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December  2019, 9(4): 445-460. doi: 10.3934/naco.2019026

System of generalized mixed nonlinear ordered variational inclusions

Salahuddin ,

Department of Mathematics, Jazan University, Jazan, 45142, KSA

* Corresponding author: Salahuddin

Received  November 2017 Revised  April 2019 Published  May 2019

In this paper, we consider a system of generalized mixed nonlinear ordered variational inclusions in partially ordered Banach spaces and suggest an algorithm for a solution of the considered system. We prove an existence and convergence result for the solution of the system of generalized mixed nonlinear ordered variational inclusions.

Keywords: Algorithm, convergence, sequences, resolvent operators, solution, system, partial ordered Banach spaces.
Mathematics Subject Classification: Primary: 49J40, 47H09; Secondary: 47J20.
Citation: Salahuddin. System of generalized mixed nonlinear ordered variational inclusions. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 445-460. doi: 10.3934/naco.2019026
References:
[1]

R. Ahmad, M. F. Khan and Sa lahuddin, Mann and Ishikawa type perturbed iterative algorithm for generalized nonlinear variational inclusions, Math. Comput. Appl., 6 (2001), 47-52.  doi: 10.3390/mca6010047.

[2]

M. K. Ahmad and Sa lahuddin, Resolvent equation technique for generalized nonlinear variational inclusions, Adv. Nonlinear Var. Inequal., 5 (2002), 91-98. 

[3]

M. K. Ahmad and Salahuddin, Perturbed three step approximation process with errors for a general implicit nonlinear variational inequalities, Int. J. Math. Math. Sci., Article ID 43818, (2006), 1–14. doi: 10.1155/IJMMS/2006/43818.

[4]

M. K. Ahmad and Salahuddin, Generalized strongly nonlinear implicit quasi-variational inequalities, J. Inequal. Appl., 2009 (2009), Article ID 124953, 1–16. doi: 10.1155/2009/124953.

[5]

H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal., 11 (1972), 346-384.  doi: 10.1016/0022-1236(72)90074-2.

[6]

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.  doi: 10.1016/j.amc.2008.12.028.

[7]

X. P. Ding and Sa lahuddin, On a system of general nonlinear variational inclusions in Banach spaces, Appl. Math. Mech., 36 (2015), 1663-1672.  doi: 10.1007/s10483-015-2001-6.

[8]

Y. P. Du, Fixed points of increasing operators in ordered Banach spaces and applications, Anal., 38 (1990), 1-20.  doi: 10.1080/00036819008839957.

[9]

Y. P. Fang, N. J. Huang and H. B. Thompson, A new system of variational inclusions with $(H, \eta)$-monotone operators in Hilbert spaces, Comput. Math. Appl., 49 (2005), 365-374.  doi: 10.1016/j.camwa.2004.04.037.

[10]

X. F. He, J. L. Lou and Z. He, Iterative methods for solving variational inclusions in Banach spaces, J. Comput. Appl. Math., 203 (2007), 80-86.  doi: 10.1016/j.cam.2006.03.011.

[11]

N. J. Huang and Y. P. Fang, Generalized $m$-accretive mappings in Banach spaces, J. Sichuan Univ., 38 (2001), 591-592. 

[12]

S. Hussain, M. F. Khan and Sa lahuddin, Mann and Ishikawa type perturbed iterative algorithms for completely generalized nonlinear variational inclusions, Int. J. Math. Anal., 3 (2006), 51-62. 

[13]

P. Junlouchai, S. Plubtieng and Sa lahuddin, On a new system of nonlinear regularized nonconvex variational inequalities in Hilbert spaces, J. Nonlinear Anal. Optim., 7 (2016), 103-115. 

[14]

M. F. Khan and Sa lahuddin, Mixed multivalued variational inclusions involving H-accretive operators, Adv. Nonlinear Var. Inequal., 9 (2006), 29-47. 

[15]

M. F. Khan and Salahuddin, Generalized co-complementarity problems in p-uniformly smooth Banach spaces, JIPAM, J. Inequal. Pure Appl. Math., 7 (2006), 1–11, Article ID 66.

