Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space
Ahmed Salem, Hashim M. Alshehri, Lamya Almaghamsi
Abstract
Abstract A new sequence space related to the space $\ell _{p}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℓ</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math> , $1\leq p<\infty $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>∞</mml:mi></mml:math> (the space of all absolutely p -summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.