MATHEMATICS

L. A. GROZA

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A LINEAR DIFFERENTIAL EQUATION OF SECOND ORDER IN A BANACH SPACE WHEN THE SMALL PARAMETER AT THE HIGHEST DERIVATIVE TENDS TO ZERO

(Presented by Academician S. L. Sobolev on 29 VI 1960)

Let \(Y\) be a Banach space. The space of linear bounded operators mapping the space \(Y\) into \(Y\) will be denoted by \(\{Y \to Y\}_{[1]}\). Consider the ordinary linear differential equation of second order in the Banach space \(Y\):

\[ \varepsilon y'' + p_0(x)y' + q(x)y = a(x). \tag{1} \]

Here \(\varepsilon > 0\) is a small real parameter tending to zero; \(y = y(x,\varepsilon)\in Y\) is the unknown function of the real variable \(x\); \(p_0(x)\ge \chi > 0\) is a real function, \(q(x)\in\{Y\to Y\}\); \(a(x)\in Y\) (the functions \(p_0,q,a\) are defined for \(x_1\le x\le x_2\)). Suppose that the functions \(p_0(x)\), \(q(x)\), \(a(x)\) are continuous, infinitely differentiable, and

\[ \|q(x)\|\le K,\qquad \|a(x)\|\le K,\qquad \|q^{(n)}(x)\|\le M(n),\qquad \|a^{(n)}(x)\|\le M(n) \]
\[ (x_1\le x\le x_2), \tag{2} \]

where \(K, M(n)\) are constants (\(M(n)\) may increase together with \(n\)). Under these assumptions it will be proved that any solution \(y(x,\varepsilon)\) of equation (1) with Cauchy initial data tends, as \(\varepsilon\to 0\), to the corresponding solution of the limiting equation

\[ p_0(x)y' + q(x)y = a(x). \tag{3} \]

The method of investigating equation (1) is based on the theory of asymptotic series in a Banach space \((^2)\). As is known \((^2)\), for an arbitrary set of elements \(C_0,C_1,C_2,\ldots\in Y\) there exists a function \(f(\varepsilon)\in Y\) (and consequently infinitely many such functions) which satisfies the asymptotic relation \(f(\varepsilon)\sim \sum_{i=0}^{\infty} C_i\varepsilon^i\). Taking an arbitrary set of functions \(C_0(x), C_1(x), C_2(x),\ldots\in Y\), one can construct such a function \(f(x,\varepsilon)\in Y\) that the relation

\[ f(x,\varepsilon)\sim \sum_{i=0}^{\infty} C_i(x)\varepsilon^i \tag{4} \]

will hold.

It is obvious that this asymptotic relation can be differentiated with respect to \(x\), if the coefficients \(C_i(x)\) are differentiable. We shall first consider the homogeneous equation

\[ \varepsilon y'' + p_0(x)y' + q(x)y = \theta. \tag{5} \]

Simultaneously with equation (5) we shall consider the equation

\[ \varepsilon \bar y'' + p_0(x)\bar y' + \bar q(x)\bar y = \bar\theta. \tag{6} \]

in the space \(\{Y \to Y\}\). Obviously, knowing some solution \(\bar y=\bar y(x,\varepsilon)\) of equation (6), we obtain a solution \(y=\bar y(x,\varepsilon)C\) of equation (5), where \(C\in Y\).

Theorem 1. Equation (6) has two independent solutions \(\bar y_1(x,\varepsilon)\), \(\bar y_2(x,\varepsilon)\), satisfying the relations

\[ \bar y_1(x,\varepsilon)\sim \exp\left[-\frac{1}{\varepsilon}\int_{x_0}^{x} p_0(t)\,dt\right] \sum_{i=0}^{\infty}\bar u_{1i}(x)\varepsilon^i, \tag{7} \]

\[ (x_1\leqslant x_0\leqslant x\leqslant x_2) \]

\[ \bar y_2(x,\varepsilon)\sim \sum_{i=0}^{\infty}\bar u_{2i}(x)\varepsilon^i, \tag{8} \]

where the functions \(\bar u_{1i}(x)\), \(\bar u_{2i}(x)\in\{Y\to Y\}\) are continuous, infinitely differentiable, and moreover

\[ \bar y_1(x_0,\varepsilon)=I,\qquad \bar y_2(x_0,\varepsilon)=I \quad (I\text{ is the identity operator}). \tag{9} \]

The solutions \(\bar y_1,\bar y_2\) are independent in the sense that the function

\[ y(x,\varepsilon)=\bar y_1(x,\varepsilon)C_1+\bar y_2(x,\varepsilon)C_2 \quad (C_1,C_2\in Y;\; x_0\leqslant x\leqslant x_2) \tag{10} \]

will be the general solution of equation (5).

