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.