×

zbMATH — the first resource for mathematics

On periodic solutions of nonlinear evolution equations in Banach spaces. (English) Zbl 1029.34045
Here, the existence of \(T\)-periodic solutions to the nonlinear evolution equation \((*)\quad x'(t)+A(t,x(t))=f(t,x(t)), \quad t\in (0,T)=:I\), is proved under the following conditions: Let \(V\hookrightarrow{H}\hookrightarrow{V^*}\) be an evolution triple, where \(H\) is a real separable Hilbert space, \(V\) is a dense subspace of \(H\) and \(V^*\) is the topological dual space of \(V.\) Then \(A:I\times V\to V^*\) is such that for each \(t\in I\) the operator \(A(t,\cdot)\) is uniformly monotone and coercive. The nonlinearity \(f(t,x)\) is a Carathéodory function which is Hölder continuous with respect to \(x\in H\) and with exponent \(\alpha\in(0,1]\) uniformly in \(t.\) After transforming \((*)\) into an operator equation, the authors use the Leray-Schauder fixed-point theorem and prove the existence of periodic solutions. An application is given to a quasi-linear parabolic partial differential equation of even order.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34C25 Periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Becker, R.R., Periodic solutions of semilinear equations of evolution of compact type, J. math. anal. appl., 82, 33-48, (1981) · Zbl 0465.34014
[2] Browder, F., Existence of periodic solutions for nonlinear equations of evolution, Proc. nat. acad. sci. USA, 53, 1110-1113, (1965)
[3] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1980), Springer-Verlag New York · Zbl 0691.35001
[4] Hirano, N., Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. amer. math. soc., 120, 185-192, (1994) · Zbl 0795.34051
[5] Hu, S.; Papageorgiou, N.S., On the existence of periodic solutions for a class of nonlinear evolution inclusions, Boll. un. mat. ital., 7B, 591-605, (1993) · Zbl 0795.34009
[6] Kandilakis, D.; Papageorgiou, N.S., Periodic solutions for nonlinear evolution inclusions, Arch. math. (Brno), 32, 195-209, (1996) · Zbl 0908.34043
[7] Prüss, J., Periodic solutions of semilinear evolution equations, Nonlinear anal., 3, 221-235, (1979)
[8] Shioji, N., Existence of periodic solutions for nonlinear evolution equations with pseudo-monotone operators, Proc. amer. math. soc., 125, 2921-2929, (1997) · Zbl 0883.47051
[9] Vrabie, I., Periodic solutions for nonlinear evolution equations in Banach space, Proc. amer. math. soc., 109, 653-661, (1990) · Zbl 0701.34074
[10] Zeidler, E., Nonlinear functional analysis and its applications, II, (1990), Springer-Verlag New York
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.