Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument
We consider the difference equation with continuous argument x(t+2)−2λx(t+1)+λ²x(t)=f(t,x(t)), where λ > 0, t ∈ [0, ∞), and f: [0, ∞) × R → R. Conditions for the existence and uniqueness of continuous asymptotically periodic solutions of this equation are given. We also prove the following re...
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| Date: | 2004 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2004
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| Series: | Український математичний журнал |
| Subjects: | |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument / S. Stevic // Український математичний журнал. — 2004. — Т. 56, № 8. — С. 1095–1100. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | We consider the difference equation with continuous argument
x(t+2)−2λx(t+1)+λ²x(t)=f(t,x(t)),
where λ > 0, t ∈ [0, ∞), and f: [0, ∞) × R → R. Conditions for the existence and uniqueness of continuous asymptotically periodic solutions of this equation are given. We also prove the following result: Let x(t) be a real continuous function such that
limt→∞(x(t+2)−(1−α)x(t+1)−αx(t))=0
for some α ∈ R. Then it always follows from the boundedness of x(t) that
limt→∞(x(t+1)−x(t))=0
t → ∞ if and only if α ∈ R {1}. |
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