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UDC 517.949.2
ON THE STABILITY OF DIFFERENCE EQUATIONS
L. V. MASLOVSKAYA
The article considers questions of boundedness and stability of solutions of difference equations. The theorems proved in the first part of the article are difference analogues of the corresponding theorems of L. F. Rakhmatullina for differential equations [1]. In the second part of the article, the results of the first part are applied to difference equations arising in the approximation of differential equations.
I. Consider the system of difference equations
\[ \Delta z = A(t)z + f(t,z), \qquad t=0,1,\ldots \tag{1} \]
and the initial conditions
\[ z(0)=z_0, \tag{2} \]
where \(z=\{z_1(t),\ldots,z_n(t)\}\) is an \(n\)-dimensional vector; \(A(t)=(a_{ij}(t))\) is a square matrix of order \(n\); \(f(t,z)=\{f_1(t,z_1,\ldots,z_n),\ldots,f_n(t,z_1,\ldots,z_n)\}\) is a vector-function. In addition, consider the difference equation
\[ x(t)=\sum_{t_1=0}^{t-1} K(t,t_1,x(t_1))+\varphi(t), \qquad t=1,2,\ldots, \tag{3} \]
\[ x(0)=\varphi(0). \]
Denote by \(x_\varphi\) the solution of this equation.
Lemma. Let \(K(t,t_1,x)\) be defined for \(t=0,1,\ldots,\ t_1=0,1,\ldots\) and for all \(x\); for fixed \(t\) and \(t_1\), \(K(t,t_1,x)\) is a nondecreasing function of \(x\).
If the function \(F(t)\) satisfies the inequality
\[ F(t)\leq \sum_{t_1=0}^{t-1} K(t,t_1,F(t_1))+\varphi(t), \]
\[ F(0)\leq \varphi(0), \]
then \(F(t)\leq x_\varphi(t),\ t=0,1,\ldots\).
Proof. \(F(0)\leq \varphi(0)=x_\varphi(0)\).
Suppose that the inequality \(F(t)\leq x_\varphi(t)\) does not hold for all \(t=1,2,\ldots\). Then there exists such a \(t_0\) that \(F(t)\leq x_\varphi(t)\) for \(t=0,1,\ldots,t_0-1\), but \(F(t_0)>x_\varphi(t_0)\), i.e. \(x_\varphi(t_0)-F(t_0)<0\).
On the other hand,
\[ x_\varphi(t_0)-F(t_0)\geq \sum_{t_1=0}^{t-1} \left[ K(t_0,t_1,x_\varphi(t_1))-K(t_0,t_1,F(t_1)) \right]\geq 0. \]
Thus, we have obtained a contradiction.
The lemma is proved.
The lemma is a difference analogue of Chaplygin’s theorem on differential inequalities.
Remark. The lemma is also valid in the case when the domain of variation of the discrete argument is \(t=0, 1, \ldots, N\), where \(N\) is a natural number.
Consider the linear homogeneous system
\[ \Delta y=A(t)y \tag{4} \]
and the initial conditions
\[ y(0)=z_0. \tag{5} \]
Let \(\varphi(t)\) and \(K(t,t_1)\) be nonnegative functions satisfying the inequalities
\[ \varphi(t)\geqslant \|y(t)\|,\qquad K(t,t_1)\geqslant \|Y(t)Y^{-1}(t_1+1)\|, \]
where \(y(t)\) is the solution of system (4) satisfying condition (5); \(Y(t)\) is the fundamental matrix of system (4), such that \(Y(0)\) is the identity matrix.
Theorem 1. Let \(f(t,u)\) be defined for \(t=0,1,\ldots\), arbitrary \(u\), and satisfy the condition \(\|f(t,u)\|\leqslant g(t,\|u\|)\), where \(g(t,v)\) is a nonnegative function nondecreasing in \(v\).
If the solution \(x_\varphi\) of the equation
\[ x(t)=\sum_{t_1=0}^{t-1} K(t,t_1)g(t_1,x(t_1))+\varphi(t),\quad t=1,2,\ldots, \tag{6} \]
\[ x(0)=\varphi(0) \]
is bounded, then the solution \(z(t)\) of system (1), satisfying condition (2), is also bounded.
