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Caristi’s fixed point theorem and selections of set-valued contractions. (English) Zbl 0916.47044
Let $(X,d)$ be a metric space and $T:X\to X$ a map which need not be continuous but satisfies $d(x,Tx)\le\varphi(x)-\varphi(Tx)$ for some lower semicontinuous function $\varphi:[0,\infty)\to[0,\infty)$. Caristi proved this result using transfinite induction. {\it W. A. Kirk} [Colloq. Math. 36, 81-86 (1976; Zbl 0353.53041)] defined a partial ordering on $X$ by $x\le_\varphi y$ iff $d(x,y)\le\varphi(x)-\varphi(y)$ in order to prove this theorem. His proof uses Zorn’s lemma. {\it F. E. Browder} [in: Fixed point theorem, Appl. Proc. Sem. Halifax 1975, 23-27 (1976; Zbl 0379.54016)] gave a constructive proof using the axiom of choice only for countable families. {\it R. Mańka} [Rep. Math. Logic 22, 15-19 (1988; Zbl 0687.04003)] then gave a constructive proof based on Zermelo’s theorem. The present author gives a simple derivation of Caristi’s theorem from Zermelo’s theorem in case $T$ is continuous. On the other hand, the author describes examples of set-valued contractions which admit (not necessarily continuous) selections which satisfy the assumptions of Caristi’s theorem. Finally, the author answers a question posed by W. A. Kirk by proving the following result: Let $\eta:[0,\infty)\to[0,\infty)$ be a function satisfying $\eta(0)=0$. Then the right hand lower Dini derivative of $\eta$ at $0$ (i.e., $\liminf_{s\to t^+}[\eta(s)-\eta(t)]/[s-t]$) vanishes if and only if there is a complete metric space $(X,d)$, a continuous and asymptotically regular mapping $T:X\to X$ which has no fixed points and a continuous function $\varphi:[0,\infty)\to[0,\infty)$ such that $\eta(d(x,Tx))\le\varphi(x)-\varphi(Tx)$ for all $x\in X$.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
47H04Set-valued operators
03E25Axiom of choice and related propositions (logic)
Full Text: DOI
[1] Abian, A.: A fixed point theorem equivalent to the axiom of choice. Arch. math. Logik 25, 173-174 (1985) · Zbl 0615.04008
[2] Bell, H.: Review of ”Proceedings of the seminar on fixed point theory and its applications”. Math. rev. 57 (1976)
[3] Boyd, D. W.; Wong, J. S. W.: On nonlinear contractions. Proc. amer. Math. soc. 20, 458-464 (1969) · Zbl 0175.44903
[4] Brøndsted, A.: On a lemma of Bishop and phelps. Pacific J. Math. 55, 335-341 (1974) · Zbl 0248.46009
[5] F. E. Browder, On a theorem of Caristi and Kirk, in, Proceedings of the Seminar on Fixed Point Theory and Its Applications, June 1975, 23, 27, Academic Press, New York, 1976
[6] Brunner, N.: Topologische maximalprinzipien. Z. math. Logik grundlagen math. 33, 135-139 (1987) · Zbl 0639.49006
[7] Büber, T.; Kirk, W. A.: A constructive proof of a fixed point theorem of soardi. Math. japon. 41, 233-237 (1995) · Zbl 0824.54025
[8] Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. amer. Math. soc. 215, 241-251 (1976) · Zbl 0305.47029
[9] Covitz, H.; Jr., S. B. Nadler: Multi-valued contraction mappings in generalized metric spaces. Israel J. Math. 8, 5-11 (1970) · Zbl 0192.59802
[10] Daffer, P. Z.; Kaneko, H.; Li, W.: On a conjecture of S. Reich. Proc. amer. Math. soc. 124, 3159-3162 (1996) · Zbl 0866.47040
[11] Daneš, J.: A geometric theorem useful in nonlinear functional analysis. Boll. un. Mat. ital. 6, 369-375 (1972) · Zbl 0236.47053
[12] Daneš, J.: Equivalence of some geometric and related results of nonlinear functional analysis. Comment. math. Univ. carolin. 26, 443-454 (1985) · Zbl 0656.47050
[13] Downing, D.; Kirk, W. A.: A generalization of caristi’s theorem with applications to nonlinear mapping theory. Pacific J. Math. 69, 339-345 (1977) · Zbl 0357.47036
