Strong convergence of an iterative algorithm for a class of nonlinear set-valued variational inclusions
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Abstract
In this communication, we introduce an Ishikawa type iterative algorithm for finding the approximate solutions of a class of nonlinear set valued variational inclusion problems. We also establish a characterization of strong convergence of this iterative techniques.
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References
[1] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, 1980. Google Scholar
[2] J. P. Aubin, Mathematical methods of game theory and economics, North Hol- land, Amsterdam, The Netherlands, 1982. Google Scholar
[3] H. Brezis, Operateurs maximaux monotone er semi groupes de contractions dans les espaces de Hilbert, North-Holland Mathematices Studies 5 Notes de Matem- atica (50) North-Holland, Amsterdam, 1973. Google Scholar
[4] R. E. Bruck and S. Reich, Accretive operators, Banach limits and dual ergodic theorems, Bull. Acad. Polon Sci. 12 (1981), 585–589 Google Scholar
[5] R. Glowinski, J. L. Lions and R. Tremolieres, Numerical analysis of variational inequalities, North-Holland, Amsterdam, 1981. Google Scholar
[6] F. Giannessi and A. Mugeri, Variational inequalities and network equilibrium problems, Plenum Press, New York NY USA, 1995. Google Scholar
[7] L. Gioranescu, Geometry of Banach spaces, duality mapping and nonlinear prob- lems, kluwer Acad. Press, Amsterdam, 1990. Google Scholar
[8] N. Kikuchi and J. T. Oden, Contact problems in elasticity, SIAM, Philadelphia, 1988. Google Scholar
[9] P. D. Panagiotoupoulos and G. E. Stavroulakis, New types of variational principles based on the notion of quasi differentiability, Acta Mech. 94 (1992), 171–194. Google Scholar
[10] A. Hassouni and A. Moudafi, A perturbed algorithms for variational inequalities, J. Math. Anal. Appl. 185 (1994), 706–712. Google Scholar
[11] X. P. Ding, Perturbed proximal point algorithms for generalized quasi variational inclusions, J. Math. Appl. Appl. 201 (1997), 88–101. Google Scholar
[12] X. P. Ding, Proximal point algorithm with errors for generalized strongly non-linear quasi-variational inclusions, Appl. Math. Mech. 19 (7) (1998), 637–643. Google Scholar
[13] X. P. Ding, On a class of generalized nonlinear implicit quasivariational inclusions, Appl. Math. Mech. 20 (10) (1999), 1087–1098. Google Scholar
[14] X. P. Ding, Generalized implicit quasivariational inclusions with fuzzy set-valued mappings, Comput. Math. Appl. 38 (1) (1999), 71–79. Google Scholar
[15] X. P. Ding, Perturbed proximal point algorithms for general quasi-variational-like inclusions, J. Computat. Appl. Math. 113 (2000), 153–165. Google Scholar
[16] X. P. Ding, Generalized quasi-variational-like inclusions with nonconvex functionals, Appl. Math. Comput. 122 (2001), 267–282. Google Scholar
[17] X. P. Ding, Perturbed Ishikawa Type Iterative Algorithm for Generalized Quasi-variational Inclusions, Appl. Math. Comput. 14 (2003), 359–373. Google Scholar
[18] X. P. Ding, Algorithms of solutions for completely generalized mixed implicit quasivariational inclusions, Appl. Math. Comput. 148 (1) (2004), 47–66. Google Scholar
[19] X. P. Ding, Predictor-Corrector iterative algorithms for solving generalized mixed quasi-variational-like inclusion, J. Comput. Appl. Math. 182 (1) (2005), 1–12. Google Scholar
[20] 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
[21] X. P. Ding and H. R. Feng, The p-step iterative algorithm for a system of gen- eralized mixed quasi-variational inclusions with (A,η)-accretive operators in q- uniformly smooth Banach spaces, J. Comput. Appl. Math. 220 (1-2) (2008), 163–174. Google Scholar
