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We shall consider the Sturm-Liouville boundary value problem \(y^{(m)}(t)+\lambda F (t,y(t),y'(t),\ldots, y^{(q)}(t) )=0\), \(t\in (0,1)\), \(y^{(k)}(0)=0\), \(0\leq k\leq m-3\), \(\zeta y^{(m-2)}(0)-\theta y^{(m-1)}(0)=0\), \(\rho y^{(m-2)}(1)+\delta y^{(m-1)}(1)=0 \) where \(m\geq3\), \(1\leq q\leq m-2\), and \(\lambda>0\). It is noted that the boundary value problem considered has a derivative-dependent nonlinear term, which makes the investigation much more challenging. In this paper we shall develop a new technique to characterize the eigenvalues λ so that the boundary value problem has a positive solution. Explicit eigenvalue intervals are also established. Some examples are included to dwell upon the usefulness of the results obtained.

Eigenvalues of higher order Sturm-Liouville boundary value problems with derivatives in nonlinear terms