Smallest eigenvalues for boundary value problems of two term fractional differential operators depending on fractional boundary conditions

Abstract

Let \(n\ge 2\) be an integer, and let \(n-1<\alpha \le n\). We consider eigenvalue problems for two point \(n-1 ,1\) boundary value problems $$\displaylines{ D_{0+}^{\alpha} u+a(t)u+\lambda p(t)u=0, \quad 0<t<1, \cr u^{(i)}(0)=0, \quad i=0,1,\dots,n-2, \quad D_{0^+}^{\beta} u(1)=0, }$$ where \(0\le \beta \le n-1\) and \(D_{0+}^{\alpha}\) and \(D_{0+}^{\beta}\) denote standard Riemann-Liouville differential operators. We prove the existence of smallest positive eigenvalues and then obtain comparisons of these smallest eigenvalues as functions of both \(p\) and \(\beta\). For more information see https://ejde.math.txstate.edu/Volumes/2021/62/abstr.html