Abstract
In the paper, we establish a Lyapunov inequality and two Lyapunov-type inequalities for a higher-order fractional boundary value problem with a controllable nonlinear term. Two applications are discussed. One concerns an eigenvalue problem, the other a Mittag-Leffler function.
Highlights
For the following boundary value problem (BVP for short):y (t) + q(t)y(t) =, a < t < b, ( . )y(a) = y(b) =, where q is a real and continuous function, Lyapunov [ ] proved that if ( . ) has a nontrivial solution, b q(s) ds > . ( . ) a b–aThe expression in ( . ) is called the Lyapunov inequality
The Lyapunov inequality proved to be very useful in various problems related with differential equations
Many improvements and generalizations of the inequality ( . ) for integer-order BVP have appeared in the literature, and here we omit these detailed conclusions but only refer the reader to a summary reference [ ] given by Tiryaki in, in which research results about Lyapunov inequality were summarized
Summary
For the following boundary value problem (BVP for short):. ). y(a) = y(b) = , where q is a real and continuous function, Lyapunov [ ] proved that if The Lyapunov inequality proved to be very useful in various problems related with differential equations. ) for integer-order BVP have appeared in the literature, and here we omit these detailed conclusions but only refer the reader to a summary reference [ ] given by Tiryaki in , in which research results about Lyapunov inequality were summarized. New results for the integer-order boundary value problem appeared continuously; see [ – ]. The research of Lyapunov inequality for fractional BVP has begun in which a fractional derivative (Riemann-Liouville derivative and Caputo derivative) is used instead of the classical ordinary derivative in differential equation. We refer the reader to Ferreira [ – ], Jleli and Samet [ ], Rong and Bai [ ], Arifi et al [ ], and so on
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