### Abstract

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u′))′ + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)φ(p(t)u_{1}), , q(t)φ(p(t)u_{n})), where u = (u_{1}, ..., u_{n}), and φ covers the two important cases φ (u) = u and φ (u) = |u| ^{p-2}u, p > 1, h(t) = diag[h_{1}(t), ..., h_{n}(t)] and f(u) = (f^{1}(u), ..., f^{n} (u)). Assume that f ^{i} and h_{i} are nonnegative continuous. For u = (u _{1}, , u_{n}), let f_{0}^{i} = lim _{∥u∥→0} f^{i}(u)/φ(∥u∥),f _{∞}^{i} = lim _{∥u∥→∞} f ^{i}(u)/φ(∥u∥), (i=1,...,n), f_{0} = max{f _{0}^{1}, , f_{0}^{n}} and f_{∞} = max{f_{∞}^{1}, , f_{∞}^{n}}. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f_{0} and f_{∞} is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.

Original language | English (US) |
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Pages (from-to) | 1941-1949 |

Number of pages | 9 |

Journal | Computers and Mathematics with Applications |

Volume | 49 |

Issue number | 11-12 |

DOIs | |

State | Published - Jun 1 2005 |

### Keywords

- Existence
- Fixed index theorem
- p-Laplacian

### ASJC Scopus subject areas

- Modeling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

*Computers and Mathematics with Applications*,

*49*(11-12), 1941-1949. https://doi.org/10.1016/j.camwa.2003.08.015