##
**Locally coercive nonlinear equations, with applications to some periodic solutions.**
*(English)*
Zbl 0571.47051

The equation
\[
(NL)\quad Au=f,
\]
where \(u\in Y\) is the unknown and \(f\in Y^*\) is a given element is considered. Namely, the setting of the problem is the following:

i) \(\{Y,Y^*\}\) is a pair of real Banach spaces in duality. This means that there is a nondegenerate continuous bilinear form \(<\;,\;>\) on \(Y\times Y^*\). Moreover Y is reflexive and separable.

ii) There is another pair \(\{V,V^*\}\) of Banach spaces in duality, with V separable such that \(V\subset Y\) and \(V^*\supset Y^*\), with the injections continuous and dense. Moreover, the duality \(<\;,\;>\) on \(V\times V^*\) is compatible with that of \(\{Y,Y^*\}.\)

iii) There is a bounded, closed and convex subset K of Y, containing the origin O as an internal point, and weakly sequentially continuous map A of K into \(V^*\), such that \(<v,Av>\geq \beta \geq 0\) for all \(v\in V\cap \hat K\), where \(\hat K\) denotes the set of bounding points of K.

Then the main result is the following

Theorem. Under the assumptions i), ii), iii), we have \(AK\supset \beta K^ 0\), where \(K^ 0\subset Y^*\) is the polar set of K. In other words, (NL) has a solution \(u\in K\) for every \(f\in \beta K^ 0.\)

Another theorem proves under some added assumptions uniqueness of the solution u. In the rest of the paper these theorems are used to prove existence, uniqueness and continuous dependence for periodic solutions to certain nonlinear PDE’s.

i) \(\{Y,Y^*\}\) is a pair of real Banach spaces in duality. This means that there is a nondegenerate continuous bilinear form \(<\;,\;>\) on \(Y\times Y^*\). Moreover Y is reflexive and separable.

ii) There is another pair \(\{V,V^*\}\) of Banach spaces in duality, with V separable such that \(V\subset Y\) and \(V^*\supset Y^*\), with the injections continuous and dense. Moreover, the duality \(<\;,\;>\) on \(V\times V^*\) is compatible with that of \(\{Y,Y^*\}.\)

iii) There is a bounded, closed and convex subset K of Y, containing the origin O as an internal point, and weakly sequentially continuous map A of K into \(V^*\), such that \(<v,Av>\geq \beta \geq 0\) for all \(v\in V\cap \hat K\), where \(\hat K\) denotes the set of bounding points of K.

Then the main result is the following

Theorem. Under the assumptions i), ii), iii), we have \(AK\supset \beta K^ 0\), where \(K^ 0\subset Y^*\) is the polar set of K. In other words, (NL) has a solution \(u\in K\) for every \(f\in \beta K^ 0.\)

Another theorem proves under some added assumptions uniqueness of the solution u. In the rest of the paper these theorems are used to prove existence, uniqueness and continuous dependence for periodic solutions to certain nonlinear PDE’s.

Reviewer: J.Siška

### MSC:

47J05 | Equations involving nonlinear operators (general) |

47H05 | Monotone operators and generalizations |

35B10 | Periodic solutions to PDEs |

### Keywords:

space in duality; higher order equations; polar; periodic solutions to certain nonlinear PDE’s
Full Text:
DOI

### References:

[1] | F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces , Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), Amer. Math. Soc., Providence, R. I., 1976, pp. 1-308. · Zbl 0327.47022 |

[2] | P. Hess, On nonlinear mappings of monotone type with respect to two Banach spaces , J. Math. Pures Appl. (9) 52 (1973), 13-26. · Zbl 0222.47019 |

[3] | T. Kato, Nonlinear equations of evolution in Banach spaces , Proc. Amer. Math. Soc. Summer Institute, to appear, 1983. · Zbl 0606.35049 |

[4] | T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow , J. Funct. Anal. 56 (1984), no. 1, 15-28. · Zbl 0545.76007 |

[5] | J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I , Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 265-315. · Zbl 0144.18202 |

[6] | P. H. Rabinowitz, A curious singular perturbation problem , Differential Equations (Birmingham, Ala., 1983) eds. I. W. Knowles and R. T. Lewis, North-Holland Math. Stud., vol. 92, North-Holland, Amsterdam, 1984, pp. 455-464. · Zbl 0585.34033 |

[7] | P. H. Rabinowitz, A rapid convergence method for a singular perturbation problem , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 1, 1-17. · Zbl 0547.35047 |

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