Proof. Let \(x_\varphi(t)\) be a bounded solution of equation (4), such that \(x_\varphi(0)=\varphi(0)\). Hence, \(x_\varphi(t)\leqslant M,\ t=0,1,\ldots,\ \|z(0)\|=\|y(0)\|\leqslant \varphi(0)\leqslant M\). We shall prove that \(\|z(t)\|\leqslant M,\ t=1,2,\ldots\).
The solution \(z(t)\) of system (1), satisfying condition (2), can be represented in the form [2]
\[ z(t)=\sum_{t_1=0}^{t-1} Y(t)Y^{-1}(t_1+1)f(t_1,z(t_1))+y(t), \]
where \(y(t)\) is the solution of (4) satisfying condition (5):
\[ \|z(t)\|\leqslant \sum_{t_1=0}^{t-1} K(t,t_1)g(t_1,\|z(t_1)\|)+\varphi(t), \]
and, by the lemma,
\[ \|z(t)\|\leqslant x_\varphi(t)\leqslant M. \]
Remark. If all solutions of system (4) are bounded, then it is convenient to choose as \(K(t,t_1)\) a function of one variable \(K(t,t_1)\equiv K(t_1)\), and to regard \(\varphi(t)\) as constant. Then equation (6), which is called the comparison equation, has the form
\[ x(t)=C+\sum_{t_1=0}^{t-1} K(t_1)g(t_1,x(t_1)),\quad t=1,2,\ldots, \]
\[ x(0)=C \]
or
\[ \Delta x = K(t)g(t,x(t)),\quad t=0,1,\ldots, \]
\[ x(0)=C, \]
and from the boundedness of the solution of this equation follows the boundedness of the solution of system (1) satisfying condition (2).
Let
\[ \|Y(t)Y^{-1}(t_1+1)\|\leq K=\mathrm{const}. \]
This will be the case, in particular, if the matrix \(A\) is constant and all solutions of system (4) are bounded. Indeed, in this case
\[ Y(t)Y^{-1}(t_1+1)=Y(t-t_1-1), \]
\[ \|Y(t-t_1-1)\|\leq K=\mathrm{const}. \]
Moreover, suppose that the function \(f(t,z)\) satisfies the condition
\[ \|f(t,z)\|\leq g(t)\|z\|. \]
Then the comparison equation takes the form
\[ \Delta x = Kg(t)x,\quad t=0,1,\ldots, \tag{7} \]
\[ x(0)=C. \]
Its solution is
\[ x(t+1)=C\prod_{t_1=0}^{t}\bigl(1+Kg(t_1)\bigr). \tag{8} \]
This solution is bounded if
\[ \left\|\prod_{t_1=0}^{t}\bigl(1+Kg(t_1)\bigr)\right\| \]
is bounded for \(t=0,1,\ldots\), in particular, if
\[ \prod_{t=0}^{\infty}\bigl(1+Kg(t)\bigr) \]
converges, i.e., if the series
\[ \sum_{t=0}^{\infty} g(t) \]
converges.
Consider the system
\[ \Delta z=(A+B(t))z,\quad t=0,1,\ldots \tag{9} \]
and the initial conditions
\[ z(0)=z_0, \]
where the matrix \(B(t)\to 0\) as \(t\to\infty\). For it, under the assumption of boundedness of the solutions of (4), the comparison equation has the form
\[ \Delta x=K\|B(t)\|x,\quad t=0,1,\ldots, \]
\[ x(0)=C. \]
Its solution
\[ x(t+1)=C\prod_{t_1=0}^{t}\bigl(1+K\|B(t)\|\bigr) \]
is bounded if the series
\[ \sum_{t=0}^{\infty}\|B(t)\| \]
converges.
