ON CERTAIN PROPERTIES OF SOLUTIONS OF LINEAR DIFFERENCE EQUATIONS APPROXIMATING DIFFERENTIAL EQUATIONS IN THE INTERVAL OF NON-OSCILLATION

A. L. TEPTIN

As is known, the question of the limits of applicability of the theorem on differential inequalities for the Vallée-Poussin boundary-value problem is closely connected with the study of the non-oscillatory properties of solutions of linear homogeneous differential equations [1—6]. Non-oscillatory properties of solutions play an analogous role for difference equations as well [7—10]. The theorems on difference inequalities obtained as a result of studying these properties can be used in the investigation of differential equations (see, for example, [8, 10]). In doing so one has to consider difference equations close to differential ones. Naturally the question arises whether the above-mentioned properties of solutions are preserved in passing from the differential equation to the corresponding difference equation. The present paper considers this question, and also gives an estimate, uniform with respect to the step of the equation, for the solution of a difference boundary-value problem and presents an example of the application of the results obtained to the investigation of the Vallée-Poussin boundary-value problem for a nonlinear differential equation.

Below we use the following notation and definitions:

1) \(g_R\) is the set of integers \(x = 0, 1, \ldots, R\).

2) \(G\) is an \(n+1\)-dimensional domain: \(a \le t \le b,\ A_i \le y_i \le B_i\) \((i = 1, \ldots, n;\ a, b, A_i, B_i\) are given numbers).

3)
\[ h=\frac{b-a}{M}\quad (M>0 \text{ is an integer}). \]
Below, everywhere \(h\) and \(M\) are assumed to be connected by this relation.

4) \(t_x = a + hx\ (x \in g_M)\).

5) \(u_x^h\) is a function of the arguments \(x\) and \(h\), defined for \(x \in g_M\),

\[ h=\frac{b-a}{M}; \]

6)
\[ \Delta u_x^h=\Delta^1 u_x^h=u_{x+1}^h-u_x^h,\quad \Delta^k u_x^h=\Delta[\Delta^{k-1}u_x^h]\ (k=2,3,\ldots),\quad \Delta^0 u_x^h=u_x^h. \]

7)
\[ \Delta y(t)=\Delta^1 y(t)=y(t+h)-y(t),\quad \Delta^k y(t)=\Delta[\Delta^{k-1}y(t)]\ (k=2,3,\ldots), \]
\[ \Delta^0 y(t)=y(t). \]

8)
\[ \bar y=\{y_1,\ldots,y_n\}\ \text{is a vector},\quad \|\bar y\|=\sum_{k=1}^{n}|y_k|. \]

9)
\[ \bar u_x^h=\{u_{x1}^h,\ldots,u_{xn}^h\} \]
is a vector-function of the arguments \(x\) and \(h\), defined for \(x \in g_M,\ h=\dfrac{b-a}{M}\).

10) \(\bar f(t,\bar y)=\{f_1(t,y_1,\ldots,y_n),\ldots,f_n(t,y_1,\ldots,y_n)\}\) is a vector function defined in the domain \(G\) and satisfying, with respect to \(y\), a Lipschitz condition with constant \(K\), i.e., in this domain

\[ \|\bar f(t,\bar y_1)-\bar f(t,\bar y_2)\|\leq K\|\bar y_1-\bar y_2\| \tag{1} \]

for any two \(n\)-dimensional vectors \(\bar y_1\) and \(\bar y_2\).

11) \(L[y]\equiv y^{(n)}-\displaystyle\sum_{k=0}^{n-1}p_k(t)y^{(k)}\) is a linear differential operation with coefficients piecewise continuous on \([a,b]\).

12) \(L_h[u_x^h]\equiv \dfrac{\Delta^n u_x^h}{h^n}-\displaystyle\sum_{k=0}^{n-1}p_k(t_x)\dfrac{\Delta^k u_x^h}{h^k}\) is a linear difference operation approximating the operation \(L[y]\).

13) We shall say that \(u_x^h\) \((\bar u_x^h)\), as \(h\to0\), converges uniformly with respect to \(x\in g_M\) to the function \(u(t)\) (the vector function \(\bar u(t)\)) defined on the interval \([a,b]\), if

\[ \lim_{h\to0}\max_{x\in g_M}|u_x^h-u(t_x)|=0 \quad \left( \lim_{h\to0}\max_{x\in g_M}\|\bar u_x^h-\bar u(t_x)\|=0 \right). \]

14) A solution \(u_x^h\) of the equation \(L_h[u_x^h]=f(t_x)\) will be called corresponding to the solution \(y(t)\) of the equation \(L[y]=f(t)\), if

\[ \frac{\Delta^k u_0^h}{h^k}=y^{(k)}(a)\quad (k=0,1,\ldots,n-1) \]

for any sufficiently small \(h\).

15) We agree to assign to the value \(u_{x_0}^h=0\) \((x_0\in g_M)\) the sign opposite to the sign of \(u_{x_0+1}^h\), if \(u_0^h=u_1^h=\cdots=u_{x_0}^h=0\); in all other cases we assign to the value \(u_{x_0}^h=0\) the sign opposite to the sign of \(u_{x_0-1}^h\). We shall say that \(u_x^h\) has a change of sign at the point \(x^*\in g_M\), if the signs of \(u_{x^*}^h\) and \(u_{x^*+1}^h\) are opposite.

16) We shall call the equation \(L_h[u_x^h]=0\) non-oscillatory on the set \(g_M\) if every nontrivial solution of it has on this set no more than \(n-1\) changes of sign.

17) The interval \([a,b]\) will be called an interval of non-oscillation \([1,2,4]\) for the equation \(L[y]=0\), if every nontrivial solution of this equation has on \([a,b]\) no more than \(n-1\) zeros, zeros being counted according to their multiplicity.

18) All functions of \(t\) occurring below are, at points of discontinuity, agreed to be extended by continuity from the right.

§ 1.

Consider two systems of difference equations in vector form

\[ \frac{\Delta \bar u_x^h}{h}=\bar f(t_x,\bar u_x^h), \tag{2} \]

\[ \frac{\Delta \bar v_x^h}{h}=\bar f_h(t_x,\bar v_x^h), \tag{3} \]

where \(\bar f_h(t,\bar y)\) is a vector function, for each \(h\) defined in the domain \(G\). It holds that

Lemma 1. Suppose that for every \(h<h_0\)

\[ \left\|\bar f(t_x,\bar y)-\bar f_h(t_x,\bar y)\right\|\leqslant \delta \tag{4} \]

for all \((t_x,\bar y)\in G,\ x\in g_{M-1}\), with the exception of a finite number of hypersurfaces \(t_x=c_i^h\) \((i=1,\ldots,m)\), on which

\[ \left\|\bar f(c_i^h,\bar y)-\bar f_h(c_i^h,\bar y)\right\|\leqslant \frac{\delta}{h} \tag{5} \]

\((i=1,\ldots,m;\ \delta>0\) and \(m\) are numbers independent of \(h\)).
If for all \(h<h_0\) on the set \(g_M\) there exist solutions \(\bar u_x^h\) and \(\bar v_x^h\) of systems (2) and (3), respectively, and moreover

\[ \left\|\bar u_0^h-\bar v_0^h\right\|\leqslant \delta, \tag{6} \]

then

\[ \left\|\bar u_x^h-\bar v_x^h\right\|\leqslant e^{K(b-a)}(b-a+m+1)\delta \quad (x\in g_M). \]

Proof. Write systems (2) and (3) in the form

\[ \bar u_{x+1}^h=\bar u_x^h+h\bar f(t_x,\bar u_x^h), \]

\[ \bar v_{x+1}^h=\bar v_x^h+h\bar f_h(t_x,\bar v_x^h) \]

and subtract the second equality term by term from the first:

\[ \bar u_{x+1}^h-\bar v_{x+1}^h = \bar u_x^h-\bar v_x^h +h\left[\bar f(t_x,\bar u_x^h)-\bar f(t_x,\bar v_x^h)\right]+ \]

\[ +h\left[\bar f(t_x,\bar v_x^h)-\bar f_h(t_x,\bar v_x^h)\right]. \]

Passing to the norm and denoting

\[ \left\|\bar u_x^h-\bar v_x^h\right\|=\eta_x^h,\qquad \left\|\bar u_0^h-\bar v_0^h\right\|=\beta_h, \]

\[ \alpha(t,h)= \begin{cases} \delta, & \text{for } t\ne c_i^h\ (i=1,\ldots,m),\\[4pt] \dfrac{\delta}{h}, & \text{for } t=c_i^h, \end{cases} \tag{7} \]

by virtue of (1), (4), (5), we obtain

\[ \eta_{x+1}^h\leqslant (1+hK)\eta_x^h+h\alpha(t_x,h), \qquad \eta_0^h=\beta_h. \tag{8} \]

