characterization of weak-anodd set-valued map- pings with applications to approximate solution of gnmoqv inclusions involving operator in ordered Banach space, Fixed Point Theory Appl. 2013 (2013), 241. doi:10.1186/1687-1812-2013-241. Google Scholar

[24] H. G. Li, L. P. Li and M. M. Jin, A class of nonlinear mixed ordered inclusion problems for oredered (αa,λ)-ANODM set-valued mapping with strong comparison mapping, Fixed Point Theory Appl. 2014 (2014), 79. Google Scholar

[25] H. G. Li, A nonlinear inclusion problem involving (α, λ)-NODM set-valued mappings in ordered Hilbert space, Appl. Math. Lett. 25 (2012), 1384–1388. Google Scholar

[26] H. G. Li, Approximation solution for general nonlinear ordered variational inequalities and ordered equations in ordered Banach space, Nonlinear Anal. Forum 13 (2) (2008), 205–214. Google Scholar

[27] J. W. Peng and D. L. Zhu, A system of variational inclusions with p-η-accretive operators, J. Comput. Appl. Math. 216 (2008), 198–209. Google Scholar

[28] H. H. Schaefer, Banach Lattices and Positive Operators Springer, Berlin (1994). Google Scholar

[29] Salahuddin, Regularized equilibrium problems in Banach spaces, Korean Math. J. 24 (1) (2016), 51–63. Google Scholar

[30] A. H. Siddiqi, M. K. Ahmad and Salahuddin, Existence results for generalized nonlinear variational inclusions, Appl. Math. Lett. 18 (8) (2005), 859–864. Google Scholar

[31] Y. K. Tang, S. S. Chang and Salahuddin, A system of nonlinear set valued variational inclusions, SpringerPlus 2014, 3:318, Doi:10.1186/2193-180-3-318. Google Scholar

[32] R. U. Verma, Projection methods, algorithms and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41 (7-8) (2001), 1025–1031. Google Scholar

[33] R. U. Verma, M. F. Khan and Salahuddin, Fuzzy generalized complementarity problems in Banach spaces, PanAmer. Math. J. 17 (4) (2007), 71–80. Google Scholar

[34] R. U. Verma and Salahuddin, Extended systems of nonlinear vector quasi variational inclusions and extended systems of nonlinear vector quasi optimization problems in locally FC-spaces, Commun. Appl. Nonlinear Anal. 23 (1) (2016), 71–88. Google Scholar

[35] F. Q. Xia and N. J. Huang, Variational inclusions with a general H-monotone operator in Banach spaces, Comput. Math. Appl. 54 (2007), 24–30. Google Scholar