Existence of weighted bounded solutions for nonlinear discrete-time fractional equations. Issue 10 (26th July 2020)

• Record Type: Journal Article
• Authors: Leal, Claudio
• Is Part Of: Applicable analysis. Volume 99:Issue 10(2020)
• Page Start: 1780
• Page End: 1794
• Publication Date: 2020-07-26
• Subjects: Fractional differences -- weighted sequence space -- Banach space -- α-resolvent sequences -- fixed point; Primary: 35R11 -- 34A08 -- Secondary: 39A14 -- 65Q10 -- 47B39; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2018.1546001
• Languages: English
• ISSNs: 0003-6811

Abstract

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 ! < ∞ .