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

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

∆(a

_{n}|∆_{X}_{n}|^{α}sgn ∆x_{n}) + b_{n}|x_{n+q}|^{α}sgn x_{n+q }= 0,where ∆ is the forward difference operator ∆

_{Un}= u_{n+1 }- u_{n}, a = {a_{n }} , b = {b_{n}} 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∆(a

_{n}|∆y_{n}|^{α}sgn ∆y_{n}) + b_{n}|y_{n+1}|^{α}sgn y_{n+1 }= 0.As a result, we obtain necessary and sufficient conditions for the existence of intermediate solutions, i.e., eventually positive solutions x satisfying lim

_{n}x_{n}= + ∞ , lim_{n}a_{n}∆x_{n}I^{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.