Singular right focal boundary value problem with given maximal values

Main Article Content

Yu Tian
Weigao Ge

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.

Article Details

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
Section
Papers
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