Proof. Substituting (7), (8) into (6), we obtain:

\[ -p_0\bar u'_{10}+q\bar u_{10}=\bar\theta,\qquad -p_0\bar u'_{11}+q\bar u_{11}+\bar u''_{10}=\bar\theta,\ldots, -p_0\bar u'_{1i}+q\bar u_{1i}+\bar u''_{1\,i-1}=\bar\theta,\ldots \tag{11} \]

\[ p_0\bar u'_{20}+q\bar u_{20}=\bar\theta,\qquad p_0\bar u'_{21}+q\bar u_{21}+\bar u''_{20}=\bar\theta,\ldots, p_0\bar u'_{2i}+q\bar u_{2i}+\bar u''_{2\,i-1}=\bar\theta,\ldots \tag{12} \]

We shall assume that the solutions \(\bar u_{1i}(x)\), \(\bar u_{2i}(x)\) \((i=0,1,2,\ldots)\) of equations (11), (12) satisfy the initial conditions: for \(x=x_0\)

\[ \bar u_{k0}=I,\qquad \bar u_{ki}=\bar\theta \quad (k=1,2;\; i=1,2,\ldots). \tag{13} \]

On the basis of (4) we obtain: there exist such functions \(\bar y_1^*,\bar y_2^*\) that

\[ \bar y_1^*(x,\varepsilon)\sim \exp\left[-\frac{1}{\varepsilon}\int_{x_0}^{x}p_0(t)\,dt\right] \sum_{i=0}^{\infty}\bar u_{1i}(x)\varepsilon^i,\qquad \bar y_2^*(x,\varepsilon)\sim \sum_{i=0}^{\infty}\bar u_{2i}(x)\varepsilon^i, \tag{14} \]

and moreover \(\bar y_1^*(x_0,\varepsilon)=\bar y_2^*(x_0,\varepsilon)=I\). The functions \(\bar y_1^*(x,\varepsilon)\), \(\bar y_2^*(x,\varepsilon)\) in general are not solutions of equation (6), but with their help we shall show the existence of the desired solutions \(\bar y_1(x,\varepsilon)\), \(\bar y_2(x,\varepsilon)\). To prove this, introduce

\[ \bar z_k=\bar y_k-\bar y_k^*(x,\varepsilon)\quad (k=1,2), \tag{15} \]

where \(\bar y_k\) is a solution of equation (6) satisfying the relations

\[ \bar y_k(x_0,\varepsilon)=I,\qquad \bar y'_k(x_0,\varepsilon)=\bar y_k^{*\,\prime}(x_0,\varepsilon). \tag{16} \]

Substituting \(\bar y_k=\bar z_k+\bar y_k^*\) into (6), we obtain for \(\bar z_k\) the differential equation

\[ \varepsilon \bar z_k''+p_0(x)\bar z_k'+q(x)\bar z_k=\bar\alpha_k(x,\varepsilon), \tag{17} \]

where

\[ \bar\alpha_k(x,\varepsilon) =-\varepsilon \bar y_k^{*\,\prime\prime}-p_0(x)\bar y_k^{*\,\prime}-q(x)\bar y_k^*, \tag{18} \]

moreover, \(\bar z_k\), on the basis of (15), (16), satisfies the initial conditions: for \(x=x_0\)

\[ \bar z_k=\bar\theta,\qquad \bar z'_k=\bar\theta . \tag{19} \]

Since \(\bar y_k^{\,*}(x,\varepsilon)\) has the asymptotic expansion (14), which satisfies (6) formally, it follows from (18) that \(\bar\alpha_k(x,\varepsilon)\sim \bar\theta\), or

\[ \|\bar\alpha_k(x,\varepsilon)\|\leqslant \varepsilon^{N+m} \tag{20} \]

for arbitrary positive numbers \(N,m\) and all sufficiently small \(\varepsilon>0\) \((\varepsilon\to0)\). We write the solution \(\bar z_k\) of equation (17) under condition (19) in the form of a series

\[ \bar z_k(x,\varepsilon)=\sum_{m=1}^{\infty}\bar u_{km}(x,\varepsilon), \tag{21} \]