Since \(1+hK>0\), for the equation

\[ z_{x+1}^h=(1+hK)z_x^h+h\alpha(t_x,h), \qquad z_0^h=\beta_h \]

there is valid the theorem on difference inequalities \([11,12]\), analogous to the well-known theorem of S. A. Chaplygin on differential inequalities \([13]\), i.e., from inequality (8) it follows that

\[ \eta_x^h\leqslant z_x^h \quad (x\in g_M). \tag{9} \]

But

\[ z_x^h=\sum_{s=1}^{x}(1+hK)^{x-s}h\alpha(t_{s-1},h) +(1+hK)^x\beta_h \ [14,\ 11,\ 12]. \]

Thus, by virtue of (6), (7), (9),

\[ \| \bar u_x^h-\bar v_x^h \|\le z_x^h \le \sum_{s=1}^{M}(1+hK)^M h\,\alpha(t_{s-1},h)+(1+hK)^M\beta_h \le \]

\[ \le \sum_{s=1}^{M}(1+hK)^M h\delta +m(1+hK)^M\delta+(1+hK)^M\delta = \]

\[ =(1+hK)^M(Mh+m+1)\delta \quad (x\in \mathfrak g_M). \]

Since \(M=\dfrac{b-a}{h}\) and \((1+hK)^{\frac{b-a}{h}}<e^{K(b-a)}\) for any \(h>0\), it follows that

\[ \|\bar u_x^h-\bar v_x^h\| < e^{K(b-a)}(b-a+m+1)\delta \quad (x\in \mathfrak g_M). \]

The lemma is proved.

Lemma 1 makes it easy to prove the convergence of the solution of the system of difference equations

\[ \frac{\Delta \bar u_x^h}{h}=\bar f(t_x,\bar u_x^h), \qquad \bar u_0^h=\bar A \tag{10} \]

(\(\bar A\) is a vector independent of \(h\)) to the solution of the system of differential equations

\[ \bar y'=\bar f(t,\bar y), \qquad \bar y(a)=\bar A, \tag{11} \]

where \(\bar f(t,\bar y)\) in the domain \(G\) satisfies condition (1) and is continuous in \(t\), except for a finite number of hyperplanes \(t=c_i\) \((i=1,\ldots,m)\), on which it may have only discontinuities of the first kind.

The solution of system (11) will be understood in the extended sense.

Lemma 2. If system (11) has a solution \(\bar y(t)\) on the interval \([a,b]\), and for every \(h<h_0\) system (10) has a solution \(\bar u_x^h\) on the set \(\mathfrak g_M\), then

\[ \lim_{h\to 0}\bar u_x^h=\bar y(t) \]

uniformly with respect to \(x\in \mathfrak g_M\).

Proof. Put

\[ \frac{\Delta \bar y(t)}{h}-\bar f(t,\bar y(t))=\bar a(t,h), \]

\[ \bar f(t,\bar u)+\bar a(t,h)=\bar f_h(t,\bar u). \]

Then \(\bar y(t)\) satisfies the system of difference equations

\[ \frac{\Delta \bar y(t)}{h}=\bar f_h(t,\bar y(t)). \]

Since

\[ h\bar a(t,h)=\Delta \bar y(t)-h\bar f(t,\bar y(t)) \]

and \(\bar y(t)\) is continuous on \([a,b]\), we have

\[ \lim_{h\to 0} h\bar a(t,h)=0 \]

uniformly with respect to \(t\in [a,b-h]\). Hence, for any \(\delta>0\) there exists a number \(h_1>0\) such that, for every \(h<h_1\),

\[ \|\overline a(t,h)\| < \frac{\delta}{h}\quad (t \in [a,b-h]). \tag{12} \]

\(\overline y(t)\) on the interval \([a,b]\) has a derivative, continuous everywhere except at the points \(t=c_i\ (i=1,\ldots,m)\), at which discontinuities of the first kind are possible. Therefore, if the interval \([t,t+h]\) does not contain a point \(t=c_i\), then

\[ \overline a(t,h)=\frac{\Delta \overline y(t)}{h}-\overline y'(t), \]

and, by Lagrange’s theorem,

\[ \|\overline a(t,h)\|=\sum_{k=1}^{n}\left|y'_k(\xi_k)-y'_k(t)\right| \]

\[ (t<\xi_k<t+h,\quad k=1,\ldots,n;\quad y_k(t)\text{ are the coordinates of the vector-function } \overline y(t)). \]

It is easy to see that \(y'_k(t)\ (k=1,\ldots,n)\) are uniformly continuous in each of the intervals not containing the points \(t=c_i\ (i=1,\ldots,m)\). And since there are only finitely many of these intervals, for any \(\delta>0\) there exists a number \(h_2>0\) such that for every \(h<h_2\)

\[ \|\overline a(t_x,h)\|<\delta \tag{13} \]

for all \(x\in g_{M-1}\), except for those values \(x=x_i\) for which
\(t_{x_i}\leq c_i<t_{x_i}+h\ (i=1,\ldots,m)\).

By virtue of (12) and (13), for any \(\delta>0\) and the corresponding \(h_1\) and \(h_2\), for all \(h<\min\{h_0,h_1,h_2\}\) the inequality

\[ \|\overline f(t_x,\overline y)-\overline f_h(t_x,\overline y)\| =\|\overline a(t_x,h)\|<\delta \]

holds for all \((t_x,\overline y)\in G,\ x\in g_{M-1}\), except for the hyperplanes \(t_x=t_{x_i}\)
\((i=1,\ldots,m)\), on which

\[ \|\overline f(t_{x_i},\overline y)-\overline f_h(t_{x_i},\overline y)\| =\|\overline a(t_{x_i},h)\|<\frac{\delta}{h}. \]

Moreover, \(\overline y(t_0)=\overline y(a)=\overline u_0^h\). But then, by Lemma 1,

\[ \|\overline u_x^h-\overline y(t_x)\| <e^{K(b-a)}(b-a+m+1)\delta\quad (x\in g_M) \]

for any \(h<\min\{h_0,h_1,h_2\}\). In view of the arbitrariness of \(\delta\), this means that

\[ \lim_{h\to 0}\max_{x\in g_M}\|\overline u_x^h-\overline y(t_x)\|=0. \]

The lemma is proved.

Corollary. If \(y(t)\) is a solution of the differential equation

\[ L[y]=f(t), \tag{14} \]

where \(f(t)\) is a piecewise-continuous function, and \(u_x^h\) is the corresponding solution of the difference equation

\[ L_h[u_x^h]=f(t_x), \tag{15} \]

then

\[ \lim_{h\to 0}\frac{\Delta^k u_x^h}{h^k}=y^{(k)}(t)\quad (k=0,1,\ldots,n-1). \tag{16} \]

uniformly with respect to \(x \in g_M\), and

\[ \lim_{h\to 0}\frac{\Delta^k u^h_{M+m}}{h^k} = y^{(k)}(b) \quad (m=0,1,\ldots,n-k;\ k=0,1,\ldots,n-1). \tag{17} \]

Remark. Since the coefficients and the free term of equation (15) are defined on the set \(g_M\), its solution on the set \(g_M\) can be extended to the set \(x=M+1,\ldots,M+n\), and therefore each difference quotient
\[ \frac{\Delta^k u_x^h}{h^k} \]
is defined for \(x\in g_{M+n-k}\) \((k=0,1,\ldots,n)\).

Proof. To verify equality (16), it suffices to write equations (14) and (15) in the form of systems. From (16) and (15), in view of the continuity of \(y^{(k)}(t)\) \((k=0,1,\ldots,n-1)\) on the interval \([a,b]\), there follows the uniform boundedness with respect to \(h\) of all
\[ \frac{\Delta^k u_x^h}{h^k} \]
\((k=0,1,\ldots,n)\) on the set \(g_M\). Hence, by virtue of (16) and the relation

\[ \frac{\Delta^k u^h_{M+m}}{h^k} = \sum_{s=0}^{m}\binom{m}{s} \frac{\Delta^{k+s}u^h_M}{h^{k+s}}\,h^s \quad (m=0,1,\ldots,n-k; \]

\[ k=0,1,\ldots,n-1) \quad [15] \]

we obtain equality (17).

On the basis of the corollary to Lemma 2, the main assertion of the present section is proved.

Theorem 1. If \([a,b]\) is an interval of non-oscillation of the equation

\[ L[y]=0, \tag{18} \]

then, for every sufficiently small \(h\), the equation

\[ L_h[u_x^h]=0 \tag{19} \]

is non-oscillatory on the set \(g_{M+n}\).

Proof. Extend the coefficients of equation (18) to the interval \([c,a)\) \((c<a)\) so that they are continuous on \([c,a]\).