[16]

M. F. Khan and Sa lahuddin, Generalized multivalued nonlinear co-variational inequalities in Banach spaces, Funct. Diff. Equat., 14 (2007), 299-313. 

[17]

B. S. Lee and Sa lahuddin, Fuzzy general nonlinear ordered random variational inequalities in ordered Banach spaces, East Asian Math. J., 32 (2016), 685-700. 

[18]

B. S. Lee, M. F. Khan and Sa lahuddin, Generalized nonlinear quasi-variational inclusions in Banach spaces, Comput. Math. Appl., 56 (2008), 1414-1422.  doi: 10.1016/j.camwa.2007.11.053.

[19]

B. S. Lee, M. F. Khan and Sa lahuddin, Hybrid-type set-valued variational-like inequalities in Reflexive Banach spaces, J. Appl. Math. Inform., 27 (2009), 1371-1379. 

[20]

H. G. Li, L. P. Li, J. M. Zheng and M. M. Jin, Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with $(\alpha, \lambda)$-nodsm mappings in ordered Banach spaces, Fixed Point Theory Appl., 2014 (2014), 122. doi: 10.1186/1687-1812-2014-122.

[21]

H. G. Li, D. Qui and Y. Zou, Characterization of weak-anodd set-valued mappings with applications to approximate solution of gnmoqv inclusions involving $\oplus$ operator in ordered Banach spaces, Fixed Point Theory Appl., 2013 (2013), 241. doi: 10.1186/1687-1812-2013-241.

[22]

H. G. Li, L. P. Li and M. M. Jin, A class of nonlinear mixed ordered inclusion problems for oredered $(\alpha_a, \lambda)$-ANODM set-valued mappings with strong comparison mapping, Fixed Point Theory Appl., 2014 (2014), 79. doi: 10.1186/1687-1812-2014-79.

[23]

H. G. Li, A nonlinear inclusion problem involving $(\alpha, \lambda)$-NODM set-valued mappings in ordered Hilbert space, Appl. Math. Lett., 25 (2012), 1384-1388.  doi: 10.1016/j.aml.2011.12.007.

[24]

H. G. Li, Approximation solution for general nonlinear ordered variational inequalities and ordered equations in ordered Banach space, Nonlinear Anal. Forum, 13 (2008), 205-214. 

[25]

H. G. Li, D. Qiu and M. M. Jin, GNM ordered variational inequality system with ordered Lipschitz continuous mappings in an ordered Banach space, J. Inequal. Appl., 2013 (2013), 514. doi: 10.1186/1029-242X-2013-514.

[26]

H. G. Li, X. B. Pan, Z. Y. Deng and C. Y. Wang, Solving GNOVI frameworks involving $(\gamma_g, \lambda)$-weak-GRD set-valued mappings in positive Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 146. doi: 10.1186/1687-1812-2014-146.

[27]

H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1994.

[28]

Sa lahuddin, Regularized equilibrium problems in Banach spaces, Korean J. Math., 24 (2016), 51-63.  doi: 10.11568/kjm.2016.24.1.51.

[29]

Sa lahuddin, Solvability for a system of generalized nonlinear ordered variational inclusions in ordered Banach spaces, Korean J. Math., 25 (2017), 359-377. 

[30]

Sa lahuddin and S. S. Irfan, Proximal methods for quasi-variational inequalities, Math. Computat. Appl., 9 (2004), 165-172.  doi: 10.3390/mca9020165.

[31]

A. H. Siddiqi, M. K. Ahmad and Sa lahuddin, Existence results for generalized nonlinear variational inclusions, Appl. Math. Lett., 18 (2005), 859-864.  doi: 10.1016/j.aml.2004.08.015.

[32]

Y. K. Tang, S. S. Chang and Salahuddin, A system of nonlinear set valued variational inclusions, SpringerPlus, 3 (2014), 318.

