2,092 research outputs found
Interval Oscillation Criteria for Second-Order Forced Functional Dynamic Equations on Time Scales
This paper is concerned with oscillation of second-order forced functional dynamic equations of the form (r(t)(xΔ(t))γ)Δ+∑i=0nqi(t)|x(δi(t))|αisgn x(δi(t))=e(t) on time scales. By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation criteria which handle some cases not covered by known criteria
Oscillation of forced impulsive differential equations with -Laplacian and nonlinearities given by Riemann-Stieltjes integrals
In this article, we study the oscillation of second order forced impulsive differential equation with -Laplacian and nonlinearities given by Riemann-Stieltjes integrals of the form
\begin{equation*}
\left( p(t)\phi _{\gamma }\left( x^{\prime }(t)\right) \right) ^{\prime}+q_{0}\left( t\right) \phi _{\gamma }\left( x(t)\right)+\int_{0}^{b}q\left( t,s\right) \phi _{\alpha \left( s\right) }\left(x(t)\right) d\zeta \left(s\right) =e(t), t\neq \tau _{k},
\end{equation*}
with impulsive conditions
\begin{equation*}
x\left( \tau _{k}^{+}\right) =\lambda _{k}~x\left( t_{k}\right), x^{\prime }\left( \tau _{k}^{+}\right) =\eta _{k}~x^{\prime }\left( \tau_{k}\right),
\end{equation*}
where \phi _{\gamma }\left( u\right) :=\left\vert u\right\vert ^{\gamma } \mbox{{\rm sgn}\,}u, is strictly increasing such that , and is the the impulsive moments sequence. Using the Riccati transformation technique, we obtain sufficient conditions for this equation to be oscillatory
Qualitative analysis of dynamic equations on time scales
In this article, we establish the Picard-Lindelof theorem and approximating
results for dynamic equations on time scale. We present a simple proof for the
existence and uniqueness of the solution. The proof is produced by using
convergence and Weierstrass M-test. Furthermore, we show that the Lispchitz
condition is not necessary for uniqueness. The existence of epsilon-approximate
solution is established under suitable assumptions. Moreover, we study the
approximate solution of the dynamic equation with delay by studying the
solution of the corresponding dynamic equation with piecewise constant
argument. We show that the exponential stability is preserved in such
approximations.Comment: 13 page
Asymptotic behavior of third order functional dynamic equations with -Laplacian and nonlinearities given by Riemann-Stieltjes integrals
In this paper, we study the third-order functional dynamic equations with -Laplacian and nonlinearities given by Riemann-Stieltjes integrals
\begin{equation*}
\left\{ r_{2}\left( t\right) \phi _{\gamma _{2}}\left( \left[
r_{1}\left( t\right) \phi _{\gamma _{1}}\left( x^{\Delta }\left(
t\right) \right) \right] ^{\Delta }\right) \right\} ^{\Delta
}+\int_{a}^{b}q\left( t,s\right) \phi _{\alpha \left( s\right)
}\left( x(g\left( t,s\right) )\right) d\zeta \left( s\right) =0,
\end{equation*}
on an above-unbounded time scale , where and denotes the Riemann-Stieltjes integral of the function on with respect to . Results are obtained for the asymptotic and oscillatory behavior of the solutions. This work extends and improves some known results in the literature on third order nonlinear dynamic equations
Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities
We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form (r(t)Phi(alpha)(x(Delta)))(Delta) + f(t,x(sigma)) = e(t), t is an element of [t(0), infinity)(T) with f (t, x) = q(t) Phi(alpha)(x) + Sigma(n)(i=1)q(i)(t)Phi(beta i)(x), Phi(*)(u) = vertical bar u vertical bar*(-1) u, where [t(0), infinity)(T) is a time scale interval with t(0) is an element of T, the functions r, q, q(i), e : [t(0), infinity)(T) -> R are right-dense continuous with r > 0, sigma is the forward jump operator, x(sigma) (t) := x(sigma(t)), and beta(1) > ... > beta(m) > alpha > beta(m+1) > ... beta(n) > 0. All results obtained are new even for T = R and T = Z. In the special case when T = R and alpha = 1 our theorems reduce to (Y. G. Sun and J. S. W. Wong, Journal of Mathematical Analysis and Applications. 337 (2007), 549-560). Therefore, our results in particular extend most of the related existing literature from the continuous case to arbitrary time scale. Copyright (C) 2009 R. P. Agarwal and A. Zafer.Publisher's Versio
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