For the system
\[ \Delta z=(A+B(t))z+f(t,z),\quad t=0,1,\ldots, \tag{10} \]
\[ z(0)=z_0, \]
where the matrix \(B(t)\to 0\) as \(t\to\infty\), and \(f(t,z)\) satisfies the condition \(\|f(t,z)\|\leq g(t)\|z\|\), the comparison equation, under the assumption of boundedness of the solutions of (4), has the form
\[ \Delta x=K(\|B(t)\|+g(t))x,\quad t=0,1,\ldots, \]
\[ x(0)=C. \]
Its solution
\[ x(t+1)=C\prod_{t_1=0}^{t}\bigl(1+K\|B(t)\|+Kg(t)\bigr) \]
is bounded if the series
\[ \sum_{t=0}^{\infty}(\|B(t)\|+g(t)). \]
converges.
By stability of systems (1) and (4) we shall mean stability with respect to the initial data, i.e., we shall call system (1) stable if, for any solutions \(z(t)\), \(\widetilde z(t)\) of this system, from \(\|\widetilde z(0)-z(0)\|<\delta\) it follows that \(\|\widetilde z(t)-z(t)\|<c\delta\) (\(c=\mathrm{const}\)) for \(0<\delta<\delta_0\) and \(t=1,2,\ldots\). If, in addition, \(\lim_{t\to\infty}(\widetilde z(t)-z(t))=0\), then system (1) is called asymptotically stable.
Equation (3) will be regarded as stable if from \(|\varphi(t)|<\delta\) it follows that \(|x_\varphi(t)|<c_1\delta\) (\(c_1=\mathrm{const}\)), \(t=1,2,\ldots\). If, in addition, \(\lim_{t\to\infty}x_\varphi(t)=0\), then equation (3) is called asymptotically stable.
Theorem 2. Let \(f(t,u)\) be defined for \(t=0,1,\ldots\) and arbitrary \(u\), and satisfy the conditions
\[ f(t,0)=0,\quad \|f(t,u_1)-f(t,u_2)\|\leq g(t,\|u_1-u_2\|), \]
where \(g(t,v)\) is a nonnegative function, nondecreasing in \(v\). Then from the stability of system (4) and the stability (asymptotic stability) of the comparison equation
\[ x(t)=\sum_{t_1=0}^{t-1} K(t,t_1)g(t_1,x(t_1))+\varphi(t), \]
\[ x(0)=\varphi(0) \]
there follows the stability (asymptotic stability) of system (1).
Proof. Let \(\widetilde z(t)\) be a solution of the system
\[ \Delta \widetilde z=A(t)\widetilde z+f(t,\widetilde z), \]
satisfying the initial conditions
\[ \widetilde z(0)=\widetilde z_0. \]
The corresponding linear system is
\[ \Delta \widetilde y=A(t)\widetilde y,\quad \widetilde y(0)=\widetilde z_0. \]
Then
\[ \widetilde z(t)=\sum_{t_1=0}^{t-1} Y(t)Y^{-1}(t_1+1) f(t_1,\widetilde z(t_1))+\widetilde y(t), \]
\[ \widetilde z(t)-z(t)=\sum_{t_1=0}^{t-1} Y(t)Y^{-1}(t_1+1) \bigl[f(t_1,\widetilde z(t_1))-f(t_1,z(t_1))\bigr]+\widetilde y(t)-y(t), \]
\[ \|\widetilde z(t)-z(t)\|\leq \sum_{t_1=0}^{t-1} K(t,t_1)g(t_1,\|\widetilde z(t_1)-z(t_1)\|) +\|\widetilde y(t)-y(t)\|. \]
Hence, on the basis of the lemma,
\[ \|\widetilde z(t)-z(t)\|\leq x_{\|\widetilde y(t)-y(t)\|}, \tag{11} \]
where \(x_{\|\widetilde y(t)-y(t)\|}\) is the solution of the equation
\[ x(t)=\sum_{t_1=0}^{t-1} K(t,t_1)g(t_1,x(t_1))+\|\widetilde y(t)-y(t)\|,\qquad t=1,2,\ldots, \]
\[ x(0)=\|\widetilde y(0)-y(0)\|. \]
By virtue of the stability of system (4),
\[ \|\widetilde y(t)-y(t)\|<c\delta \quad \text{for } \|\widetilde y(0)-y(0)\|<\delta . \]
By virtue of the stability of the comparison equation,
\(x_{\|\widetilde y(t)-y(t)\|}<c_1\delta\). Thus,
\[ \|\widetilde z(t)-z(t)\|<c_1\delta, \]
as soon as \(\|\widetilde z(0)-z(0)\|<\delta\), i.e., system (1) is stable.
