Singular right focal boundary value problem with given maximal values

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.