Let \(y_1(t,s),\ldots,y_n(t,s)\) be functions which, for fixed \(s\) \((c\le s\le b)\), form a fundamental system of solutions of equation (18), satisfying the initial conditions:
\[ y_i^{(k)}(s,s)=\delta_{i,n-k} \quad (k=0,1,\ldots,n-1;\ i=1,\ldots,n;\ \delta_{i,j}=0\ \text{for } i\ne j,\ \delta_{i,i}=1; \]
\(y_i^{(k)}(t,s)\) is the \(k\)-th derivative with respect to the variable \(t\)).

Denote

\[ W_1(t,s)=y_1(t,s), \]

\[ W_i(t,s)= \left| \begin{array}{cccc} y_1(t,s) & \cdots & y_i(t,s)\\ y'_1(t,s) & \cdots & y'_i(t,s)\\ \cdots & \cdots & \cdots\\ y_1^{(i-1)}(t,s) & \cdots & y_i^{(i-1)}(t,s) \end{array} \right| \quad (i=2,\ldots,n). \]

It is easy to show that none of the Wronskians \(W_i(t,a)\) \((i=1,\ldots,n)\) has zeros in the interval \((a,b]\). Indeed, if \(W_i(\xi,a)=0\), \(\xi\in(a,b]\), then the system of equations

\[ \sum_{k=1}^{i} c_k y_k^{(j)}(\xi, a)=0 \quad (j=0, 1, \ldots, i-1) \]

has a nonzero solution \(\{c_1,\ldots,c_i\}\), and then \(y(t)=\sum_{k=1}^{i} c_k y_k(t,a)\) is a nontrivial solution of equation (18), having at the points \(t=a\) and \(t=\xi\) zeros of multiplicities \(n-i\) and \(i\), respectively. But the existence of such a solution contradicts the definition of the interval of nonoscillation. Hence all \(W_i(t,a)\) \((i=1,\ldots,n)\) preserve their sign in the interval \((a,b]\).

In view of the continuous dependence of solutions on the initial conditions, it is easy to verify the existence of a point \(d<a\) such that each of the Wronskians \(W_i(t,d)\) \((i=1,\ldots,n)\) preserves its sign in \((d,b]\). And since all \(W_i(t,d)\) \((i=1,\ldots,n)\) are continuous on \([c,b]\), we have

\[ \min_{t\in[a,b]} |W_i(t,d)|>0 \quad (i=1,\ldots,n). \]

Let \(u_{x1}^h,\ldots,u_{xn}^h\) be a system of solutions of equation (19) corresponding to the solutions \(y_1(t,d),\ldots,y_n(t,d)\) of equation (18). Put: \(W_{x,1}^h=D_{x,1}^h=u_{x1}^h\),

\[ W_{x,i}^h= \left| \begin{array}{ccc} u_{x1}^h & \cdots & u_{xi}^h\\[4pt] \dfrac{\Delta u_{x1}^h}{h} & \cdots & \dfrac{\Delta u_{xi}^h}{h}\\ \cdots & \cdots & \cdots\\[4pt] \dfrac{\Delta^{\,i-1}u_{x1}^h}{h^{i-1}} & \cdots & \dfrac{\Delta^{\,i-1}u_{xi}^h}{h^{i-1}} \end{array} \right|, \qquad D_{x,i}^h= \left| \begin{array}{ccc} u_{x1}^h & \cdots & u_{xi}^h\\ u_{x+1,1}^h & \cdots & u_{x+1,i}^h\\ \cdots & \cdots & \cdots\\ u_{x+i-1,1}^h & \cdots & u_{x+i-1,i}^h \end{array} \right| \quad (i=2,\ldots,n). \]

By virtue of the corollary to Lemma 2,

\[ \lim_{h\to 0} W_{x,i}^h=W_i(t,d) \quad (i=1,\ldots,n) \]

uniformly with respect to \(x\in g_M\), and

\[ \lim_{h\to 0} W_{M+m,i}^h=W_i(b,d) \quad (m=0,1,\ldots,n-i+1;\ i=1,\ldots,n). \]

Consequently, there exists a number \(h_0>0\) such that, for all \(h<h_0\), each of the determinants \(W_{x,i}^h\) preserves its sign on the set \(g_{M+n-i+1}\) \((i=1,\ldots,n)\). But by elementary properties of determinants,

\[ W_{x,i}^h=h^{-\frac{i(i-1)}{2}}D_{x,i}^h. \]

Hence, for any \(h<h_0\), each of the determinants \(D_{x,i}^h\) preserves its sign on the set \(g_{M+n-i+1}\). From this, by the results of [7, 8], it follows that, for \(h<h_0\), equation (19) is nonoscillatory on the set \(g_{M+n}\).

The theorem is proved.

Remark. It should be noted that there is no one-to-one correspondence between the solutions of equations (18) and (19), since to each solution of equation (18) one can assign a set of convergent

converging to it, and, moreover, there exists a set of solutions of equation (19) that do not converge to any solutions of equation (18). Nevertheless, as has just been proved, in the passage from equation (18) to equation (19) the nonoscillatory properties of the solutions are completely preserved.

§ 2

Let us additionally introduce the following notation:

1) \(\alpha_1 < \alpha_2 < \cdots < \alpha_m\) are points of the interval \([a,b]\).

2) \(x_i \in g_M\) \((i=1,\ldots,m)\) are numbers chosen so that

\[ \lim_{h\to 0} t_{x_i}=\alpha_i \quad (i=1,\ldots,m). \tag{20} \]

3) \(r_i\) \((i=1,\ldots,m)\) are integers such that

\[ \sum_{i=1}^{m} r_i=n. \]

4) \(y_1(t),\ldots,y_n(t)\) is a fundamental system of solutions of equation (18), and \(u^h_{x1},\ldots,u^h_{xn}\) is the corresponding system of solutions of equation (19).

5)

\[ W= \left| \begin{array}{cccc} y_1(\alpha_1) & \cdots & y_n(\alpha_1)\\ \cdot & \cdot & \cdot & \cdot\\ y_1^{(r_1-1)}(\alpha_1) & \cdots & y_n^{(r_1-1)}(\alpha_1)\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ y_1(\alpha_m) & \cdots & y_n(\alpha_m)\\ \cdot & \cdot & \cdot & \cdot\\ y_1^{(r_m-1)}(\alpha_m) & \cdots & y_n^{(r_m-1)}(\alpha_m) \end{array} \right|, \]

\[ W_h= \left| \begin{array}{cccc} u^h_{x_1 1} & \cdots & u^h_{x_1 n}\\ \cdot & \cdot & \cdot & \cdot\\ \dfrac{\Delta^{r_1-1}u^h_{x_1 1}}{h^{r_1-1}} & \cdots & \dfrac{\Delta^{r_1-1}u^h_{x_1 n}}{h^{r_1-1}}\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ u^h_{x_m 1} & \cdots & u^h_{x_m n}\\ \cdot & \cdot & \cdot & \cdot\\ \dfrac{\Delta^{r_m-1}u^h_{x_m 1}}{h^{r_m-1}} & \cdots & \dfrac{\Delta^{r_m-1}u^h_{x_m n}}{h^{r_m-1}} \end{array} \right|. \]

6) \(G^h_{x,s}\) is the Green’s function of the difference boundary-value problem

\[ L_h[u^h_x]=\varphi^h_x,\qquad \frac{\Delta^{k_i}u^h_{x_i}}{h^{k_i}}=0 \quad (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m), \tag{21} \]

i.e., the function defined for \(x\in g_{M+n}\), \(s\in g_M\), \(h=\dfrac{b-a}{M}\), and, for any fixed \(s\) and \(h\), satisfying with respect to \(x\) the boundary-value problem

\[ L_h[G^h_{x,s}]=\frac{\delta_{x,s}}{h} \quad (\delta_{x,s}=0 \text{ for } x\ne s,\ \delta_{s,s}=1), \tag{22} \]

\[ \frac{\Delta_x^{k_i} G_{x,s}^h}{h^{k_i}}=0\quad (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m)^*. \tag{23} \]

Here and below
\[ \Delta_x G_{x,s}^h=G_{x+1,s}^h-G_{x,s}^h,\qquad \Delta_x^k G_{x,s}^h=\Delta_x\left[\Delta_x^{k-1}G_{x,s}^h\right]\ (k=2,3,\ldots). \]

It is easy to verify that, in the case of existence of \(G_{x,s}^h\), the solution of problem (21) can be represented in the form

\[ u_x^h=\sum_{s=0}^{M}G_{x,s}^h\varphi_s^h h\quad \text{(cf. [8])}. \tag{24} \]

Below we give estimates, uniform in \(h\), for the Green’s function and for the solutions of difference boundary-value problems. For the proof of these estimates we shall need the following

Lemma 3. If \([a,b]\) is an interval of non-oscillation of equation (18), then there exist numbers \(h_0>0\) and \(A>0\), independent of \(h\), such that \(|W_h|>A\) for all \(h<h_0\).

