A survey on stationary problems , Green ’ s functions and spectrum of Sturm – Liouville problem with nonlocal boundary conditions ∗

In the theory of differential equations, the basic concepts have been formulated studying the problems of classical mathematical physics. However, the modern problems motivate to formulate and investigate the new ones, for example, a class of nonlocal problems. Nonlocal conditions arise when we cannot measure data directly at the boundary. In this case, the problem is formulated, where the value of the solution and/or a derivative is linked to a few points or the whole interval. A review on differential equations with more general boundary conditions (BC) involving also Stieltjes measures has been written by Whyburn [232]. In 1963, Cannon [26] formulated new problem with BCs, which are now called nonlocal. The term “nonlocal boundary value problem” most likely to have been used by Beals in 1964 [12, 13]. He investigated elliptic type differential equations with nonclassical BCs. A parabolic problem with integral boundary condition


Introduction
In the theory of differential equations, the basic concepts have been formulated studying the problems of classical mathematical physics.However, the modern problems motivate to formulate and investigate the new ones, for example, a class of nonlocal problems.Nonlocal conditions arise when we cannot measure data directly at the boundary.In this case, the problem is formulated, where the value of the solution and/or a derivative is linked to a few points or the whole interval.
A review on differential equations with more general boundary conditions (BC) involving also Stieltjes measures has been written by Whyburn [232].In 1963, Cannon [26] formulated new problem with BCs, which are now called nonlocal.The term "nonlocal boundary value problem" most likely to have been used by Beals in 1964 [12,13].He investigated elliptic type differential equations with nonclassical BCs.A parabolic problem with integral boundary condition was investigated by Kamynin in 1964 [109].Samarskii and Bitsadze investigated nonlocal problem for general elliptic equation and, in the case of rectangular space domain and Laplace operator, applying methods of the theory of integral equations.They proved the existence and uniqueness of solution in 1969 [19].Now some one-dimensional nonlocal boundary conditions (NBC) are called as Bitsadze-Samarskii type NBCs.Differential equations (for example, ordinary, elliptic, parabolic etc.) with various types of nonlocal conditions were investigated by many scientists.Two-points NBCs we can find in [90].Il'in [78] and Moiseev [80,81] investigated multi-points NBCs (the study of multi-point BCs was initiated by Picone [159]) Ionkin [87] considered parabolic problem with u(0, t) = ν(t), In [91], he investigate stability FDS for a parabolic equation with NBCs u(0, t) = 0, u x (1, t) = u x (0, t) (Samarskii-Ionkin type NBCs).Day investigated the heat equation subject to linear thermoelasticity with integral BCs [43,44] u(0, t) = γ 0 1 Henderson proved existence theorems for BVPs for nth-order nonlinear difference equations [72].A survey of numerical methods for the one-dimensional parabolic equation subject to nonlocal conditions was done by Dehghan [45].Webb [229,230], Infante [85,86] investigated quasilinear problems with Bitsadze-Samarskii type NBCs and had proved results about the properties of positive solutions.We can find more results on the existence and multiplicity of solutions of nonlocal BVP involving the second-order ordinary differential equations in surveys of Ma [125] and Ntouyas [144].In 2011, special issue for Nonlocal Boundary Conditions (27 articles) was published in the journal Boundary Value Problems [52].Articles in this special issue deal with BVPs with NCs for ordinary, discrete, impulsive, neutral, parabolic, fractional and time scales equations.Existence, nonexistence, multiplicity, asymptotic behavior and approximation of solutions are investigated by using several methods as fixed point theorems, fixed point index, variational methods, iterative techniques, bifurcation theory, lower and upper solutions.
One of the objectives of this study is to describe the results of Lithuanian mathematicians in the field of differential and numerical problems with NBCs.Prof. M. Sapagovas was not only a pioneer in the study of such problems, but also the founder of the scientific school in Vilnius.The first problem with NBCs came from the applications, and it was the investigation of a mercury droplet in electric contact, given the droplet volume [174][175][176][178][179][180].Difference scheme for two-dimensional elliptic problem with an integral condition was constructed in [177].Scientific supervisor of Sapagovas (Kiev, 1963-1965) Prof. V. Makarov also began investigating problems with NBCs [127][128][129].Sapagovas and his doctoral student (Vilnius, 1982(Vilnius, -1985) ) Čiegis investigated elliptic and parabolic problems with integral and Bitsadze-Samarskii type NBCs and finite-difference schemes for them [31,181,182].They published some new results about numerical solutions for problems with NBCs in [32,183,190,191].
