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 FIRST AND SECOND DERIVATIVES TENDS TO ZERO

(Presented by Academician S. L. Sobolev on 23 X 1961)

Consider, in a Banach space (Y), the ordinary linear homogeneous differential equation of second order

[
\mu y''+\mu p(x)y'=y
\tag{1}
]

with a small parameter (\mu>0) tending to zero. The unknown function (y=y(x,\mu)\in Y) is a function of the real variable (x); (p(x)) is a given function of the variable (x) with values in the space of linear bounded operators ({Y\to Y}) ((^1)), mapping the space (Y) into (Y). We study the behavior of any solution of equation (1) with Cauchy initial data as (\mu) tends to zero. We note that the limiting equation for equation (1) has the solution (y_0(x)\equiv 0).

Assume that the function (p(x)) is continuous, infinitely differentiable, and

[
|p(x)|\leq K,\qquad |p^{(n)}(x)|\leq M(n)
\tag{2}
]

for (x_1\leq x\leq x_2), where (K), (M(n)) are constants ((M(n)) may increase together with (n)). Under these assumptions it will be proved that the norm of any solution (y(x,\mu)) of equation (1), as (\mu\to 0), tends to infinity of order (\exp(1/\mu)). The behavior of the derivative (y'(x,\mu)) is analogous.

Putting (\mu=\varepsilon^2) in (1), we shall have

[
\varepsilon^2 y''+\varepsilon^2 p(x)y'=y.
\tag{3}
]

Simultaneously with equation (3), consider the equation ((^2))

[
\varepsilon^2 \bar y''+\varepsilon^2 p(x)\bar y'=\bar y.
\tag{4}
]

in the space ({Y\to Y}).

Theorem 1. Equation (4) has two independent solutions (\bar y_1(x,\varepsilon)), (\bar y_2(x,\varepsilon)), satisfying the asymptotic relations

[
\bar y_1(x,\varepsilon)\sim
\exp\left(-\frac{x-x_0}{\varepsilon}\right)
\sum_{i=0}^{\infty}\bar u_{1i}(x)\varepsilon^i,
\tag{5}
]

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

[
\bar y_2(x,\varepsilon)\sim
\exp\left(\frac{x-x_0}{\varepsilon}\right)
\sum_{i=0}^{\infty}\bar u_{2i}(x)\varepsilon^i,
\tag{6}
]

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
\tag{7}
]

((I) is the identity operator().)

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

[
\bar y(x,\varepsilon)=\bar y_1(x,\varepsilon)C_1+\bar y_2(x,\varepsilon)C_2,
\tag{8}
]

where $C_1$, $C_2$ are arbitrary constant elements of $Y$, is the general solution of equation (3).

Proof. Substituting (5), (6) into (4), we obtain

[
2\bar u'{10}+p(x)\bar u=\bar\theta,\ldots,\quad
2\bar u'{1i}+p(x)\bar u=\bar u''{1,i-1}+p(x)\bar u',\ldots;
\tag{9}
]

[
2\bar u'{20}+p(x)\bar u=\bar\theta,\ldots,\quad
2\bar u'{2i}+p(x)\bar u+\bar u''{2,i-1}+p(x)\bar u'=\bar\theta,\ldots
\tag{10}
]

Assume that the solutions $\bar u_{1i}(x)$, $\bar u_{2i}(x)$ $(i=0,1,2,\ldots)$ of equations (9), (10) satisfy the initial conditions:

[
\text{for } x=x_0 \qquad \bar u_{k0}=I,\qquad \bar u_{ki}=\bar\theta
\quad (k=1,2;\ i=1,2,\ldots).
\tag{11}
]

It is known (${}^2$) that there exist functions $\bar y_1^{}(x,\varepsilon)$, $\bar y_2^{}(x,\varepsilon)$ such that

[
\bar y_1^{*}(x,\varepsilon)\sim
\exp\left(-\frac{x-x_0}{\varepsilon}\right)
\sum_{i=0}^{\infty}\bar u_{1i}(x)\varepsilon^i,
]

[
\bar y_2^{*}(x,\varepsilon)\sim
\exp\left(\frac{x-x_0}{\varepsilon}\right)
\sum_{i=0}^{\infty}\bar u_{2i}(x)\varepsilon^i,
\tag{12}
]

and, moreover, $\bar y_1^{}(x_0,\varepsilon)=\bar y_2^{}(x_0,\varepsilon)=I$. The functions $\bar y_1^{}$, $\bar y_2^{}$ are not, in general, solutions of equation (4), but with their aid we shall show the existence of the desired solutions $\bar y_1(x,\varepsilon)$, $\bar y_2(x,\varepsilon)$. Introduce

[
\bar z_k=\bar y_k-\bar y_k^{*}(x,\varepsilon)\quad (k=1,2),
\tag{13}
]

where $\bar y_k$ is a solution of equation (4) satisfying the conditions

[
\bar y_k(x_0,\varepsilon)=I,\qquad
\bar y'_k(x_0,\varepsilon)=\bar y_k^{*\,\prime}(x_0,\varepsilon).
\tag{14}
]

Substituting $\bar y_k=\bar z_k+\bar y_k^{*}$ into (4), we obtain for $\bar z_k$ the differential equation

[
\varepsilon^2\bar z''_k+\varepsilon^2\bar z'_k
=\bar z_k+\bar a_k(x,\varepsilon),
\tag{15}
]

where

[
\bar a_k(x,\varepsilon)=\bar y_k^{}-\varepsilon^2\bar y_k^{\,\prime\prime}
-\varepsilon^2p(x)\bar y_k^{*\,\prime},
\tag{16}
]

and, moreover, $\bar z_k$, on the basis of (13), (14), satisfies the initial conditions:

[
\text{for } x=x_0 \qquad \bar z_k=\bar\theta,\qquad \bar z'_k=\bar\theta.
\tag{17}
]

Since $\bar y_k^{*}(x,\varepsilon)$ has the asymptotic expansion (12), which satisfies (4) formally, it follows from (16) that $\bar a_k(x,\varepsilon)\sim\bar\theta$, i.e.

