Lyapunov-type inequalities for a nonlinear fractional boundary value problem
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Lyapunov-type inequalities for a nonlinear fractional boundary value problem Aidyn Kassymov1,2,3 · Berikbol T. Torebek1,3,4 Received: 7 July 2020 / Accepted: 22 October 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper, we obtain a Lyapunov-type and a Hartman–Wintner-type inequalities for a nonlinear fractional hybrid equation with left Riemann–Liouville and right Caputo fractional derivatives of order 1/2 < α ≤ 1, subject to Dirichlet boundary conditions. It is also shown that failure of the Lyapunov-type and Hartman–Wintner-type inequalities, corresponding nonlinear boundary value problem has only trivial solutions. In the case α = 1, our results coincide with the classical Lyapunov and Hartman–Wintner inequalities, respectively. Keywords Lyapunov inequality · Hartman–Wintner inequality · Fractional hybrid equation · Caputo derivative · Riemann–Liouville derivative Mathematics Subject Classification Primary 26D10; Secondary 26A33 · 35A23
Contents 1 Introduction . . . . . . . . . . . . . . . . 2 Green’s function and some of its properties 3 Main results . . . . . . . . . . . . . . . . 3.1 Lyapunov-type inequality . . . . . . . 3.2 Hartman–Wintner-type inequality . . . References . . . . . . . . . . . . . . . . . . .
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Berikbol T. Torebek [email protected] Aidyn Kassymov [email protected]
1
Al–Farabi Kazakh National University, Al–Farabi Ave. 71, 050040 Almaty, Kazakhstan
2
Institute of Mathematics and Mathematical Modeling, 125 Pushkin Street, 050010 Almaty, Kazakhstan
3
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
4
RUDN University, 6 Miklukho-Maklay Street, 117198 Moscow, Russia 0123456789().: V,-vol
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Page 2 of 10
A. Kassymov, B. T. Torebek
1 Introduction The classical Lyapunov inequality is an outstanding result in mathematics with many applications. In [12] Lyapunov proved that, if q ∈ C ([a, b]; R) , then the necessary condition for the boundary value problem u (t) + q(t)u(t) = 0, a < t < b, (1.1) u(a) = u(b) = 0, to have a nontrivial classical solution is given by b |q(s)| ds > a
4 . b−a
(1.2)
In [8], Hartman and Wintner proved that, if (1.1) has a nontrivial solution, then b
(b − s)(s − a)q + (s)ds > b − a,
(1.3)
a
where q + (s) = max{q(s), 0}. Observe that the Lyapunov inequality (1.2) can be deduced 2 from (1.3) using the identity that max (b − s)(s − a) = (b−a) 4 . a≤s≤b
In [4,5] Ferreira investigated a Lyapunov-type inequality for the fractional boundary value problems with Riemann–Liouville and
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