Comparison theorems and asymptotic behavior of solutions of discrete fractional equations

Consider the following $\nu$-th order nabla and delta fractional difference equations \begin{equation} \begin{aligned} \nabla^\nu_{\rho(a)}x(t)&=c(t)x(t),\quad \quad t\in\mathbb{N}_{a+1},\\ x(a)&>0. \end{aligned}\tag{$\ast$} \end{equation} and \begin{equation} \begin{aligned} \Delta^\nu_{a+\nu-1}x(t)&=c(t)x(t+\nu-1),\quad \quad t\in\mathbb{N}_{a},\\ x(a+\nu-1)&>0. \end{aligned}\tag{$\ast\ast$} \end{equation} We establish comparison theorems by which we compare the solutions $x(t)$ of ($\ast$) and ($\ast\ast$) with the solutions of the equations $\nabla^\nu_{\rho(a)}x(t)=bx(t)$ and $\Delta^\nu_{a+\nu-1}x(t)=bx(t+\nu-1),$ respectively, where $b$ is a constant. We obtain four asymptotic results, one of them extends the recent result [F. M. Atici, P. W. Eloe, Rocky Mountain J. Math. 41(2011) 353--370]. These results show that the solutions of two fractional difference equations $\nabla^\nu_{\rho(a)}x(t)=cx(t),\ 0<\nu<1$, and $\Delta^\nu_{a+\nu-1}x(t)=cx(t+\nu-1),\ 0<\nu<1$, have similar asymptotic behavior with the solutions of the first order difference equations $\nabla x(t)=cx(t),\ |c|<1$ and $\Delta x(t)=cx(t)$, $|c|<1$, respectively

An extended Halanay inequality of integral type on time scales

In this paper, we obtain a Halanay-type inequality of integral type on time scales which improves and extends some earlier results for both the continuous and discrete cases. Several illustrative examples are also given

Comparison theorems and asymptotic behavior of solutions of discrete fractional equations

Consider the following n-th order nabla and delta fractional difference equations rn r (a)x(t) = c(t)x(t), t 2 Na+1, x(a) \u3e 0. and Va+v-1x(t) = c(t)x(t + v - 1), t 2 Na, x(a + n - 1) \u3e 0 We establish comparison theorems by which we compare the solutions x(t) of (*) and (**) with the solutions of the equations rn r(a)x(t) = bx(t) and Dn a+v-1x(t) = bx(t + v -1), respectively, where b is a constant. We obtain four asymptotic results, one of them extends the recent result [F. M. Atici, P. W. Eloe, Rocky Mountain J. Math. 41(2011), 353–370]. These results show that the solutions of two fractional difference equations vp(a)x(t) = cx(t), 0 \u3c n \u3c 1, and Dn a+v-1x(t) = cx(t + v - 1), 0 \u3c n \u3c 1, have similar asymptotic behavior with the solutions of the first order difference equations rx(t) = cx(t), jcj \u3c 1 and Dx(t) = cx(t), jcj \u3c 1, respectively

SOME RELATIONS BETWEEN THE CAPUTO FRACTIONAL DIFFERENCE OPERATORS AND INTEGER-ORDER DIFFERENCES

In this article, we are concerned with the relationships between the sign of Caputo fractional differences and integer nabla differences. In particular, we show that if N -1 \u3c v \u3c N. f : Na -N + 1 -\u3e R, va * f(t) \u3e O, for t - Na +1 and N-1f(a) \u3e 0, then N -1 f(t) \u3e 0 for t- Na +1, then va* f(t) \u3e 0, for each t - Na +1. As applications of these two results, we get that if 1 \u3c vR, va*f(t) \u3e 0 for t - Na +1 and f(a) \u3e f(a-1), then f(t) is an increasing function for t- Na -1. Conversely if 0 \u3c vR and f is an increasing function for t - Na, then va*f(t) \u3e 0, for each t - Na +1. We also give a counterexample to show that the above assumption f(a) \u3e f(a-1) in the last result is essential. These results demonstrate that, in some sense, the positivity of the v-th order Caputo fractional difference has a strong connection to the monotonicity of f(t)

Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain

AbstractWe are concerned with proving existence of one or more than one positive solution of a general two point boundary value problem for the nonlinear equation Lx(t)≔−[p(t)xΔ(t)]Δ+q(t)xσ(t)=λa(t)f(t,xσ(t)). We shall also obtain criteria which leads to nonexistence of positive solutions. Here the independent variable t is in a “measure chain”. We will use fixed point theorems for operators on a Banach space

THEOREMS OF KIGURADZE-TYPE AND BELOHOREC-TYPE REVISITED ON TIME SCALES

This article concerns the oscillation of second-order nonlinear dynamic equations. By using generalized Riccati transformations, Kiguradzetype and Belohorec-type oscillation theorems are obtained on an arbitrary time scale. Our results cover those for differential equations and difference equations, and provide new oscillation criteria for irregular time scales. Some examples are given to illustrate our results

Oscillation of Certain Emden-Fowler Dynamic Equations on Time Scales

The theory of time scales has attracted a great deal of attention since it was first introduced by Hilger [1] in order to unify continuous and discrete analysis. For completeness, we recall the following concepts related to the notion of time scales; see [2, 3] for more details. A time scale T is an arbitrary nonempty closed subset of the real numbers R. In this paper, since we shall be concerned with the oscillatory behavior of solutions, we shall also assume that sup T = ∞.We define the time scale interval [0,∞)T by [0,∞)T := [0,∞)∩T.The forward and backward jump operators are defined b