Dynamic Systems and Applications 30 (2021) No.9, 1479-1501 EXISTENCE RESULTS FOR A SELF-ADJOINT COUPLED SYSTEM OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH NONLOCAL NON-SEPARATED INTEGRAL BOUNDARY CONDITIONS AHMED ALSAEDI a , AMAL ALMALKI a , SOTIRIS K. NTOUYAS b,a , BASHIR AHMAD a , AND RAVI P. AGARWAL c,a a Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. b Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece. c Department of Mathematics, Texas A& M University, Kingsville, Texas 78363-8202, USA. ABSTRACT. In this article, we develop the existence theory for a self-adjoint coupled system of nonlinear ordinary differential equations supplemented with nonlocal non-separated integral bound- ary conditions on an arbitrary domain. The two existence results depend on the Leray-Schauder alternative and Schauder fixed point theorem, while the uniqueness result relies on the Banach contraction mapping principle. For the illustration of the obtained result, we constructed several examples in the last section. AMS (MOS) Subject Classification. 34B10, 34B15, 47H10. 1. INTRODUCTION Consider the following self-adjoint coupled system of nonlinear second-order or- dinary differential equations on an arbitrary domain complemented with nonlocal non-separated integral boundary conditions of the form: (1.1)                            ( p(t)u ′ (t) ) ′ = µ 1 f (t,u(t),v(t)),t ∈ [a,b], ( q(t)v ′ (t) ) ′ = µ 2 g(t,u(t),v(t)),t ∈ [a,b], α 1 u(a)+ α 2 u(b)= λ 1  η a v(s)ds, α 3 u ′ (a)+ α 4 u ′ (b)= λ 2  η a v ′ (s)ds, β 1 v(a)+ β 2 v(b)= λ 3  b ξ u(s)ds, β 3 v ′ (a)+ β 4 v ′ (b)= λ 4  b ξ u ′ (s)ds, Received July 1, 2021 ISSN1056-2176(Print); ISSN 2693-5295 (online) 15.00 Dynamic Publishers, Inc. www.dynamicpublishers.org; https://doi.org/10.46719/dsa202130.09.06.