Nonlocal boundary value problem in terms of flow for Sturm-Liouville operator in differential and difference statements

Abstract Sturm-Liouville operator with second kind of nonlocal boundary value conditions is considered. For the classical solution, a priori estimate is established and unique existence is proved. Associated finite-difference scheme is proposed on uniform mesh and second-order accuracy for approximation is proved. An application of obtained results to nonlocal boundary problems with weight integral conditions is provided.


Introduction
For the first time, nonlocal boundary value problems (NLBVP) were considered by M. Picone [1], A. Sommerfeld [2], R. Mises [3] and J. D. Tamarkin [4] at the beginning of 20th century. Modern research on NLBVP for last fifty years was strongly motivated by two-page joint article of A. V. Bitsadze and A. A. Samarskii [5]. V. A. Il'in and E. I. Moiseev, in their joint works [6,7], considered differential and difference statements of NLBVP for the Sturm-Liouville operator with first kind of nonlocal boundary (NLB) conditions where NLB conditions are written in terms of flow function Π(x) = k(x)u ′ (x), respectively.
First and second kinds of NLBVP, formulated in [6,7], were considered under assumption that nonlocality factors α k have the same signs. For classical solution of differential NLBVP, a priori estimations were established and theorems on unique existence were proved. Associated finite-difference schemes for numerical approximation of above mentioned NLBVP were proposed on uniform mesh with second-order of accuracy. V. A. I'in and E. I. Moiseev in [8] investigated criteria for a priori estimations of solutions for a wide class of conjugate nonlocal problems for the Sturm-Liouville operator with first, second, mixed and integral kinds of NLB conditions. Subsequently to [6,7], some variations for first kind NLBVP for ordinary secondorder linear differential equations with different signs on nonlocality factors in NLB conditions were considered by the author [9]. Other variations of first and second kinds of NLBVP with associated finite-difference schemes for secondorder ordinary linear differential equations were also considered by the author [10].
Other researches on NLBVP for second-order differential equations with multipoint, integral and functional boundary conditions were conducted in [11][12][13][14][15] with respect to existence and uniqueness criteria, conditions for the well-posedness, difference interpretation and physical applications.
The references for NLBVP listed in article [10] and some recent works [16][17][18] show an unceasing interest in NLBVP for elliptic, parabolic, hyperbolic and mixed kinds of differential equations.
The present paper is stimulated by work of V. A. Il'in and E. I. Moiseev [7] and concerns the specific variations of NLBVP regarding the Sturm-Liouville operator with NLB conditions when nonlocality factors have different signs. In the current paper, we get a priori estimations and prove unique existence for classical solution of the differential problem. We propose the second-order of accuracy difference scheme on uniform mesh for approximation of smooth solution in difference metrics C, W 1 2 and W 2 2 .

Note 2.
To prove a priori estimation (3) under assumption n k=1 α k − m l=1 β l = 1 while ζ n < η 1 we require the limitation q(x) ≥ m 1 > 0 in particular. It is easy to show that this bound on q(x) is essential, too. Actually, if α − β = 1, then the homogenous equation u ′′ (x) = 0 with NLB conditions (2) has the non-trivial solution u(x) = cx ∀ c = 0. It means that without this essential bound a priori estimation (3) is not valid in general. Note 3. By the way, for the homogenous equation u ′′ (x) = 0 with NLB conditions (2) the non-trivial solution u(x) = cx also exists if α > 1 and 0 < β = α−1. It means that for α > 1 the estimation (3) is not valid in general, too. Proof. The uniqueness of the solution follows from a priori estimation which was proved in Theorem 2.1.
Following to [7, p. 1425], for an arbitrary function F (x) we consider the following problem The solution of this problem is defined by where For the expression in square brackets of (16) the following is true: Indeed, from MV property for continuous function P (x) we have some pointŝ ζ ∈ [ζ 1 , ζ n ] andη ∈ [η 1 , η m ] which satisfy the equality part in (17). Further, for these points we have the following cases: (ii) 0 <η <ζ < 1, 0 < α ≤ 1 when otherwise. Since the function P (x), defined by (15), is continuous, positive and strictly increasing, the following inequalities are valid for case (i): and the following inequalities are valid for case (ii): Then inequality (17) is true; therefore, constant A defined by (16) always exists for u(x) given by expression (14). (14) makes the problem (1),(2) equivalent to corresponding Fredholm integral equation of the second kind where Second kind of NLBVP for Sturm-Lioville operator 45 for k = 1, ..., n, for l = 1, ..., m and From defined expressions for kernels K 0 (x, t), K 1 (x, t), Z k (x, t) (k = 1, ..., n) and H l (x, t) (l = 1, ..., m) it follows that each kernel is a continuous function on the square 0 ≤ x ≤ 1, 0 ≤ t ≤ 1; therefore, K(x, t) is also continuous on this square. So, the Fredholm alternative is applicable for integral equation (18) in Hilbert space L 2 (0, 1). Suppose thatf (x) ≡ 0 on [0, 1] for integral equation (18). On this assumption we will show that f(x) ≡ 0 on [0, 1].