Proof. Any solution of equation (18) has the form

\[ y(t)=\sum_{k=1}^{n} c_k y_k(t). \]

Since in \([a,b]\) there cannot exist a nontrivial solution of equation (18) having \(n\) zeros, the solution of this equation satisfying the boundary conditions

\[ y^{(k_i)}(\alpha_i)=0\quad (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m), \]

is identically equal to zero. Consequently, the system

\[ \sum_{k=1}^{n} c_k y_k^{(k_i)}(\alpha_i)=0\quad (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m) \]

has only the zero solution, i.e. its determinant \(W\) is different from zero.

By virtue of the corollary of Lemma 2, the continuity of \(y_j^{(k)}(t)\) \((j=1,\ldots,n;\ k=0,1,\ldots,n-1)\), and equality (20),

\[ \lim_{h\to 0} W_h=W\ne 0. \]

Hence the assertion of the present lemma follows.

Theorem 2. If \([a,b]\) is an interval of non-oscillation of equation (18), then there exist numbers \(h_0>0\) and \(B>0\), independent of \(h\), such that for all \(h<h_0\) there exists a Green’s function \(G_{x,s}^h\) of the boundary-value problem (21), and moreover

\[ \left|\frac{\Delta_x^k G_{x,s}^h}{h^k}\right|<B\quad (x\in g_{M+n-k},\ s\in g_M;\ k=0,1,\ldots,n-1). \]

Proof. Let, as before, \(y_1(t),\ldots,y_n(t)\) be a fundamental system of solutions of equation (18), and let \(y_{x1}^h,\ldots,y_{xn}^h\) be the corresponding system of solutions of equation (19).

\[ \text{* In works [7, 8] the function } h^{-(n-1)}G_{x,s}^h \text{ is called the Green’s function.} \]

Consider the function

\[ K_{x,s}^{h}= \left| \begin{array}{ccc} u_{s1}^{h} & \cdots & u_{sn}^{h}\\ \cdot & \cdots & \cdot\\ \dfrac{\Delta^{\,n-2}u_{s1}^{h}}{h^{n-2}} & \cdots & \dfrac{\Delta^{\,n-2}u_{sn}^{h}}{h^{n-2}}\\ u_{x1}^{h} & \cdots & u_{xn}^{h} \end{array} \right| : W_{s,n}^{h}, \]

where

\[ W_{x,n}^{h}= \left| \begin{array}{ccc} u_{x1}^{h} & \cdots & u_{xn}^{h}\\ \cdot & \cdots & \cdot\\ \dfrac{\Delta^{\,n-2}u_{x1}^{h}}{h^{n-2}} & \cdots & \dfrac{\Delta^{\,n-2}u_{xn}^{h}}{h^{n-2}}\\ \dfrac{\Delta^{\,n-1}u_{x1}^{h}}{h^{n-1}} & \cdots & \dfrac{\Delta^{\,n-1}u_{xn}^{h}}{h^{n-1}} \end{array} \right|. \]

It is easy to see that \(K_{x,s}^{h}\), as a function of \(x\), for any fixed \(s,h\), satisfies equation (19) and the initial conditions

\[ \left(\frac{\Delta_x^{k}K_{x,s}^{h}}{h^{k}}\right)_{x=s} =0\ (k=0,1,\ldots,n-2), \qquad \left(\frac{\Delta_x^{\,n-1}K_{x,s}^{h}}{h^{n-1}}\right)_{x=s} =1. \]

Thus, \(K_{x,s}^{h}\) is a difference analogue of the Cauchy function of the differential equation (18) [16, 17].

Since the Wronskian

\[ W_n(t)= \left| \begin{array}{ccc} y_1(t) & \cdots & y_n(t)\\ \cdot & \cdots & \cdot\\ y_1^{(n-1)}(t) & \cdots & y_n^{(n-1)}(t) \end{array} \right| \]

is continuous and does not vanish on \([a,b]\), we have
\(\min\limits_{t\in[a,b]} |W_n(t)| \ge C>0\).

By virtue of the corollary to Lemma 2,

\[ \lim_{h\to 0} W_{x,n}^{h}=W_n(t) \]

uniformly with respect to \(x\in g_M\), and

\[ \lim_{h\to 0} W_{M+1,n}^{h}=W_n(b). \]

Hence it follows that there exist numbers \(h_1>0\) and \(D>0\), independent of \(h\), such that

\[ |W_{x,n}^{h}|>D \quad (x\in g_{M+1}) \tag{25} \]

for all \(h<h_1\).

By virtue of (16), (17) and the continuity of \(y_j^{(k)}(t)\) \((j=1,\ldots,n;\ k=0,1,\ldots,n-1)\), there exist numbers \(h_2>0\) and \(E>0\), independent of \(h\), such that

\[ \left|\frac{\Delta^{k}u_{xj}^{h}}{h^{k}}\right|<E \quad (x\in g_{M+n-k};\ k=0,1,\ldots,n-1;\ j=1,\ldots,n) \tag{26} \]

for every \(h<h_2\).

From (25) and (26) it follows that

\[ \left|\frac{\Delta_x^{k}K_{x,s}^{h}}{h^{k}}\right| < \frac{n!E^{n}}{D} =F \quad (x\in g_{M+n-k},\ s\in g_{M+1};\ k=0,1,\ldots,n-1). \tag{27} \]

for all \(h<\min\{h_1,h_2\}\).

Put

\[ g^h_{x,s}= \begin{cases} K^h_{x,s+1}, & \text{for } x>s,\\ 0, & \text{for } x\le s \quad (x\in g_{M+n},\ s\in g_M). \end{cases} \]

It is obvious that

\[ \frac{\Delta_x^k g^h_{x,s}}{h^k}= \begin{cases} \dfrac{\Delta_x^k K^h_{x,s+1}}{h^k}, & \text{for } x>s,\\ 0, & \text{for } x\le s \end{cases} \]

\[ (x\in g_{M+n-k},\ s\in g_M;\ k=0,1,\ldots,n-1), \]

\[ \frac{\Delta_x^n g^h_{x,s}}{h^n}= \begin{cases} \dfrac{\Delta_x^n K^h_{x,s+1}}{h^n}, & \text{for } x>s,\\[6pt] \dfrac{1}{h}, & \text{for } x=s,\\[6pt] 0, & \text{for } x<s \quad (x\in g_M,\ s\in g_M). \end{cases} \]

Thus, \(g^h_{x,s}\), as a function of \(x\), for any fixed \(s,h\), satisfies equation (22) and, moreover, by virtue of (27),

\[ \left|\frac{\Delta_x^k g^h_{x,s}}{h^k}\right|<F \quad (x\in g_{M+n-k},\ s\in g_M;\ k=0,1,\ldots,n-1) \tag{28} \]

for any \(h<\min\{h_1,h_2\}\).

Since, by Lemma 3, \(W_h\ne0\) for all sufficiently small \(h\), the function

\[ G^h_{x,s}= \left| \begin{array}{cccc} g^h_{x,s} & u^h_{x1} & \cdots & u^h_{xn}\\ g^h_{x_1,s} & u^h_{x_1 1} & \cdots & u^h_{x_1 n}\\ \cdots & \cdots & \cdots & \cdots\\ \dfrac{\Delta_x^{r_1-1}g^h_{x_1,s}}{h^{r_1-1}} & \dfrac{\Delta^{r_1-1}u^h_{x_1 1}}{h^{r_1-1}} & \cdots & \dfrac{\Delta^{r_1-1}u^h_{x_1 n}}{h^{r_1-1}}\\ \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots\\ g^h_{x_m,s} & u^h_{x_m 1} & \cdots & u^h_{x_m n}\\ \cdots & \cdots & \cdots & \cdots\\ \dfrac{\Delta_x^{r_m-1}g^h_{x_m,s}}{h^{r_m-1}} & \dfrac{\Delta^{r_m-1}u^h_{x_m 1}}{h^{r_m-1}} & \cdots & \dfrac{\Delta^{r_m-1}u^h_{x_m n}}{h^{r_m-1}} \end{array} \right| : W_h \]

is defined for \(x\in g_{M+n}\), \(s\in g_M\), and sufficiently small \(h\). It is easy to see that, for fixed \(s\) and \(h\), this function satisfies, with respect to \(x\), problem (22)—(23), i.e. is the Green’s function of problem (21).

Finally, by Lemma 3 and inequalities (26), (28), there exists a number \(h_0\), \(0<h_0\le \min\{h_1,h_2\}\), such that

\[ \left|\frac{\Delta_x^k G^h_{x,s}}{h^k}\right| < \frac{(n+1)!\,F E^n}{A} =B \quad (x\in g_{M+n-k},\ s\in g_M;\ k=0,1,\ldots,n-1) \]

for all \(h<h_0\).

