×

A fixed point theorem of Krasnoselskii-Schaefer type. (English) Zbl 0896.47042

The authors focus on three fixed point theorems and an integral equation. Schaefer’s fixed point theorem yields a \(T\)-periodic solution of \[ x(t)= a(t)+ \int^t_{t- h}D(t, s)g(s, x(s))ds\tag{1} \] if \(D\) and \(g\) satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder’s theorem (known as Krasnoselskii’s theorem) yields a \(T\)-periodic solution of \[ x(t)= f(t, x(t))+ \int^t_{t- h}D(t,s)g(s, x(s))ds\tag{2} \] if \(f\) defines a contraction and if \(D\) and \(g\) are small enough.
The authors prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer’s theorem which yields a \(T\)-periodic solution of (2) when \(f\) defines a contraction mapping, while \(D\) and \(g\) satisfy the aforementioned sign conditions.

MSC:

47H10 Fixed-point theorems
45M15 Periodic solutions of integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Burton, Tohoku Math. J. 46 pp 207– (1994)
[2] Burton, Differential Equations and Dynamical Systems 1 pp 161– (1993)
[3] Burton, Nachr. 147 pp 175– (1990) · Zbl 0725.45006
[4] Burton, Mat. pura appl. pp 271– (1992)
[5] Burton, Dynamic Systems and Applications 3 pp 583– (1994)
[6] : Integral Equations and Applications, Cambridge University Press, Cambridge, 1991
[7] , and : Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990 · Zbl 0695.45002
[8] Krasnosblskii, Amer. Math. Soc. Trans. (2) 10 pp 345– (1958) · Zbl 0080.10403
[9] : Introductory Functional Analysis with Applications, Wiley, New York, 1978 · Zbl 0368.46014
[10] Levin, Amer. Math. Soc. 14 pp 534– (1963)
[11] Levin, J. Math. Anal. Appl. 39 pp 458– (1972)
[12] Levin, J. Math. Anal. Appl. 8 pp 31– (1964)
[13] Schaefer, Ann. 129 pp 415– (1955)
[14] Schauder, Ann. 106 pp 661– (1932)
[15] : Fixed Point Theorems, Cambridge University Press, Cambridge, 1980 · Zbl 0521.54028
[16] : Leçons sur la Théorie Mathématique de la Lutte pour la Vie, Gauthier-Villars, Paris, 1931
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.