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On the Well-Posedness Concept in the Sense of Tykhonov

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  • Mircea Sofonea

    (University of Electronic Science and Technology of China
    University of Perpignan Via Domitia)

  • Yi-bin Xiao

    (University of Electronic Science and Technology of China)

Abstract

We introduce a general concept of well-posedness in the sense of Tykhonov for abstract problems formulated on metric spaces and characterize it in terms of properties for a family of approximating sets. Then, we illustrate these results in the study of some relevant particular problems with history-dependent operators: a fixed point problem, a nonlinear operator equation, a variational inequality and a hemivariational inequality, both formulated in the framework of real normed spaces. For each problem, we clearly indicate the approximating sets, characterize its well-posedness by using our abstract results, then we state and prove specific results which guarantee the well-posedness under appropriate assumptions on the data. For part of the problems, we provide the continuous dependence of the solution with respect to the data and/or present specific examples.

Suggested Citation

  • Mircea Sofonea & Yi-bin Xiao, 2019. "On the Well-Posedness Concept in the Sense of Tykhonov," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 139-157, October.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:1:d:10.1007_s10957-019-01549-0
    DOI: 10.1007/s10957-019-01549-0
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    References listed on IDEAS

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    1. Fang, Ya-Ping & Huang, Nan-Jing & Yao, Jen-Chih, 2010. "Well-posedness by perturbations of mixed variational inequalities in Banach spaces," European Journal of Operational Research, Elsevier, vol. 201(3), pages 682-692, March.
    2. Yi-bin Xiao & Xinmin Yang & Nan-jing Huang, 2015. "Some equivalence results for well-posedness of hemivariational inequalities," Journal of Global Optimization, Springer, vol. 61(4), pages 789-802, April.
    3. X. X. Huang, 2001. "Extended and strongly extended well-posedness of set-valued optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 53(1), pages 101-116, April.
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    Cited by:

    1. Mircea Sofonea & Yi-bin Xiao, 2021. "Well-Posedness of Minimization Problems in Contact Mechanics," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 650-672, March.

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