Inequalities For Boundary Value Problems Research Articles

On Lyapunov-type inequalities for boundary value problems of fractional Caputo-Fabrizio derivative

In this study, Lyapunov-type inequalities for fractional boundary value problems involving the fractional Caputo Fabrizio differential equation with mixed boundary conditions when the fractional order of $\beta \in (1,2]$ and Dirichlet-type boundary condition when the fractional order of $\sigma \in (2,3]$ have been derived. Some consequences of the results related to the fractional Sturm?Liouville eigenvalue problems have also been given. Additionally, we examine the nonexistence of the solution of the fractional boundary value problem.

The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order

In this paper, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.

Lyapunov-type inequalities and solvability of second-order ODEs across multi-resonance

We present some new Lyapunov-type inequalities for boundary value problems of the form \begin{document}$y''+u(x)y=0$\end{document} , \begin{document}$y(0)=0=y(1)$\end{document} , where \begin{document}$-A≤ u(x)≤ B$\end{document} and there are many resonance points lying inside the interval \begin{document}$[-A, B]$\end{document} . The classical Lyapunov's inequality and its reverse are improved by using Pontryagin's maximum principle. As applications, we establish two readily verifiable unique solvability criteria for general \begin{document}$u(x)$\end{document} . Some relevant examples are given to illustrate our results. Variants of Lyapunov-type inequalities for nonlinear BVPs are discussed at the end of the paper.

Optimal Lyapunov inequalities for boundary value problems

This work is devoted to review some recent results on Lp Lyapunov-type inequalities (1 p ∞ ) for resonant differential equations. In the case of Ordinary Differential Equations, we consider Neumann boundary conditions and an explicit optimal result is obtained. Moreover, it is also treated the case in which the resonance appears at higher eigenvalues. We also study mixed boundary conditions. From this study, and under some natural restrictions on the linear coefficient, the relation between Neumann boundary conditions and disfocality arises in a natural way. For Partial Differential Equations it is proved that the relation between the quantities p and N/2 plays a crucial role in order to obtain Lp Lyapunov-type inequalities, for resonant linear problems with Neumann boundary conditions on a bounded domain Ω⊂ RN . This fact shows a deep difference with respect to the ordinary case. Combining these linear results with Schauder fixed point theorem, we can obtain some new results about the existence and uniqueness of solutions for resonant nonlinear problems. Finally, we comment some conclusions on systems of equations. Mathematics subject classification (2000): 34B05, 34B15, 35J25, 35J65.