Existence of positive solutions for multipoint boundary value problem with \(p\)-Laplacian on time scales
The authors are concerned with the existence of positive solutions for the following four-point \(p-\)Laplacian boundary value problem set on a time scale \(\mathbb{T}\) and subjected to Sturm-Liouville boundary conditions: \[ \begin{cases}(\varphi_p(u^\Delta(t)))^\Delta+h(t)f(u(t))=0, \quad t\in[a,b]_{\mathbb{T}},\\ \alpha u(a)-\beta u^\Delta(\xi)=0;\;\gamma u(\sigma^2(b))+\delta u^\Delta(\eta)=0, \quad u^\Delta(\theta)=0, \end{cases} \] where \(\varphi_p(s)=| s|^{p-2}s,\;p>1\) is the \(p-\)Laplacian, \(f\) is a non-negative continuous function on \([0,\infty)\) and \(h:[a,\sigma^2(b)]\to[0,\infty)\) is rd-continuous. The constants \(\alpha, \beta, \gamma, \delta\) are nonnegative with \(\alpha,\gamma>0\) while \(a<\xi<\theta<\eta<b\). Essentially, two existence results (and some other versions) of solutions, in a positive cone of the Banach space rd-continuous functions, are proved using the classical Krasnosel'ski fixed point theorem of cone compression and expansion. The assumptions on \(f\) involve the limits \[ f_0=\lim_{u\to0}(f(u)/\varphi_p(u)) \quad\text{and}\quad f_\infty=\lim_{u\to\infty}(f(u)/\varphi_p(u)) \] and cover the sub-linear and super-linear growth cases
scientific article; zbMATH DE number 5629306