Singular right focal boundary value problem with given maximal values

Authors

  • Yu Tian
  • Weigao Ge

DOI:

https://doi.org/10.5556/j.tkjm.37.2006.146

Abstract

In this paper, we prove existence results for the singular problem $ (-1)^{n-p}(\Phi_m x^{(n-1)})' $ $ (t)=\mu f(t, x(t), \ldots, x^{(n-1)}(t)), $ for $ 0<1$, $x^{(i)}(0)=0 $, $ i=0, 1, \ldots, p-1 $, $ x^{(i)}(1)=0$, $i=p $, $ p+1, \ldots, n-1 $, $ \max\{x(t):t\in [0, 1]\}=A $. The paper presents conditions which guarantee that for any $ A>0 $ there exists $ \mu_A>0 $ such that the above problem with $ \mu=\mu_{A} $ has a solution $ x\in C^{n-1}([0, 1]) $ with $ \Phi_m(x^{(n-1)})\in AC([0, 1]) $ which is positive on $ (0, 1) $. Here the positive Carath\'edory function $ f $ may be singular at the zero value of all its phase variables. Proofs are based on the Leray-Schauder degree and Vitali's convergence theorem.

Author Biographies

Yu Tian

Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China.

Weigao Ge

Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China

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Published

2006-12-31

How to Cite

Tian, Y., & Ge, W. (2006). Singular right focal boundary value problem with given maximal values. Tamkang Journal of Mathematics, 37(4), 317-332. https://doi.org/10.5556/j.tkjm.37.2006.146

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Papers