Displaying similar documents to “Oscillation theorems for third order nonlinear delay difference equations”

Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type

Anna Andruch-Sobiło, Andrzej Drozdowicz (2008)

Mathematica Bohemica

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In the paper we consider the difference equation of neutral type Δ 3 [ x ( n ) - p ( n ) x ( σ ( n ) ) ] + q ( n ) f ( x ( τ ( n ) ) ) = 0 , n ( n 0 ) , where p , q : ( n 0 ) + ; σ , τ : , σ is strictly increasing and lim n σ ( n ) = ; τ is nondecreasing and lim n τ ( n ) = , f : , x f ( x ) > 0 . We examine the following two cases: 0 < p ( n ) λ * < 1 , σ ( n ) = n - k , τ ( n ) = n - l , and 1 < λ * p ( n ) , σ ( n ) = n + k , τ ( n ) = n + l , where k , l are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as n with a weaker assumption on q than the usual assumption i = n 0 q ( i ) = that is used in literature.

Necessary and sufficient conditions for oscillations of delay partial difference equations

Bing Gen Zhang, Shu Tang Liu (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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This paper is concerned with the delay partial difference equation (1) A m + 1 , n + A m , n + 1 - A m , n + Σ i = 1 u p i A m - k i , n - l i = 0 where p i are real numbers, k i and l i are nonnegative integers, u is a positive integer. Sufficient and necessary conditions for all solutions of (1) to be oscillatory are obtained.

Oscillatory properties of third-order semi-noncanonical nonlinear delay difference equations

Govindasamy Ayyappan, George E. Chatzarakis, Thaniarasu Kumar, Ethiraj Thandapani (2023)

Mathematica Bohemica

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We study the oscillatory properties of the solutions of the third-order nonlinear semi-noncanonical delay difference equation D 3 y ( n ) + f ( n ) y β ( σ ( n ) ) = 0 , where D 3 y ( n ) = Δ ( b ( n ) Δ ( a ( n ) ( Δ y ( n ) ) α ) ) is studied. The main idea is to transform the semi-noncanonical operator into canonical form. Then we obtain new oscillation theorems for the studied equation. Examples are provided to illustrate the importance of the main results.

On oscillation of solutions of forced nonlinear neutral differential equations of higher order II

N. Parhi, R. N. Rath (2003)

Annales Polonici Mathematici

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Sufficient conditions are obtained so that every solution of [ y ( t ) - p ( t ) y ( t - τ ) ] ( n ) + Q ( t ) G ( y ( t - σ ) ) = f ( t ) where n ≥ 2, p,f ∈ C([0,∞),ℝ), Q ∈ C([0,∞),[0,∞)), G ∈ C(ℝ,ℝ), τ > 0 and σ ≥ 0, oscillates or tends to zero as t . Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed that 0 Q ( t ) d t = . Through examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.

Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients

Arun K. Tripathy, Shyam S. Santra (2021)

Mathematica Bohemica

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In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type ( r ( t ) ( z ' ( t ) ) γ ) ' + i = 1 m q i ( t ) x α i ( σ i ( t ) ) = 0 , t t 0 , where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) . Under the assumption ( r ( η ) ) - 1 / γ d η = , we consider two cases when γ > α i and γ < α i . Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

On oscillatory nonlinear fourth-order difference equations with delays

Arun K. Tripathy (2018)

Mathematica Bohemica

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In this work, oscillatory behaviour of solutions of a class of fourth-order neutral functional difference equations of the form Δ 2 ( r ( n ) Δ 2 ( y ( n ) + p ( n ) y ( n - m ) ) ) + q ( n ) G ( y ( n - k ) ) = 0 is studied under the assumption n = 0 n r ( n ) < . New oscillation criteria have been established which generalize some of the existing results in the literature.

Delay-dependent stability conditions for fundamental characteristic functions

Hideaki Matsunaga (2023)

Archivum Mathematicum

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This paper is devoted to the investigation on the stability for two characteristic functions f 1 ( z ) = z 2 + p e - z τ + q and f 2 ( z ) = z 2 + p z e - z τ + q , where p and q are real numbers and τ > 0 . The obtained theorems describe the explicit stability dependence on the changing delay τ . Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.

Oscillation criteria for two dimensional linear neutral delay difference systems

Arun Kumar Tripathy (2023)

Mathematica Bohemica

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In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form Δ x ( n ) + p ( n ) x ( n - m ) y ( n ) + p ( n ) y ( n - m ) = a ( n ) b ( n ) c ( n ) d ( n ) x ( n - α ) y ( n - β ) are established, where m > 0 , α 0 , β 0 are integers and a ( n ) , b ( n ) , c ( n ) , d ( n ) , p ( n ) are sequences of real numbers.

Forced oscillation of third order nonlinear dynamic equations on time scales

Baoguo Jia (2010)

Annales Polonici Mathematici

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Consider the third order nonlinear dynamic equation x Δ Δ Δ ( t ) + p ( t ) f ( x ) = g ( t ) , (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation Δ ³ x ( n ) + n α | x | γ s g n ( n ) = ( - 1 ) n c , where α ≥ -1, γ > 0, c > 3, is oscillatory.

Oscillation properties for a scalar linear difference equation of mixed type

Leonid Berezansky, Sandra Pinelas (2016)

Mathematica Bohemica

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The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type Δ x ( n ) + k = - p q a k ( n ) x ( n + k ) = 0 , n > n 0 , where Δ x ( n ) = x ( n + 1 ) - x ( n ) is the difference operator and { a k ( n ) } are sequences of real numbers for k = - p , ... , q , and p > 0 , q 0 . We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.