where

\[ \bar u_{k1}(x,\varepsilon)=\frac{1}{\varepsilon}\int_{x_0}^{x}\int_{x_0}^{\tau_1}\bar\alpha(\tau_2,\varepsilon)\,d\tau_2d\tau_1,\ldots \]

\[ \ldots,\bar u_{k,m+1}(x,\varepsilon) =-\frac{1}{\varepsilon}\int_{x_0}^{x}\int_{x_0}^{\tau_1} \bigl[p_0(\tau_2)\bar u_{km}(\tau_2,\varepsilon)+q(\tau_2)\bar u_{km}(\tau_2,\varepsilon)\bigr]\,d\tau_2d\tau_1,\ldots \tag{22} \]

Hence, and from (2), (20), we obtain

\[ \|\bar u_{k,m+1}\|\leqslant (2K)^m\frac{(x-x_0)^{m+1}}{(m+1)!}\varepsilon^N . \]

Consequently, the solution (21) satisfies the inequality \(\|\bar z_k(x,\varepsilon)\|\leqslant L\varepsilon^N\) \((L=\mathrm{const})\) for all sufficiently small \(\varepsilon>0\), i.e.

\[ \bar z_k(x,\varepsilon)\sim \bar\theta,\qquad x_1\leqslant x\leqslant x_2\quad (k=1,2). \tag{23} \]

Hence (7) and (8) follow from (14).

Let \(y_0,y'_0\) be any prescribed elements of \(Y\). For the solution \(y=y(x,\varepsilon)\) of equation (5), from (10) we have: for \(x=x_0\)

\[ y=C_1+C_2=y_0,\qquad y'=\bar y'_1(x_0,\varepsilon)C_1+\bar y'_2(x_0,\varepsilon)C_2=y'_0 . \tag{24} \]

Hence

\[ [\bar y'_2(x_0,\varepsilon)-\bar y'_1(x_0,\varepsilon)]C_1 =\bar y'_2(x_0,\varepsilon)y_0-y'_0, \]

\[ [\bar y'_2(x_0,\varepsilon)-\bar y'_1(x_0,\varepsilon)]C_2 =y'_0-\bar y'_1(x_0,\varepsilon)y_0 . \tag{25} \]

From (7), (8) we have

\[ \bar y'_1(x_0,\varepsilon)=-\frac{1}{\varepsilon}p_0(x)+\bar u'_{10}(x_0)+\bar\eta_1(\varepsilon), \]

\[ \bar y'_2(x_0,\varepsilon)=\bar u'_2(x_0)+\bar\eta_2(\varepsilon), \]

where \(\bar\eta_1(\varepsilon),\bar\eta_2(\varepsilon)\) tend to \(\bar\theta\) as \(\varepsilon\to0\). Consequently, for all sufficiently small \(\varepsilon\) the operator \(\bar y'_2(x_0,\varepsilon)-\bar y'_1(x_0,\varepsilon)\) has an inverse

\[ [\bar y'_2(x_0,\varepsilon)-\bar y'_1(x_0,\varepsilon)]^{-1} =\varepsilon\bar\gamma(\varepsilon), \]

where \(\bar\gamma(\varepsilon)\) is a bounded function, with \(\bar\gamma(\varepsilon)\to 1/p_0(x_0)\) as \(\varepsilon\to0\). Hence, from (25), we obtain

\[ C_1=\varepsilon\bar\gamma(\varepsilon)(\bar y'_2(x_0,\varepsilon)y_0-y'_0), \]

\[ C_2=\varepsilon\bar\gamma(\varepsilon)(y'_0-\bar y'_1(x_0,\varepsilon)y_0), \]

and \(C_1\to\theta,\ C_2\to y_0\) as \(\varepsilon\to0\).

Theorem 2. The solution \(y(x,\varepsilon)\) of equation (5) \((x_0\leqslant x\leqslant x_2)\), satisfying the initial conditions (24), tends as \(\varepsilon\to0\) to the solution \(y_0(x)\) of the degenerate equation \(p_0(x)y'+q(x)y=0\), satisfying the condition \(y_0(x_0)=y_0\).

