대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제36권1호
  • /
  • Pages.103-119
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  • 2021
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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SYSTEM OF GENERALIZED SET-VALUED PARAMETRIC ORDERED VARIATIONAL INCLUSION PROBLEMS WITH OPERATOR ⊕ IN ORDERED BANACH SPACES

  • Akram, Mohammad (Department of Mathematics Faculty of Science Islamic University of Madinah) ;
  • Dilshad, Mohammad (Department of Mathematics Faculty of Science University of Tabuk)
  • 투고 : 2020.04.30
  • 심사 : 2020.08.31
  • 발행 : 2021.01.31
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초록

In this article, we study a system of generalized set-valued parametric ordered variational inclusion problems with operator ⊕ in ordered Banach spaces. We introduce the concept of the resolvent operator associated with (α, λ)-ANODSM set-valued mapping and establish the existence theorem of solution for the system of generalized set-valued parametric ordered variational inclusion problems in ordered Banach spaces. In order to prove the existence of solution, we suggest an iterative algorithm and discuss the convergence analysis under some suitable mild conditions.

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참고문헌

  1. R. Ahmad, I. Ahmad, I. Ali, S. Al-Homidan, and Y. H. Wang, H(·, ·)-orderedcompression mapping for solving XOR-variational inclusion problem, J. Nonlinear Convex Anal. 19 (2018), no. 12, 2189-2201.
  2. R. Ahmad, M. Akram, and M. Dilshad, Graph convergence for the H(·, ·)-co-accretive mapping with an application, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 4, 1481-1506. https://doi.org/10.1007/s40840-014-0103-z
  3. R. Ahmad, J. Iqbal, S. Ahmed, and S. Husain, Solving a variational inclusion problem with its corresponding resolvent equation problem involving XOR-operation, Nonlinear Funct. Anal. Appl. 24 (2019), no. 3, 562-582.
  4. I. Ahmad, S. S. Irfan, M. Farid, and P. Shukla, Nonlinear ordered variational inclusion problem involving XOR operation with fuzzy mappings, J. Inequal. Appl. 2020. https: //doi.org/10.1186/s13660-020-2308-z
  5. M. Akram, J. W. Chen, and M. Dilshad, Generalized Yosida approximation operator with an application to a system of Yosida inclusions, J. Nonlinear Funct. Anal. (2018), 1-20.
  6. H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Functional Analysis 11 (1972), 346-384. https://doi.org/10.1016/0022-1236(72)90074-2
  7. C. Baiocchi and A. Capelo, Variational and quasivariational inequalities, translated from the Italian by Lakshmi Jayakar, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.
  8. H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973.
  9. S. S. Chang, Set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl. 248 (2000), no. 2, 438-454. https://doi.org/10.1006/jmaa.2000.6919
  10. 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, J. Math. Anal. Appl. 268 (2002), no. 1, 89-108. https://doi.org/10.1006/jmaa.2001.7800
  11. S. S. Chang, C. F. Wen, and J. C. Yao, Zero point problem of accretive operators in Banach spaces, Bull. Malays. Math. Sci. Soc. 42 (2019), no. 1, 105-118. https://doi.org/10.1007/s40840-017-0470-3
  12. B. C. Dhage, A coupled hybrid fixed point theorem for sum of two mixed monotone coupled operators in a partially ordered Banach space with applications, Tamkang J. Math. 50 (2019), no. 1, 1-36. https://doi.org/10.5556/j.tkjm.50.2019.2502
  13. Y. H. Du, Fixed points of increasing operators in ordered Banach spaces and applications, Appl. Anal. 38 (1990), no. 1-2, 1-20. https://doi.org/10.1080/00036819008839957
  14. Y.-P. Fang and N.-J. Huang, H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput. 145 (2003), no. 2-3, 795-803. https://doi.org/10.1016/S0096-3003(03)00275-3
  15. Y.-P. Fang and N.-J. Huang, Approximate solutions for non-linear variational inclusions with (H, η)-monotone operator, Research Report, Sichuan University, 2003.
  16. Y.-P. Fang and N.-J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004), no. 6, 647-653. https://doi.org/10.1016/S0893-9659(04)90099-7
