Fixed point results for generalized contractions in gauge spaces and applications

Frigon, M.

doi:10.1090/S0002-9939-00-05838-X

Fixed point results for generalized contractions in gauge spaces and applications
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Proc. Amer. Math. Soc. 128 (2000), 2957-2965 Request permission

Abstract:

In this paper, we present fixed point results for generalized contractions defined on a complete gauge space $\mathbb {E}$. Also, we consider families of generalized contractions $\{f^t : X \to \mathbb {E}\}_{t \in [0,1]}$ where $X \subset \mathbb {E}$ is closed and can have empty interior. We give conditions under which the existence of a fixed point for some $f^{t_0}$ imply the existence of a fixed point for every $f^t$. Finally, we apply those results to infinite systems of first order nonlinear differential equations and to integral equations on the real line.

References

G. L. Cain Jr. and M. Z. Nashed, Fixed points and stability for a sum of two operators in locally convex spaces, Pacific J. Math. 39 (1971), 581–592. MR 322606
G. Dore, Sul problema di Cauchy per equazioni differenziali ordinarie in spazi localmente convessi, Rend. Mat. Appl. (7) 1 (1981), 237–247.
J. Dugundji. Topology, Wm. C. Brown Publ., Dubuque, 1989.
Marlène Frigon and Andrzej Granas, Résultats du type de Leray-Schauder pour des contractions multivoques, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 197–208 (French). MR 1321812, DOI 10.12775/TMNA.1994.026
M. Frigon and A. Granas, Résultats de type Leray-Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (1998), 161–168.
G. Herzog, On ordinary linear differential equations in $\mathbb {C}^J$, Demonstratio Math. 28 (1995), 383–398.
Gerd Herzog, On Lipschitz conditions for ordinary differential equations in Fréchet spaces, Czechoslovak Math. J. 48(123) (1998), no. 1, 95–103. MR 1614009, DOI 10.1023/A:1022587812606
Ronald J. Knill, Fixed points of uniform contractions, J. Math. Anal. Appl. 12 (1965), 449–455. MR 192484, DOI 10.1016/0022-247X(65)90012-0
Roland Lemmert and Ägidius Weckbach, Charakterisierungen zeilenendlicher Matrizen mit abzählbarem Spektrum, Math. Z. 188 (1984), no. 1, 119–124 (German). MR 767369, DOI 10.1007/BF01163880
Roland Lemmert, On ordinary differential equations in locally convex spaces, Nonlinear Anal. 10 (1986), no. 12, 1385–1390. MR 869547, DOI 10.1016/0362-546X(86)90109-4
Krzysztof Moszyński and Andrzej Pokrzywa, Sur les systèmes infinis d’équations différentielles ordinaires dans certains espaces de Fréchet, Dissertationes Math. (Rozprawy Mat.) 115 (1974), 37 (French). MR 390415
Sam B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488. MR 254828
E. Tarafdar, An approach to fixed-point theorems on uniform spaces, Trans. Amer. Math. Soc. 191 (1974), 209–225. MR 362283, DOI 10.1090/S0002-9947-1974-0362283-5
W. W. Taylor, Fixed-point theorems for nonexpansive mappings in linear topological spaces, J. Math. Anal. Appl. 40 (1972), 164–173. MR 322612, DOI 10.1016/0022-247X(72)90040-6
B. N. Sadovskiĭ, Measures of noncompactness and condensing operators, Problemy Mat. Anal. Slož. Sistem 2 (1968), 89–119 (Russian). MR 0301582
B. N. Sadovskiĭ, Limit-compact and condensing operators, Uspehi Mat. Nauk 27 (1972), no. 1(163), 81–146 (Russian). MR 0428132
T. Sengadir, D. V. Pai, and A. K. Pani, A Leray-Schauder type theorem and applications to boundary value problems for neutral equations, Nonlinear Anal. 28 (1997), no. 4, 701–719. MR 1420386, DOI 10.1016/0362-546X(95)00181-T