Some generalizations of Caristi’s fixed point theorem with applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem. (English) Zbl 1222.47081

Amini-Harandi, A.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 12, 4661-4665 (2010).

Author’s abstract: We first prove some generalizations of Caristi’s fixed point theorem. Then we give some applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem.

Reviewer: Ioan A. Rus (Cluj-Napoca)

MSC:

47H10	Fixed-point theorems
54H25	Fixed-point and coincidence theorems (topological aspects)

Keywords:

Caristi’s fixed point theorem; partially ordered set; minimal element; weakly contractive set-valued map; Takahashi minimisation theorem

References:

[1]	Caristi, J., Fixed point theory and inwardness condition, Appl. nonlinear anal., 483-497, (1979)
[2]	Takahashi, W., Existence theorems generalizing fixed point theorems for multivalued mappings, (), 397-406 · Zbl 0760.47029
[3]	Qiu, J.H., Ekeland’s variational principle in Fréchet spaces and the density of extremal points, Studia math., 168, 81-94, (2005) · Zbl 1061.49013
[4]	Qiu, J.H., A generalized Ekeland vector variational principle and its applications in optimization, Nonlinear anal., 71, 4705-4717, (2009) · Zbl 1170.58004
[5]	Khamsi, M.A., Remarks on caristi’s fixed point theorem, Nonlinear anal. TMA, 70, 4341-4349, (2009) · Zbl 1175.54056
[6]	Rhodes, B.E., Some theorems on weakly contractive maps, Nonlinear anal., 47, 2683-2693, (2001) · Zbl 1042.47521