  17. Y.-P. Fang, N.-J. Huang, and H. B. Thompson, A new system of variational inclusions with (H, η)-monotone operators in Hilbert spaces, Comput. Math. Appl. 49 (2005), no. 2-3, 365-374. https://doi.org/10.1016/j.camwa.2004.04.037
  18. D. J. Guo, Fixed points of mixed monotone operators with applications, Appl. Anal. 31 (1988), no. 3, 215-224. https://doi.org/10.1080/00036818808839825
  19. D. J. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11 (1987), no. 5, 623-632. https://doi.org/10.1016/0362-546X(87)90077-0
  20. A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185 (1994), no. 3, 706-712. https://doi.org/10.1006/jmaa.1994.1277
  21. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, 88, Academic Press, Inc., New York, 1980.
  22. H.-Y. Lan, Y. J. Cho, and R. U. Verma, Nonlinear relaxed cocoercive variational inclusions involving (A, η)-accretive mappings in Banach spaces, Comput. Math. Appl. 51 (2006), no. 9-10, 1529-1538. https://doi.org/10.1016/j.camwa.2005.11.036
  23. H. G. Li, Approximation solution for generalized nonlinear ordered variational inequality and ordered equation in ordered Banach space, Nonlinear Anal. Forum 13 (2008), no. 2, 205-214.
  24. H. G. Li, A nonlinear inclusion problem involving (α, λ)-NODM set-valued mappings in ordered Hilbert space, Appl. Math. Lett. 25 (2012), no. 10, 1384-1388. https://doi.org/10.1016/j.aml.2011.12.007
  25. H. G. Li, L. P. Li, and M. M. Jin, A class of nonlinear mixed ordered inclusion problems for ordered (αA, λ)-ANODM set-valued mappings with strong comparison mapping A, Fixed Point Theory Appl. 2014 (2014), 79, 9 pp. https://doi.org/10.1186/1687-1812-2014-79
  26. H. G. Li, L. P. Li, J. M. Zheng, and M. M. Jin, Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with (α, λ)-NODSM mappings in ordered Banach spaces, Fixed Point Theory Appl. 2014 (2014), 122, 12 pp. https://doi.org/10.1186/1687-1812-2014-122
  27. H. G. Li, X. B. Pan, Z. Y. Deng, and C. Y. Wang, Solving GNOVI frameworks involving (γG, λ)-weak-GRD set-valued mappings in positive Hilbert spaces, Fixed Point Theory Appl. 2014 (2014), 146, 9 pp. https://doi.org/10.1186/1687-1812-2014-146
  28. H. G. Li, D. Qiu, and M. Jin, GNM ordered variational inequality system with ordered Lipschitz continuous mappings in an ordered Banach space, J. Inequal. Appl. 2013 (2013), 514, 11 pp. https://doi.org/10.1186/1029-242X-2013-514
  29. H. G. Li, D. Qiu, and Y. Zou, Characterizations of weak-ANODD set-valued mappings with applications to an approximate solution of GNMOQV inclusions involving ⊕ operator in ordered Banach spaces, Fixed Point Theory Appl. 2013 (2013), 241, 12 pp. https://doi.org/10.1186/1687-1812-2013-241
  30. H. G. Li, Y. Q. Yong, M. M. Jin, and Q. S. Zhang, Set-valued fixed point theorem based on the super-trajectory in complete super Hausdorff metric spaces, Nonlinear Funct. Anal. Appl. 21 (2016), no. 2, 195-213. https://doi.org/10.22771/NFAA.2016.21.02.02
  31. B. Martinet, Regularisation d'inequations variationnelles par approximations successives, Rev. Francaise Informat. Recherche Operationnelle 4 (1970), Ser. R-3, 154-158.
  32. A. Nagurney, Network economics: a variational inequality approach, Advances in Computational Economics, 1, Kluwer Academic Publishers Group, Dordrecht, 1993. https://doi.org/10.1007/978-94-011-2178-1
  33. M. A. Noor, K. I. Noor, and R. Kamal, General variational inclusions involving difference of operators, J. Inequal. Appl. 2014 (2014), 98, 16 pp. https://doi.org/10.1186/1029-242X-2014-98
  34. M. Patriksson, Nonlinear programming and variational inequality problems, Applied Optimization, 23, Kluwer Academic Publishers, Dordrecht, 1999. https://doi.org/10.1007/978-1-4757-2991-7
  35. A. H. Siddiqi and Q. H. Ansari, An algorithm for a class of quasivariational inequalities, J. Math. Anal. Appl. 145 (1990), no. 2, 413-418. https://doi.org/10.1016/0022-247X(90)90409-9
  36. G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.
  37. S. Wang, On fixed point and variational inclusion problems, Filomat 29 (2015), no. 6, 1409-1417. https://doi.org/10.2298/FIL1506409W
  38. D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim. 6 (1996), no. 3, 714-726. https://doi.org/10.1137/S1052623494250415