It follows from inequality (11) that, if instead of stability of the comparison equation one requires asymptotic stability, then system (1) will also be asymptotically stable.
Corollaries. If \(f(t,z)\) satisfies the conditions
\[ f(t,0)=0,\qquad \|f(t,z_1)-f(t,z_2)\|\leq g(t)\|z_1-z_2\|, \]
and the matrix \(A\) is constant, then from the stability of system (4) there follows the stability of system (1), provided the series \(\sum_{t=0}^{\infty} g(t)\) converges.
This follows from the fact that the comparison equation (7), whose solution is written in the form (8), is stable.
If in this case (4) is asymptotically stable, then (1) will also be asymptotically stable; moreover, here it is not necessary to require convergence of the series \(\sum_{t=0}^{\infty} g(t)\); it is sufficient that \(g(t)\to 0\) as \(t\to\infty\), or even that \(a+Cg(t)<1\), where \(a\) is greater than the spectral radius of the matrix \(A+E\).
This follows from the fact that in this case the comparison equation has the form
\[ x(t)=C\sum_{t_1=0}^{t-1} a^{t-t_1-1}g(t_1)x(t_1)+Ca^t,\qquad t=1,2,\ldots, \]
\[ x(0)=C, \]
and its solution
\[ x(t+1)=C\prod_{t_1=0}^{t}\bigl(a+Cg(t_1)\bigr). \]
Consider system (9). The stability of this system follows from the stability of system (4), if the series \(\sum_{t=0}^{\infty}\|B(t)\|\) converges.
In the case of asymptotic stability of system (4), system (9) is also asymptotically stable, even if, instead of the convergence of the series \(\sum_{t=0}^{\infty}\|B(t)\|\), one requires that \(B(t)\to 0,\ t\to\infty\), or \(a+C\|B(t)\|<1\), where \(a\) is greater than the spectral radius of the matrix \(A+E\).
Consider system (10). Its stability follows from the stability of system (4), if the series converges
\[ \sum_{t=0}^{\infty}\bigl(\|B(t)\|+g(t)\bigr). \]
If (4) is asymptotically stable, then system (10) will also be asymptotically stable, if \(B(t)\to 0,\ g(t)\to 0\) as \(t\to\infty\), or if \(a+C\|B(t)\|+Cg(t)<1\), where \(a\) is greater than the spectral radius of the matrix \(A+E\).
II. Consider the differential equation
\[ \frac{d^{m}z}{dx^{m}}+p_1(x)\frac{d^{m-1}z}{dx^{m-1}}+\cdots+p_m(x)z=f(x,z) \tag{12} \]
and the initial conditions
\[ z(0)=z_0,\quad \frac{dz}{dx}(0)=z'_0,\ \ldots,\ \frac{d^{m-1}z}{dx^{m-1}}(0)=z_0^{m-1}. \]
\(p_1(x), p_2(x), \ldots, p_m(x)\) are bounded for \(0\le x\le X\), and \(f(x,z)\) satisfies the Lipschitz condition with respect to the variable \(z\), i.e.
\[ \|f(x,z_1)-f(x,z_2)\|\le L\|z_1-z_2\|,\quad L=\mathrm{const},\quad f(x,0)=0. \]
Divide the segment \([0,X]\) into \(N\) equal parts by the points \(th,\ t=0,1,\ldots,N\), where \(h\) is the step, \(h=X/N\).