[33]

R. U. Verma, Projection methods, algorithms and a new system of nonlinear variational inequalities, Comput. Math. Appl., 41 (2001), 1025-1031.  doi: 10.1016/S0898-1221(00)00336-9.

[34]

R. U. Verma, M. F. Khan and Sa lahuddin, Fuzzy generalized complementarity problems in Banach spaces, PanAmer. Math. J., 17 (2007), 71-80. 

[35]

R. U. Verma and Sa lahuddin, 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 (2016), 71-88. 

show all references

References:
[1]

R. Ahmad, M. F. Khan and Sa lahuddin, Mann and Ishikawa type perturbed iterative algorithm for generalized nonlinear variational inclusions, Math. Comput. Appl., 6 (2001), 47-52.  doi: 10.3390/mca6010047.

[2]

M. K. Ahmad and Sa lahuddin, Resolvent equation technique for generalized nonlinear variational inclusions, Adv. Nonlinear Var. Inequal., 5 (2002), 91-98. 

[3]

M. K. Ahmad and Salahuddin, Perturbed three step approximation process with errors for a general implicit nonlinear variational inequalities, Int. J. Math. Math. Sci., Article ID 43818, (2006), 1–14. doi: 10.1155/IJMMS/2006/43818.

[4]

M. K. Ahmad and Salahuddin, Generalized strongly nonlinear implicit quasi-variational inequalities, J. Inequal. Appl., 2009 (2009), Article ID 124953, 1–16. doi: 10.1155/2009/124953.

[5]

H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal., 11 (1972), 346-384.  doi: 10.1016/0022-1236(72)90074-2.

[6]

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.  doi: 10.1016/j.amc.2008.12.028.

[7]

X. P. Ding and Sa lahuddin, On a system of general nonlinear variational inclusions in Banach spaces, Appl. Math. Mech., 36 (2015), 1663-1672.  doi: 10.1007/s10483-015-2001-6.

[8]

Y. P. Du, Fixed points of increasing operators in ordered Banach spaces and applications, Anal., 38 (1990), 1-20.  doi: 10.1080/00036819008839957.

[9]

Y. P. Fang, N. J. Huang and H. B. Thompson, A new system of variational inclusions with $(H, \eta)$-monotone operators in Hilbert spaces, Comput. Math. Appl., 49 (2005), 365-374.  doi: 10.1016/j.camwa.2004.04.037.

[10]

X. F. He, J. L. Lou and Z. He, Iterative methods for solving variational inclusions in Banach spaces, J. Comput. Appl. Math., 203 (2007), 80-86.  doi: 10.1016/j.cam.2006.03.011.

[11]

N. J. Huang and Y. P. Fang, Generalized $m$-accretive mappings in Banach spaces, J. Sichuan Univ., 38 (2001), 591-592. 

[12]

S. Hussain, M. F. Khan and Sa lahuddin, Mann and Ishikawa type perturbed iterative algorithms for completely generalized nonlinear variational inclusions, Int. J. Math. Anal., 3 (2006), 51-62. 

[13]

P. Junlouchai, S. Plubtieng and Sa lahuddin, On a new system of nonlinear regularized nonconvex variational inequalities in Hilbert spaces, J. Nonlinear Anal. Optim., 7 (2016), 103-115. 

[14]

M. F. Khan and Sa lahuddin, Mixed multivalued variational inclusions involving H-accretive operators, Adv. Nonlinear Var. Inequal., 9 (2006), 29-47. 

[15]

M. F. Khan and Salahuddin, Generalized co-complementarity problems in p-uniformly smooth Banach spaces, JIPAM, J. Inequal. Pure Appl. Math., 7 (2006), 1–11, Article ID 66.

[16]

M. F. Khan and Sa lahuddin, Generalized multivalued nonlinear co-variational inequalities in Banach spaces, Funct. Diff. Equat., 14 (2007), 299-313. 

[17]

B. S. Lee and Sa lahuddin, Fuzzy general nonlinear ordered random variational inequalities in ordered Banach spaces, East Asian Math. J., 32 (2016), 685-700. 

