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On the Well-Posedness Concept in the Sense of Tykhonov

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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.

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Acknowledgements

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH. This research was also supported by the National Natural Science Foundation of China (11771067) and the Applied Basic Project of Sichuan Province (2019YJ0204).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Mircea Sofonea & Yi-bin Xiao

  2. Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan, France

    Mircea Sofonea

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  1. Mircea Sofonea
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  2. Yi-bin Xiao
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Correspondence to Yi-bin Xiao.

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Sofonea, M., Xiao, Yb. On the Well-Posedness Concept in the Sense of Tykhonov. J Optim Theory Appl 183, 139–157 (2019). https://doi.org/10.1007/s10957-019-01549-0

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