Replace the derivatives in equation (12) and in the initial conditions by difference relations according to the formulas of numerical differentiation [3]
\[ \left.\frac{d^{i}z}{dx^{i}}\right|_{th} = \frac{1}{h^{i}}\sum_{l=-N_1^{i}}^{l=N_2^{i}}q_l^{i}z(th+lh) = \frac{1}{h^{i}}\sum_{l=-N_1^{i}}^{l=N_2^{i}}q_l^{i}z_h(t+l), \]
\[ i=1,2,\ldots,m, \]
\[ N_1^m\ge N_1^{m-1}\ge\cdots\ge N_1^1,\quad N_2^m\ge N_2^{m-1}\ge\cdots\ge N_2^1. \]
The difference equation that is thereby obtained can be reduced to the system
\[ \Delta z_h=(A+B_h(t))z_h+f_h(t,z_h),\quad t=0,1,\ldots, \tag{13} \]
and the initial conditions to the form
\[ z_h(0)=z_{0h}, \]
where \(z_h=\{z_{1h}(t), \ldots, z_{nh}(t)\}\) is an \(n\)-dimensional vector; \(A=(a_{ij})\), \(B_h(t)=(b_{ijh}(t))\) are square matrices of order \(n\); \(f_h(t,z_h)=\{f_{1h}(t,z_{1h},\ldots,z_{nh}), \ldots, f_{nh}(t,z_{1h},\ldots,z_{nh})\}\) is a vector function; \(h\) plays the role of a parameter; the argument \(t=0,1,\ldots,N\).
By stability of system (13) we shall mean stability with respect to the initial data, as defined in [4], i.e., we shall call system (13) stable if, for any solutions \(\tilde z_h(t)\), \(z_h(t)\) of this system, from \(\|\tilde z_h(0)-z_h(0)\|<\delta\) it follows that \(\|\tilde z_h(t)-z_h(t)\|<c\delta\) as the number of partition points increases without bound \((t\to\infty)\), i.e., as the step decreases without bound. Thus, as \(t\to\infty\), \(h\to0\) at the same rate, while the quantity \(th\) remains bounded.
If, moreover, \(\lim(\tilde z_h(t)-z_h(t))=0\) as the number of partition points increases without bound, then system (13) is called asymptotically stable. With this definition of stability, Theorems 1 and 2 hold for system (13). Here it is only necessary to take into account that, as \(t\to\infty\), \(h\to0\).
The matrix \(A\) of the system is constant, while the matrix \(B_h(t)\to0\) as \(t\to\infty\), \(h\to0\) at the same rate as \(h\to0\), or even faster.
We shall understand the norm of a vector to be the maximum of the moduli of its components; then the norm of a matrix is the maximum of the sums of the moduli of the elements in its rows. In this case
\[ \|B_h(t)\|\le d_1h,\quad \|f_h(t,z_h)\|\le d_2h\|z_h\|,\quad d_1,\ d_2 \text{ are constants.} \]
The comparison equation for system (13) has the form
\[ \Delta x_h=K\bigl(\|B_h(t)\|+d_2h\bigr)x_h,\quad t=0,1,\ldots,\quad x(0)=C. \]
Its solution is
\[ x_h(t)=C\prod_{t_1=0}^{t-1}\bigl[1+K\bigl(\|B_h(t)\|+d_2h\bigr)\bigr]. \]
The comparison equation is stable, since
\[ \prod_{t_1=0}^{t-1}\bigl[1+K\|B_h(t)\|+d_2h\bigr] \le [1+(d_1+d_2)h]^t \le \]
\[ \le [1+(d_1+d_2)h]^{X/h}<e^{X(d_1+d_2)}. \]
Thus, the asymptotic stability of system (13) follows from the asymptotic stability of the system
\[ \Delta y=Ay. \tag{14} \]
The coefficients of system (14) are linear combinations of the coefficients of the formula of numerical differentiation for
\[ \left.\frac{d^m z}{dx^m}\right|_{t_h}, \]
and the elements of the matrix \(B_h(t)\) do not depend on the latter. Therefore, the stability of system (13) is not destroyed when the method of replacing the lower derivatives in equation (12) by difference quotients is changed.
In conclusion, the author thanks the staff of the Izhevsk Mathematical Seminar for valuable advice in discussing the present work.
References
- Rakhmatullina L. F. Izv. vuzov, Matematika, No. 2 (9), 1959.
- Bellman R. Trans. Amer. Math. Soc., 62, 3, 1947.
- Berezin I. S., Zhidkov N. P. Methods of Computation, Vols. I, II. Fizmatgiz, 1960.
- Ryabenkii V. S., Filippov A. F. On the Stability of Difference Equations. GITTL, Moscow, 1956.
Received by the editors
June 29, 1965
Odessa State University
named after I. I. Mechnikov