Skip to main content
Log in

On the characterization of strong convergence of an iterative algorithm for a class of multi-valued variational inclusions

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

This paper introduces an Ishikawa type iterative algorithm for finding approximating solutions of a class of multi-valued variational inclusion problems. Characterization of strong convergence of this iterative method is established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Bruck RE, Reich S (1981) Accretive operators, Banach limits and dual ergodic theorems. Bull Acad Polon Sci 12: 585–589

    MathSciNet  Google Scholar 

  • Chang SS (2000) Set-valued variational inclusions in Banach spaces. J Math Anal Appl 248: 438–454

    Article  MATH  MathSciNet  Google Scholar 

  • Chang SS, Kim JK, Kim KH (2002) On the existence and iterative approximation problems of solutions for set-valued variational inclusions in Banach spaces. J Math Anal Appl 268: 89–108

    Article  MATH  MathSciNet  Google Scholar 

  • Chang SS, Cho YJ, Lee BS, Jung IJ (2000) Generalized set-valued variational inclusions in Banach spaces. J Math Anal Appl 246: 409–422

    Article  MATH  MathSciNet  Google Scholar 

  • Chang SS, Cho YJ, Lee BS, Jung IJ, Kang SM (1998) Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces. J Math Anal Appl 224: 149–165

    Article  MATH  MathSciNet  Google Scholar 

  • Chidume CE, Zegeye H, Kazmi KR (2004) Existence and convergence theorems for a class of multi-valued variational inclusions in Banach spaces. Nonlinear Anal 59: 649–656

    MATH  MathSciNet  Google Scholar 

  • Cho YJ, Zhou HY, Kang SM, Kim SS (2001) Approximations for fixed points of \({\phi}\)-hemicontractive mappings by the Ishikawa iterative process with mixed errors. Math Comput Model 34: 9–18

    Article  MATH  MathSciNet  Google Scholar 

  • Demyanov VF, Stavroulakis GE, Polyakova LN, Panagiotoupoulos PD (1996) Quasidifferentiability and nonsmooth modelling in mechanics, engineering and economics. Kluwer, Dordrecht

    Google Scholar 

  • Ding XP (1997) Perturbed proximal point algorithm for generalized quasivariational inclusions. J Math Anal Appl 210: 88–101

    Article  MATH  MathSciNet  Google Scholar 

  • Giannessi F, Maugeri A (1995) Variational inequalities and network equilibrium problems. Plenum Press, New York

    MATH  Google Scholar 

  • Gioranescu L (1990) Geometry of banach spaces, duality mapping and nonlinear problems. Kluwer, Amsterdam

    Google Scholar 

  • Glowinski R (1984) Numerical methods for nonlinear variational problems. Springer, Berlin

    MATH  Google Scholar 

  • Glowinski R, Lions JL, Tremolieres R (1981) Numerical analysis of variational inequalities. North-Holland, Amsterdam

    MATH  Google Scholar 

  • Hassouni A, Moudafi A (1994) A perturbed algorithm for variational inequalities. J Math Anal Appl 185: 706–712

    Article  MATH  MathSciNet  Google Scholar 

  • He X (2003) On \({\phi}\)-strongly accretive mapping and some set-valued variational problems. J Math Anal Appl 277(2): 504–511

    Article  MATH  MathSciNet  Google Scholar 

  • Huang NJ (1997) On the generalized implicit quasivariational inequalities. J Math Anal Appl 216: 197–210

    Article  MATH  MathSciNet  Google Scholar 

  • Jung JS, Morales CH (2001) The Mann process for perturbed m-accretive operators in Banach spaces. Nonlinear Anal 46(2): 231–243

    Article  MATH  MathSciNet  Google Scholar 

  • Kazmi KR (1997) Mann and Ishikawa types perturbed iterative algorithms for generalized quasivariational inclusions. J Math Anal Appl 209: 584

    Article  MathSciNet  Google Scholar 

  • Kikuchi N, Oden JT (1988) Contact problems in elasticity. SIAM, Philadelphia

    MATH  Google Scholar 

  • Nadler SB (1969) Multi-valued contraction mappings. Pacific J Math 30: 475–488

    MATH  MathSciNet  Google Scholar 

  • Panagiotoupoulos PD, Stavroulakis GE (1992) New types of variational principles based on the notion of quasidifferentiability. Acta Mech 94: 171–194

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. C. Yao.

Additional information

L. C. Ceng’s research partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai.

S. Schaible’s research partially supported by the National Science Council of Taiwan.

This research was partially supported by the grant NSC 96-2628-E-110-014-MY3.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ceng, L.C., Schaible, S. & Yao, J.C. On the characterization of strong convergence of an iterative algorithm for a class of multi-valued variational inclusions. Math Meth Oper Res 70, 1–12 (2009). https://doi.org/10.1007/s00186-008-0227-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-008-0227-8

Keywords

Mathematics Subject Classification (2000)

Navigation