The theorem is proved.

Theorem 3. If \([a,b]\) is a nonoscillation interval of equation (18), then there exists a number \(h_0>0\) such that, for arbitrary \(\varphi_x^h\), \(A_{ik_i}\) \((k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m)\) and \(h<h_0\), the boundary-value problem

\[ L_h\left[u_x^h\right]=\varphi_x^h, \]

\[ \frac{\Delta^{k_i}u_{x_i}^h}{h^{k_i}}=A_{ik_i} \quad (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m) \tag{29} \]

has a unique solution \(u_x^h\), and moreover

\[ \max_{x\in g_{M+n-k}} \left| \frac{\Delta^k u_x^h}{h^k} \right| < N_1\max_{x\in g_M}\left|\varphi_x^h\right| + N_2 \sum_{i=1}^m \sum_{k_i=0}^{r_i-1} \left|A_{ik_i}\right| \quad (k=0,1,\ldots,n), \tag{30} \]

where the numbers \(N_1\) and \(N_2\) do not depend on \(h\), \(\varphi_x^h\), or \(A_{ik_i}\).

Proof. Let \(y_1(t),\ldots,y_n(t)\) be a fundamental system of solutions of equation (18), and let \(u_{x1}^h,\ldots,u_{xn}^h\) be the corresponding system of solutions of equation (19).

By virtue of Theorem 2, for sufficiently small \(h\) there exists a Green’s function \(G_{x,s}^h\) of the boundary-value problem (21). It is obvious that the function

\[ u_x^h=\sum_{j=1}^n c_j u_{xj}^h+\sum_{s=0}^{M}G_{x,s}^h\varphi_s^h h, \tag{31} \]

where

\[ c_j= \left| \begin{array}{cccccc} u_{x_1 1}^h & \cdots & u_{x_1 j-1}^h & A_{10} & u_{x_1 j+1}^h & \cdots \\[0.4em] \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\[0.4em] \dfrac{\Delta^{r_1-1}u_{x_1 1}^h}{h^{r_1-1}} & \cdots & \dfrac{\Delta^{r_1-1}u_{x_1 j-1}^h}{h^{r_1-1}} & A_{1r_1-1} & \dfrac{\Delta^{r_1-1}u_{x_1 j+1}^h}{h^{r_1-1}} & \cdots \\[0.6em] \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\[0.4em] u_{x_m 1}^h & \cdots & u_{x_m j-1}^h & A_{m0} & u_{x_m j+1}^h & \cdots \\[0.4em] \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\[0.4em] \dfrac{\Delta^{r_m-1}u_{x_m 1}^h}{h^{r_m-1}} & \cdots & \dfrac{\Delta^{r_m-1}u_{x_m j-1}^h}{h^{r_m-1}} & A_{mr_m-1} & \dfrac{\Delta^{r_m-1}u_{x_m j+1}^h}{h^{r_m-1}} & \cdots \\[0.6em] \cdots & \cdots & u_{x_1 n}^h & \cdots & \cdots & \cdots \\[0.4em] \cdots & \cdots & \dfrac{\Delta^{r_1-1}u_{x_1 n}^h}{h^{r_1-1}} & \cdots & \cdots & \cdots \\[0.6em] \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\[0.4em] \cdots & \cdots & u_{x_m n}^h & \cdots & \cdots & \cdots \\[0.4em] \cdots & \cdots & \dfrac{\Delta^{r_m-1}u_{x_m n}^h}{h^{r_m-1}} & \cdots & \cdots & \cdots \end{array} \right| :W_h \tag{32} \]

\[ (j=1,\ldots,n), \]

is a solution of problem (29). This solution is unique, for the difference of two distinct solutions of problem (29) would be a nontrivial solution of equation (19) having \(n\) changes of sign (in the sense of definition (15)) on the set \(g_{M+n}\) for any small \(h\), which contradicts Theorem 1.

By virtue of (26), (31), (32), Lemma 3, and Theorem 2, there exists a number \(h_0>0\) such that

\[ \max_{x\in g_{M+n-k}} \left| \frac{\Delta^k u_x^h}{h^k} \right| < \frac{n!E^n}{A} \sum_{i=1}^{m}\sum_{k_i=0}^{r_i-1} |A_{ik_i}| + B(b-a)\max_{x\in g_M}|\varphi_x^h| \tag{33} \]

\[ (k=0,1,\ldots,n-1) \]

for any \(h<h_0\).

Further, since \(p_k(t)\) \((k=0,1,\ldots,n-1)\) are piecewise continuous on \([a,b]\), there exists a number \(H>0\) such that

\[ |p_k(t)|\le H \quad (k=0,1,\ldots,n-1;\ t\in[a,b]). \tag{34} \]

From equation (29), by virtue of (33) and (34), we obtain

\[ \max_{x\in g_M} \left| \frac{\Delta^n u_x^h}{h^n} \right| < \frac{n!nE^nH}{A} \sum_{i=1}^{m}\sum_{k_i=0}^{r_i-1} |A_{ik_i}| + [BHn(b-a)+1]\max_{x\in g_M}|\varphi_x^h| \]

for any \(h<h_0\).

Thus, inequalities (29) are satisfied for

\[ N_1=\max\{B(b-a),\,BHn(b-a)+1\}, \]

\[ N_2=\max\left\{\frac{n!E^n}{A},\,\frac{n!nE^nH}{A}\right\}. \]

The theorem is proved.

Remark. Using the terminology of [18] and choosing in the proper way the norms of \(u_x^h\), \(\varphi_x^h\), and \(A_{ik_i}\), Theorem 3 can be formulated as follows: if \([a,b]\) is a nonoscillation interval of equation (18), then the equation \(L_h[u_x^h]=\varphi_x^h\) with boundary conditions

\[ \frac{\Delta^{k_i}u_{x_i}^h}{h^{k_i}}=A_{ik_i} \]

\[ (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m) \]

is stable.

§ 3

As was already noted above, the results obtained may be used in the study of differential equations. As an example we consider the boundary-value problem

\[ N[y]\equiv L[y]-f(t,y)=0, \tag{35} \]

\[ y^{(k_i)}(\alpha_i)=A_{ik_i} \quad (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m;\ \sum_{i=1}^{m}r_i=n;\ a=\alpha_1<\cdots<\alpha_m=b), \tag{36} \]

where the coefficients \(p_k(t)\) \((k=0,1,\ldots,n-1)\) of the operation \(L[y]\) are continuous on \([a,b]\), and the function \(f(t,y)\) is continuous in the domain \(Q:\ a\le t\le b,\ A\le y\le B\) and satisfies there, with respect to \(y\), a Lipschitz condition with constant \(K\). From the Lipschitz condition there follows the existence of functions \(p(t)\) and \(q(t)\), continuous on \([a,b]\), \(-K\le p(t)\le q(t)\le K\), such that

\[ p(t)(y_1-y_2)\leq f(t,y_1)-f(t,y_2)\leq q(t)(y_1-y_2)\quad (t\in [a,b]) \tag{37} \]

for any pair of numbers \(y_1\) and \(y_2\), \(A\leq y_2\leq y_1\leq B\).

In papers [1—3, 5, 6, 10] theorems on differential inequalities for the boundary-value problem (35)—(36) were considered under \(p_k(t)\equiv 0\) \((k=0,1,\ldots,n-1)\). In paper [6] the possibility was also noted of extending its results to the case where \(p_k(t)\not\equiv 0\) \((k=0,1,\ldots,n-1)\), if Theorem 2 of paper [4] is extended to differential equations with summable coefficients. Below a method is proposed for proving a theorem on differential inequalities for problem (35)—(36), different from those used in the works cited above. This method is based on a limiting passage from a difference equation to a differential one. In order to show possible applications of the results obtained in the study of difference equations, the author has not sought to prove the theorem on differential inequalities in its most complete form. The theorem formulated below can be considerably strengthened.

Theorem 4. Let \([a,b]\) be an interval of non-oscillation of the equations

\[ L[y]-p(t)y=0,\quad L[y]-q(t)y=0. \tag{38} \]

If there exist functions \(z_1(t)\) and \(z_2(t)\), \(n\) times continuously differentiable, satisfying the boundary conditions (36) and the inequalities

\[ A\leq z_j(t)\leq B\quad (j=1,2), \tag{39} \]

\[ N[z_1]>0,\quad N[z_2]<0\quad (t\in [a,b]), \tag{40} \]

then problem (35)—(36) has a solution \(y(t)\), and moreover

\[ \operatorname*{sign}_{t\in(\alpha_i,\alpha_{i+1})}[z_j(t)-y(t)] = (-1)^{\,n-\sum_{k=1}^{i} r_{k+j-1}} \quad (j=1,2;\ i=1,\ldots,m-1) \]

everywhere where \(z_j(t)\ne y(t)\).

