×

zbMATH — the first resource for mathematics

Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces. (English) Zbl 0795.34051
Let \(H\) be a Hilbert space and \(A\subset H\times H\) be an \(m\)-accretive operator. In the paper there is considered the existence of \(T\)-periodic solutions for equations \[ du/dt+ Au\ni g(t,u), \qquad t\in \mathbb{R}^ 1, \quad \text{where}\tag{1} \] \(g: \mathbb{R}^ 1\times H\to H\) is a Carathéodory function. The main result asserts: Let hold (a) For some \(\lambda>0\), \((I+\lambda^{-1} A)^{-1}\) is a compact mapping on \(H\); (b) \(g\) satisfies that for some \(M_ 1, M_ 2>0\) \(\| g(t,v)\|\leq M_ 1\| v\|+ M_ 2\) for all \(t\in\mathbb{R}^ 1\), \(v\in H\). Assume further that \(g\) is \(T\)-periodic with respect to the first variable and satisfies that there exist positive constants \(a\) and \(b\) such that \(\langle z- g(t,v), v\rangle\geq a\| v\|^ 2-b\) for all \(v\in D(A)\), \(z\in Av\). Then (1) has at least one \(T\)-periodic mild solution.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34C25 Periodic solutions to ordinary differential equations
35K55 Nonlinear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. · Zbl 0328.47035
[2] Ronald I. Becker, Periodic solutions of semilinear equations of evolution of compact type, J. Math. Anal. Appl. 82 (1981), no. 1, 33 – 48. · Zbl 0465.34014
[3] Haïm Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 1, 115 – 175 (French). · Zbl 0169.18602
[4] -, Operateurs maximaux monotones, North-Holland, Amsterdam, 1973.
[5] Felix E. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1100 – 1103. · Zbl 0135.17601
[6] Klaus Deimling, Periodic solutions of differential equations in Banach spaces, Manuscripta Math. 24 (1978), no. 1, 31 – 44. · Zbl 0373.34032
[7] Norimichi Hirano, Existence of multiple periodic solutions for a semilinear evolution equation, Proc. Amer. Math. Soc. 106 (1989), no. 1, 107 – 114. · Zbl 0729.35006
[8] Jan Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979), no. 5, 601 – 612. · Zbl 0419.34061
[9] Ioan I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), no. 3, 653 – 661. · Zbl 0701.34074
[10] Ioan I. Vrabie, The nonlinear version of Pazy’s local existence theorem, Israel J. Math. 32 (1979), no. 2-3, 221 – 235. · Zbl 0406.34064
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.