zbMATH — the first resource for mathematics

On the connectedness of the set of fixed points of a compact operator in the Fréchet space \(C^ m(\langle b,\infty),\mathbb{R}^ n\). (English) Zbl 0793.47055
The authors prove an Aronshejn-type result on the connectedness of the fixed point set of a nonlinear operator in a Fréchet space satisfying some technical conditions. The abstract theorem is illustrated by means of an application to an initial value problem for the functional- differential equation \(y'(t)=\omega(t)F(y(t)+c)\) in the Fréchet space \(C^ m([b,\infty),\mathbb{R}^ n)\) equipped with the usual “exhausting” sequence of seminorms.

47H10 Fixed-point theorems
46A04 Locally convex Fréchet spaces and (DF)-spaces
Full Text: EuDML
[1] Akhmerov, R. R., Kamenski, M. I., Potapov, A. S.: Mery nekompaktnosti i uplotnyayushchie operatory. Nauka, Novosibirsk, 1986. · Zbl 0623.47070
[2] Aroszajn, N.: Le correspondant topologique de l’unicité dans la théorie des équation différentielles. Annals of Mathematics 43 (1942), 730-738. · Zbl 0061.17106
[3] Belohorec, Š.: Generalization of a certain theorem of N. Aroszajn and its application in the theory of functional differential equations. Manuscript.
[4] Browder, F. E., and Gupta, G. P.: Topological degree and non-linear mappings of analytic type in Banach spaces. Math. Anal. Appl. 26 (1969), 390-402. · Zbl 0176.45401
[5] Czarnowski, K., and Pruszko, T.: On the structure of fixed point sets of compact maps in \(B_0\) spaces with applications to integral and differential equations in unbounded domain. J. Math. Anal. Appl. 54 (1991), 151-163. · Zbl 0729.47054
[6] Deimling, K.: Nichtlineare Gleichungen und Abbildungsgrade. Springer-Verlag, Berlin Heidelberg New York, 1974. · Zbl 0281.47033
[7] Hukuhara, M.: Sur une généralization d’un thèoreme de Kneser. Proc. Japan Acad. 29 (154-155). · Zbl 0051.29704
[8] Krasnoselski, M. A., Perov, A. I.: O sushchestvovanii resheni u nekotorykh nelinenykh operatornykh uravneni. Doklady AN SSSR 126 (1959), 15-18.
[9] Krasnoselski, M. A., Zabreko, P. P.: Geometricheskie metody nelinenogo analiza. Nauka, Moskva, 1975.
[10] Kubáček, Z.: A generalization of N. Aronszajn’s theorem on connectedness of the fixed point set of a compact mapping. Czechoslovak Math. J. 37 (112) (1987), 415-423. · Zbl 0638.54037
[11] Kuratowski, C.: Topologie, Volume II. Pol. Tow. Mat., Warszawa, 1952. · Zbl 0049.39704
[12] Morales, P.: Topological properties of the set of global solutions for a class of semilinear evolution equations in a Banach space. Atti del Convegno celebrativo del \(1^0\) centenario del Circolo Matematico di Palermo. Suppl. di Rendiconti del Circolo Matematico di Palermo, Serio II 8 (1985), 379-397. · Zbl 0627.47028
[13] Sadovski, V. N.: Predelno kompaktnye i uplotnyayushchie operatory. Uspekhi mat. nauk 27:1 (1972), 81-146.
[14] Stampacchia, G.: Le transformazioni che presentano il fenomeno di Peano. Rend. Accad. Naz Lincei 7 (1949), 80-84. · Zbl 0041.23302
[15] Szufla, S.: On the structure of solution sets of differential and integral equations in Banach spaces. Ann. Polon. Math XXXIV (1977), 165-177. · Zbl 0384.34038
[16] Szufla, S.: On Volterra integral equations in Banach spaces. Funkcialaj Ekvacioj 20 (1977), 247-258. · Zbl 0379.45025
[17] Szufla, S.: Sets of fixed points of nonlinear mappings in function spaces. Funkcialaj Ekvacioj 22 (1979), 121-126. · Zbl 0419.47025
[18] Szufla, S.: On the existence of \(L^p\)-solutions of Volterra integral equations in Banach spaces. Funkcialaj Ekvacioj 27 (1984), 157-172. · Zbl 0553.45006
[19] Vidossich, G.: On the structure of the set of solutions of nonlinear equations. J. Math. Anal. Appl. 34 (1971), 602-617. · Zbl 0211.17401
[20] Zeidler, E.: Vorlesungen über nichtlineare Funktionalanalysis I - Fixpunktsätze. Teubner Verlagsgesellschaft, Leipzig, 1976. · Zbl 0326.47053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.