An itertive algorithm with error terms for solving a system of implicit n-variational inclusions
AbstractA new system of implicit n-variational inclusions is considered. We propose a new algorithm with error terms for computing the approximate solutions of our system. The convergence of the iterative sequences generated by the iterative algorithm is also discussed. Some special cases are also discussed.
R. P. Agarwal, Multiple positive solutions of singular and non-singular discrete problems via variational methods, J. Nonlin. Anal.,vol. 58, pp. 69-73, 2004.
I. Ahmad, M. Rahaman, and R. Ahmad, Relaxed resolvent operator for solving a variational inclusion problem, Stat. Optim. Inf.Comput., vol. 4, pp. 183-193, 2016.
I. Ahmad, R. Ahmad, and M. Rahaman, Implicit resolvent equation problem in Hilbert spaces, Math. Sci. Lett., vol. 6, no. 2, pp.1-7, 2017.
I. Ahmad, V. N. Mishra, R. Ahmad, and M. Rahaman, An Iterative algorithm for a system of generalized implicit variational inclusions,, Spring Plus, vol. 5, no. 1283, 2016.
I. Ahmad, R. Ahmad, and M. Rahaman, A resolvent approach for solving a set-valued variational inclusion problem using weak-RRDset-valued mapping, Korean J. Math., vol. 24, no. 2, pp. 199-213, 2016.
Q. H. Ansari, S. Schaible, and J. C. Yao, System of vector equillibrium problems and its applications, J. Optim. Theory Appl., vol.107, pp. 547-557, 2000.
R. Ahmad, M. Ishtyak, M. Rahaman and I. Ahmad, Graph convergence and generalized Yosida approximation operator with an application, Math. Sci., vol. 11, no. 2, pp. 155-163, 2017.
M. Bianchi, Pseudo P-monotone Operators and variational inequalities, Report 6, Istitute di econometria e Matematica per ledecisioni economiche, Universita Cattolica del Cuore,Milan, 1993.
S. S. Chang, J. K. Kim, and K. H. Kim, On the existence and iterative approximation problems of solutions for set-valued variational inclusions in Banach Spaces, Comput. Math. Appl., vol. 49, pp. 365-374, 2005.
Y. J. Cho, Y. P. Fang, and N. J. Huang, Algorithms for systems of nonlinear variational inequalities, J. Korean Math. Soc., vol. 41,pp. 489-499, 2004.
G. Cohen, and F. Chaplais, Nested monotony for variatonal inequalities over a product of spaces and convergence of iterative algorithms, J. Optim. Theory Appl., vol. 59, pp. 360-390, 1998.
X. P. Ding, Existence and algorithms of solutions for nonlinear mixed variational-like inequalities in Banach spaces, J. Comput.Appl, Math., vol. 157, pp. 419-434, 2003.
Y. P. Fang, and N. J. Huang. Existence results for systems of strongly implicit vector variational inequalities, Acta Math. Hung.,vol. 103, pp. 265-277, 2004.
Y. P. Fang, N. J. Huang, and H. B. Thompson, A new system of variational inclusions with (H; )-monotone operators in Hilbert spaces, Comput. Math. Appl., vol. 49, no. 2-3, pp. 365-374, 2005.
N. J. Huang, Generalized nonlinear variational inclusions with noncompact valued mapping, Appl. Math. Lett., vol. 9, no. 3, pp.25-29, 1996.
N. J. Huang, A new completely general class of variational inclusions with non compact valued mappings, Comput. Math. Appl.,vol. 35, no. 10, pp. 9-14, 1998.
N. J. Huang, A new class of generalized variational inclusions involving maximal -monotone mappings, Publ. Math., 62, Debrecen,2003.
S. Hussain, An Ishikawa type ierative algorithm for a generalised variational inclusions, J. Inter. Math. Forum, vol. 25, pp. 1221-1228, 2009.
S. S. Irfan, Generalized Variational-like Inclusion in Fuzzy Environment with -Proximal Operator J. Math. Anal., vol. 10, no. 3,pp. 89-99, 2019.
J. K. Kim, and D. S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Korean Math.Soc., vol. 11, no. 1, pp. 203-210, 2004.
H. Y. Lan, J. H. Kim, and Y. J. Cho, On a new system of nonlinear monotone multivalued variational inclusions, J. Math. Anal.Appl., vol. 327, pp. 481-493, 2007.
S. B. Nadler, Multivalued contraction mappings, Pacific J. Math., vol. 475-488, 1969.
J. W. Peng, and D. Zhu, A new system of generalised mixed quasivariational inclusions with (H; )-monotone operators, J. Math.Anal. Appl., vol. 327, pp. 175-187, 2007.
M. Rahaman, R. Ahmad, M. Dilshad, and I. Ahmad, Relaxed -proximal operator for solving a variational-like inclusion problem,Math. Model. Anal., vol. 20 (6), pp. 819-835, 2015.
M. Rahaman, Y.C. Liou, R. Ahmad, and I. Ahmad, Convergence theorems for split equality generalized mixed equilibrium problems for demi-contractive mappings, J. Inequal. Appl., vol. 2015 (1), pp. 1-25, 2015.
N. K. Sahu, N. K. Mahato, and R. N. Mohapatra, System of nonlinear variational inclusion problems with (A; )-maximal monotonicity in Banach spaces, Stat Optim Inf Comput, vol. 5, pp. 244-261, 2017.
R. U. Verma, Nonlinear variational and constrained hemivariational inequalities involving relaxed operators, J. Appl. Math. Mech.,vol. 77, no. 7, pp. 387-391, 1999.
J. Wu, and G. Yu, On the convergence and O(1/N) complexity of a class of nonlinear proximal point algorithms for monotonic variational inequalities, Stat Optim Inf Comput, vol. 2, pp. 105-113, 2014.
F. Q. Xia, and N. J. Huang, Algorithm for solving a new class of general mixed variational inequalities in Banach spaces, J. Comput.Math. App., vol. 220, pp. 632-642, 2007.
- 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).