International Journal of Nonlinear Analysis and Applications
$H(.,.,.,.)$-$\varphi$-$\eta$-cocoercive Operator with an Application to Variational Inclusions
Document Type : Research Paper
Authors
Department of Mathematics, University of Jammu, Jammu-180006, India
Abstract
In this work, we study generalized $ H(.,.,.,.)$-$\varphi$-$\eta-$ cocoercive operator to find the solution of variational like inclusion involving an infinite family of set-valued mappings in semi-inner product spaces via resolvent equation approach. Furthermore, we established an equivalence between the set-valued variational-like inclusion problem and fixed point problem by employing generalized resolvent operator technique involving generalized $H(.,.,.,.)$-$\varphi$-$\eta$-cocoercive operator. Using the equivalent formulation of set-valued variational-like inclusion problem and resolvent equation problem, an iterative algorithm is developed that approximate the uniqueness of solution of the resolvent equation problem.
Keywords
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[15] J.R. Giles, Classes of semi-inner product spaces, Trans. Amer. Math. Soc. 129 (1963) 436-446.
[16] S. Gupta, M. Singh, Variational-like inclusion involving infinite family of set-valued mappings governed by resolvent equations, J. Math. Comput. Sci. 11(1) (2021) 874-892.
[2] R. Ahmad, M. Dilshad, Application of H(., .)-cocoercive operators for solving a set-valued variational inclusion problem via a resolvent equation problem, Indian J. Ind. Appl. Math. 4(2) (2011) 160-169.
[3] J.P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Berlin (1984).
[4] D. Aussel, N. Hadjisavvas, On quasimonotone variational inequalities, Journal of Optimization Theory and Applications 121(2) (2004) 445-450.
[5] M.I. Bhut, B. Zahoor, Existence of solution and iterative approximation of a system of generalized variational-like inclusion problems in semi-inner product spaces, Filomat 31(19) (2017) 6051-6070.
[6] M.I. Bhat, B. Zahoor, Approximation solvability for a system of variational-like inclusions involving generalized (H, ϕ)-η-monotone operators, Int. J. Modern Math. Sci. 15(1) (2017) 30-49.
[7] S.S. Chang, J.K. Kim, K.H. Kim, On the existence and iterative approximation problems of solutions for set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl. 268(1) (2002) 89-108.
[8] Y.P. Fang, N.J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Applied Mathematics Letters 17(6) (2004) 647-653.
[9] Y.P. Fang, N.-J. Huang, H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput. 145(2-3) (2003) 795-803.
[10] Y.P. Fang, N.J. Huang, Approximate solutions for nonlinear operator inclusions with (H, η)-monotone operator, Research Report, Sichuan University (2003).
[11] Y.P. Fang, N.-J. Huang, H. B. Thompson, A new system of variational inclusions with (H, η)-monotone operators in Hilbert spaces, Computers & Mathematics with Applications 49(2-3) (2005) 365-374.
[12] A. Farajzadeh, P. Chuasuk, A. Kaewcharoen, M. Mursaleen, An iterative process for a hybrid pair of generalized I-asymptotically nonexpansive single-valued mappings and generalized nonexpansive multi-valued mappings in Banach spaces, Carpathian Journal of Mathematics 34(1) (2018) 31-45.
[13] A.P. Farajzadeh, M. Mursaleen, A. Shafie, On mixed vector equilibrium problems, Azerbaijan Journal of Mathematics 6(2) (2016) 87-102.
[14] H.R. Feng, X.P. Ding, A new system of generalized nonlinear quasi-variational-like inclusions with Amonotone operators in Banach spaces, J. Comput. Appl. Math. 225 (2009) 365-373.
[15] J.R. Giles, Classes of semi-inner product spaces, Trans. Amer. Math. Soc. 129 (1963) 436-446.
[16] S. Gupta, M. Singh, Variational-like inclusion involving infinite family of set-valued mappings governed by resolvent equations, J. Math. Comput. Sci. 11(1) (2021) 874-892.
[17] N. Hadjisavvas, Continuity and maximality properties of pseudomonotone operators, Journal of Convex Analysis, 10(2) (2003) 465-475.
[18] N. Hadjisavvas, S. Schaible, On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality, Optimization 55(5-6) (2006) 593-604.
[19] S. Husain, S. Gupta, V.N. Mishra, Graph convergence for the H(., .)-mixed mapping with an application for solving the system of generalized variational inclusions, Fixed Point Theory Appl. 2013, Article ID 304 (2013).
[20] S. Husain, S. Gupta, V.N. Mishra, Generalized H(., ., .)-η-cocoercive operators and generalized set-valued variational-like inclusions, J. Math. 2013, Article ID 738491 (2013).
[21] S. Husain, H. Sahper, S. Gupta, H(., ., .)-η-Proximal-Point Mapping with an Application, in: J.M. Cushing, M. Saleem, H.M. Srivastava, M.A. Khan, M. Merajuddin (Eds.), Applied Analysis in Biological and Physical Sciences, Springer India, New Delhi (2016) 351-372.