[18]

B. S. Lee, M. F. Khan and Sa lahuddin, Generalized nonlinear quasi-variational inclusions in Banach spaces, Comput. Math. Appl., 56 (2008), 1414-1422.  doi: 10.1016/j.camwa.2007.11.053.

[19]

B. S. Lee, M. F. Khan and Sa lahuddin, Hybrid-type set-valued variational-like inequalities in Reflexive Banach spaces, J. Appl. Math. Inform., 27 (2009), 1371-1379. 

[20]

H. G. Li, L. P. Li, J. M. Zheng and M. M. Jin, Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with $(\alpha, \lambda)$-nodsm mappings in ordered Banach spaces, Fixed Point Theory Appl., 2014 (2014), 122. doi: 10.1186/1687-1812-2014-122.

[21]

H. G. Li, D. Qui and Y. Zou, Characterization of weak-anodd set-valued mappings with applications to approximate solution of gnmoqv inclusions involving $\oplus$ operator in ordered Banach spaces, Fixed Point Theory Appl., 2013 (2013), 241. doi: 10.1186/1687-1812-2013-241.

[22]

H. G. Li, L. P. Li and M. M. Jin, A class of nonlinear mixed ordered inclusion problems for oredered $(\alpha_a, \lambda)$-ANODM set-valued mappings with strong comparison mapping, Fixed Point Theory Appl., 2014 (2014), 79. doi: 10.1186/1687-1812-2014-79.

[23]

H. G. Li, A nonlinear inclusion problem involving $(\alpha, \lambda)$-NODM set-valued mappings in ordered Hilbert space, Appl. Math. Lett., 25 (2012), 1384-1388.  doi: 10.1016/j.aml.2011.12.007.

[24]

H. G. Li, Approximation solution for general nonlinear ordered variational inequalities and ordered equations in ordered Banach space, Nonlinear Anal. Forum, 13 (2008), 205-214. 

[25]

H. G. Li, D. Qiu and M. M. Jin, GNM ordered variational inequality system with ordered Lipschitz continuous mappings in an ordered Banach space, J. Inequal. Appl., 2013 (2013), 514. doi: 10.1186/1029-242X-2013-514.

[26]

H. G. Li, X. B. Pan, Z. Y. Deng and C. Y. Wang, Solving GNOVI frameworks involving $(\gamma_g, \lambda)$-weak-GRD set-valued mappings in positive Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 146. doi: 10.1186/1687-1812-2014-146.

[27]

H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1994.

[28]

Sa lahuddin, Regularized equilibrium problems in Banach spaces, Korean J. Math., 24 (2016), 51-63.  doi: 10.11568/kjm.2016.24.1.51.

[29]

Sa lahuddin, Solvability for a system of generalized nonlinear ordered variational inclusions in ordered Banach spaces, Korean J. Math., 25 (2017), 359-377. 

[30]

Sa lahuddin and S. S. Irfan, Proximal methods for quasi-variational inequalities, Math. Computat. Appl., 9 (2004), 165-172.  doi: 10.3390/mca9020165.

[31]

A. H. Siddiqi, M. K. Ahmad and Sa lahuddin, Existence results for generalized nonlinear variational inclusions, Appl. Math. Lett., 18 (2005), 859-864.  doi: 10.1016/j.aml.2004.08.015.

[32]

Y. K. Tang, S. S. Chang and Salahuddin, A system of nonlinear set valued variational inclusions, SpringerPlus, 3 (2014), 318.

[33]

R. U. Verma, Projection methods, algorithms and a new system of nonlinear variational inequalities, Comput. Math. Appl., 41 (2001), 1025-1031.  doi: 10.1016/S0898-1221(00)00336-9.

[34]

R. U. Verma, M. F. Khan and Sa lahuddin, Fuzzy generalized complementarity problems in Banach spaces, PanAmer. Math. J., 17 (2007), 71-80. 

[35]

R. U. Verma and Sa lahuddin, 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 (2016), 71-88. 

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