[14] Downing, D.; Kirk, W. A.: Fixed point theorems for set-valued mappings in metric and Banach spaces. Math. japon. 22, 99-112 (1977) · Zbl 0372.47030
[15] Dugundji, J.; Granas, A.: Fixed point theory I. (1982) · Zbl 0483.47038
[16] Ekeland, I.: On the variational principle. J. math. Anal. appl. 47, 324-353 (1974) · Zbl 0286.49015
[17] Fuchssteiner, B.: Iterations and fixpoints. Pacific J. Math. 68, 73-80 (1977) · Zbl 0339.26007
[18] Gajek, L.; Zagrodny, D.: Geometric variational principle. Different aspects of differentiability, 55-71 (1995) · Zbl 0858.49007
[19] Guillerme, J.: Coincidence theorems in complete spaces. Rev. mat. Apl. 15, 43-61 (1994) · Zbl 0840.54032
[20] Hille, E.; Phillips, R. S.: Functional analysis and semigroups. American mathematical society colloquium publications 31 (1957) · Zbl 0078.10004
[21] Jachymski, J. R.: On reich’s question concerning fixed points of multimaps. Boll. un. Math. ital A (7) 9, 453-460 (1995) · Zbl 0863.54042
[22] Jachymski, J. R.: Fixed point theorems in metric and uniform spaces via the knaster--Tarski principle. Nonlinear anal. 32, 225-233 (1998)
[23] J. R. Jachymski, Some consequences of the Tarski--Kantorovitch ordering theorem in metric fixed point theory, Quaestiones Math. · Zbl 0924.47040
[24] Kada, O.; Suzuki, T.; Takahashi, W.: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. japon. 44, 381-391 (1996) · Zbl 0897.54029
[25] Khamsi, M. A.; Misane, D.: Compactness of convexity structures in metric spaces. Math. japon. 41, 321-326 (1995) · Zbl 0824.54026
[26] Kirk, W. A.: Caristi’s fixed point theorem and metric convexity. Colloq. math. 36, 81-86 (1976) · Zbl 0353.53041
[27] Kirk, W. A.; Caristi, J.: Mapping theorems in metric and Banach spaces. Bull. acad. Polon. sci. 23, 891-894 (1975) · Zbl 0313.47041
[28] Kneser, H.: Eine direkte ableitung des zornschen lemmas aus dem auswahlaxiom. Math. Z. 53, 110-113 (1950) · Zbl 0037.31902
[29] Mańka, R.: Some forms of the axiom of choice. Jbuch. kurt-Gödel-ges. 1, 24-34 (1988)
[30] Mańka, R.: Turinici’s fixed point theorem and the axiom of choice. Rep. math. Logic 22, 15-19 (1988) · Zbl 0687.04003
[31] Mizoguchi, N.; Takahashi, W.: Fixed point theorems for multivalued mappings on complete metric spaces. J. math. Anal. appl. 141, 177-188 (1989) · Zbl 0688.54028
[32] Jr., S. B. Nadler: Multi-valued contraction mappings. Pacific J. Math. 30, 475-488 (1969) · Zbl 0187.45002
[33] Pasicki, L.: A short proof of the caristi theorem. Comment. math. Prace mat. 20, 427-428 (1978) · Zbl 0396.54038
[34] Penot, J. -P.: A short constructive proof of caristi’s fixed point theorem. Publ. math. Pau 10, 1-3 (1976)
[35] Penot, J. -P.: The drop theorem, the petal theorem and Ekeland’s variational principle. Nonlinear anal. 10, 813-822 (1986) · Zbl 0612.49011
[36] Reich, S.: Fixed points of contractive functions. Boll. un. Mat. ital. 5, 26-42 (1972) · Zbl 0249.54026
[37] Smithson, R. E.: Fixed points of order preserving multifunctions. Proc. amer. Math. soc. 28, 304-310 (1971) · Zbl 0238.06003
[38] Takahashi, W.: Existence theorems generalizing fixed point theorems for multivalued mappings. Pitman research notes in mathematics 252, 397-406 (1991) · Zbl 0760.47029
[39] Turinici, M.: Maximal elements in ordered topological spaces. Bull. Greek math. Soc. 20, 141-148 (1979) · Zbl 0452.54024
[40] Wong, C. S.: On a fixed point theorem of contractive type. Proc. amer. Math. soc. 57, 283-284 (1976) · Zbl 0329.54042
[41] Yu-Qing, C.: On a fixed point problem of reich. Proc. amer. Math. soc. 124, 3085-3088 (1996) · Zbl 0874.47027
[42] Zermelo, E.: Neuer beweis für die möglichkeit einer wohlordnung. Math. ann. 65, 107-128 (1908) · Zbl 38.0096.02