[22] K. R. Kazmi, F. A. Khan, Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P-η-accretive mappings, Journal of Mathematical Analysis and Applications 337(2) (2008) 1198-1210.
[23] K. R. Kazmi, F. A. Khan, Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: a P-η-proximal-point mapping approach, Journal of Mathematical Analysis and Applications 325(1) (2007) 665-674.
[24] K. R. Kazmi, N. Ahmad, M. Shahzad, Convergence and stability of an iterative algorithm for a system of generalized implicit variational-like inclusions in Banach spaces, Applied Mathematics and Computation 218(18) (2012) 9208-9219.
[25] K. R. Kazmi, F. A. Khan, M. Shahzad, A system of generalized variational inclusions involving generalized H(., .)- accretive mapping in real q-uniformly smooth Banach spaces, Applied Mathematics and Computation 217(23) (2011) 9679-9688.
[26] K. R. Kazmi, M. I. Bhat, N. Ahmad, An iterative algorithm based on M-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces, Journal of Computational and Applied Mathematics 233(2) (2009) 361-371.
[27] F. A. Khan, A. A. Alatawi, M. Dilshad, Parametric generalized mixed multi-valued implicit quasi-variational inclusion problems, Adv. Fixed Point Theory 8(3) (2018) 287-302.
[28] H.-Y. Lan, Y. J. Cho, R. U. Verma, Nonlinear relaxed cocoercive variational inclusions involving (A, η)-accretive mappings in Banach spaces, Computers & Mathematics with Applications 51(9-10) (2006) 1529-1538.
[29] J. Lou, X.F. He, Z. He, Iterative methods for solving a system of variational inclusions H-η-monotone operators in Banach spaces, Comput. Math. Appl. 55 (2008) 1832-1841.
[30] G. Lumer, Semi-inner product spaces, Trans. Amer. Math. Soc. 100 (1961) 29-43.
[31] X.P. Luo, N.J. Huang, (H, φ)-η-monotone operators in Banach spaces with an application to variational inclusions, Appl. Math. Comput. 216 (2010) 1131-1139.
[32] T. L. Magnanti, G. Perakis, Convergence condition for variational inequality algorithms, Working Paper OR-282- 93, Massachusetts Institute of Technology, (1993).
[33] J.-W. Peng, D.L. Zhu, A new system of generalized mixed quasi-vatiational inclusions with (H, η)-monotone operators, J. Math. Anal. Appl. 327 (2007) 175-187.
[34] T. Ram, Parametric generalized nonlinear quasi-variational inclusion problems, Int. J. Math. Arch. 3(3) (2012) 1273-1282.
[35] N.K. Sahu, R.N. Mohapatra, C. Nahak, S. Nanda, Approximation solvability of a class of A-monotone implicit variational inclusion problems in semi-inner product spaces, Appl. Math. Comput. 236 (2014) 109-117.
[36] N.K. Sahu, C. Nahak, S. Nanda, Graph convergence and approximation solvability of a class of implicit variational inclusion problems in Banach spaces, J. Indian Math. Soc. 81(12) (2014) 155-172.
[37] P. Tseng, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Mathematical Programming 48(2) (1990) 249-263.
[38] R.U. Verma, Approximation solvability of a class of nonlinear set-valued inclusions involving (A, η)-monotone mappings, J. Math. Appl. Anal. 337 (2008) 969-975.
[39] R.U. Verma, General class of implicit variational inclusions and graph convergence on A-maximal relaxed monotonicity, J. Optim. Theory Appl. 155(1) (2012) 196-214.
[40] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16(12) (1991) 1127-1138.
[41] Z. Xu, Z. Wang, A generalized mixed variational inclusion involving (H(., .), η)-monotone operators in Banach spaces, Journal of Mathematics Research 2(3) (2010) 47-56.
[42] Y.H. Wang, The infinite family of generalized set-valued quasi-variation inclusions in Banach spaces, Acta Anal. Funct. Appl. 10 (2008) 1009-1327.
[43] D. L. Zhu, P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variationalinequalities, SIAM Journal on Optimization 6(3) (1996) 714-726.
[44] Y.Z. Zou, N.J. Huang, H(., .)-accretive operator with an application for solving variational inclusions in Banach spaces, Applied Mathematics and Computation 204(2) (2008) 809-816.
[18] N. Hadjisavvas, S. Schaible, On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality, Optimization 55(5-6) (2006) 593-604.
[19] S. Husain, S. Gupta, V.N. Mishra, Graph convergence for the H(., .)-mixed mapping with an application for solving the system of generalized variational inclusions, Fixed Point Theory Appl. 2013, Article ID 304 (2013).
[20] S. Husain, S. Gupta, V.N. Mishra, Generalized H(., ., .)-η-cocoercive operators and generalized set-valued variational-like inclusions, J. Math. 2013, Article ID 738491 (2013).
