On uniqueness and successive approximations in the Darboux problem for the equation
B. Palczewski (1965)
Annales Polonici Mathematici
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B. Palczewski (1965)
Annales Polonici Mathematici
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N. Parhi, R. N. Rath (2003)
Annales Polonici Mathematici
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Sufficient conditions are obtained so that every solution of 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 . Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed that . Through examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.
Irina Astashova (2015)
Mathematica Bohemica
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For the equation existence of oscillatory solutions is proved, where is an arbitrary point and is a periodic non-constant function on . The result on existence of such solutions with a positive periodic non-constant function on is formulated for the equation
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 be defined on L₂[0,1]. We prove that 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 always equals 1.
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 where is a constant and is positive continuous function on , are given in terms of an increasing continuously differentiable function from to which satisfies .
Pavla Hofmanová (2016)
Commentationes Mathematicae Universitatis Carolinae
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Let be a weight on . Assume that is continuous on . Let the operator be given at measurable non-negative function on by We characterize weights on for which there exists a positive constant such that the inequality holds for every . Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.
D.E. Edmunds, J. Lang (2008)
Bollettino dell'Unione Matematica Italiana
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Let , let , let and be positive functions with e and let be the Hardy-type operator given by We show that the asymptotic behavior of the eigenvaluesof the non-linear integral system (where, for example,is given by Here, is an explicit constant depending only on and , , where stands for the set of all eigenvalues corresponding to eigenfunctions with zeros.
Takanao Kanemitsu, Satoshi Tanaka (2017)
Archivum Mathematicum
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The second order linear differential equation is considered, where , , , for . Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near without the Hartman–Wintner condition.
Manabu Naito (1998)
Czechoslovak Mathematical Journal
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The neutral differential equation (1.1) is considered under the following conditions: , , , is nonnegative on and is nondecreasing in , and as . It is shown that equation (1.1) has a solution such that (1.2) Here, is an integer with . To prove the existence of a solution satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.
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) uniformly in a neighborhood of the origin, where m > 1, ; (c) . Let x₀,x₁ ∈ (0,α) and , n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: .