Three positive solutions for delay differential equation BVPs with \(p\)-Laplacian on infinite interval
The authors investigated the existence of three positive solutions for the boundary value problems of a delay differential equation with \(p\)-Laplacian on an infinite interval \[ (\phi_p(x'(t)))'+a(t)f(t, x, x', x_{t})=0, 0<t<\infty, \] \[ x_0=\xi, \alpha x(0)-\beta x'(0)=0, x'(\infty)=0, \] where \(\phi_p(s)=|s|^{p-2}s\), \(p>1\), \(\xi\in C([-r, 0]\), \({\mathbb R}^{+})\), both \(f: {\mathbb R}^{+}\times {\mathbb R}^{+}\times {\mathbb R}^{+}\times C([-r, 0], {\mathbb R}^{+})\to {\mathbb R}^{+}\) and \(a: {\mathbb R}^{+}\to {\mathbb R}^{+}\) are continuous, with \({\mathbb R}^{+}=[0, \infty)\), \(\alpha>0\), \(\beta\geq 0\), \(x'(\infty)=\lim_{t\to +\infty}x'(t)\). By using a fixed point point theorem in a cone introduced by Avery and Peterson, the existence of at least three positive solutions is obtained under suitable growth conditions imposed on the nonlinear term. An example is given to illustrate the feasibility of the main result.
scientific article; zbMATH DE number 5728286
scientific article