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
Dipartimento di Matematica
Target audience: 
University community
Contact person: 
Prof. Marco Sabatini
Contact details: 
Staff Dipartimento di Matematica
Serena Matucci (Università di Firenze)


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].

[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.