Gulin and Mokin have similar results for NBCs [58,138] u(0, t) = 0, Stability of a family of difference schemes for the Samarskii-Ionkin problem with variable coefficient is obtained in [139].Sapagovas with co-authors began to investigate eigenvalues for Bitsadze-Samarskii type and integral type NBCs [41,184,185,197] They showed that there exists eigenvalues, which do not depend on parameters γ 0 or γ 1 in boundary conditions (see Eqs. ( 2) and ( 6)) and complex eigenvalues may exist. .In [38] and in doctoral dissertation of Tumanova (scientific supervisor Čiegis) [223], the one-dimensional parabolic equation with three types of integral NBCs was approximated by the implicit Euler FDS and stability analysis was done.Some new results published by doctoral students (scientific supervisor Sapagovas): Jesevičiūtė (Jokšienė) investigate stability of the FDS for parabolic equations subject to integral conditions with applications for thermoelasticity [106,107], Jakubėlienė investigates twodimensional problems with double integral in BC [100,102,193], Jachimavičienė investigates pseudo-parabolic equation with NBCs and FDS [96,97,99].The stability of explicit three-layer FDS for pseudo-parabolic problems were investigated in papers [ In this survey, we present recent results on stationary problems (Section 2), Green's functions (Section 3) and spectrum of SLP with NBCs (Section 4) and selected problems with NBCs (Section 5).Some of them were obtained by Sapagovas scientists or scientific school, which he created.

Stationary problem with nonlocal boundary conditions and characteristic curve
One of the directions of Sapagovas scientific school is the development of a general theory for problems with NBCs.The various linear NBCs (1)-( 11) can be written as linear functional (for classical BC, see [42]).The first results in this direction were obtained in [35].In this paper, the second-order linear stationary equation with nonlocal boundary conditions was investigated where k 0 and k 1 are linear functionals: Note that given expressions are examples of functionals and the results were proved in a general case.For a parabolic equation (in the case α 0 = α 1 = 0), Ekolin [49] proved the convergence of the forward and backward Euler methods.The integrals in the nonlocal boundary conditions were approximated by the trapezoidal rule.The convergence of these methods is proved under the assumption that where (•, •) denotes the standard inner product (f, g) = 1 0 f (x)g(x) dx.For the Crank-Nicolson method, the convergence is proved under the assumption that where f := (f, f ).Using Galerkin method, Fairweather and Lopez-Marcos [51] solved semilinear parabolic problem with integral boundary conditions under the assumptions In papers [35,36], analogous results for FDS and for implicit FDS for parabolic equation were established.In papers [35,36] and in Suboč doctoral dissertation (scientific supervisor Čiegis) [220], the dependence of solution on parameters γ 0 and γ 1 was investigated.The main result of these papers is that the solution for differential problem with NBCs ( 15)-( 16) or solution for FDS with NBC ( 21)-( 22) exists and is unique for all γ 0 0 and γ 1 0 except points on hyperbola or line(s).So, sufficient and necessary conditions for existence of unique solution for stationary problem were found.In [36], the stability region in the plane of parameters was found and the stability of FDS stationary problem and implicit FDS for a parabolic equation with NBCs were proved.These results used for investigation of FDS for nonlinear parabolic equation [34].
In [38], the one dimensional parabolic equation with three types of integral nonlocal boundary conditions is approximated by the implicit Euler finite difference scheme.Stability analysis is done in the maximum norm and it is proved that the radius of the stability region depends on the signs of coefficients in the nonlocal boundary condition.
The analysis of stationary problem with one classical boundary condition (γ 0 = 0) and another NBC [197,214,218] and investigation of auxiliary stationary problems [107,187,198] shows that restrictions γ 0 0 and γ 1 0 are not necessary, and, in general case, we can take γ 0 , γ 1 ∈ R. The main research tools in [35,36] were the maximum principle and comparison theorems.Existence and uniqueness of a solution for the stationary problem is equivalent to existence of zero eigenvalue.So, we formulate SLP with two nonlocal boundary conditions (as functional conditions): We can write many problems with NBC in this form, where k i , u is a classical part and n i , u , i = 0, 1, is a nonlocal part of boundary conditions.For example, the functionals n i , i = 0, 1, can describe the multi-point or integral NBCs and the functional k i , i = 0, 1, can describe the local (classical) boundary conditions.If γ 0 = γ 1 = 0, then problem ( 23)-( 25) becomes classical.
