Mixed boundary value problem for an ordinary differential equation with fractional derivatives with different origins

Решается смешанная краевая задача для обыкновенного дифференциального уравнения, содержащего композицию лево- и правосторонних операторов дробного дифференцирования Римана-Лиувилля и Капуто. Задача эквивалентно редуцирована к интегральному уравнению Фредгольма второго рода, для которого найдено достаточное условие однозначной разрешимости. В качестве следствия,для исследуемой задачи доказано неравенство Ляпунова A mixed boundary value problem is solved for an ordinary differential equation containing a composition of left- and right-sided Riemann-Liouville and Caputo fractional differentiation operators. The problem is equivalently reduced to a Fredholm integral equation of the second kind, for which a sufficient condition for unique solvability is found. As a consequence, the Lyapunov inequality is proved for the problem under study.

Lyapunov-type inequality for higher order left and right fractional p-Laplacian problems

In this paper, we consider a p-Laplacian eigenvalue boundary value problem involving both right Caputo and left Riemann-Liouville types fractional derivatives. To prove the existence of solutions, we apply the Schaefer’s fixed point theorem. Furthermore, we present the Lyapunov inequality for the corresponding problem.

Fiedler vector analysis for particular cases of connected graphs

In this paper, we consider a p-Laplacian eigenvalue boundary value problem involving both right Caputo and left Riemann-Liouville types fractional derivatives. To prove the existence of solutions, we apply the Schaefer’s fixed point theorem. Furthermore, we present the Lyapunov inequality for the corresponding problem.

LYAPUNOV-TYPE INEQUALITY FOR EXTREMAL PUCCI’S EQUATIONS

In this article, we establish a Lyapunov-type inequality for the following extremal Pucci’s equation: $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}{\mathcal{M}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC}}^{+}(D^{2}u)+b(x)|Du|+a(x)u=0 & \text{in}~\unicode[STIX]{x1D6FA},\\ u=0 & \text{on}~\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$ where $\unicode[STIX]{x1D6FA}$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 2$. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci’s equations with gradient nonlinearity.

Analogue of Lyapunov’s inequality for the second-order ordinary differential equation with continuously distributed differentiation operator

For an ordinary second-order differential equation with an operator of continuously distributed differentiation, an analogue of the Lyapunov inequality of the Dirichlet problem is proved.

LYAPUNOV INEQUALITY FOR AN EQUATION WITH FRACTIONAL DERIVATIVES WITH DIFFERENT ORIGINS

В работе рассмотрено обыкновенное дифференциальное уравнение дробного порядка, содержащее композицию дробных производных с различными началами, являющееся модельным уравнением движения во фрактальной среде. Для рассматриваемого уравнения найдено необходимое условие существования нетривиального решения однородной задачи Дирихле. Условие имеет форму интегральной оценки для потенциала и является аналогом неравенства Ляпунова We consider an ordinary differential equation of fractional order with the composition of left and rightsided fractional derivatives, which is a model equation of motion in fractal media. We find a necessary condition for existence of nontrivial solution of homogeneous Dirichlet problem for the equation under consideration. The condition has the form of integral estimate for the potential and is an analog of Lyapunov inequality.

Green’s functions, positive solutions, and a Lyapunov inequality for a caputo fractional-derivative boundary value problem

We consider a nonlinear boundary problem whose highest-order derivative is a Caputo derivative of order α with 1 < α < 2. Properties of its associated Green’s function are derived. These properties enable us to deduce sufficient conditions for the existence of a positive solution to the boundary value problem and to prove a Lyapunov inequality for the problem. Our results sharpen and extend earlier results of other authors.