Existence of positive strongly decaying solutions of second-order nonlinear q-difference equations

Abstract

This paper investigates the existence of positive strongly decaying solutions of second-order nonlinear q-difference equation $D_q(a(t){\mathrm{\Phi }}_{\alpha }(D_q(x(t))))=b(t){\mathrm{\Phi }}_{\beta }(x(qt)),\alpha >\beta >0,$ with q-regularly varying coefficients a and b. By means of a fixed point theorem in partially ordered Banach space, necessary and sufficient conditions for the existence of solutions satisfying $x(t)\to 0$ and $a(t){\mathrm{\Phi }}_{\alpha }(D_q(x(t)))\to 0$ as $t\to \mathrm{\infty }$ are established. Moreover, with the help of q-regular varying theory, precise asymptotic behaviour of such solutions is described.

Keywords:

• q-difference equation
• nonoscillatory solution
• strongly decaying solutions
• asymptotic behaviour
• regular variation
• q-regular variation

MSC2010:

• 26A12
• 39A13
• 39A22
• 34N05

Acknowledgments

Authors would like to express their sincere thanks to referees for a number of useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 $\mu (t)=\sigma (t)-t,\sigma (t)=\text{inf}\{s>t:s\in \mathbb{T}\}$.

Additional information

Funding

Authors acknowledge financial support through the Ministry of Education and Science of Republic of Serbia agreement no. 451-03-68/2020-14/200124.