The example of NBCs is Sometimes instead of NBCs ( 26), the following conditions (one or both) are used for SLP: Formally, we can say that such cases are realized for γ 0 = ∞ or γ 1 = ∞.More general problem will be if we consider NBCs Now left-hand side of these BC is "classical" and right-hand side "nonlocal". www.mii.lt/NA The condition for existence of zero eigenvalue is where u = [u 0 , u 1 ] is any fundamental system of stationary equation (23) and We call the solution of equation ( 29) a characteristic curve for problem ( 23)-( 25) and denote a set of it's points in plane R 2 γ0,γ1 by the letter C.
The two main results on characteristic curve are the following lemmas [215].
Lemma 2. On torus, Characteristic Curve for problem (23)-( 25) can be one of the following three types: , then the curve is homeomorphic to a circle, and this curve winds around the torus one time (one time in one direction and one in the other direction); , then the curve is the union of two circles (strictly "latitudinal" and strictly "longitudinal") with one common point; (iii) Otherwise (i.e., A = O), the curve is whole torus.

These lemmas are valid for the discrete SLP
where (γ 0 , γ Green's functions for problems with classical boundary conditions were described by Stakgold [212,213], Duffy [47].We can find Green's functions for problems with difference operator in the books of Samarskii and Nikolaev [172,173] and the textbook of Bahvalov, Zhidkov and Kobel'kov [9].The investigation of the semilinear problems with NBCs and the existence of positive solutions are based on the investigation of Green's function for the linear problems with NBCs.Green's functions for the second-and higherorder boundary problems with various NCs were constructed by Anderson [3,4], Webb and Infante [82,83,85,86,226,227], Ma and An [121,124,126], Sun [221], Truong, Ngoc and Long [222], Yang [236], Zhao [240,241], Xie, Liu and Bai [233] and other scientists.
In their works, authors considered the existence and multiplicity of solutions by applying various methods: lower and upper solution method, Leggett-Williams fixed-point theorem, Guo-Krasnoselskii fixed-point theorem, Leray-Schauder continuation principle, Avery-Peterson fixed-point theorem.Henderson and Ntouyas [77], Guo, Sun and Zhao [67], Liu and O'Regan [117] have considered the third-and higher-order differential equations with various NBCs and investigate the existence of solutions.Bai [10] proved the existence of one or two positive solutions for the nonlocal fourth-order BVP L. Kong and Q. Kong [113] considered nonlinear BVP with nonhomogeneous multi-point BCs.Hao, Liu, Wub and Sun [69] considered nth-order singular nonlocal BVP with Riemann-Stieltjes integral in one BC and proved existence of positive solutions.Zhao et al. [239] investigated the second-order BVP with four-point BC.The existence results of multiple monotone and convex positive solutions for some fourth-order multipoint BVP was established by Liu, Weiguo and Chunfang in [118].Ma [122] studied the existence and nonexistence of positive solutions of nonlinear periodic BVP and Green's function for corresponding linear problem.Henderson and Luca in [74,75] presented some results for positive solutions of a system of nonlinear the second-order ODEs subject to multi-point BCs.Wang and An [224] investigated the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous m-point BVP with two parameters.Under conditions weaker than those used by Ma, Zhang and Ge [238] established various results on the existence and nonexistence of symmetric positive solutions to fourthorder BVP with integral BC.By using a fixed point theorem in a cone and the nonlocal third-order BVP Green's function, the existence of at least one positive solution for the third-order BVP with the integral BC was considered by Guo and Yang in [68].A class of second-order three-point integral BVP at resonance was investigated in [116] by Liu and Ouyang.Jankowski established the existence of at least three nonnegative solutions of some nonlocal BVP to third-order differential equations with advanced arguments [103].He considered nonlocal BVP for systems of second-order differential equations with dependence on the first-order derivatives and deviating arguments [104].In these papers, nonlinear differential equations has nonlinearity in the right-hand side only (see ( 33)-( 34) as examples).Green's functions were constructed for simple linear differential operator.They were used for investigation of properties of solutions nonlinear differential equations.
