Some results on the fractional order Sturm-Liouville problems
Some results on the fractional order Sturm-Liouville problems
In this work, we introduce some new results on the Lyapunov inequality, uniqueness and multiplicity results of nontrivial solutions of the nonlinear fractional Sturm-Liouville problems $$\textstyle\begin{cases} D_{0^{+}}^{q} (p(t)u'(t))+\Lambda(t)f(u(t))=0,\quad1 < q\leq2, t\in (0,1), \\ \alpha u(0)-\beta p(0)u'(0)=0,\qquad\gamma u(1)+\delta p(1)u'(1)=0, \end{cases} $$ where α, β, γ, δ are constants satisfying $0\neq \vert\beta\gamma+\alpha\gamma\int_{0}^{1}\frac{1}{p(\tau)}\,d\tau …