Oscillatory properties of fourth order self-adjoint differential equations

summary:Oscillation and nonoscillation criteria for the self-adjoint linear differential equation \[ (t^\alpha y^{\prime \prime })^{\prime \prime }-\frac{\gamma _{2,\alpha }}{t^{4-\alpha }}y=q(t)y,\quad \alpha \notin \lbrace 1, 3\rbrace \,, \] where \[ \gamma _{2,\alpha }=\frac{(\alpha -1)^2(\alpha -3)^2}{16}\] and $q$ is a real and continuous function, are established. It is proved, using these criteria, that the equation \[\left(t^\alpha y^{\prime \prime }\right)^{\prime \prime }-\left(\frac{\gamma _{2,\alpha }}{t^{4-\alpha }} + \frac{\gamma }{t^{4-\alpha }\ln ^2 t}\right)y = 0\] is nonoscillatory if and only if $\gamma \le \frac{\alpha ^2-4\alpha +5}{8}$