The investigations of Green's functions and relations between them there were one of the main task of the group of researchers in Vilnius university.Linear NBCs can be written in the form ( 24)- (25).The first results were obtained for the second-order differential problem in [165,166].These results were generalised in [216] for BVP In this paper, the second-order linear differential equation with two additional conditions was investigated and Green's function was constructed.The relation between two Green's functions for two such problems with different additional conditions was derived and general formulae were applied to a stationary problem with NBCs ( 24)-( 25).Green's function for a problem with NBCs can be expressed per Green's function for a problem with classical BCs.The differential equation ( 35) can be approximated by the difference equation where a 2 , a 0 = 0. Let S be a two-dimensional linear space of solutions for Eq.(35) or Eq.(37).Then the following lemma is valid for the differential problem [216] and discrete problem [168] with two additional conditions (36).
Lemma 3. Let {u 1 , u 2 } be the basis of the linear space S. Then the following propositions are equivalent: Green's function for problem (37) with homogeneous conditions L 1 , u = 0, L 2 , u = 0, where the functionals L 1 and L 2 are linearly independent, is .
For the theoretical investigation of problems with NBCs, the next result about the relations between Green's functions G u ij and G v ij of two nonhomogeneous discrete problems is useful [168]. .
Let us investigate Green's function for the problem with nonlocal boundary conditions If γ 1 , γ 2 = 0, then problem ( 38)-( 39) becomes classical.Suppose that there exists Green's function G cl ij for the classical case.If ϑ := D(L)[u] = 0, then Green's function exists for problem (37)- (39).The following formula is valid: (θ = 0 is equation of the characteristic curve) and Analogous results are valid in differential case [216] and for the third-order linear differential equation with three additional conditions [167].In the case of mth-order differential equation with m additional conditions or discrete mth-order problem formulae for Green's functions were derived in [164,217].These results are summarized in Roman doctoral dissertation [163].

Generalized Green's functions
In the case D[L](u) = 0, Green's function do not exist.For investigation of solutions, we can construct generalized Green's function [14].Most authors have developed a theory of generalized Green's matrix for systems of differential equations with two-points BCs. Brown used generalized Green's functions and generalized inverses for linear differential systems with Stieltjes BCs [23,24].Locker investigated the generalized Green's function for nth-order linear differential operator and k linearly independent boundary values as a kernel of the Moore-Penrose inverse that describes the minimum norm least squares solution for compatible BVP [119,120].Now we will review recent articles related to generalized Green's function for differential equations and some applications.In [20], linear Fredholm's BVPs for systems of ODE with constant coefficients and a single delay were investigated, assuming that these solutions satisfy the initial and boundary conditions.A method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions.Some commonly used interpolation algorithms were analysed in [46].Among all of the methods, biharmonic spline interpolation was considered, which is based on Green's function.In [135], a new class of generalized inverses on semigroups was introduced by means of Green's relations.The classical generalized inverses (group inverse, Drazin inverse and Moore-Penrose inverse) belong to this class.In [1], Abu-Saman reviews the mathematical considerations behind the generalized inverse of a matrix.Simple derivations for the determination of different types of generalized inverses of a matrix are presented.These include results of the generalized inverse of singular and rectangular matrices.It also includes applications of the generalized inverse to solution of a set of linearly dependent equations.In [115], Li and Zhang extend and generalize the standard spectral graph theory on undirected graphs to digraphs.In particular, they introduce and define a normalized digraph Laplacian for digraphs, and prove that its Moore-Penrose pseudoinverse is the discrete Green's function of the Diplacian matrix as an operator on digraphs.Xu and Yau proved an explicit formula of Chung-Yau's discrete Green's functions as well as hitting times of random walks on graphs [234].Maroncelli and Rodríguez investigated least squares solutions for a linear nonhomogeneous BVP with impulses [132].The analysis of least squares solution of minimal norm is directly related to generalized inverses and generalized Green's functions for current BVP.The explicit formula of least squares solution is given.
Paukštaitė investigated a generalized Green's function that describes the minimum norm least squares solution of every second-order discrete problem with two nonlocal conditions.In [149][150][151], properties of generalized Green's function that are analogous to properties of ordinary Green's function were proved.She investigated the nullity of discrete problem and presented its classifications.Null spaces of discrete problem and its adjoint problem were also analysed, explicit formulas of bases of null spaces were found.Moreover, the necessary and sufficient existence conditions of exact solutions were given.