Proof. Choose arbitrary numbers \(\omega>0\) and \(\sigma>0\). Let \(f^{*}(t,y)\) be the function defined in the domain \(Q^{*}: a\leq t\leq b,\ A-\omega\leq y\leq B+\omega\) by the equalities

\[ f^{*}(t,y)= \begin{cases} f(t,y), & \text{for } (t,y)\in Q,\\ f(t,B)+\lambda(t)(y-B), & \text{for } a\leq t\leq b,\ B<y\leq B+\omega,\\ f(t,A)+\lambda(t)(y-A), & \text{for } a\leq t\leq b,\ A-\omega\leq y<A, \end{cases} \]

where \(\lambda(t)\) is a continuous function satisfying the inequalities \(p(t)\leq \lambda(t)\leq q(t)\) \((t\in [a,b])\). It is obvious that \(f^{*}(t,y)\) is continuous in the domain \(Q^{*}\) and satisfies there, with respect to \(y\), a Lipschitz condition with constant \(K\) and condition (37).

Consider the difference boundary-value problem

\[ N_h[u_x^h]\equiv L_h[u_x^h]-f^{*}(t_x,u_x^h)=0, \tag{41} \]

\[ \frac{\Delta^{k_i}u_{x_i}^{h}}{h^{k_i}}=A_{ik_i}\quad (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m), \tag{42} \]

where \(x_i\in G_M\) are such that

\[ x_i\leq \frac{\alpha_i-a}{h}<x_i+1\quad (i=1,2,\ldots,m). \]

Extend the functions \(z_1(t)\) and \(z_2(t)\) to the interval \((b,b+\sigma]\) so that they are \(n\) times continuously differentiable and satisfy inequalities (39) on \([a,b+\sigma]\).

Using Taylor’s formula it is easy to show that

\[ \frac{\Delta^n z_j(t)}{h^n} -\sum_{k=0}^{n-1} p_k(t)\frac{\Delta^k z_j(t)}{h^k} - \]

\[ - f(t,z_j(t))=N[z_j]+\alpha_j(t,h)\quad (j=1,2), \tag{43} \]

\[ \frac{\Delta^k z_j(t_{x_i})}{h^k} = z_j^{(k)}(\alpha_i)+\beta^{h}_{jik} \quad (j=1,2;\ k=0,1,\ldots,n;\ i=1,\ldots,m), \tag{44} \]

where

\[ \lim_{h\to 0}\alpha_j(t,h)=0\quad (j=1,2) \tag{45} \]

uniformly with respect to \(t\in [a,b]\), and

\[ \lim_{h\to 0}\beta^{h}_{jik}=0 \quad (j=1,2;\ k=0,1,\ldots,n;\ i=1,\ldots,m). \tag{46} \]

Since \(z_j(t)\ (j=1,2)\) satisfy the boundary conditions (36), it follows from (44) that

\[ \frac{\Delta^{k_i}z_j(t_{x_i})}{h^{k_i}} = A_{ik_i}+\beta^{h}_{jik_i} \quad (j=1,2;\ k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m). \tag{47} \]

Let \(v^h_{xj}\ (j=1,2)\) be the solution of the boundary-value problem

\[ L_h[v^h_x]-p(t_x')v^h_x=0, \]

\[ \frac{\Delta^{k_i}v^h_{x_i}}{h^{k_i}} = -\beta^{h}_{jik_i} \quad (j=1,2;\ k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m). \tag{48} \]

Since \([a,b]\) is an interval of non-oscillation for equations (38), by Theorem 3 there exists a number \(N_2\), independent of \(h\) and of \(\beta^{h}_{jik_i}\), such that, for all sufficiently small \(h\),

\[ \max_{x\in g_{M+n-k}} \left| \frac{\Delta^k v^h_{xj}}{h^k} \right| < N_2 \sum_{i=1}^{m}\sum_{k_i=0}^{r_i-1} |\beta^{h}_{jik_i}| \quad (j=1,2;\ k=0,1,\ldots,n). \]

Hence, by (46), it follows that

\[ \lim_{h\to 0}\frac{\Delta^k v^h_{xj}}{h^k}=0 \quad (j=1,2;\ k=0,1,\ldots,n) \tag{49} \]

uniformly with respect to \(x\in g_{M+n-k}\).

Construct the functions

\[ z^h_{xj}=z_j(t_x)+v^h_{xj}\quad (j=1,2). \]

By (39) and (49), for all sufficiently small \(h\),

\[ A-\omega\le z^h_{xj}\le B+\omega \quad (x\in g_{M+n};\ j=1,2). \tag{50} \]

Thus the expressions \(f^*(t_x,z^h_{xj})\ (j=1,2)\) are defined for \(x\in g_M\).

Denote

\[ f^*(t_x,z^h_{xj})-f^*(t_x,z_j(t_x))=\gamma^h_{xj}\quad (j=1,2). \tag{51} \]

By the Lipschitz condition,

\[ |\gamma^h_{xj}|\le K|z^h_{xj}-z_j(t_x)|=K|v^h_{xj}|\quad (j=1,2). \]

Hence, by virtue of (49),

\[ \lim_{h \to 0} \gamma_{xj}^{h}=0 \quad (j=1,2) \tag{52} \]

uniformly with respect to \(x \in g_M\).

Next, taking into account (43) and (51), we obtain

\[ N_h[z_{xj}^{h}] = N_h[z_j(t_x)] + L_h[v_{xj}^{h}] - \gamma_{xj}^{h} = N[z_j]_{t=t_x}+\delta_{x,j}^{h}, \]

where

\[ \delta_{x,j}^{h} = a_j(t_x,h)+L_h[v_{xj}^{h}]-\gamma_{xj}^{h} \quad (j=1,2). \]

By virtue of (45), (49), and (52),

\[ \lim_{h \to 0}\delta_{x,j}^{h}=0 \quad (j=1,2) \]

uniformly with respect to \(x \in g_M\). But then, in view of (40), for sufficiently small \(h\),

\[ N_h[z_{x1}^{h}]>0,\quad N_h[z_{x2}^{h}]<0 \quad (x \in g_M). \]

Moreover, by virtue of (47) and (48), \(z_{xj}^{h}\) \((j=1,2)\) satisfy the boundary conditions (42).

Finally, since \(f^{*}(t,y)\) satisfies condition (37), and the operation \(L_h[u_x^{h}]\) is linear, the operation \(N_h[u_x^{h}]\) satisfies, in \(u_x^{h}, u_{x+1}^{h}, \ldots, u_{x+n-1}^{h}\), conditions \(L_1\) and \(L_2\) [8, 10], i.e., for any pair of functions \(u_{x1}^{h}\) and \(u_{x2}^{h}\), \(A-\omega \leq u_{x2}^{h} \leq u_{x1}^{h} \leq B+\omega\) \((x \in g_{M+n})\), the inequalities

\[ L_h[u_{x1}^{h}-u_{x2}^{h}] - q(t_x)(u_{x1}^{h}-u_{x2}^{h}) \leq N_h[u_{x1}^{h}]-N_h[u_{x2}^{h}] \leq \]

\[ \leq L_h[u_{x1}^{h}-u_{x2}^{h}] - p(t_x)(u_{x1}^{h}-u_{x2}^{h}) \quad (x \in g_M), \]

hold; moreover, by virtue of Theorem 1 and the condition of the present theorem, the linear equations

\[ L_h[u_x^{h}]-q(t_x)u_x^{h}=0,\quad L_h[u_x^{h}]-p(t_x)u_x^{h}=0, \]

corresponding to conditions \(L_2\) and \(L_1\), are nonoscillatory on the set \(g_{M+n}\) for any sufficiently small \(h\).

Thus, there exists an \(h_0>0\) such that for all \(h<h_0\) the boundary-value problem (41)—(42) and the functions \(z_{xj}^{h}\) \((j=1,2)\) satisfy the conditions of Theorem 3 of [10]. By this theorem there exists a unique solution \(u_x^{h}\) of problem (41)—(42), and

\[ \operatorname*{sign}_{x_i<x<x_{i+1}} [z_{xj}^{h}-u_x^{h}] = (-1)^{\,n-\sum_{k=1}^{i} r_k+j-1} \quad (j=1,2;\ i=1,\ldots,m-1) \tag{53} \]

everywhere where \(z_{xj}^{h}\ne u_x^{h}\). Hence, by virtue of (50), it follows that

\[ A-\omega \leq u_x^{h}\leq B+\omega \quad (x \in g_M). \]

It is easy to see that \(u_x^{h}\) can be extended to the set

\[ x=M+1,\ldots,M+n. \]

Thus, the difference ratios

\[ \frac{\Delta^k u_x^{h}}{h^k} \]

are each defined respectively on the set \(g_{M+n-k}\) \((k=0,1,\ldots,n)\).

