A family of nonlinear difference equations: Existence, uniqueness, and asymptotic behavior of positive solutions

Alsulami, Saud M.; Nevai, Paul; Szabados, J.; Assche, Walter Van

doi:10.1016/j.jat.2014.04.012

Cited by 13 publications

(25 citation statements)

References 10 publications

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“…Lew and Quarles [13], Nevai [16], and Bonan and Nevai [3] showed that there exists a unique solution of (1.6) with x 0 = 0 for which x n > 0 for all n ≥ 1 and this solution corresponds to x 1 = 2Γ( 3 4 )/Γ( 1 4 ). This result was recently also proved for a family of non-linear recurrence relations generalizing (1.6), see [1].…”

Section: Introductionmentioning

confidence: 65%

Unique positive solution for an alternative discrete Painlevé I equation

Clarkson

Loureiro

Assche

2016

Journal of Difference Equations and Applications

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We show that the alternative discrete Painlevé I equation (alt-dP I ) has a unique solution which remains positive for all n ≥ 0. Furthermore, we identify this positive solution in terms of a special solution of the second Painlevé equation (P II ) involving the Airy function Ai(t). The special-function solutions of P II involving only the Airy function Ai(t) therefore have the property that they remain positive for all n ≥ 0 and all t ≥ 0, which is a new characterization of these special solutions of P II and alt-dP I .

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Section: Introductionmentioning

confidence: 65%

Unique positive solution for an alternative discrete Painlevé I equation

Clarkson

Loureiro

Assche

2016

Journal of Difference Equations and Applications

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“…The existence of β 1 (t; λ) > 0 such that (5.2) is a positive sequence follows immediately from [2,Thm. 4.1].…”

Section: Existence and Uniqueness Of Positive Solutionsmentioning

confidence: 89%

“…Several papers, including [38,52,61] provide an answer for the case where t = 0. In a recent paper by Alsulami et al [2], existence and uniqueness of a positive solution are discussed for the nonlinear second-order difference equations of the type β n (σ n,1 β n+1 + σ n,0 β n + σ n,−1 β n−1 ) + κ n β n = n (5.1)…”

Section: Existence and Uniqueness Of Positive Solutionsmentioning

confidence: 99%

Properties of generalized Freud polynomials

Clarkson¹,

Jordaan²

2018

Journal of Approximation Theory

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We consider the semiclassical generalized Freud weight functionWe analyse the asymptotic behaviour of the sequences of monic polynomials that are orthogonal with respect to w λ (x; t), as well as the asymptotic behaviour of the recurrence coefficient, when the degree, or alternatively, the parameter t, tend to infinity. We also investigate existence and uniqueness of positive solutions of the nonlinear discrete equation satisfied by the recurrence coefficients and prove properties of the zeros of the generalized Freud polynomials.

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“…In the paper [1], Alsulami et al discussed the existence and uniqueness of a positive solution for the nonhomogeneous nonlinear second order difference equations of the form, n = x n (σ n,1 x n+1 + σ n,0 x n + σ n,−1 β n−1 ) + κ n x n (7.1)…”

Section: Existence Uniqueness Of Positive Solutionsmentioning

confidence: 99%

“…with the initial conditions x 0 ∈ R, x 1 ≥ 0, whilst κ n ∈ R, σ n,j > 0, j ∈ {0, ±1}, or {σ n,0 > 0, σ n,−1 ≥ 0, σ n,1 ≥ 0}. Following the results presented in [1], the solution of Eq. (4.9) is obtained.…”

Section: Existence Uniqueness Of Positive Solutionsmentioning

confidence: 99%

On properties of a deformed Freud weight

Zhu¹,

Chen²

2018

Random Matrices: Theory Appl.

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We study the recurrence coefficients of the monic polynomials P n (z) orthogonal with respect to the deformed (also called semi-classical) Freud weightwith parameters α > −1, N > 0, s ∈ [0, 1]. We show that the recurrence coefficients β n (s) satisfy the first discrete Painlevé equation (denoted by dP I ), a differential-difference equation and a second order nolinear ordinary differential equation (ODE) in s. Here n is the order of the Hankel matrix generated by w α (x; s, N ). We describe the asymptotic behavior of the recurrence coefficients in three situations, (i) s → 0, n, N finite, (ii) n → ∞, N finite, (iii) n, N → ∞, such that the radio r := n N is bounded away from 0 and closed to 1. We also investigate the existence and uniqueness for the positive solutions of the dP I .Further more, we derive, using the ladder approach, a second order linear ODE satisfied by the polynomials P n (z). It is found as n → ∞, the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, D n (s), associated with w α (x; s, N ) when n tends to infinity.

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A family of nonlinear difference equations: Existence, uniqueness, and asymptotic behavior of positive solutions

Cited by 13 publications

References 10 publications

Unique positive solution for an alternative discrete Painlevé I equation

Unique positive solution for an alternative discrete Painlevé I equation

Properties of generalized Freud polynomials

On properties of a deformed Freud weight

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