[22] X. P. Ding and H. R. Feng, Algorithm for solving a new class of generalized nonlinear implicit quasi-variational inclusions in Banach spaces, Appl. Math. Comput. 208 (2009), 547–555. Google Scholar
[23] X. P. Ding and Z. B. Wang, Sensitivity analysis for a system of parametric gen- eralized mixed quasi-variational inclusions involving (K, η)-monotone mappings, Appl. Math. Comput. 214 (2009) 318–327. Google Scholar
[24] X. P. Ding, Z. B. Wang, Auxiliary principle and algorithm for a system of gen- eralized set-valued mixed variational-like inequality problems in Banach spaces, J. Comput. Appl. Math. 223 (2010), 2876–2883. Google Scholar
[25] Y. P. Fang and N. J. Huang, H-accretive operators and resolvent operators technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (6) (2004), 647–653. Google Scholar
[26] X. He, On φ-strongly accretive mapping and some set valued variational problems, J. Math. Anal. Appl. 277 (2) (2003), 504–511. Google Scholar
[27] N. J. Huang, On the generalized implicit quasi variational inequalities, J. Math. Anal. Appl. 216 (1997), 197–210. Google Scholar
[28] J. S. Jung and C. H. Morales, The Mann process for perturbed m-accretive operators in Banach spaces, Nonlinear Anal. 46 (20) (2001), 231–243. Google Scholar
[29] S. S. Chang, Set valued variational inclusions in Banach Spaces, J. Math. Anal. Appl. 248 (2000), 438–454. Google Scholar
[30] S. S. Chang, J. K. Kim and H. K. Kim, On the existence and iterative approx- imation problems of solutions for set valued variational inclusions in Banach spaces, J. Math. Anal. Appl. 268 (2002), 89–108. Google Scholar
[31] S. S. Chang, Y. J. Cho, B. S. Lee and I. J. Jung, Generalized set valued varia- tional inclusions in Banach spaces, J. Math. Anal. Appl. 246 (2000), 409–422. Google Scholar
[32] S. S. Chang, Salahuddin and Y. K. Tang, A system of nonlinear set valued variational inclusions, SpringerPlus 2014, 3:318, Doi:10.1186/2193-180-3-318. Google Scholar
[33] S. S. Chang, Y. J. Cho, B. S. Lee, I. J. Jung and S. M. Kang, Iterative approxi- mations of fixed points and solutions for strongly accretive and strongly pseudo contractive mappings in Banach Spaces, J. Math. Anal. Appl. 224 (1998), 149– 165. Google Scholar
[34] L. C. Ceng, S. S. Schaible and J. C. Yao, On the characterization of strong convergence of an iterative algorithm for a class of multivalued variational in- clusions, Math. Math. Oper. Res. 70 (2009), 1–12. Google Scholar
[35] Y. J. Cho, H. Y. Zhou, S. M. Kang, S. S. Kim, Approximations for fixed points of φ-hemicontractive mappings by the Ishikawa iterative process with mixed errors, Math. Comput. Model. 34 (2001), 9–18. Google Scholar
[36] C. E. Chidume, H. Zegeye and K.R. Kazmi, Existence and convergence theorem for a class of multivalued variational inclusions in Banach space, Nonlinear Anal. 59 (2004), 649–656. Google Scholar
[37] M. F. Khan and Salahuddin, Generalized multivalued nonlinear co-variational inequalities in Banach spaces, Funct. Diff. Equations 14 (2-4) (2007), 299–313. Google Scholar
[38] M. K. Ahmad and Salahuddin, Stable perturbed algorithms for a new class of generalized nonlinear implicit quasi variational inclusions in Banach spaces, Advances in Pure Math. 2 (2) (2012), 139–148. Google Scholar
[39] M. F. Khan and Salahuddin, Generalized co-complementarity problems in p-uniformly smooth Banach spaces, JIPAM, J. Inequal. Pure Appl. Math. 7 (2), Article 66, (2006), 11 pages. Google Scholar
[40] 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
[41] Jr. S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475–487. Google Scholar