Since \(f^{*}(t,y)\) is continuous in the domain \(Q^{*}\) and \(p(t)\) is continuous on \([a,b]\), there exists a number \(C>0\) such that

\[ |f^{*}(t,y)-p(t)y|\leq C \]

everywhere in \(Q^*\).

Thus,

\[ \left|f^*(t_x,u_x^h)-p(t_x)u_x^h\right|\leq C \quad (x\in g_M), \]

whence, by virtue of (41),

\[ \left|L_h[u_x^h]-p(t_x)u_x^h\right|\leq C \quad (x\in g_M). \]

But then, by Theorem 3, there exist numbers independent of \(h\), \(h_1\leq h_0,\ N_1>0\), and \(N_2>0\), such that

\[ \max_{x\in g_{M+n-k}}\left|\frac{\Delta^k u_x^h}{h^k}\right| < N_1C+N_2\sum_{i=1}^{m}\sum_{k_i=0}^{r_i-1}|A_{ik_i}|=D \quad (k=0,1,\ldots,n) \tag{54} \]

for all \(h<h_1\).

For each \(h<h_1\) construct continuous functions \(u_{hl}(t)\) on \([a,\ b+(n-l)h]\), defined by the equalities

\[ u_{hl}(t)=\frac{\Delta^{l+1}u_x^h}{h^{l+1}}(t-t_x)+\frac{\Delta^l u_x^h}{h^l} \quad \text{for } t\in [t_x,t_{x+1}] \tag{55} \]

\[ (x=0,1,\ldots,M+n-l-1;\ l=0,1,\ldots,n). \]

By virtue of (54),

\[ |u_{hl}(t)|\leq D \quad (t\in [a,\ b+(n-l)h];\ l=0,1,\ldots,n). \tag{56} \]

It is easy to see that

\[ \frac{\Delta u_{hl}(t)}{h}=u_{h\,l+1}(t) \quad (l=0,1,\ldots,n-1). \tag{57} \]

Each of the functions \(u_{hl}(t)\) \((l=0,1,\ldots,n-1)\) has on \([a,b]\) a piecewise constant derivative, and, by virtue of (55),

\[ u'_{hl}(t)=\frac{\Delta^{l+1}u_x^h}{h^{l+1}} \quad \text{for } t\in (t_x,t_{x+1}) \tag{58} \]

\[ (x=0,1,\ldots,M-1;\ l=0,1,\ldots,n-1). \]

Therefore, for any position of the points \(t',t''\in [a,b]\),

\[ u_{hl}(t'')-u_{hl}(t')=\int_{t'}^{t''}u'_{hl}(t)\,dt \quad (l=0,1,\ldots,n-1), \tag{59} \]

whence, by virtue of (54) and (58),

\[ |u_{hl}(t'')-u_{hl}(t')|\leq D\,|t''-t'| \quad (l=0,1,\ldots,n-1). \tag{60} \]

Thus, for \(h<h_1\), the functions of each of the families \(\{u_{hl}(t)\}\) \((l=0,1,\ldots,n-1)\) are uniformly bounded and equicontinuous on \([a,b]\). Consequently, by Arzelà’s theorem, from any infinite subset of each family one can extract an infinite sequence of functions uniformly converging to some function continuous on \([a,b]\).

It is easy to see that there exists a sequence \(\{h_s\}\) tending to zero such that, for each \(l\), the sequence of functions \(\{u_{h_s l}(t)\}\), as \(h_s\to 0\), converges uniformly to a function \(\varphi_l(t)\) continuous on \([a,b]\) \((l=0,1,\ldots,n-1)\).

We shall show that the functions \(\varphi_l(t)\) \((l=0,1,\ldots,n-2)\) are continuously differentiable on \([a,b]\).

Let \(t', t'' \in [a,b]\) be any two points. We have

\[ \left|\frac{\varphi_l(t'')-\varphi_l(t')}{t''-t'}-\varphi_{l+1}(t')\right| \leq \left|\frac{\varphi_l(t'')-\varphi_l(t')}{t''-t'}-\frac{u_{h_s l}(t'')-u_{h_s l}(t')}{t''-t'}\right| + \]

\[ + \left|\frac{u_{h_s l}(t'')-u_{h_s l}(t')}{t''-t'}-u_{h_s l+1}(t_x)\right| +\left|u_{h_s l+1}(t_x)-u_{h_s l+1}(t')\right| + \]

\[ +\left|u_{h_s l+1}(t')-\varphi_{l+1}(t')\right|, \tag{61} \]

where \([t_x,t_{x+1}]\) is the interval containing \(t'\) for the given \(h_s\). But by virtue of (55), (58), (59), (60), and the theorem on estimating an integral,

\[ \left|\frac{u_{h_s l}(t'')-u_{h_s l}(t')}{t''-t'}-u_{h_s l+1}(t_x)\right| = \left|\frac{\displaystyle\int_{t'}^{t''} u'_{h_s l}(t)\,dt}{t''-t'}-u_{h_s l+1}(t_x)\right| \leq \]

\[ \leq \left|\sup_{t\in[t^*,t^{**}]} u_{h_s l+1}(t)- \inf_{t\in[t^*,t^{**}]} u_{h_s l+1}(t)\right| \leq D|t^{**}-t^*|= \]

\[ = D|t''-t'|+Dh_s, \tag{62} \]

where \(t^*=\min\{t',t''\}-h_s\) for \(t'>a\) and \(t^*=t'\) for \(t'=a\), \(t^{**}=\max\{t',t''\}\). From (60), (61), and (62) we obtain

\[ \left|\frac{\varphi_l(t'')-\varphi_l(t')}{t''-t'}-\varphi_{l+1}(t')\right| \leq \frac{|\varphi_l(t'')-u_{h_s l}(t'')|+|\varphi_l(t')-u_{h_s l}(t')|}{|t''-t'|} + \]

\[ +\left|u_{h_s l+1}(t')-\varphi_{l+1}(t')\right| +D|t''-t'|+2Dh_s \quad (l=0,1,\ldots,n-2). \tag{63} \]

Since

\[ \lim_{h_s\to 0} u_{h_s l}(t)=\varphi_l(t) \quad (l=0,1,\ldots,n-1) \tag{64} \]

uniformly with respect to \(t\in[a,b]\), it follows from (63) that for every \(\varepsilon>0\) there exists a \(\delta>0\) such that

\[ \left|\frac{\varphi_l(t'')-\varphi_l(t')}{t''-t'}-\varphi_{l+1}(t')\right| <\varepsilon \quad (l=0,1,\ldots,n-2), \]

as soon as \(|t''-t'|<\delta\) \((t''\ne t')\), since in (63) \(h_s\) may be taken arbitrarily small. Hence,

\[ \lim_{t''\to t'}\frac{\varphi_l(t'')-\varphi_l(t')}{t''-t'} =\varphi_{l+1}(t'), \]

i.e. \(\varphi_l(t)\) has a continuous derivative on \([a,b]\), and

\[ \varphi'_l(t)=\varphi_{l+1}(t) \quad (l=0,1,\ldots,n-2). \]

Thus the function \(\varphi_0(t)\) is \(n-1\) times continuously differentiable on \([a,b]\), and

\[ \varphi_0^{(k)}(t)=\varphi_k(t) \quad (k=0,1,\ldots,n-1). \tag{65} \]

Finally, let us show that \(\varphi_0(t)\) has a continuous \(n\)-th derivative on \([a,b]\) and satisfies equation (35). Let \(t',t''\in[a,b]\) \((t'\ne t'')\) be two arbitrary points.

