# Iterative Algorithms for a Generalized System of Mixed Variational-Like Inclusion Problems and Altering Points Problem

### Abstract

In this article, we introduce and study a generalized system of mixed variational-like inclusion problems involving αβ-symmetric η-monotone mappings. We use the resolvent operator technique to calculate the approximate common solution of the generalized system of variational-like inclusion problems involving αβ-symmetric η-monotone mappings and a fixed point problem for nonlinear Lipchitz mappings. We study strong convergence analysis of the sequences generated by proposed Mann type iterative algorithms. Moreover, we consider an altering points problem associated with a generalized system of variational-like inclusion problems. To calculate the approximate solution of our system, we proposed a parallel S-iterative algorithm and study the convergence analysis of the sequences generated by proposed parallel S-iterative algorithms by using the technique of altering points problem. The results presented in this paper may be viewed as generalizations and refinements of the results existing in the literature.### References

Ahmad, I., Rahaman, M., Ahmad, R.: Relaxed resolvent operator for solving a variational inclusion problem, Statistics, Optimization & Information Computing, Vol. 4, no. 2, pp. 183-193, (2016)

Agrawal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschizian-type Mappings with Applications. Springer ScienceBusiness Media, (2009)

Ames, W.F.: Numerical Methods for Partial Differential Equations. Academic Press, New York (1992)

Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. John Wiley and Sons, New York (1984)

Baiocchi, C., Capelo, A.: Variational and Quasi-variational Inequalities. J. Wiley and Sons, New York, London (1984)

Chang, S. S., Lee, H.W., Chan, C.K.: Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces. Appl. Math. Lett. Vol. 20, pp. 329-334, (2007)

Cubiotti, P.: Existence of solutions for lower semicontinuous quasi-equilibrium problems. Comput. Math. Appl. Vol. 30, no. 12, pp.11-22, (1995)

Ding, X.P.: Generalized quasi-variational-like inclusions with nonconvex functionals. Appl. Math. Comput. Vol. 122, no. 3, pp. 267-282, (2001)

Ding, X.P., Lou, C.L.: Perturbed proximal point algorithms for general quasi-variational-like inclusions. J. Comput. Appl. Math. Vol. 113, no. 12, pp. 153-165, (2000)

Eckstein, J., Bertsekas, B.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. Vol. 55, no. 1, pp. 293-318, (1992)

Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. Vol. 53, pp. 99-110, (1992)

Giannessi, F. Maugeri, A.: Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York (1995)

Glowinski, R., Lions J.L. Tremolieres, R.: Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam (1981)

Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, Berlin (1985)

Gursoy, F. A., Khan, R., Ertrk, M., Karakaya, V.: Convergence and data dependency of normal S-iterative method for discontinuous operators on Banach space. Numer. Funct. Anal. Optim. Vol. 39, pp. 322-345, (2018)

Gursoy, F., Ertrk, M., Abbas, M.: A Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings. Numer. Algorithms https://doi.org/10.1007/s11075-019-00706-w (2019)

Hartmann, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta Math. Vol. 115, no. 1, pp. 271-310, (1966)

He, Z.H., Gu, F.: Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces. Appl. Math. Comput. Vol. 214, pp. 26-30, (2009)

Kazmi, K. R., Khan, F.A.: Auxiliary problems and algorithm for a system of generalized variational-like inequality problems. Appl. Math. Comput. Vol. 187, pp. 789-796, (2007)

Kazmi, K.R., Ahmad, N., Shahzad, M.: Convergence and stability of an iterative algorithm for a system of generalized implicit variational-like inclusions in Banach spaces. Appl. Math. Comput. Vol. 218, pp. 9208-9219, (2012)

Lee, C.H., Ansari, Q.H., Yao, J.C.: A perturbed algorithm for strongly nonlinear variational-like inclusions. B. Aust. Math. Soc. Vol. 62, pp. 417-426, (2000)

Li, X. and Li, X.: A new system of multivalued mixed variational inequality problem. Abstr. Appl. Anal. Volume 2014, Article ID982606, 7 pages, http://dx.doi.org/10.1155/2014/982606 (2014)

Mann, W.R.: Mean value methods in iterations. Proc. Amer. Math. Soc. Vol. 4, pp. 506-510, (1953)

Noor, M. A.: General nonlinear variational inequalities. J. Math. Anal. Appl. Vol. 158, pp. 78-84, (1987)

Noor, M. A.: General auxiliary principle for variational inequalities. PanAmer. Math. J. Vol. 4, no. 1, pp. 27-44, (1994)

Parida, J., Sahoo, M., Kumar, A.: A variational-like inequality problem. B. Aust. Math. Soc. Vol. 39, no. 2, pp. 225-231, (1989)

Petrot, N.: A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems. Appl. Math. Lett. Vol. 23, pp. 440-445, (2010)

Sahu, D.R.: Application of the S-iteration process to constrained minimization problem and split feasibility problems. Fixed Point Theory, Vol. 12, pp. 187-204, (2011)

Sahu, D.R.: Altering points and applications. Nonlinear Stud. Vol. 21, pp. 349-365, (2014)

Sahu, D.R., Yao, J.C., Singh, V. K., Kumar, V.: Semilocal convergence analysis of S-iteration process of Newton–Kantorovich like in Banach spaces. J. Optim. Theory. Appl. Vol. 172, pp. 102-127, (2017)

Sahu, N. K., Mahato, N. K., Mohapatra, R. N.: System of nonlinear variational inclusion problems with (A, η)-maximal monotonicity in Banach space, Statistics, Optimization & Information Computing, Vol. 5, no. 3, pp. 244-261, (2017)

Tian, G.: Generalized quasi-variational-like inequality problem. Math. Oper. Res. Vol. 18, pp. 752-764, (1993)

Verma, R. U.: General convergence analysis for two-step projection methods and applications to variational problems. Appl. Math. Lett. Vol. 18, pp. 1286-1292, (2005)

Verma, M., Shukla, K.K.: A new accelerated proximal technique for regression with high-dimensional datasets. Knowl. Inf. Syst. Vol. 53, pp. 423-438, (2017)

Yao, J.C.: The generalized quasi-variational inequality problem with applications. J. Math. Anal. Appl. Vol. 158, pp. 139-160, (1991)

Zhang, S., Guo, X., Luan, D.: Generalized system for relaxed cocoercive mixed variational inequalities and iterative algorithms in Hilbert spaces. An. St. Univ. Ovidius Constanta Vol. 20, no. 3, pp. 131-140, (2012)

Zhao, X., Sahu, D.R., Wen, C.F.: Iterative methods for system of variational inclusions involving accretive operators and applications. Fixed Point Theory, Vol. 19, no. 2, pp. 801-822, (2018)

Zhou, X.J., Chen, G.: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. J. Math. Anal. Appl. Vol. 132, pp. 213-225, (1988)

*Statistics, Optimization & Information Computing*,

*8*(2), 549-564. https://doi.org/10.19139/soic-2310-5070-884

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).