Sturm-Liouville problem
An eigenvalue problem for the second-order ordinary differential operator with nonlocal condition was formulated and analysed by Ionkin [88,89] and by Cahlon, Kulkarni, Shi [25].In these works, eigenvalue problem is associated with the stability of difference schemes for one-dimensional parabolic equations.In Sapagovas article [185], the eigenvalue problem is associated with iterative methods for finite-difference schemes for problems with NBCs.Shkalikov [204] investigated the properties of eigenfunctions.We describe the results of Gulin with co-authors in Section 1.Note that the eigenvalue for problems with NBCs is the separate part of the general nonselfadjoint operator theory (see the article of Il'in [79] and the book of Mennicken and Möller [136]).
The structure of spectrum remains uncertain for problems with nonlocal boundary conditions and variable coefficients.However, even for the simplest operator − d 2 / dx 2 interesting results on the spectrum were obtained.Čiupaila, Jesevičiūtė and Sapagovas [41] investigated eigenvalue problem For this problem, real eigenvalues exist only in the following cases: if γ < 2, then all eigenvalues are positive; if γ 2, then all eigenvalues are positive, except one negative eigenvalue (γ > 2) or zero eigenvalue (γ = 2).The same result is valid for NBCs Ionkin and Valikova [92] for NBCs (7) prove that all nonzero eigenvalues are not simple, i.e., for each such eigenvalue, there exist eigenfunction and generalized eigenfunction.Such (multiply) eigenvalues (see [41]) exist for NBCs too.In [197], Sapagovas and Štikonas investigate the eigenvalue problem with one Bitsadze-Samarskii type NBC They found that some eigenvalues do not depend on parameter γ and for some γ complex eigenvalues exist.More results about real eigenvalues for this problem are in [214].SLP ( 40)-( 41) with one of the two cases of integral type NBCs were investigated in [156,157].All eigenvalues (countable set) of problem ( 40)-( 41), (45b) are real and simple.If , then there exist one negative eigenvalue.In the case of the boundary condition (45a), multiple and complex eigenvalues may exist for some values of parameter γ.SLP ( 40)-( 41) with one of the three cases of two-point NBCs were investigated in [153].In the case of boundary (46c), two negative eigenvalues may exist for some values of parameter γ.Properties of eigenfunctions for such problems were obtained in [154].These results are part of doctoral dissertation of Pečiulytė [152].The properties of negative eigenvalues and negative critical points of a characteristic function for such problems were investigated in [158].Some results about all critical points are described in [155].
In [218], the characteristic function method for investigation of the spectrum for the problem ( 40), ( 41), (44) were used.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 NBC.Some new results on characteristic functions are proved.A definition of constant eigenvalues and the characteristic function is introduced for the SLP with general NBCs.This method of the characteristic function used for investigation of complex eigenvalues and their qualitative behaviour (dynamics) if parameters γ and ξ in NBC (45) are changed [205].The dynamics of complex eigenvalues for analogues discrete problems is shown in [207] (NBC (45) are approximated by the trapezoidal rule or by Simpson's rule) and in [207] (two-points NBCs).In [206], for SLP (40), (41) with NBC characteristic functions in the neighbourhood constant eigenvalue point and in the neighbourhood of the critical point of the second order were investigated.For NBC analogous investigation was done in [208].The paper of Bandyrskii, Lazurchak, Makarov and Sapagovas [11] deals with numerical methods for eigenvalue problem for the second-order ordinary differential operator with variable coefficient u + λ − q(x) u = 0, x ∈ (0, 1), subject to NBC (43).FD method (functional-discrete method) is derived and analysed for calculating of eigenvalues, particularly complex eigenvalues.Jesevičiūtė and Sapagovas [107] investigated the stability of FDS for parabolic equation with integral type NBCs with variable weight functions and analysed discrete SLP in the case of various weights α 0 and α 1 (for approximation integrals the trapezoidal rule was used).In [105], an eigenvalue problem for a differential operator with nonlocal integral conditions, when variable coefficients arise in nonlocal Nonlinear Anal.Model.Control, 2014, Vol. 19, No. 3, 301-334 integral conditions, was investigated and was found how eigenvalues depend on the parameters occurring in the nonlocal boundary conditions.Discrete SLP was analysed by Jachimavičienė, Jesevičiūtė and Sapagovas [99].These results are part of doctoral dissertation of Jesevičiūtė (Jokšienė) [106].Instead BC (50), Jachimavičienė obtained few new results [95,96] about spectrum for NBCs Jakubėlienė analysed