General solutions of two functional inequalities and converses to contraction theorems

• Autorzy: Jachymski J.
• Języki publikacji: EN

Abstrakty

EN

We give necessary and sufficient conditions for the existence of some general solutions of the following two functional inequalities with an unknown real function phi: phi(Fx) is less than or equal to eta(phi(x)) and gamma(phi(Fx)) is less than or equal to phi(x). As an application we establish reciprocals to fixed point theorems of Matkowski and Wong. This extends an earlier result of Bessaga on a converse to the Banach contraction principle.

Słowa kluczowe

EN

fixed-point property; periodic point; nonlinear contractive condition; functional inequalities

PL

punkt stały; punkt okresowy; nierówność funkcyjna

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2003
• Tom: Vol. 51, no 2
• Strony: 147--156
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Jachymski J.

• Institute of Mathematics, Technical University of Łódź, Żwirki 36, 90-924 Łódź, Poland,

Bibliografia

• [1] C. Bessaga, On the converse of the Banach fixed-point principle, Colloq. Math., 7 (1959) 41-43.
• [2] J. Jachymski, A short proof of the converse to the contraction principle, and some related results, Topol. Methods Nonlinear Anal., 15 (2000) 179-186.
• [3] J. Matkowski, Integrable solutions of functional equations, Dissertationes Math. (Rozprawy Mat.), 127 (1975).
• [4] J. Dugundji, A. Granas, Fixed point theory I, PWN, Warszawa 1982.
• [5] C. S. Wong, Characterizations of certain maps of contractive type, Pacific J. Math., 68 (1977) 293-296.
• [6] J. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc., 125 (1997) 2327-2335.
• [7] W. I. Opojcev, The converse of the contraction mapping principle (in Russian), Uspekhi Mat. Nauk, 31 (1976) 169-198.
• [8] A. J. Levin, E. A. Lifshitz, On generalized contraction principle of M. A. Krasnoselskiĭ (in Russian), Problemy Matem. Anal. Sloz. Sist. (Voronezh), 1 (1967) 49-52.
• [9] L. Janos, A converse of Banach’s contraction theorem, Proc. Amer. Math. Soc., 16 (1967) 287-289.
• [10] P. R. Meyers, A converse to Banach’s contraction theorem, J. Res. Nat. Bur. Stand. Sect. B, 71 (1967) 73-76.
• [11] V. G. Angelov, A converse to a contraction mapping theorem in uniform spaces, Nonlinear Anal., 12 (1988) 989-996.