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Abstract. In this paper, we present a survey of recent results on the Green's functions and on spectrum for stationary problems with nonlocal boundary conditions. Results of Lithuanian mathematicians in the field of differential and numerical problems with nonlocal boundary conditions are described.

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Abstract. This paper presents some new results on a spectrum in a complex plane for the second order stationary differential equation with one Bitsadze-Samarskii type nonlocal boundary condition. In this paper, we survey the characteristic function method for investigation of the spectrum of this problem. Some new results on characteristic functions are proved. Many results of this investigation are presented as graphs of characteristic functions. A definition of constant eigenvalues and the characteristic function is introduced for the Sturm-Liouville problem with general nonlocal boundary conditions.

The Sturm‐Liouville problem with various types of nonlocal integral boundary conditions is considered in this paper. In the first part of paper we investigate Sturm‐Liouville problem with two cases of nonlocal integral boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such problem in the complex case. In the second part we investigate real eigenvalues case. The spectrum depends of these problems on boundary condition parameters is analyzed. Qualitative behaviour of all eigenvalues subject to nonlocal boundary condition parameters is described. Šiame straipsnyje nagrinejamas Šturmo‐Liuvilio uždavinys su viena nelokaliaja integralinio tipo kraštine salyga. Pirmoje straipsnio dalyje tiriamas Šturmo‐Liuvilio uždavinys su dvieju tipu integraline nelokaliaja salyga. Irodytos tikriniu funkciju ir tikriniu reikšmiu bendrosios savybes komplesineje plokštumoje. Antroje dalyje plačiau ištirtas realiuju tikriniu reikšmiu atvejis. Straipsnyje nagrinejama kaip Šturmo‐Liuvilio uždavinio spektras priklauso nuo kraštiniu salygu parametru. Priklausomai nuo nelokaliuju kraštiniu salygu parametru, aprašytas kokybinis tikriniu reikšmiu pasiskirstymas.

In this article a stationary problems with general nonlocal boundary conditions is considered. The differential problems and finite difference schemes for solving this problem are investigated. Stability estimates are proved in the maximum norm and the non‐negativity of the solution is investigated. All theoretical results are illustrated by representative examples.

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The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.