Lyapunov’s inequality on timescales

Stability criteria for linear Hamiltonian dynamic systems on time scales

“…Recently, Agarwal et al [7], Jiang and Zhou [8], Wong et al [9] and He et al [10] established some Lyapunov-type inequalities for dynamic equations on time scales, which generalize the corresponding results on differential and difference equations. Lyapunov-type inequalities are very useful in oscillation theory, stability, disconjugacy, eigenvalue problems and numerous other applications in the theory of differential and difference equations.…”

“…(1.10) holds), then Φ(t + ω) is also a fundamental matrix solution for (1.1) ( see [18]). Therefore, it follows from the uniqueness of solutions of system (1.1) with initial condition ( see [9,18,19]…”

Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives

“…This inequality is considered to be the best reachable one in the sense that the constant 4 in ( 1.3 ) cannot be replaced by any other larger constant (see [ 6 ] and [ 8 , Theorem 5.1]). In his remarkable book, Hartman [ 8 ] was beyond this estimate and obtained a generalized version as follows: The study of this problem on various types of differential and difference equations over the last years has resulted in different versions of Lyapunov-type inequalities; the reader may consult the papers [ 9 – 13 ] and the first chapter in [ 14 ] for a complete view. In parallel to the intensive investigation tendency amongst researchers, Agarwal et al in [ 15 ] has recently considered the mixed non-linear BVP of the form where the non-linearities satisfy and no sign restrictions are imposed on the real-valued integrable potential functions , , and obtained the following Hartman-type and Lyapunov-type inequalities.…”

Lyapunov-type inequality for a higher order dynamic equation on time scales

“…Wong et al. ( 2006 ) investigated the following dynamic equation on time scale T under the assumptions ( with ) and is monotone and , and showed that if is a solution of ( 4 ), then where .…”