an eigenvalue problem for the system with boundary conditions and analogous discrete SLP in [100,102].The spectrum and eigenfunctions for SLP ( 49), ( 51)-( 52) were investigated for all values of parameters γ 1 and γ 2 in [143].For this problem: 1) all eigenvalues are simple and real; 2 Chanane [30] used the regularized sampling method introduced recently to compute the eigenvalues of Sturm-Liouville problems In [108], the second-order singular Sturm-Liouville integral boundary value problem was concerned and the existence of at least one positive solution was proved by Jiang, Liu, Wub.Ma and An [123] considered the nonlinear eigenvalue problems They investigated the global structure of positive solutions by using global bifurcation techniques.Hao, Liu, Wub, Sun [70] similar n-order problem considered under some conditions concerning the first eigenvalue corresponding to the relevant linear operator.The existence of positive solutions is obtained by means of the fixed point index theory in cones.Hao, Liu, Wub, Xu in [71] considered a class of singular n-order nonlocal BVPs in Banach spaces.The existence of multiple positive solutions for the problem was obtained by using the fixed point index theory of strict set contraction operators.Karulina [111] considered the Sturm-Liouville problem with symmetric boundary conditions and an integral condition and estimated the first eigenvalue of this problem for different values of the parameters.Nizhnik solved the inverse spectral problem for a class of Sturm-Liouville operators with singular nonlocal potentials and NBCs on a star graph [142].By employing known Guo-Krasnoselskii fixed point theorem, Wang, Liu and Zhang investigated the eigenvalue interval for the existence and non existence of at least one positive solution of nonlinear fractional differential equation with integral boundary conditions [225].Yan determined the principal eigenvalue of the linear and nonlinear fourth-order eigenvalue problems with integral type NBCs [235].
5 Some other recent differential and discrete problems with NBCs Note that many of such articles are connected with various applications when NBCs have clear physical or mechanical meaning [27][28][29]237].In this section, we mention some articles on BVP with NCs.Hyperbolic problems with integral type NBCs were investigated in [147] by Panejah and Panejah.Various hyperbolic equations with NBCs were investigated by a lot of authors: Duzheva considered BVP with integral BC of the first order [48]; Khaleel and Khtan in [112] used numerical method for solving the initial value problem that consists of the multi-dimensional hyperbolic equation with nonlinear integral NBCs.This Nonlinear Anal.Model.Control, 2014, Vol. 19, No. 3, 301-334 method depends on Crank-Nikolson FDS and Taylor's expansion; Martín-Vaquero and Wade [134] derived a new family of high-order stable three-level algorithms to solve the wave equation.Pulkina analysed BVPs for a hyperbolic equation with NBCs of the first and the second kind [160,161].The unique existence of classical solution of initialboundary value problem for wave equation with a special integral NBC is proved in the work of Korzyuk [114].Ashyralyev and Prenov studied FDS for initial-boundary value problem for a hyperbolic system with NBCs [8], Ashyralyev and Ozdemir analysed FDS for hyperbolic-parabolic equations with multi-point NBCs [7].In [33] Cheniguel presented new technique for solving wave equation with NBCs.
Pao and Wang considered a class of fourth-order nonlinear elliptic equation with multi-point NBCs in [148].They considered a second-order elliptic equation with NBC and the usual multi-point boundary problem in ODEs, too.The aim of the paper was to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions.The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the FDM or the FEM.The paper of D. Gordeziani, E. Gordeziani, Davitashvili and Meladze [53] deals with the formulation and analysis of a generalized nonlocal problem for the elliptic equations with variable coefficients.Berikelashvili and Khomeriki in [15,16] considered a nonlocal BVP for the Poisson equation in a rectangular domain.Dirichlet and Neumann conditions are posed on a pair of adjacent sides of a rectangle, and integral constraints are given instead of BCs on the other pair.The corresponding difference scheme is constructed and investigated.
In paper [211], the existence was proved for a solution of the nonlocal problem with integral conditions for linear PDE of the third order by Sopuev and Arkabaev.Moldojarov used the method of integral equations and the contraction mapping was used for proof unique solvability of nonlocal problem with integral conditions for a nonlinear PDE of the third order [140].Smirnov investigated the third-order nonlinear BVP with twopoint NBC [209,210].An estimation of the number of solutions to BVP and their nodal structure are established.
