Well-posedness of a nonlocal boundary value difference elliptic problem^★,★★

¹ Department of Mathematics, Near East University, Lefkosa, Mersin 10, Turkey.
² Peoples’ Friendship University of Russia (RUDN University), Ul Miklukho Maklaya 6, Moscow 117198, Russia.
³ Institute of Mathematics and Mathematical Modeling, 050010 Almaty, Kazakhstan.
⁴ Department of Mathematics, Near East University, Nicosia, Mersin 10, Turkey.
⁵ Department of Mathematics, Omar Al-Mukhtar University, El-Beida, Libya.

^*** Corresponding author: e-mail: ayman2952000@gmail.com

Received: 17 September 2018
Accepted: 27 November 2019

Abstract

The second order of approximation two-step difference scheme for the numerical solution of a nonlocal boundary value problem for the elliptic differential equation $$-v^{\prime \prime }\left(t\right)+\mathit{Av}\left(t\right)=f\left(t\right)\left(0\le t\le T\right),v\left(0\right)=v\left(T\right)+\phi ,\underset{0}{\overset{T}{\int }}v\left(s\right)\mathrm{\text{d}}s=\psi$$ in an arbitrary Banach space E with the positive operator A is presented. The well-posedness of the difference scheme in Banach spaces is established. In applications, the stability, almost coercive stability and coercive stability estimates in maximum norm in one variable for the solutions of difference schemes for numerical solution of two type elliptic problems are obtained.