Then

\[ \left| \frac{\varphi_{n-1}(t'')-\varphi_{n-1}(t')}{t''-t'} -\sum_{k=0}^{n-1}p_k(t')\varphi_k(t')-f^*(t',\varphi_0(t')) \right|\leq \]

\[ \begin{aligned} \leq{}& \left| \frac{\varphi_{n-1}(t'')-\varphi_{n-1}(t')}{t''-t'} - \frac{u_{h_s n-1}(t'')-u_{h_s n-1}(t')}{t''-t'} \right| + \left| \frac{u_{h_s n-1}(t'')-u_{h_s n-1}(t')}{t''-t'} -\frac{\Delta^n u_x^h}{h_s^n} \right| \\ &+ \left| \frac{\Delta^n u_x^{h_s}}{h_s^n} -\sum_{k=0}^{n-1}p_k(t_x)\frac{\Delta^k u_x^{h_s}}{h_s^k} -f^*(t_x,u_x^{h_s}) \right| \\ &+ \left| \sum_{k=0}^{n-1}p_k(t_x) \left[ \frac{\Delta^k u_x^{h_s}}{h_s^k}-u_{h_s k}(t') \right] \right| + \left| \sum_{k=0}^{n-1}u_{h_s k}(t')\,[p_k(t_x)-p_k(t')] \right| \\ &+ \left| \sum_{k=0}^{n-1}p_k(t')\,[u_{h_s k}(t')-\varphi_k(t')] \right| + \left|f^*(t_x,u_x^h)-f^*(t_x,u_{h_s 0}(t'))\right| \\ &+ \left|f^*(t_x,u_{h_s 0}(t'))-f^*(t',u_{h_s 0}(t'))\right| + \left|f^*(t',u_{h_s 0}(t'))-f^*(t',\varphi_0(t'))\right|, \tag{66} \end{aligned} \]

where again \([t_x,t_{x+1}]\) is the interval containing \(t'\) for the given \(h_s\). But by virtue of the theorem on the estimate of an integral, relations (41), (54), (55), (58), (59), (60), and the Lipschitz condition,

\[ \left| \frac{u_{h_s n-1}(t'')-u_{h_s n-1}(t')}{t''-t'} -\frac{\Delta^n u_x^h}{h_s^n} \right| = \left| \frac{\displaystyle\int_{t'}^{t''}u'_{h_s n-1}(t)\,dt}{t''-t'} -\frac{\Delta^n u_x^{h_s}}{h_s^n} \right| \leq \]

\[ \begin{aligned} \leq{}& \left| \sup_{t_x\in[t^*,t^{**}]}\frac{\Delta^n u_x^{h_s}}{h_s^n} - \inf_{t_x\in[t^*,t^{**}]}\frac{\Delta^n u_x^{h_s}}{h_s^n} \right| \\ ={}& \left| \left[ \sum_{k=0}^{n-1}p_k(t_{x^{**}}) \frac{\Delta^k u_{x^{**}}^{h_s}}{h_s^k} + f^*(t_{x^{**}},u_{x^{**}}^{h_s}) \right] - \left[ \sum_{k=0}^{n-1}p_k(t_{x^*}) \frac{\Delta^k u_{x^*}^{h_s}}{h_s^k} + f^*(t_{x^*},u_{x^*}^{h_s}) \right] \right| \leq \\ \leq{}& \left| \sum_{k=0}^{n-1}p_k(t_{x^{**}}) \left[ \frac{\Delta^k u_{x^{**}}^{h_s}}{h_s^k} - \frac{\Delta^k u_{x^*}^{h_s}}{h_s^k} \right] \right| + \left| \sum_{k=0}^{n-1}\frac{\Delta^k u_{x^*}^{h_s}}{h_s^k} [p_k(t_{x^{**}})-p_k(t_{x^*})] \right| \\ &+ \left|f^*(t_{x^{**}},u_{x^{**}}^{h_s})-f^*(t_{x^{**}},u_{x^*}^{h_s})\right| + \left|f^*(t_{x^{**}},u_{x^*}^{h_s})-f^*(t_{x^*},u_{x^*}^{h_s})\right| \\ \leq{}& HDn\,|t^{**}-t^*| + D\sum_{k=0}^{n-1}|p_k(t_{x^{**}})-p_k(t_{x^*})| + KD\,|t^{**}-t^*| \\ &+ \left|f^*(t_{x^{**}},u_{x^*}^{h_s})-f^*(t_{x^*},u_{x^*}^{h_s})\right|, \tag{67} \end{aligned} \]

where \(t^*=\min\{t',t''\}-h_s\) when \(t'>a\) and \(t^*=t'\) when \(t'=a\), \(t^{**}=\max\{t',t''\}\), \(t_{x^*},t_{x^{**}}\in[t^*,t^{**}]\), and \(H\) is the number satisfying inequalities (34). From (66), by virtue of (34), (41), (55), (56), (60), (67), and the Lipschitz condition, we obtain

\[ \left| \frac{\varphi_{n-1}(t'')-\varphi_{n-1}(t')}{t''-t'} -\sum_{k=0}^{n-1}p_k(t')\varphi_k(t')-f^*(t',\varphi_0(t')) \right|\leq \]

\[ \leq \frac{ |\varphi_{n-1}(t'')-u^h_{s\,n-1}(t'')| + |\varphi_{n-1}(t')-u^h_{s\,n-1}(t')| }{|t''-t'|} + D\sum_{k=0}^{n-1}|p_k(t_x)-p_k(t')|+ \]

\[ + D\sum_{k=0}^{n-1}|p_k(t_{x^{**}})-p_k(t_{x^*})| + |f^*(t_x,u^h_{s\,0}(t'))-f^*(t',u^h_{s\,0}(t'))|+ \]

\[ + |f^*(t_{x^{**}},u^h_{x^{**}s})-f^*(t_{x^*},u^h_{x^*s})| + H\sum_{k=0}^{n-1}|u^h_{s\,k}(t')-\varphi_k(t')|+ \]

\[ + K|u^h_{s\,0}(t')-\varphi_0(t')| + D(Hn+K)|t''-t'|+2D(Hn+K)h_s. \]

Hence, by virtue of (64), the uniform continuity of the functions \(p_k(t)\) \((k=0,1,\ldots,n-1)\) on the interval \([a,b]\) and of the function \(f^*(t,y)\) in the domain \(Q^*\), and the fact that \(|t_x-t'|\leq h_s,\ |t_{x^{**}}-t_{x^*}|\leq |t''-t'|+h_s\), for every \(\varepsilon>0\) there follows the existence of such a \(\delta>0\) that

\[ \left| \frac{\varphi_{n-1}(t'')-\varphi_{n-1}(t')}{t''-t'} -\sum_{k=0}^{n-1}p_k(t')\varphi_k(t')-f^*(t',\varphi_0(t')) \right|<\varepsilon, \]

as soon as \(|t''-t'|<\delta\), since \(h_s\) can be chosen arbitrarily small. Thus,

\[ \lim_{t''\to t'} \frac{\varphi_{n-1}(t'')-\varphi_{n-1}(t')}{t''-t'} = \sum_{k=0}^{n-1}p_k(t')\varphi_k(t')+f^*(t',\varphi_0(t')), \]

i.e. \(\varphi_{n-1}(t)\) has a continuous derivative on \([a,b]\), and moreover

\[ \varphi'_{n-1}(t)=\sum_{k=0}^{n-1}p_k(t)\varphi_k(t)+f^*(t,\varphi_0(t)). \]

By virtue of (65), this means that the function \(\varphi_0(t)\) is \(n\) times continuously differentiable on \([a,b]\) and satisfies the equation

\[ \varphi_0^{(n)}(t)=\sum_{k=0}^{n-1}p_k(t)\varphi_0^{(k)}(t)+f^*(t,\varphi_0(t)). \tag{68} \]

Since

\[ \lim_{h\to 0} z^h_{xj}=z_j(t)\quad (j=1,2) \]

uniformly with respect to \(x\in g_M\), and

\[ \lim_{h\to 0} t_{x_i}=a_i\quad (i=1,\ldots,m), \tag{69} \]

it follows from relation (53), by virtue of (55) and (64), that

\[ \operatorname{sign}_{t\in(\alpha_i,\alpha_{i+1})}\,[z_j(t)-\varphi_0(t)] = (-1)^{\,n-\sum_{k=1}^{i} r_k+j-1} \qquad (j=1,2;\ i=1,\ldots,m-1) \tag{70} \]

everywhere where \(z_j(t)\ne\varphi_0(t)\). Hence, by virtue of (39),

\[ A<\varphi_0(t)<B \qquad (t\in[a,b]). \]

And since in the domain \(Q\) the function \(f^*(t,y)\) coincides with \(f(t,y)\), it follows from (68) that \(\varphi_0(t)\) satisfies the equation

\[ L[\varphi_0]=f(t,\varphi_0). \]

Finally, by virtue of (55), (64), (65), (69), and the continuity of \(\varphi_0^{(k)}(t)\),

\[ \lim_{h_s\to 0}\frac{\Delta^k u^{h_s}_{x_i}}{h_s^k} = \varphi_0^{(k)}(\alpha_i) \qquad (k=0,1,\ldots,n-1;\ i=1,\ldots,m), \]

whence, by virtue of (42),

\[ \varphi_0^{(k_i)}(\alpha_i)=A_{ik_i} \qquad (k_i=0,1,\ldots,r_i-1;\ i=1,\ldots,m). \]

Thus, \(\varphi_0(t)\) is a solution of the boundary-value problem (35)—(36) and satisfies the relations (70).

The theorem is proved.

Remark. By complicating the proof, the strict inequalities (40) can be replaced by non-strict ones, and one can also prove uniqueness of the solution of problem (35)—(36).

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Received by the editors
October 8, 1964

Izhevsk Mechanical Institute