Displaying similar documents to “Asymptotic behavior of solutions of a 2 n t h order nonlinear differential equation”

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 → ∞.

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 .

On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov (2008)

Studia Mathematica

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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator V ϕ : f ( x ) 0 ϕ ( x ) f ( t ) d t be defined on L₂[0,1]. We prove that V ϕ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator V ϕ always equals 1.

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 < .

A weighted inequality for the Hardy operator involving suprema

Pavla Hofmanová (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let u be a weight on ( 0 , ) . Assume that u is continuous on ( 0 , ) . Let the operator S u be given at measurable non-negative function ϕ on ( 0 , ) by S u ϕ ( t ) = sup 0 < τ t u ( τ ) ϕ ( τ ) . We characterize weights v , w on ( 0 , ) for which there exists a positive constant C such that the inequality 0 [ S u ϕ ( t ) ] q w ( t ) d t 1 q 0 [ ϕ ( t ) ] p v ( t ) d t 1 p holds for every 0 < p , q < . Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

Asymptotics for Eigenvalues of a Non-Linear Integral System

D.E. Edmunds, J. Lang (2008)

Bollettino dell'Unione Matematica Italiana

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Let I = [ a , b ] , let 1 < q , p < , let u and v be positive functions with u L p ( I ) e v L q ( I ) and let T : L p ( I ) L q ( I ) be the Hardy-type operator given by ( T f ) ( x ) = v ( x ) a x f ( t ) u ( t ) d t , x I . We show that the asymptotic behavior of the eigenvalues λ of the non-linear integral system g ( x ) = ( T F ) ( x ) ( f ( x ) ) ( p ) = λ ( T * g ( p ) ) ) ( x ) (where, for example, t ( p ) = | t | p - 1 sgn ( t ) is given by lim n n λ ^ n ( T ) = c p , q I ( u v ) r ) 1 / r d t 1 / r , for 1 < p < q < lim n n λ ˇ n ( T ) = c p , q I ( u v ) r d t 1 / r for 1 < q < p < Here r = 1 p + 1 p , c p , q is an explicit constant depending only on p and q , λ ^ ( T ) = max ( s p n ( T , p , q ) ) , λ ˇ n ( T ) = min ( s p n ( T , p , q ) ) where s p n ( T , p , q ) stands for the set of all eigenvalues λ corresponding to eigenfunctions g with n zeros.

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.

An asymptotic theorem for a class of nonlinear neutral differential equations

Manabu Naito (1998)

Czechoslovak Mathematical Journal

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The neutral differential equation (1.1) d n d t n [ x ( t ) + x ( t - τ ) ] + σ F ( t , x ( g ( t ) ) ) = 0 , is considered under the following conditions: n 2 , τ > 0 , σ = ± 1 , F ( t , u ) is nonnegative on [ t 0 , ) × ( 0 , ) and is nondecreasing in u ( 0 , ) , and lim g ( t ) = as t . It is shown that equation (1.1) has a solution x ( t ) such that (1.2) lim t x ( t ) t k exists and is a positive finite value if and only if t 0 t n - k - 1 F ( t , c [ g ( t ) ] k ) d t < for some c > 0 . Here, k is an integer with 0 k n - 1 . To prove the existence of a solution x ( t ) satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.

Asymptotic behavior of a sequence defined by iteration with applications

Stevo Stević (2002)

Colloquium Mathematicae

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We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) f ( x , y ) = p x + ( 1 - p ) y - s = m s ( x , y ) uniformly in a neighborhood of the origin, where m > 1, s ( x , y ) = i = 0 s a i , s x s - i y i ; (c) ( 1 , 1 ) = i = 0 m a i , m > 0 . Let x₀,x₁ ∈ (0,α) and x n + 1 = f ( x , x n - 1 ) , n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: x ( ( 2 - p ) / ( ( m - 1 ) i = 0 m a i , m ) ) 1 / ( m - 1 ) 1 / n m - 1 .