Mokin considered the eigenvalue problem for a nonselfadjoint difference operator with nonconstant coefficient [137].Multiplicity of eigenvalues is discussed and a region, where all eigenvalues reside is defined.Stability and convergence of FDS approximating a two-parameter nonlocal BVP for time-fractional diffusion equation were studied by Alikhanov [2].El-Shahed and Shammakh investigated the nonlinear fractional nonlocal BVP [50].They obtain the results on the existence of one and two positive solution by utilizing the results of Webb and Lan [231] involving comparison with the principal characteristic value of a related linear problem to the fractional case.We then use the theory worked out by Webb and Infante in [228,229] to study the general NBCs.www.mii.lt/NAHenderson and Luca [76] investigated the existence of positive solutions for systems of singular nonlinear higher-order differential equations subject to multi-point BCs.Bolojan-Nica, Infante and Pietramala [21] studied the existence of solutions for nonlinear firstorder impulsive systems with NBCs.Bolojan-Nica, Infante and Precup in [22] were developed an existence theory for first-order differential systems with coupled NCs given by Stieltjes integrals.The approach was based on the fixed point theorems of Perov, Schauder and Schaefer and on a vector method for treating systems, which uses matrices having spectral radius less than one.
Green's functionals and reproducing kernel method was used for BVP with NBCs in the articles of Niu [141], Özen and Oruçoglu [145,146].Sergejeva investigated Fučìk type problem with Bitsadze-Samarskii type NBC [201] and with a damping term and integral type NBC [203].Sajavičius in his doctoral dissertation [170] analysed numerical solution of PDEs with nonlocal conditions (stability of FDS, radial basis function method).

Numerical analysis for boundary value problems with nonlocal conditions
This Special Issue contains a variety of contributions within this area of research.The following articles deal with BVPs with NBCs for ordinary, discrete, impulsive, elliptic, parabolic, pseudo-parabolic, hyperbolic and nonlinear equations.
Ashyralyev and Agirseven investigate the inverse problem of a delay parabolic equation with NBCs [5].The stability estimates are established.
Ashyralyev and Ashyralyyev study the BVP of determining the parameter of an elliptic equation in an arbitrary Banach space [6].
Berikelashvili and Khomeriki consider the Poisson equation in a rectangular domain with an integral constraints [17].The corresponding FDS scheme is constructed.
Čiegis, Suboč and Bugajev investigate three-dimensional parabolic and pseudo-parabolic equations with classical, periodic and NBCs [37].Equations are approximated by the full approximation backward Euler method, locally one dimensional and Douglas ADI splitting schemes.
Henderson is concerned with differentiating solutions of BVPs with respect to boundary data for the nth-order ordinary differential equation, satisfying the Dirichlet and multistrip integral boundary conditions [73].
Infante and Pietramala study the existence of nonnegative solutions for a system of impulsive differential equations subject to nonlinear, NBCs [84].The system has a coupling in the differential equations and in the boundary conditions.
Makarov, Sytnyk and Vasylyk study a nonlocal-in-time evolutional problem for the first-order differential equation in Banach space [130].It results in the necessary and sufficient conditions for the existence of a generalized solution to the given nonlocal problem.
Martín-Vaquero numerically study polynomial-based mean weighted residuals methods for the solution of elliptic problems with NBCs in rectangular domains [133].
Rutkauskas studies Dirichlet type problem in a bounded domain for the system of linear elliptic equations of the second order, which degenerate into the first-order system at a line crossing the domain [169].The existence and uniqueness of a solution are proved without any additional condition at line of degeneracy.
Serbina considers the nonlocal initial and BVP for Lavrentiev-Bitsadze equation [200].This problem models the nonstationary one-dimensional motion of a groundwater with horizontal stopping.
Sergejeva considers the Fučík spectrum for the second-order BVP with NBC [202].The explicit formulas for the spectrum of this problem are given.
Štikonienė, Sapagovas and Čiupaila consider convergence of iterative processes for elliptic differential problem with nonlocal conditions [219].
We would like to thank the authors for their contributions and the anonymous referees, the Editorial Board of Nonlinear Analysis: Modelling and Control for the realization of this volume.
It is our great pleasure to thank Professor F. Ivanauskas, Editor-in-Chief of Nonlinear Analysis: Modelling and Control, for giving us the opportunity to prepare this volume.