[21] S. Husain, H. Sahper, S. Gupta, H(., ., .)-η-Proximal-Point Mapping with an Application, in: J.M. Cushing, M. Saleem, H.M. Srivastava, M.A. Khan, M. Merajuddin (Eds.), Applied Analysis in Biological and Physical Sciences, Springer India, New Delhi (2016) 351-372.
[22] K. R. Kazmi, F. A. Khan, Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P-η-accretive mappings, Journal of Mathematical Analysis and Applications 337(2) (2008) 1198-1210.
[23] K. R. Kazmi, F. A. Khan, Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: a P-η-proximal-point mapping approach, Journal of Mathematical Analysis and Applications 325(1) (2007) 665-674.
[24] K. R. Kazmi, N. Ahmad, M. Shahzad, Convergence and stability of an iterative algorithm for a system of generalized implicit variational-like inclusions in Banach spaces, Applied Mathematics and Computation 218(18) (2012) 9208-9219.
[25] K. R. Kazmi, F. A. Khan, M. Shahzad, A system of generalized variational inclusions involving generalized H(., .)- accretive mapping in real q-uniformly smooth Banach spaces, Applied Mathematics and Computation 217(23) (2011) 9679-9688.
[26] K. R. Kazmi, M. I. Bhat, N. Ahmad, An iterative algorithm based on M-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces, Journal of Computational and Applied Mathematics 233(2) (2009) 361-371.
[27] F. A. Khan, A. A. Alatawi, M. Dilshad, Parametric generalized mixed multi-valued implicit quasi-variational inclusion problems, Adv. Fixed Point Theory 8(3) (2018) 287-302.
[28] H.-Y. Lan, Y. J. Cho, R. U. Verma, Nonlinear relaxed cocoercive variational inclusions involving (A, η)-accretive mappings in Banach spaces, Computers & Mathematics with Applications 51(9-10) (2006) 1529-1538.
[29] J. Lou, X.F. He, Z. He, Iterative methods for solving a system of variational inclusions H-η-monotone operators in Banach spaces, Comput. Math. Appl. 55 (2008) 1832-1841.
[30] G. Lumer, Semi-inner product spaces, Trans. Amer. Math. Soc. 100 (1961) 29-43.
[31] X.P. Luo, N.J. Huang, (H, φ)-η-monotone operators in Banach spaces with an application to variational inclusions, Appl. Math. Comput. 216 (2010) 1131-1139.
[32] T. L. Magnanti, G. Perakis, Convergence condition for variational inequality algorithms, Working Paper OR-282- 93, Massachusetts Institute of Technology, (1993).
[33] J.-W. Peng, D.L. Zhu, A new system of generalized mixed quasi-vatiational inclusions with (H, η)-monotone operators, J. Math. Anal. Appl. 327 (2007) 175-187.
[34] T. Ram, Parametric generalized nonlinear quasi-variational inclusion problems, Int. J. Math. Arch. 3(3) (2012) 1273-1282.
[35] N.K. Sahu, R.N. Mohapatra, C. Nahak, S. Nanda, Approximation solvability of a class of A-monotone implicit variational inclusion problems in semi-inner product spaces, Appl. Math. Comput. 236 (2014) 109-117.
[36] N.K. Sahu, C. Nahak, S. Nanda, Graph convergence and approximation solvability of a class of implicit variational inclusion problems in Banach spaces, J. Indian Math. Soc. 81(12) (2014) 155-172.
[37] P. Tseng, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Mathematical Programming 48(2) (1990) 249-263.
[38] R.U. Verma, Approximation solvability of a class of nonlinear set-valued inclusions involving (A, η)-monotone mappings, J. Math. Appl. Anal. 337 (2008) 969-975.
[39] R.U. Verma, General class of implicit variational inclusions and graph convergence on A-maximal relaxed monotonicity, J. Optim. Theory Appl. 155(1) (2012) 196-214.
[40] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16(12) (1991) 1127-1138.
[41] Z. Xu, Z. Wang, A generalized mixed variational inclusion involving (H(., .), η)-monotone operators in Banach spaces, Journal of Mathematics Research 2(3) (2010) 47-56.
[42] Y.H. Wang, The infinite family of generalized set-valued quasi-variation inclusions in Banach spaces, Acta Anal. Funct. Appl. 10 (2008) 1009-1327.
[43] D. L. Zhu, P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variationalinequalities, SIAM Journal on Optimization 6(3) (1996) 714-726.
[44] Y.Z. Zou, N.J. Huang, H(., .)-accretive operator with an application for solving variational inclusions in Banach spaces, Applied Mathematics and Computation 204(2) (2008) 809-816.
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Volume 13, Issue 1
March 2022Pages 1311-1327
- Receive Date: 25 April 2021
- Revise Date: 13 August 2021
- Accept Date: 21 August 2021