On the Well-Posedness Concept in the Sense of Tykhonov

Mircea Sofonea () and Yi-bin Xiao ()
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Mircea Sofonea: University of Electronic Science and Technology of China
Yi-bin Xiao: University of Electronic Science and Technology of China

Journal of Optimization Theory and Applications, 2019, vol. 183, issue 1, No 8, 139-157

Abstract: 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.

Keywords: Tykhonov well-posedness; History-dependent operator; Nonlinear equation; Variational inequality; Hemivariational inequality; 49J40; 47J20; 49J45; 35M86 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-019-01549-0

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