Indeed, in (10), \(\bar y_1(x,\varepsilon)C_1\sim\theta\), \(\bar y_2(x,\varepsilon)C_2\to\bar u_{20}(x)y_0=y_0(x)\), and

\[ y_0(x_0)=\bar u_{20}(x_0)y_0=y_0 . \]

Theorem 3. The solution \(y(x,\varepsilon)\) of equation (1), satisfying the initial conditions (24), tends, as \(\varepsilon\to 0\), to the solution \(y_0(x)\) of the degenerate equation (3), satisfying the condition \(y_0(x_0)=y_0\). The derivatives of the solution \(y(x,\varepsilon)\) behave analogously for \(x_0<x\leqslant x_2\) \((x\ne x_0)\).

Proof. Obviously, if \(\tilde y(x,\varepsilon)\) is a particular solution of equation (1), satisfying the conditions \(\tilde y(x_0,\varepsilon)=\theta\), \(\tilde y'(x_0,\varepsilon)=\tilde y_0\) (\(\tilde y_0\) is some element of \(Y\)), then

\[ y(x,\varepsilon)=\bar y_1(x,\varepsilon)C_1+\bar y_2(x,\varepsilon)C_2+\tilde y(x,\varepsilon) \tag{26} \]

will be the general solution of equation (1). Substituting \(\sum_{i=0}^{\infty}\tilde u_i(x)\varepsilon^i\) into (1), we obtain the system

\[ p_0\tilde u_0' + q\tilde u_0 = a(x),\qquad p_0\tilde u_1' + q\tilde u_1 + \tilde u_0''=\theta,\ldots,\quad p_0\tilde u_i' + q\tilde u_i + \tilde u_{i-1}''=\theta,\ldots \tag{27} \]

with \(\tilde u_i(x_0)=\theta\) \((i=0,1,2,\ldots)\). Analogously to the preceding case (Theorem 1), it is proved that there exists a solution \(\tilde y(x,\varepsilon)\) of equation (1), satisfying the condition

\[ \tilde y(x,\varepsilon)\sim \sum_{i=0}^{\infty}\tilde u_i(x)\varepsilon^i, \tag{28} \]

and \(\tilde y(x_0,\varepsilon)=\theta\), \(\tilde y'(x_0,\varepsilon)=\tilde y_0\). \(C_1, C_2\) in (26) are determined from the equations

\[ C_1+C_2=y_0,\qquad \bar y_1'(x_0,\varepsilon)C_1+\bar y_2'(x_0,\varepsilon)C_2=y_0'-\tilde y_0'. \tag{29} \]

Further, analogously to the preceding case (Theorems 1 and 2), it is proved that in (26) \(\bar y_1(x,\varepsilon)C_1\sim\theta\), \(\bar y_2(x,\varepsilon)C_2+\tilde y(x,\varepsilon)\to y_0(x)\), i.e. \(y(x,\varepsilon)\to y_0(x)\), \(x_0\leqslant x\leqslant x_2\) (for \(x=x_0\), \(\|y'\|\to\infty\)).

Example 1. Let \(Y\) be the space \(C_{[a,b]}\) of continuous functions \(y=v(t)\). Equation (1) takes the form

\[ \varepsilon\,\frac{\partial^2 y(x,t)}{\partial x^2} +p_0(x)\,\frac{\partial y(x,t)}{\partial x} +\int_a^b Q(x,t;\tau)\,d\tau =A(x,t), \]

where the functions \(p_0(x)\geqslant \varkappa>0\), \(Q(x,t;\tau)\), \(A(x,t)\) are continuous for \(x_1\leqslant x\leqslant x_2\), \(a\leqslant t,\tau\leqslant b\), infinitely differentiable with respect to \(x\) for \(x_1\leqslant x\leqslant x_2\); \(\varepsilon\to 0\) \((\varepsilon>0)\).

Example 2. Let \(Y\) be an \(l\)-dimensional space. Equation (1) is transformed into the system

\[ \varepsilon y_j''+p_0(x)y_j' + \sum_{i=1}^{l} q_{ji}(x)y_i=a_j(x) \qquad (j=1,2,\ldots,l), \]

where the functions \(p_0(x)\geqslant \varkappa>0\), \(q_{ji}(x)\), \(a_j(x)\) \((j,i=1,2,\ldots,l)\) are continuous and infinitely differentiable for \(x_1\leqslant x\leqslant x_2\); \(\varepsilon\to 0\) \((\varepsilon>0)\).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
25 VI 1960

References

1. L. A. Lyusternik, V. I. Sobolev, Elements of Functional Analysis, Moscow–Leningrad, 1951.
2. L. A. Groza, DAN, 121, No. 6 (1958).