Finite-difference scheme
We consider a numerical statement for NLBVP (1),(2) on uniform mesh with step size h > 0, ω i = {x i = ih, i = 1, 2, ..., N − 1; x 0 = 0, x N = 1}. For differential operator (1), we use the following finite-difference approximation (see [19, p.148]) Furthermore, here we choose the step size h strictly less than a half of lowest distance between any two points from the set For NLB conditions (2), we apply the following difference approximation based on the approach from [7] y 0 = 0, Here, numbers i k (k = 1, ..., n) are defined such that i k h < ζ k ≤ (i k + 1)h for each ζ k and numbers i l (l = 1, ..., m) are defined correspondingly such that i l h < η l ≤ (i l + 1)h for each η l . Proof. Let z i = y i − u i , then z i is the numerical solution for the following difference scheme holds [7, p. 1428-1429]. Let us denote with v i being the solution of the difference scheme approximating the associated differential problem Since for the solution of difference problem (28),(29) the estimation holds [7, p. 1429],z can be considered as a solution of the following scheme Therefore, p i (z) = 2 −1 [(azx) i + (azx) i+1 ], i = 1, ..., N − 1 is a non-decreasing and non-negative mesh function. Then, for any ζ k , k = 1, ..., n and for any η l , l = 1, ..., m. Now, by linear interpolation of values of mesh functions p i (z), i = 1, ..., N −1, with add-in value p N (z) = p 0 N and arbitrary p 0 (z), for example equal to zero, we define a piecewise functionp(x) continuous on the whole interval [0, 1]. By using MV property for the continuous functionp(x) regarding the approximation (31), we get the following equality Since the interpolation functionp(x) is non-decreasing and non-negative, the following inequalities follow from (32) directly: Moreover, we have Therefore, the estimation p 0 N (z) = O(h 2 ) is valid in case (i) when −∞ < α − β < 1 withζ ≤η, as well as in case (ii) when 0 < α < 1 withζ >η. It means that the numerical solution of (30),(31) can be considered as a corresponding solution of the following difference scheme: Let us show that the estimation p 0 N (z) = O(h 2 ) holds in both cases, when α − β = 1 forζ ≤η or alternatively when α = 1 forζ >η.
Since we require q(x) ≥ m 1 > 0 for above mentioned cases (i) and (ii), from difference equation (33) we have the following inequalities for mesh function Q i = (azx) i , i = 1, ..., N − 1 (see [7, p. 1430-1431]) for some point ξ = max{ζ n , η m } and corresponding constants such that Since ξ is defined such that i ξ h < ξ ≤ (i ξ + 1)h, by using (35) we get valid estimations for values of th interpolation functionp(x) at above mentioned pointš ζ andηp Using these inequalities in (32) and taking into account (35), we obtain which is true for case (i) when α − β = 1 and case (ii) when α = 1. Therefore, the estimation p 0 N (z) = O(h 2 ) is always true for difference problem (30),(31) and hence the numerical solution of (31),(32) always can be considered as corresponding solution of (33),(34). Now, since we have proved the estimation p 0 N (z) = O(h 2 ) for the solution of (33),(34) and because the mesh function Q i is non-decreasing, we have and therefore, zx = O(h 2 ).
Further, by using difference analogue of embedding theorem [19, p. 290], we have that estimation max holds for solution of difference equation (33).
Since we have all necessary estimations forz i and the following estimations for corresponding solution v i of difference problem (28),(29) (see [7, p. 1429 by using a triangle inequality in (27) we obtain the second-order accuracy for the solution of finite-difference scheme (22),(23). Theorem 3.1 is proved.

Nonlocal weight integral condition
The approach proposed by V. A. Il'in and E. I. Moiseev in [6] is also applicable to the Sturm-Liouville operator with first kind nonlocal weight integral boundary conditions for function α(x) summable on (ζ, η) and ζ ∈ [0, 1], η ∈ [0, 1] in general. Actually, if α(x) has the same sign almost everywhere on the defined interval, then for the nonlocal differential problem mentioned above, corresponding results of [6] are applicable.
Indeed, if α(x) is not changing the sign and −∞ < η ζ α(x)dx < 1, then by using the first mean value theorem for definite integral in above-mentioned first kind nonlocal integral condition we obtain a priori estimation (3) and unique existence of solution of above-mentioned problem as solution of two-point NLBVP 3 [6]. Similarly, the approach described in [7] is applicable to second kind integral NLBVP too, i.e. for the problem with above mentioned requirements on weight function α(x). Actually, if α(x) is not changing the sign and −∞ < η ζ α(x)dx < 1, then by using the first mean value theorem to flow weighting integral condition, we obtain a priori estimation (3) and the unique existence of the solution of the second kind integral NLBVP as corresponding solution of two-point nonlocal problem 4 which was studied in [7].
Note that for difference interpretation of all above mentioned integral variations of NLBVP we have to require much more smoothness for weight functions to obtain the second-order accuracy for approximation. For example, requirements α(x) ∈ C 2 [0, 1] and β(x) ∈ C 2 [0, 1] provide the second-order accuracy when the trapezoidal method is applied for weighting integral NLB condition.
Finally, the associated finite-difference statements will be based on results in [6,7] if we have the weight function with no sign changes in the integral NLB condition. Otherwise, when we have sign-changing weighting function with finite number of sign changes, we can research finite-difference scheme by applying previous results in [9,10] for the first kind NLBVP and Theorem 3.1 for the second kind NLBVP, respectively.

Conclusion
In conclusion, the author expresses his deepest deference to deceased RAS Academy Sciences academician, Prof. V. A. Il'in, whose outstanding scientific articles stimulate the author for further research. Author expresses gratitude to RAS academician, Prof. E. I. Moiseev, whose remarkable scientific works influence the author's investigations in this area.