Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations

In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved

Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument

We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.Comment: 24 pages, 1 figur

On the asymptotic behavior for convection-diffusion equations associated to higher order elliptic operators in divergence form

We consider the linear convection-diffusion equation associated to higher order elliptic operators (ut + Ltu = aru on Rn × (0,1) u(0) = u0 2 L1(Rn), (1) where a is a constant vector in Rn, m 2 N_, n _ 1 and L0 belongs to a class of higher order elliptic operators in divergence form associated to non-smooth bounded measurable coefficients on Rn. The aim of this paper is to study the asymptotic behavior, in Lp (1 _ p _ 1), of the derivatives Du(t) of the solution of (1) when t tends to 1.We consider the linear convection-diffusion equation associated to higher order elliptic operators (ut + Ltu = aru on Rn × (0,1) u(0) = u0 2 L1(Rn), (1) where a is a constant vector in Rn, m 2 N_, n _ 1 and L0 belongs to a class of higher order elliptic operators in divergence form associated to non-smooth bounded measurable coefficients on Rn. The aim of this paper is to study the asymptotic behavior, in Lp (1 _ p _ 1), of the derivatives Du(t) of the solution of (1) when t tends to 1

Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte Type Inequalities for Convex Functions via Fractional Integrals

The aim of this paper is to establish Hermite-Hadamard, Hermite-Hadamard-Fej\'er, Dragomir-Agarwal and Pachpatte type inequalities for new fractional integral operators with exponential kernel. These results allow us to obtain a new class of functional inequalities which generalizes known inequalities involving convex functions. Furthermore, the obtained results may act as a useful source of inspiration for future research in convex analysis and related optimization fields.Comment: 14 pages, to appear in Journal of Computational and Applied Mathematic