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Generalization of sequential lower semicontinuity. (English) Zbl 1209.49010

In the present paper the authors introduce a new concept, naming it \(\inf\)-sequential lower semicontinuity, that generalizes sequential lower semicontinuity from above. By giving an example they refute the conjecture made earlier by other authors [see Y. Chen, Y. J. Cho and L. Yang, Bull. Korean Math. Soc. 39, No. 4, 535–541 (2002; Zbl 1040.49016)] that for convex functions on normed spaces, lower semicontinuity from above is equivalent to weak lower semicontinuity from above.
The main results of the paper are: (1) for convex functions on normed spaces, the introduced concept is equivalent to its weak counterpart; (2) some results, such as some sufficient conditions for the existence of minima, Ekeland’s variational principle and Caristi’s fixed point theorem, remain still true under this generalization; (3) the hypethesis of lower semicontinuity can be replaced by this generalization also in some results regarding well-posedness of minimum problems in the sense of Tykhonov or in a generalized sense. The authors also give some examples and results in which they compare sequential lower semicontinuity from above with \(\inf\)-sequential lower semicontinuity, both in the general case and in the case of convex functions.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J35 Existence of solutions for minimax problems
49K40 Sensitivity, stability, well-posedness
49J52 Nonsmooth analysis

Citations:

Zbl 1040.49016
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