This paper deals with a flux-limited chemotaxis-haptotaxis system nonlinear diffusion \begin{document}$ \begin{eqnarray*} \left\{ \begin{split}{} &u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u|\nabla v|^{p-2}\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &v_t \Delta v-v+u, &w_t -vw, \ \end{split} \right. \end{eqnarray*} $\end{document} in $ \Omega\times(0, \infty) $, where...

We prove a more general version of the Grüss inequality by using the recent theory of combined dynamic derivatives on time scales and the more general notions of diamond-α derivative and integral. For the particular case when α = 1, one gets a delta-integral Grüss inequality on time scales; for α = 0 a nabla-integral Grüss inequality. If we further restrict ourselves by fixing the time scale to...

We prove a necessary optimality condition for isoperimetric problems under nabla-differentiable curves. As a consequence, the recent results of [M.R. Caputo, A unified view of ostensibly disparate isoperimetric variational problems, Appl. Math. Lett. (2008), doi:10.1016/j.aml.2008.04.004], that put together seemingly dissimilar optimal control problems in economics and physics, are extended to ...

Existence criteria for two positive solutions to a nonlinear, even-order stacked deltanabla boundary value problem with stacked, vanishing conditions at the two endpoints are found using the method of Green’s functions. A few examples are given for standard time scales. The corresponding even-order nabla-delta problem is also discussed in detail. c © 2003 Elsevier Science Ltd. All rights reserv...

Abstract We propose a new monotone finite difference discretization for the variational p -Laplace operator, $$\Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u),$$ ? p u = div (</m...

Positivity analysis is used with some basic conditions to analyse monotonicity across all discrete fractional disciplines. This article addresses the of nabla differences Riemann–Liouville type by considering positivity ∇b0RLθg(z) combined a condition on g(b0+2), g(b0+3) and g(b0+4), successively. The ends relationship between integer type, which serves show difference ∇b0RLθg(z).

where y∇ is nabla derivative (see [4]). Exponential decay and stability of solutions of dynamic equations on time scales were investigated in recent papers [1, 5–7, 11, 12] using Lyapunov’s method. We use different approaches based on integral representations of solutions via asymptotic solutions and error estimates developed in [2, 8–10]. A time scale T is an arbitrary nonempty closed subset o...

The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-α derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen’s inequality on time scales via the diamond-α integral and present som...