Existence of solutions in a cone for nonlinear alternative problems.

*(English)*Zbl 0585.47050The existence of solutions in a given cone of a Banach space E for an abstract equation of the form \(Lu=Nu\) is studied, where L is a linear operator with nontrivial kernel and N is a continuous (nonlinear) operator. The author extends a well known result of L. Cesari and R. Kannan [ibid. 63, 221-225 (1977; Zbl 0361.47021)]. In applications, for instance, if L is a differential operator on a bounded domain D of \({\mathbb{R}}^ n\), one can take E as a subspace of \(L^ 2(D)\) and the cone of the positive functions.

##### MSC:

47J05 | Equations involving nonlinear operators (general) |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

35G30 | Boundary value problems for nonlinear higher-order PDEs |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

35J40 | Boundary value problems for higher-order elliptic equations |

##### Keywords:

alternative method; resonance; cone of a Banach space; linear operator with nontrivial kernel; continuous (nonlinear) operator
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##### References:

[1] | Lamberto Cesari, Functional analysis, nonlinear differential equations, and the alternative method, Nonlinear functional analysis and differential equations (Proc. Conf., Mich. State Univ., East Lansing, Mich., 1975) Marcel Dekker, New York, 1976, pp. 1 – 197. Lecture Notes in Pure and Appl. Math., Vol. 19. · Zbl 0343.47038 |

[2] | L. Cesari and R. Kannan, An abstract existence theorem at resonance, Proc. Amer. Math. Soc. 63 (1977), no. 2, 221 – 225. · Zbl 0361.47021 |

[3] | R. E. Gaines and Jairo Santanilla M., A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations, Rocky Mountain J. Math. 12 (1982), no. 4, 669 – 678. · Zbl 0508.34030 |

[4] | N. G. Lloyd, Degree theory, Cambridge University Press, Cambridge-New York-Melbourne, 1978. Cambridge Tracts in Mathematics, No. 73. · Zbl 0367.47001 |

[5] | J. J. Nieto, Positive solutions of operator equations, preprint 1984. |

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