Seminar
A fixed point approach to asymptotic boundary value problems
for second order nonlinear difference equations with deviating argument
18 November 2022
Start time
10:00 am
PovoZero - Via Sommarive 14, Povo (Trento)
Seminar Room -1
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Organizer:
Dipartimento di Matematica
Target audience:
University community
Attendance:
Free
Contact person:
Prof. Marco Sabatini
Speaker:
Serena Matucci (Università di Firenze)
Abstract:
In this seminar we present a fixed point result, based on the Schauder lin- earization device in the Frechét space of all sequences, which can be applied to the resolution of boundary problems on noncompact intervals associated with difference equations, see [1]. This result is based on an analogue valid in the continuous case, but takes into account some peculiarities of the discrete case and extends [2, 4]. As an application, we consider the problem of the existence of the so-called intermediate solutions for a half-linear difference equation with advance argument
∆(an|∆Xn|α sgn ∆xn) + bn|xn+q|α sgn xn+q = 0,
where ∆ is the forward difference operator ∆Un = un+1 - un, a = {an } , b = {bn } are positive sequences, α > 0 and q ∈ N. In particular, the effects of the advance argument on the existence of unbounded nonoscillatory solutions, are shown by a comparison with the half-linear equation
∆(an|∆yn|α sgn ∆yn) + bn|yn+1|α sgn yn+1 = 0.
As a result, we obtain necessary and sufficient conditions for the existence of intermediate solutions, i.e., eventually positive solutions x satisfying limn xn = + ∞ , limn an ∆xnI α = 0, for the equation with advance argument, generaliz- ing some results in [3].
References
[1] Z. Doˇsl´a, M. Marini, S. Matucci, A fixed-point approach for decaying solutions of difference equations, Phil. Trans. R. Soc. A 379 (2019), Art. Id 20190374, 13 pp.
[2] Z. Doˇsl´a, M. Marini, S. Matucci, Decaying solutions for discrete boundary value problems on the half line, J. Difference Equ. Appl. 22 (2016), 1244– 1260.
[3] M. Cecchi, Z. Doˇsl´a, M. Marini, On the growth of nonoscillatory solutions for difference equations with deviating argument, Adv. Difference Equ. 2008, Art. ID 505324, 15 pp.
[4] M. Marini, S. Matucci, P. Rˇeh`ak, Boundary value problems for functional difference equations on infinite intervals, Adv. Difference Equ. 2006, Art. 31283, 14 pp.