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Eigenvalues of higher order Sturm-Liouville boundary value problems with derivatives in nonlinear terms
We shall consider the Sturm-Liouville boundary value problem y(m)(t)+Ī»F(t,y(t),yā²(t),ā¦,y(q)(t))=0, tā(0,1), y(k)(0)=0, 0ā¤kā¤mā3, Ī¶y(mā2)(0)āĪøy(mā1)(0)=0, Ļy(mā2)(1)+Ī“y(mā1)(1)=0 where mā„3, 1ā¤qā¤mā2, and Ī»>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 Ī» 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.Published versio