Displaying similar documents to “Oscillation criteria for nonlinear differential equations with p ( t ) -Laplacian”

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.

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 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 < .

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.

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 → ∞.

L p type mapping estimates for oscillatory integrals in higher dimensions

G. Sampson (2006)

Studia Mathematica

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We show in two dimensions that if K f = ² k ( x , y ) f ( y ) d y , k ( x , y ) = ( e i x a · y b ) / ( | x - y | η ) , p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), v p ( y ) = y ( p / p ' ) ( 1 ̅ - b / a ) , then | | K f | | p C | | f | | p , v p if η + α₁ + α₂ < 2, α j = 1 - b j / a j , j = 1,2. Our methods apply in all dimensions and also for more general kernels.

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...

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.

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 ).

On asymptotic behavior of solutions to Emden-Fowler type higher-order differential equations

Irina Astashova (2015)

Mathematica Bohemica

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For the equation y ( n ) + | y | k sgn y = 0 , k > 1 , n = 3 , 4 , existence of oscillatory solutions y = ( x * - x ) - α h ( log ( x * - x ) ) , α = n k - 1 , x < x * , is proved, where x * is an arbitrary point and h is a periodic non-constant function on . The result on existence of such solutions with a positive periodic non-constant function h on is formulated for the equation y ( n ) = | y | k sgn y , k > 1 , n = 12 , 13 , 14 .

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.

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.

On Kakeya–Nikodym averages, L p -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

Matthew D. Blair, Christopher D. Sogge (2015)

Journal of the European Mathematical Society

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We extend a result of the second author [27, Theorem 1.1] to dimensions d 3 which relates the size of L p -norms of eigenfunctions for 2 < p < 2 ( d + 1 ) / d - 1 to the amount of L 2 -mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an " ϵ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] L 2 oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive...