Existence of weighted bounded solutions for nonlinear discrete-time fractional equations. Issue 10 (26th July 2020) - Leal, Claudio
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 Δ α u ( n ) = T u ( n ) + f ( n, u ( n ) ),   n ∈ N 0,   1 < α ≤ 2 ; u ( 0 ) = u 0, u ( 1 ) = u 1, where Δ α corresponds to the fractional difference operator of order α > 0 in sense of Riemann–Liouville and f : N 0 × X → 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 ∞ ( N 2 ; X ) = η : N 2 → X / sup n ≥ 2 η ( n ) n n ! < ∞ .