Lyapunov type inequalities for second-order forced dynamic equations with mixed nonlinearities on time scales

Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

“…Compared to the large number of references to continuous Lyapunov-type inequalities, little has been done for discrete Lyapunov-type inequalities. Let us list the works in which the Lyapunov-type inequality appears in a discrete context: for difference equations see [9][10][11][18][19][20][21]31], for discrete systems see [15,16,26,27,29,30,32], for fractional discrete problems see [12][13][14] and for problems on time scales (as these kind of problems include difference equations) see [1][2][3][4][5][6]17,24].…”

Lyapunov-type inequalities for sequential fractional boundary value problems using Hilfer’s fractional derivative

“…Since then, Lyapunov inequality and Lyapunov-type inequality have been studied with great interest, and they have been proved to be an effective tool in the study of differential and difference equations, such as oscillation theory, disconjugacy, eigenvalue problems, etc. (see [2][3][4][5]). In recent years, by the rise of theoretical research in fractional differential equations, there has been tremendous interest in the research of Lyapunov-type inequalities for fractional BVP, see and the references cited therein.…”

“…⎧ ⎨ ⎩ (D α a+ x)(t) + q(t)x(t) = 0, t ∈ (a, b), x(a) = x(b) = 0, (1.3) where D α a+ is the left Riemann-Liouville fractional derivative of order α, α ∈ (1,2] and q ∈ C ([a, b], R). The Lyapunov-type inequality for problem (1.3) was established as follows: It is worth noting that the study of the Hilfer fractional differential equations has received a significant amount of attention in the last few years.…”