Displaying similar documents to “Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients”

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 → ∞.

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.

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 behavior of higher order neutral differential equation with multiple functional delays under derivative operator

R.N. Rath, K.C. Panda, S.K. Rath (2022)

Archivum Mathematicum

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In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation ( y ( t ) - i = 1 k p i ( t ) y ( r i ( t ) ) ) ( n ) + v ( t ) G ( y ( g ( t ) ) ) - u ( t ) H ( y ( h ( t ) ) ) = f ( t ) oscillates or tends to zero as t , where, n 1 is any positive integer, p i , r i C ( n ) ( [ 0 , ) , )  and p i are bounded for each i = 1 , 2 , , k . Further, f C ( [ 0 , ) , ) , g , h , v , u C ( [ 0 , ) , [ 0 , ) ) , G and H C ( , ) . The functional delays r i ( t ) t , g ( t ) t and h ( t ) t and all of them approach as t . The results hold when u 0 and f ( t ) 0 . This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature. ...

Oscillation criteria for fourth order half-linear differential equations

Jaroslav Jaroš, Kusano Takaŝi, Tomoyuki Tanigawa (2020)

Archivum Mathematicum

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Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form ( | y ' ' | α sgn y ' ' ) ' ' + q ( t ) | y | α sgn y = 0 , t a > 0 , A where α > 0 is a constant and q ( t ) is positive continuous function on [ a , ) , are given in terms of an increasing continuously differentiable function ω ( t ) from [ a , ) to ( 0 , ) which satisfies a 1 / ( t ω ( t ) ) d t < .

Oscillation of deviating differential equations

George E. Chatzarakis (2020)

Mathematica Bohemica

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Consider the first-order linear delay (advanced) differential equation x ' ( t ) + p ( t ) x ( τ ( t ) ) = 0 ( x ' ( t ) - q ( t ) x ( σ ( t ) ) = 0 ) , t t 0 , where p ( q ) is a continuous function of nonnegative real numbers and the argument τ ( t ) ( σ ( t ) ) is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions lim sup t τ ( t ) t p ( s ) d s > 1 lim sup t t σ ( t ) q ( s ) d s > 1 and lim inf t τ ( t ) t p ( s ) d s > 1 e lim inf t t σ ( t ) q ( s ) d s > 1 e are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known...

Integral averaging technique for oscillation of damped half-linear oscillators

Yukihide Enaka, Masakazu Onitsuka (2018)

Czechoslovak Mathematical Journal

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This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ( a ( t ) φ p ( x ' ) ) ' + b ( t ) φ p ( x ' ) + c ( t ) φ p ( x ) = 0 , where φ p ( x ) = | x | p - 1 sgn x for x and p > 1 . A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial...

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.

Nonrectifiable oscillatory solutions of second order linear differential equations

Takanao Kanemitsu, Satoshi Tanaka (2017)

Archivum Mathematicum

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The second order linear differential equation ( p ( x ) y ' ) ' + q ( x ) y = 0 , x ( 0 , x 0 ] is considered, where p , q C 1 ( 0 , x 0 ] , p ( x ) > 0 , q ( x ) > 0 for x ( 0 , x 0 ] . Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near x = 0 without the Hartman–Wintner condition.

A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations

Manabu Naito (2024)

Mathematica Bohemica

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The half-linear differential equation ( | u ' | α sgn u ' ) ' = α ( λ α + 1 + b ( t ) ) | u | α sgn u , t t 0 , is considered, where α and λ are positive constants and b ( t ) is a real-valued continuous function on [ t 0 , ) . It is proved that, under a mild integral smallness condition of b ( t ) which is weaker than the absolutely integrable condition of b ( t ) , the above equation has a nonoscillatory solution u 0 ( t ) such that u 0 ( t ) e - λ t and u 0 ' ( t ) - λ e - λ t ( t ), and a nonoscillatory solution u 1 ( t ) such that u 1 ( t ) e λ t and u 1 ' ( t ) λ e λ t ( t ).

Neutral set differential equations

Umber Abbas, Vasile Lupulescu, Donald O&amp;#039;Regan, Awais Younus (2015)

Czechoslovak Mathematical Journal

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The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type D H X ( t ) = F ( t , X t , D H X t ) , X | [ - r , 0 ] = Ψ , where F : [ 0 , b ] × 𝒞 0 × 𝔏 0 1 K c ( E ) is a given function, K c ( E ) is the family of all nonempty compact and convex subsets of a separable Banach space E , 𝒞 0 denotes the space of all continuous set-valued functions X from [ - r , 0 ] into K c ( E ) , 𝔏 0 1 is the space of all integrally bounded set-valued functions X : [ - r , 0 ] K c ( E ) , Ψ 𝒞 0 and D H is the Hukuhara derivative. The continuous dependence of solutions on initial...

A note on the oscillation problems for differential equations with p ( t ) -Laplacian

Kōdai Fujimoto (2023)

Archivum Mathematicum

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This paper deals with the oscillation problems on the nonlinear differential equation ( a ( t ) | x ' | p ( t ) - 2 x ' ) ' + b ( t ) | x | λ - 2 x = 0 involving p ( t ) -Laplacian. Sufficient conditions are given under which all proper solutions are oscillatory. In addition, we give a-priori estimates for nonoscillatory solutions and propose an open problem.

Solutions of an advance-delay differential equation and their asymptotic behaviour

Gabriela Vážanová (2023)

Archivum Mathematicum

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The paper considers a scalar differential equation of an advance-delay type y ˙ ( t ) = - a 0 + a 1 t y ( t - τ ) + b 0 + b 1 t y ( t + σ ) , where constants a 0 , b 0 , τ and σ are positive, and a 1 and b 1 are arbitrary. The behavior of its solutions for t is analyzed provided that the transcendental equation λ = - a 0 e - λ τ + b 0 e λ σ has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.