System of nonlinear variational inclusion problems with $(A,\eta)$-maximal monotonicity in Banach spaces

Nabin Kumar Sahu, N. K. Mahato, R. N. Mohapatra

Abstract


This paper deals with a new system of nonlinear variational inclusion problems involving $(A,\eta)$-maximal relaxed monotone and relative $(A,\eta)$-maximal monotone mappings in 2-uniformly smooth Banach spaces. Using the generalized resolvent operator technique, the approximation solvability of the proposed problem is investigated. An iterative algorithm is constructed to approximate the solution of the problem. Convergence analysis of the proposed algorithm is investigated. Similar results are also proved for other system of variational inclusion problems involving relative $(A,\eta)$-maximal monotone mappings and $(H,\eta)$-maximal monotone mappings.

Keywords


Variational inclusion, Generalized resolvent operator, $2$-uniformly smooth Banach space, Semi-inner product space

References


S. Adly, Perturbed algorithm and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl., vol. 201, pp. 609–630, 1996.

R. P. Agarwal, and R. U. Verma, General system of (A; )-maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms, Commun. Nonlinear Sci. Numer. Simulat., vol. 15, pp. 238–251, 2010.

I. Ahmad, M. Rahaman, and R. Ahmad, Relaxed resolvent operator for solving variational inclusion problem, Stat. Optim. Inf. Comput., vol. 4, pp. 183–193, 2016.

Y. J. Cho, Y. P. Fang, N. J. Huang, and H. J. Hwang, Algorithms for systems of nonlinear variational inequalities, J. Korean Math. Soc., vol. 41, pp. 489–499, 2004.

X. P. Ding, Perturbed proximal point for generalized quasi-variational inclusions, J. Math. Anal. Appl., vol. 210, pp. 88–101, 1997.

X. P. Ding, and C. L. Luo, Perturbed proximal point algorithms for generalized quasi-variational like inclusions, J. Comput. Appl. Math., vol. 210, pp. 153–165, 2000.

Y. P. Fang, and N. J. Huang, H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput., vol. 145, pp. 795–803, 2003.

J. R. Giles, Classes of semi-inner product spaces, Trans. Amer. Math. Soc., vol. 129, pp. 436–446, 1967.

N. J. Huang, A new class of generalized set valued implicit variational inclusions in Banach spaces with an application, Comput. Math. Appl., vol. 41, pp. 937–943, 2001.

D. O. Koehler, A note on some operator theory in certain semi-inner product spaces, Proc. Amer. Math. Soc., vol. 30, pp. 363–366, 1971.

H. Y. Lan, J. H. Kim, and Y. J. Cho, On a new system of nonlinear A-monotone multivalued variational inclusions, J. Math. Anal. Appl., vol. 327, pp. 481–493, 2007.

C. H. Lee, Q. H. Ansari, and J. C. Yao, A perturbed algorithm for strongly nonlinear variational like inclusions, Bull. Aust. Math. Soc., vol.62, pp. 417–426, 2000.

G. Lumer, Semi-inner product spaces, Trans. Amer. Math. Soc., vol. 100, pp. 29–43, 1961.

J. Peng, and D. Zhu, A new system of generalized mixed quasi-variational inclusions with (H; )-monotone operators, J. Math. Anal. Appl., vol. 327, pp. 175–187, 2007.

N. K. Sahu, R. N. Mohapatra, C. Nahak, and S. Nanda, Approximation solvability of a class of A-monotone implicit variational inclusion problems in semi-inner product spaces, Appl. Math. Comput., vol. 236, pp. 109–117, 2014.

R. U. Verma, Approximation solvability of a class of nonlinear set-valued variational inclusions involving (A; )-monotone mappings, J. Math. Anal. Appl., vol. 337, pp. 969–975, 2008.

R. U. Verma, Sensitivity analysis for generalized strongly monotone variational inclusions based on (A; )-resolvent operator technique, Appl. Math. Lett., vol. 19, pp. 1409–1413, 2006.

H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Analysis: Theory, Methods and Applications, vol. 16, pp. 1127–1138, 1991.


Full Text: PDF

DOI: 10.19139/soic.v5i3.238

Refbacks

  • There are currently no refbacks.