##
**Bounded solutions in a given set of differential systems.**
*(English)*
Zbl 0944.34012

The authors deal with systems of ordinary differential equations of the form
\[
\dot y= Ay+ g(t,y,z),\quad \dot z= h(t,y,z),\tag{1}
\]
where \(A\) is a hyperbolic \(m\times m\)-matrix (i.e. a real constant matrix with all eigenvalues having nonzero real parts). \(g\) and \(h\) are supposed to be continuous vector functions. Using the continuation method, which was developed by M. Furi and P. Pera [Ann. Pol. Math. 47, 331-346 (1987; Zbl 0656.47052)], the authors prove the existence of at least one bounded (on \(\mathbb{R}\)) solution to (1) lying in a given set. This set is defined by means of strict bounded (on \(\mathbb{R}\)) lower and upper functions to (1), whose existence is assumed. Under the existence of more families of such strict lower and upper functions to (1) a multiplicity result is formulated.

Reviewer: I.Rachůnková (Olomouc)

### MSC:

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

34C11 | Growth and boundedness of solutions to ordinary differential equations |

### Citations:

Zbl 0656.47052
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\textit{J. Andres} and \textit{B. Krajc}, J. Comput. Appl. Math. 113, No. 1--2, 73--82 (2000; Zbl 0944.34012)

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### References:

[1] | J. Andres, Almost-periodic and bounded solutions of Carathéodory differential inclusions, Differential and Integral Equations, to appear. |

[2] | Andres, J., Multiple bounded solutions of differential inclusions: the Nielsen theory approach, Journal of differential equations, 155, 285-310, (1999) · Zbl 0940.34008 |

[3] | J. Andres, G. Gabor, L. Gorniewicz, Boundary value problems on infinite intervals, Trans. Amer. Math. Soc., to appear. · Zbl 0936.34023 |

[4] | Andres, J.; Krajc, B., Unified approach to bounded, periodic and almost periodic solutions of differential systems, Ann. math. sil., 11, 39-53, (1997) · Zbl 0899.34029 |

[5] | Cecchi, M.; Furi, M.; Marini, M., On continuity and compactness of some nonlinear operators associated with differential equations in non-compact intervals, Nonlinear anal. TMA, 9, 2, 171-180, (1985) · Zbl 0563.34018 |

[6] | B.P. Demidovitch, Lectures on the Mathematical Stability Theory, Nauka, Moscow, 1967 (in Russian). |

[7] | Fernandez, M.L.C.; Zanolin, F., On periodic solutions, in a given set, for differential systems, Riv. mat. pura appl., 8, 107-130, (1991) · Zbl 0725.34039 |

[8] | Furi, M.; Pera, M.P., A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals, Ann. polon. math., 47, 331-346, (1987) · Zbl 0656.47052 |

[9] | Gaines, R.E.; Santanilla, J.M., A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations, Rocky mountain J. math., 12, 669-678, (1982) · Zbl 0508.34030 |

[10] | M.A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. |

[11] | M.A. Krasnosel’skii, The Operator of Translation along the Trajectories of Differential Equations, Nauka, Moscow, 1966 (Russian). |

[12] | Santanilla, J., Some coincidence theorems in wedges, cones and convex sets, J. math. anal. appl., 105, 357-371, (1985) · Zbl 0576.34018 |

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