Existence of weighted bounded solutions for nonlinear discrete-time fractional equations

ABSTRACT

Let X be a Banach space and T be a bounded linear operator defined on X. In this work we study the existence of solutions of a class of nonlinear difference equation of fractional order in the form $\begin{array}{rl}{\mathrm{\Delta }}^{\alpha }u\left(n\right)& =Tu\left(n\right)+f(n,u(n\left)\right),n\in {\mathbb{N}}_0,1<\alpha \le 2;\\ u\left(0\right)& =u_0,\\ u\left(1\right)& =u_1,\end{array}$ where ${\mathrm{\Delta }}^{\alpha }$ corresponds to the fractional difference operator of order $\alpha >0$ in sense of Riemann–Liouville and $f:{\mathbb{N}}_0\times X\to X$ is a function satisfying suitable conditions. More specifically, by using operator-theoretical methods and fixed point theory, we show the existence of solutions of such class of equations on the vector-valued weighted space of sequences $l_f^{\mathrm{\infty }}({\mathbb{N}}_2;X)=\left\{\eta :{\mathbb{N}}_2\to X/\underset{n\ge 2}{sup}\frac{∥\eta \left(n\right)∥}{nn!}<\mathrm{\infty }\right\}.$

KEYWORDS:

• Fractional differences
• weighted sequence space
• Banach space
• α-resolvent sequences
• fixed point

2010 AMS MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary: 35R11
• 34A08
• Secondary: 39A14
• 65Q10
• 47B39

Disclosure statement

No potential conflict of interest was reported by the author.