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New proximal algorithms for a class of (A,η)-accretive variational inclusion problems with non-accretive set-valued mappings

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Abstract

In this work, by using Xu’s inequality, Nalder’s results, the notion of(A,η)-accretive mappings and the new resolvent operator technique associated with(A, η)- accretive mappings due to Lan et al., we study the existence of solutions for a new class of(A, η)-accretive variational inclusion problems with non-accretive set-valued mappings and the convergence of the iterative sequences generated by the algorithms in Banach spaces. Our results are new and extend, improve and unify the corresponding results in this field.

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References

  1. R. P. Agarwal, N. J. Huang and Y. J. Cho,Generalized nonlinear mixed implicit quasi- variational inclusions with set-valued mappings, J. Inequal. Appl.7(6) (2002), 807–828.

    Article  MATH  MathSciNet  Google Scholar 

  2. X. P. Ding,Generalized quasi-variational-like inclusions with nonconvex functionals, Appl. Math. Comput.122(3) (2001), 267–282.

    Article  MATH  MathSciNet  Google Scholar 

  3. K. Ding, N. J. Huang and J. K. Kim,A new system of generalized nonlinear co- complementarity problems, J. Appl. Math. & Computing21(1-2) (2006), 379–391.

    MATH  MathSciNet  Google Scholar 

  4. Y. P. Fang, N. J. Huang,H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett.17(6) (2004), 647–653.

    Article  MATH  MathSciNet  Google Scholar 

  5. Y. P. Fang and N. J. Huang,H-Monotone operator and resolvent operator technique for variatonal inclusions, Appl. Math. Comput.145 (2003), 795–803.

    Article  MATH  MathSciNet  Google Scholar 

  6. Y. P. Fang, N. J. Huang, H. B. Thompson,A new system of variational inclusions with (H,η)-monotone operators in Hilbert spaces, Comput. Math. Appl.49(2-3) (2005), 365- 374.

    Article  MATH  MathSciNet  Google Scholar 

  7. Y. P. Fang, Y. J. Cho and J. K. Kim,(H, η)-accretive operators and approximating solu- tions for systems of variational inclusions in Banach spaces, Nonlinear Anal. to appear.

  8. N. J. Huang and Y. P. Fang,A new class of general variational inclusions involving maximal η-monotone mappings, Publ. Math. Debrecen62(1-2) (2003), 83–98.

    MATH  MathSciNet  Google Scholar 

  9. N. J. Huang and Y. P. Fang,Generalized m-accretive mappings in Banach spaces, J. Sichuan Univ.38(4) (2001), 591–592.

    MATH  Google Scholar 

  10. J. U. Jeong,A system of parametric generalized nonlinear mixed quasi-variational inclu- sions in L p spaces, J. Appl. Math. and Computing19(1-2) (2005), 493–506.

    Google Scholar 

  11. H. Y. Lan, Y. J. Cho and R. U. Verma,On nonlinear relaxed cocoercive variational inclusions involving (A, η)-accretive mappings in Banach spaces, Comput. Math. Appl. 51(9-10) (2006), 1529–1538.

    Article  MathSciNet  Google Scholar 

  12. H. Y. Lan, J. K. Kim and N. J. Huang,On the generalized nonlinear quasi-variational inclusions involving non-monotone set-valued mappings, Nonlinear Funct. Anal. Appl.9(3) (2004), 451–465.

    MATH  MathSciNet  Google Scholar 

  13. H. Y. Lan,A class of nonlinear (A, η)-monotone operator inclusion problems with relaxed cocoercive mappings, Advances in Nonlinear Variational Inequal.9(2) (2006), 1–11.

    MATH  MathSciNet  Google Scholar 

  14. L. W. Liu and Y. Q. Li,On generalized set-valued variational inclusions, J. Math. Anal. Appl.261 (2001), 231–240.

    Article  MATH  MathSciNet  Google Scholar 

  15. Z. Liu, J. S. Ume and S. M. Kang,H-monotone operator and resolvent operator technique for nonlinear variational inclusions, Math. Inequal. Appl.8(4) (2005), 705–716.

    MATH  MathSciNet  Google Scholar 

  16. A. Moudafi,Proximal methods for a class of relaxed nonlinear variational inclusions, Advances in Nonlinear Variational Inequal.9(2) (2006), 59–64.

    MATH  MathSciNet  Google Scholar 

  17. S. B. Nadler,Multi-valued contraction mappings, Pacific J. Math.30 (1969), 475–488.

    MATH  MathSciNet  Google Scholar 

  18. J. W. Peng,Set-valued variational inclusions with T-accretive operators in Banach spaces, Appl. Math. Lett.19(6) (2006), 273–282.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. V. Solodov and B. F. Svaiter,A hybrid projection-proximal point algorithm, J. Convex Anal.6(1) (1999), 59–70.

    MATH  MathSciNet  Google Scholar 

  20. R. U. Verma,A-monotononicity and applications to nonlinear variational inclusion prob- lems, J. Appl. Math. Stochastic Anal.17(2) (2004), 193–195.

    Article  Google Scholar 

  21. R. U. Verma,Approximation-solvability of a class of A-monotone variational inclusion problems, J. KSIAM8(1) (2004), 55–66.

    Google Scholar 

  22. R. U. Verma,General nonlinear variational inclusion problems involving A-monotone mappings, Appl. Math. Lett.19(9) (2006), 960–963.

    Article  MATH  MathSciNet  Google Scholar 

  23. R. U. Verma,Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A, η)-resolvent operator technique, Appl. Math. Lett.19(12) (2006), 1409- 1413.

    Article  MathSciNet  Google Scholar 

  24. R. U. Verma,A-monotonicity and its role in nonlinear variational inclusions, J. Optim. Theory Appl.129(3) (2006), in press.

  25. H. K. Xu,Inequalities in Banach spaces with applications, Nonlinear Anal.16(12) (1991), 1127–1138.

    Article  MATH  MathSciNet  Google Scholar 

  26. E. Zeidler,Nonlinear Functional Analysis and its Applications II: Monotone Operators, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  27. F. R. Zhang, H. Y. Gao, Z. Q. Liu and S. M. Kang,Convergence and stability of three-step iterative scheme with errors for completely generalized strongly nonlinear quasivariational inequalities, J. Appl. Math. & Computing20(1-2) (2006), 465–478.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University of Science & Engineering, Zigong, 643000, Sichuan, P. R. China

    Heng-You Lan

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  1. Heng-You Lan
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Correspondence to Heng-You Lan.

Additional information

This work was supported by the Educational Science Foundation of Sichuan Province and and the Natural Science Foundation of Sichuan University of Science & Engineering, Zigong, Sichuan of China No. 2006ZR003.

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Lan, HY. New proximal algorithms for a class of (A,η)-accretive variational inclusion problems with non-accretive set-valued mappings. J. Appl. Math. Comput. 25, 255–267 (2007). https://doi.org/10.1007/BF02832351

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  • DOI: https://doi.org/10.1007/BF02832351

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