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1
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Opial's type inequalities on time scales and some applications

100%
Saker S.
Annales Polonici Mathematici
|
2012
|
tom 104
|
nr 3
243-260
EN
We prove some new Opial type inequalities on time scales and employ them to prove several results related to the spacing between consecutive zeros of a solution or between a zero of a solution and a zero of its derivative for second order dynamic equations on time scales. We also apply these inequalities to obtain a lower bound for the smallest eigenvalue of a Sturm-Liouville eigenvalue problem on time scales. The results contain as special cases some results obtained for second order differential equations, give some new results for difference equations and yield conditions for disfocality for second order dynamic equations on time scales.
2
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Lyapunov type inequalities for a second order differential equation with a damping term

100%
Saker S.
Annales Polonici Mathematici
|
2012
|
tom 103
|
nr 1
37-57
EN
For a second order differential equation with a damping term, we establish some new inequalities of Lyapunov type. These inequalities give implicit lower bounds on the distance between zeros of a nontrivial solution and also lower bounds for the spacing between zeros of a solution and/or its derivative. We also obtain a lower bound for the first eigenvalue of a boundary value problem. The main results are proved by applying the Hölder inequality and some generalizations of Opial and Wirtinger type inequalities. The results yield conditions for disfocality and disconjugacy. An example is considered to illustrate the main results.
3
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Asymptotic properties of third order functional dynamic equations on time scales

64%
Kubiaczyk I., Saker S.
Annales Polonici Mathematici
|
2011
|
tom 100
|
nr 3
203-222
EN
The purpose of this paper is to study the asymptotic properties of nonoscillatory solutions of the third order nonlinear functional dynamic equation $[p(t)[(r(t)x^{Δ}(t))^{Δ}]^{γ}]^{Δ} + q(t)f(x(τ(t))) = 0$, t ≥ t₀, on a time scale 𝕋, where γ > 0 is a quotient of odd positive integers, and p, q, r and τ are positive right-dense continuous functions defined on 𝕋. We classify the nonoscillatory solutions into certain classes $C_{i}$, i = 0,1,2,3, according to the sign of the Δ-quasi-derivatives and obtain sufficient conditions in order that $C_{i} = ∅$. Also, we establish some sufficient conditions which ensure the property A of the solutions. Our results are new for third order dynamic equations and involve and improve some results previously obtained for differential and difference equations. Some examples are worked out to demonstrate the main results.
4
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Oscillation of delay differential equation with several positive and negative coefficients

64%
Elabbasy E., Saker S.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2003
|
tom 23
|
nr 1
39-52
EN
Some sufficient conditions for oscillation of a first order nonautonomuous delay differential equation with several positive and negative coefficients are obtained.
5
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Oscillation of nonlinear neutral delay differential equations with applications

64%
Li W.-T., Saker S.
Annales Polonici Mathematici
|
2001
|
tom 77
|
nr 1
39-51
EN
We consider nonlinear neutral delay differential equations with variable coefficients. Finite and infinite integral conditions for oscillation are obtained. As an example, the neutral delay logistic differential equation is discussed.
6
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Oscillation and global attractivity in a discrete survival red blood cells model

64%
Kubiaczyk I., Saker S.
Applicationes Mathematicae
|
2003
|
tom 30
|
nr 4
441-449
EN
We consider the discrete survival red blood cells model (*) $N_{n+1} - Nₙ = -δₙNₙ + Pₙe^{-aN_{n-k}}$, where δₙ and Pₙ are positive sequences. In the autonomous case we show that (*) has a unique positive steady state N*, we establish some sufficient conditions for oscillation of all positive solutions about N*, and when k = 1 we give a sufficient condition for N* to be globally asymptotically stable. In the nonatonomous case, assuming that there exists a positive solution {Nₙ*}, we present necessary and sufficient conditions for oscillation of all positive solutions of (*) about {Nₙ*}. Our results can be considered as discrete analogues of the recent results by Saker and Agarwal [12] and solve an open problem posed by Kocic and Ladas [8].
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