Kannan nonexpansive maps on generalized Cesàro backward difference sequence space of non-absolute type with applications to summable equations
Awad A. Bakery, OM Kalthum S. K. Mohamed
Abstract
Abstract In this article, we investigate the notion of the pre-quasi norm on a generalized Cesàro backward difference sequence space of non-absolute type $(\Xi (\Delta,r) )_{\psi }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ξ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> under definite function ψ . We introduce the sufficient set-up on it to form a pre-quasi Banach and a closed special space of sequences (sss), the actuality of a fixed point of a Kannan pre-quasi norm contraction mapping on $(\Xi (\Delta,r) )_{\psi }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ξ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> , it supports the property (R) and has the pre-quasi normal structure property. The existence of a fixed point of the Kannan pre-quasi norm nonexpansive mapping on $(\Xi (\Delta,r) )_{\psi }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ξ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> and the Kannan pre-quasi norm contraction mapping in the pre-quasi Banach operator ideal constructed by $(\Xi (\Delta,r) )_{\psi }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ξ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> and s -numbers has been determined. Finally, we support our results by some applications to the existence of solutions of summable equations and illustrative examples.