In this work, we investigate a sequence of approximations converging to the existing unique solution of a multi-point boundary value problem(BVP) given by a linear fourth-order ordinary differential equation with variable coefficients involving nonlocal integral conditions by using reproducing kernel method(RKM). Obtaining the reproducing kernel of the reproducing kernel space by using the original conditions given directly by RKM may be troublesome and may introduce computational costs. Therefore, in these cases, initially considering more admissible conditions which will allow the reproducing kernel to be computed more easily than the original ones and then taking into account the original conditions lead us to satisfactory results. This analysis is illustrated by a numerical example. The results demonstrate that the method is still quite accurate and effective for the cases with both derivative and integral conditions even if the accuracy is less compared to the cases with just derivative conditions.

In this work, we investigate a linear completely nonhomogeneous nonlocal multipoint problem for an m-order ordinary differential equation with generally variable nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness. A system of m + 1 integro-algebraic equations called the special adjoint system is constructed for this problem. Green's functional is a solution of this special adjoint system. Its first component corresponds to Green's function for the problem. The other components correspond to the unit effects of the conditions. A solution to the problem is an integral representation which is based on using this new Green's functional. Some illustrative implementations and comparisons are provided with some known results in order to demonstrate the advantages of the proposed approach.

In this work, with the aim of determining Green's solution or generalized Green's solution, we propose a novel constructive approach by which a linear or specific nonlinear problem involving general linear nonlocal condition for a first-order functional ordinary integro-differential equation with general nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness is reduced to an integral equation. A system of two integro-algebraic equations, called "the adjoint system," is constructed for this problem. Green's functional for the problem with trivial kernel and generalized Green's functional for the problem with nontrivial kernel are the unique solutions to the specific cases of this adjoint system. Green's functional and generalized Green's functional have two components. Their first components correspond to Green's function and generalized Green's function for the problem, respectively. Some illustrative applications are provided with known and unknown results.MSC: 34B10, 34B27, 34K10, 34K30, 45J05, 47A05

In this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

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