Nonexistence criteria of solutions for a class of second order differential equations with relativistic derivative (Q2106125)

In this paper, the authors present sufficient conditions for the nonexistence of nontrivial solution for a class of second order differential equations with relativistic derivative as follows \[ -\Big(\frac{u'}{\sqrt{1-|u'|^2}}\Big)'=g(t)(u')^r+\lambda(t)u^p+\mu(t)u^q,\ t\in (a,b), \] \[ u(a)=u(b)=0, \] here \(r,p,q\geq 1\) are constants, and \(g,\lambda,\mu\in C([a,b],[0,+\infty))\). The proof is based on some new Lyapunov-type inequalities which are implied by Hölder's inequality and Sobolev's inequality. Furthermore, two examples show that the obtained criteria are more effective and complement some existing results in the literature.