American Institute of Mathematical Sciences

Advanced
  • Previous Article
    $ \theta $ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations
  • NACO Home
  • This Issue
  • Next Article
    Numerical solutions of Volterra integro-differential equations using General Linear Method
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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.   Google Scholar

[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.   Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[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.   Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.   Google Scholar

[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.   Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[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.   Google Scholar

[1]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[2]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2021, 11 (4) : 857-883. doi: 10.3934/mcrf.2020049
[3]

Felipe Alvarez, Juan Peypouquet. Asymptotic equivalence and Kobayashi-type estimates for nonautonomous monotone operators in Banach spaces. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1109-1128. doi: 10.3934/dcds.2009.25.1109
[4]

B. S. Lee, Arif Rafiq. Strong convergence of an implicit iteration process for a finite family of Lipschitz $\phi -$uniformly pseudocontractive mappings in Banach spaces. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 287-293. doi: 10.3934/naco.2014.4.287
[5]

Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232
[6]

Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131
[7]

Daliang Zhao, Yansheng Liu, Xiaodi Li. Controllability for a class of semilinear fractional evolution systems via resolvent operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 455-478. doi: 10.3934/cpaa.2019023
[8]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203
[9]

Goro Akagi, Mitsuharu Ôtani. Evolution equations and subdifferentials in Banach spaces. Conference Publications, 2003, 2003 (Special) : 11-20. doi: 10.3934/proc.2003.2003.11
[10]

Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773
[11]

Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution Equations in Banach spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002
[12]

Arnulf Jentzen, Felix Lindner, Primož Pušnik. On the Alekseev-Gröbner formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4475-4511. doi: 10.3934/dcdsb.2019128
[13]

Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91
[14]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237
[15]

Filippo Gazzola, Mirko Sardella. Attractors for families of processes in weak topologies of Banach spaces. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 455-466. doi: 10.3934/dcds.1998.4.455
[16]

Ji Gao. On the generalized pythagorean parameters and the applications in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 557-567. doi: 10.3934/dcdsb.2007.8.557
[17]

Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations & Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009
[18]

Sergiu Aizicovici, Yimin Ding, N. S. Papageorgiou. Time dependent Volterra integral inclusions in Banach spaces. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 53-63. doi: 10.3934/dcds.1996.2.53
[19]

Marcin Studniarski. Finding all minimal elements of a finite partially ordered set by genetic algorithm with a prescribed probability. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 389-398. doi: 10.3934/naco.2011.1.389
[20]

Muhammad Qiyas, Saleem Abdullah, Shahzaib Ashraf, Saifullah Khan, Aziz Khan. Triangular picture fuzzy linguistic induced ordered weighted aggregation operators and its application on decision making problems. Mathematical Foundations of Computing, 2019, 2 (3) : 183-201. doi: 10.3934/mfc.2019013

 Impact Factor: 

Tools

Metrics

  • PDF downloads (149)
  • HTML views (547)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy