Smallest Eigenvalues for a Right Focal Boundary Value Problem

Abstract

We establish the existence of smallest eigenvalues for the fractional linear boundary value problems

Eigenvalue comparison for fractional boundary value problems with the Caputo derivative

“…Recently, Eloe and Neugebauer [5] obtained these type of results for a fractional order problem with the Riemann-Liouville derivative. We would like to extend their methodology to the fractional problem with the Caputo derivative.…”

“…The methods applied here have been successfully used by several authors in comparing eigenvalues for boundary value problems for both ordinary differential equations [1,2,4,5,6,7,10,11,15] and finite difference equations [8,9]. Recently, Eloe and Neugebauer [5] obtained these type of results for a fractional order problem with the Riemann-Liouville derivative.…”

Comparison of Smallest Eigenvalues for Nabla Fractional Boundary Value Problems

“…The theories of Krein-Rutman [30] and Krasnosel'skiȋ [29] have been used by many authors to establish the existence of and then compare smallest eigenvalues of boundary value problems for differential equations [5,10,11,12,13,32], difference equations [7,16], dynamic equations on time scales [6,19], fractional differential equations [8,9,18,28], and delta fractional difference equations [17,33,34].…”

Eigenvalue problems for fractional differential equations with mixed derivatives and generalized p-Laplacian

“…In the last few years, fractional differential equations have gained attentions due to their numerous applications in various aspects of science and technology. Eigenvalue problems and their applications are gradually beginning to be studied for fractional differential equations (see, e.g., [7][8][9]11,12,16,22]). In [22], Zhao et al considered eigenvalue intervals of the following boundary value problem for nonlinear fractional differential equations: C D α 0 + u(t) = λf u(t) , 0 < t < 1, u(0) + u (0) = 0, u(1) + u (1) = 0, where 1 < α 2, C D α 0 + is the Caputo fractional derivative.…”

“…The authors studied the existence, multiplicity and nonexistence of positive solutions. In [8], Eloe et al used the theory of u 0 -positive operators to study boundary value problem for a class of linear fractional differential equations D α 0 + u(t) = λh(t)u(t), 0 < t < 1, where 1 < α 2. The existence and a comparison theorem of smallest positive eigenvalues were obtained.…”