Continuation theory for general contractions in gauge spaces


A continuation principle of Leray-Schauder type is presented for contractions with respect to a gauge structure depending on the homotopy parameter. The result involves the most general notion of a contractive map on a gauge space and in particular yields homotopy invariance results for several types of generalized contractions.


Adela Chiș
Department of Mathematics, Technical University of Cluj, Cluj-Napoca, Romania

Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania



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A. Chiș, R. Precup, Continuation theory for general contractions in gauge spaces, Fixed Point Theory and Applications 2004:3 (2004), 173-185,


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MR 2096949

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[1] R. P. Agarwal and D. O’Regan, Fixed point theory for generalized contractions on spaces with two metrics, J. Math. Anal. Appl. 248 (2000), no. 2, 402–414.
[2] J. Andres and L. Gorniewicz, ´ On the Banach contraction principle for multivalued mappings, Approximation, Optimization and Mathematical Economics (Pointe-a-Pitre, 1999), Physica, ` Heidelberg, 2001, pp. 1–23.
[3] A. Chis, Fixed point theorems for generalized contractions, Fixed Point Theory 4 (2003), no. 1, 33–48.
[4] L. B. Ciric,  A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267–273.
[5] I. Colojoara, On a fixed point theorem in complete uniform spaces, Com. Acad. R. P. R. 11 (1961), 281–283.
[6] J. Dugundji, Topology, Allyn and Bacon, Massachusetts, 1966.
[7] M. Frigon, Fixed point results for generalized contractions in gauge spaces and applications, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2957–2965.
[8] M. Frigon and A. Granas, Resultats du type de Leray-Schauder pour des contractions multivoques [Leray-Schauder-type results for multivalued contractions], Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 197–208 (French).
[9] , Resultats de type Leray-Schauder pour des contractions sur des espaces de Fr´echet [LeraySchauder-type results for contractions on Frechet spaces], Ann. Sci. Math. Quebec ´ 22 (1998), no. 2, 161–168 (French).
[10] J. A. Gatica and W. A. Kirk, Fixed point theorems for contraction mappings with applications to nonexpansive and pseudo-contractive mappings, Rocky Mountain J. Math. 4 (1974), 69–79.
[11] N. Gheorghiu, Contraction theorem in uniform spaces, Stud. Cerc. Mat. 19 (1967), 119–122 (Romanian).
[12] , Fixed point theorems in uniform spaces, An. Stiint. Univ. “Al. I. Cuza” Iasi Sect¸. I a Mat. (N.S.) 28 (1982), no. 1, 17–18.
[13] N. Gheorghiu and M. Turinici, Equations integrales dans les espaces localement convexes, Rev. Roumaine Math. Pures Appl. 23 (1978), no. 1, 33–40 (French).
[14] A. Granas, Continuation method for contractive maps, Topol. Methods Nonlinear Anal. 3 (1994), no. 2, 375–379.
[15] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76.
[16] R. J. Knill, Fixed points of uniform contractions, J. Math. Anal. Appl. 12 (1965), 449–455.
[17] M. G. Maia, Un’osservazione sulle contrazioni metriche, Rend. Sem. Mat. Univ. Padova 40 (1968), 139–143 (Italian).
[18] G. Marinescu, Spatii Vectoriale Topologice si Pseudotopologice [Topological and PseudoTopological Vector Spaces], Biblioteca Matematica, vol. IV, Editura Academiei Republicii Populare Romine, Bucharest, 1959.
[19] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Series in Mathematical Analysis and Applications, vol. 3, Gordon and Breach Science Publishers, Amsterdam, 2001.
[20] D. O’Regan and R. Precup, Continuation theory for contractions on spaces with two vector-valued metrics, Appl. Anal. 82 (2003), no. 2, 131–144.
[21] R. Precup, Le theoreme des contractions dans des espaces syntopogenes, Anal. Numer. Theor. Approx. 9 (1980), no. 1, 113–123 (French).
[22] R. Precup, A fixed point theorem of Maia type in syntopogenous spaces, Seminar on Fixed Point Theory, Preprint, vol. 88, Univ. “Babes¸-Bolyai”, Cluj-Napoca, 1988, pp. 49–70.
[23] R. Precup, Discrete continuation method for boundary value problems on bounded sets in Banach spaces, J. Comput. Appl. Math. 113 (2000), no. 1-2, 267–281.
[24] R. Precup, The continuation principle for generalized contractions, Bull. Appl. Comput. Math. (Budapest) 1927 (2001), 367–373.
[25] R. Precup, Continuation results for mappings of contractive type, Semin. Fixed Point Theory ClujNapoca 2 (2001), 23–40.
[26] R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht, 2002.
[28] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 257–290.
[29] I. A. Rus, Some fixed point theorems in metric spaces, Rend. Ist. Mat. Univ. Trieste 3 (1971), 169–172.
[30] E. Tarafdar, An approach to fixed-point theorems on uniform spaces, Trans. Amer. Math. Soc. 191 (1974), 209–225.
[27] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971), 121–124.


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