##
**Positive solutions for a nonlinear differential equation on a measure chain.**
*(English)*
Zbl 0963.34020

Summary: The authors are concerned with proving the existence of positive solutions to general two-point boundary value problems for the nonlinear equation
\[
Lx(t):= -[r(t) x^\Delta(t)]^\Delta= f(t, x(t)).
\]
They use fixed-point theorems concerning cones in a Banach space. Important results concerning Green functions for general two-point boundary value problems for
\[
Lx(t):= -[r(t) x^\Delta(t)]^\Delta= 0
\]
are given.

### MSC:

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

PDF
BibTeX
XML
Cite

\textit{L. Erbe} and \textit{A. Peterson}, Math. Comput. Modelling 32, No. 5--6, 571--585 (2000; Zbl 0963.34020)

Full Text:
DOI

### References:

[1] | Hilger, S, Analysis on measure chains—A unified approach to continuous and discrete calculus, Results in mathematics, 18, 18-56, (1990) · Zbl 0722.39001 |

[2] | Agarwal, R; Bohner, M, Basic calculus on time scales and some of its applications, Results in mathematics, 35, 3-22, (1999) · Zbl 0927.39003 |

[3] | Agarwal, R; Bohner, M, Quadratic functional for second order matrix equations on time scales, Nonlinear analysis, 33, 675-692, (1998) · Zbl 0938.49001 |

[4] | Agarwal, R; Bohner, M; Wong, P, Sturm-Liouville eigenvalue problems on time scales, Applied mathematics and computation, 99, 153-166, (1999) · Zbl 0938.34015 |

[5] | Erbe, L; Hilger, S, Sturmian theory on measure chains, Differential equations and dynamical systems, 1, 223-246, (1993) · Zbl 0868.39007 |

[6] | Erbe, L.H; Peterson, A, Green’s functions and comparison theorems for differential equations on measure chains, Dynamics of continuous, discrete and impulsive systems, 6, 121-137, (1999) · Zbl 0938.34027 |

[7] | Ahlbrandt, C; Peterson, A, Discrete Hamiltonian systems: difference equations, continued fractions, and Riccati equations, (1996), Kluwer Academic Boston · Zbl 0860.39001 |

[8] | Kelley, W; Peterson, A, Difference equations: an introduction with applications, (1991), Academic Press |

[9] | Aulbach, B; Hilger, S, Linear dynamic processes with inhomogeneous time scale, () · Zbl 0719.34088 |

[10] | Deimling, K, Nonlinear functional analysis, (1985), Springer New York · Zbl 0559.47040 |

[11] | Krasnoselskii, M, Positive solutions of operator equations, (1964), Noordhoff Groningen |

[12] | Erbe, L.H; Hu, S; Wang, H, Multiple positive solutions of some boundary value problems, Journal of mathematical analysis and applications, 184, 640-648, (1994) · Zbl 0805.34021 |

[13] | Erbe, L.H; Tang, M, Positive radial solutions to nonlinear boundary value problems for semilinear elliptic problems, (), 45-53 · Zbl 0845.35033 |

[14] | Erbe, L.H; Tang, M, Existence and multiplicity of positive solutions to nonlinear boundary value problems, Differential equations and dynamical systems, 4, 313-320, (1996) · Zbl 0868.35035 |

[15] | Wang, H, On the existence of positive solutions for semilinear elliptic equations in the annulus, Journal of differential equations, 109, 1-7, (1994) · Zbl 0798.34030 |

[16] | Kaymakcalan, B; Laksmikantham, V; Sivasundaram, S, Dynamical systems on measure chains, (1996), Kluwer Academic Boston · Zbl 0869.34039 |

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.