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- Title
On the Balanced Pantograph Equation of Mixed Type.
- Authors
Derfel, G.; van Brunt, B.
- Abstract
We consider the balanced pantograph equation (BPE) y ′ x + y x = ∑ k = 1 m p k y a k x , where ak, pk > 0 and ∑ k = 1 m p k = 1 . It is known that if K = ∑ k = 1 m p k ln a k ≤ 0 then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for K > 0 these solutions exist. In the present paper, we deal with a BPE of mixed type, i.e., a1< 1 < am, and prove that, in this case, the BPE has a nonconstant solution y and that y(x) ~ cxσ as x → ∞, where c > 0 and σ is the unique positive root of the characteristic equation P s = 1 - ∑ k = 1 m p k a k - s = 0 . We also show that y is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as x → ∞.
- Subjects
PANTOGRAPH; EQUATIONS; CATENARY
- Publication
Ukrainian Mathematical Journal, 2024, Vol 75, Issue 12, p1841
- ISSN
0041-5995
- Publication type
Article
- DOI
10.1007/s11253-024-02295-x