Weighted bounded solutions for a class of nonlinear fractional equations

Abstract Let T be a bounded linear operator defined on a Banach space X. We investigate the existence of solutions for a class of nonlinear fractional equation in the form (∗)Δαu(n)=Tu(n)+f(n,u(n)),n∈N0,0<α≤1;u(0)=x, $$ \begin{equation*} (*) \left\{ % \begin{array}{rll} \Delta^{\alpha} u(n) &= Tu(n) +f(n, u(n)), \quad n \in \mathbb{N}_0, \quad 0<\alpha \leq 1;\\ u(0) &= x, \\ \end{array} % \right. \end{equation*}$$ on the vector-valued weighted sequence space lf∞(N;X)=x:N→X/supn∈N∥x(n)∥nn!<∞. $$l_f^\infty(\mathbb{N};X)=\left\{x:\mathbb{N}\to X \,\, /\,\,\sup_{n\in\mathbb{N}}\frac{\|x(n)\|}{nn!}<\infty \right\}. $$ Our analysis relies on the fixed point theory and operator-theoretical methods.

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