[
\lim_{\varepsilon\to0}
\frac{|\bar a_k(x,\varepsilon)|}{\varepsilon^{N+2m}}=0
]

for all positive numbers $N$, $m$, or

[
|\bar a_k(x,\varepsilon)|<\varepsilon^{N+2m}
\tag{18}
]

for arbitrary positive numbers $N$, $m$ and all sufficiently small $\varepsilon>0$ $(\varepsilon\to0)$.

The solution $\bar z_k$ of equation (15) under condition (17) will be written in the form

[
\bar z_k(x,\varepsilon)=\sum_{m=1}^{\infty}\bar v_{km}(x,\varepsilon),
\tag{19}
]

where

[
\overline{v}{k1}(x,\varepsilon)=\frac{1}{\varepsilon^{2}}\int}^{x}\int_{x_0}^{\tau_1
\overline{\alpha}_{k}(\tau_2,\varepsilon)\,d\tau_2\,d\tau_1,\ldots
]

[
\ldots,\ \overline{v}{k,m+1}(x,\varepsilon)=\frac{1}{\varepsilon^{2}}\int}^{x}\int_{x_0}^{\tau_1
\left(\overline{v}{km}(\tau_2,\varepsilon)-\varepsilon^{2}p(\tau_2)\overline{v}'(\tau_2,\varepsilon)\right)
\,d\tau_2\,d\tau_1,\ldots
\tag{20}
]

Hence, from (2), (18), we obtain

[
|\overline{v}_{k,m+1}(x,\varepsilon)|\leq (2K)^{m+1}\frac{(x-x_0)^{m+1}}{(m+1)!}\varepsilon^{N}.
\tag{21}
]

It follows from (21) that

[
\overline{z}{k}(x,\varepsilon)=\sum}^{\infty}\overline{v}_{km}(x,\varepsilon)\to \overline{\theta
\quad \text{as } \varepsilon\to 0.
]

Similarly we obtain

[
\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}\overline{z}{k}(x,\varepsilon)=\overline{\theta},\ldots,\quad
\lim,\ldots,}\frac{1}{\varepsilon^{n}}\overline{z}_{k}(x,\varepsilon)=\overline{\theta
]

i.e.,

[
\overline{z}_{k}(x,\varepsilon)\sim \overline{\theta},\qquad
x_1\leq x\leq x_2,\quad k=1,2.
\tag{22}
]

Hence, from (12), (13), (5), (6) follow.

Let (y_0,y'_0) be arbitrary prescribed elements of (Y). For the solution (y=y(x,\varepsilon)) of equation (3), from (7), (8) we have:

[
\text{for } x=x_0\qquad
y=C_1+C_2=y_0,\qquad
y'=\overline{y}'_1(x_0,\varepsilon)C_1+\overline{y}'_2(x_0,\varepsilon)C_2=y'_0.
\tag{23}
]

Hence

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

[
[\overline{y}'_2(x_0,\varepsilon)-\overline{y}'_1(x_0,\varepsilon)]C_2
=
y'_0-\overline{y}'_1(x_0,\varepsilon)y_0.
\tag{24}
]

From (5), (6) we have

[
\overline{y}'1(x_0,\varepsilon)=-\frac{1}{\varepsilon}+\overline{u}'}(x_0)+\overline{\eta1(\varepsilon),\qquad
\overline{y}'_2(x_0,\varepsilon)=\frac{1}{\varepsilon}+\overline{u}'_2(\varepsilon),}(x_0)+\overline{\eta
]

where (\overline{\eta}_1(\varepsilon),\overline{\eta}_2(\varepsilon)) tend to (\overline{\theta}) as (\varepsilon\to 0). Consequently, the operator
(\overline{y}'_2(x_0,\varepsilon)-\overline{y}'_1(x_0,\varepsilon)), for all sufficiently small (\varepsilon>0), has an inverse
([\overline{y}'_2(x_0,\varepsilon)-\overline{y}'_1(x_0,\varepsilon)]^{-1}\to \varepsilon\overline{Y}(\varepsilon)), where
(\overline{Y}(\varepsilon)\to \frac12 I) as (\varepsilon\to 0). Hence, from (24), we obtain
(C_1\to y_0/2,\ C_2\to y_0/2) as (\varepsilon\to 0).

Theorem 2. The solution (y(x,\varepsilon)) of equation (3), satisfying the initial conditions (23), tends in norm to infinity as (\varepsilon\to 0) with order (\exp(1/\varepsilon)). The derivative (y'(x,\varepsilon)) behaves analogously.

Indeed, in (8) (\overline{y}_1(x,\varepsilon)C_1\sim \theta), while
(|\overline{y}_2(x,\varepsilon)C_2|) tends to (\infty) as (\varepsilon\to 0) with order (\exp(1/\varepsilon)).

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

[
\mu\frac{\partial^{2}y(x,t)}{\partial x^{2}}
+
\mu\int_{a}^{b}P(x,t;\tau)\frac{\partial y(x,\tau)}{\partial x}\,d\tau
=
y(x,t),
]

where the function (P(x,t;\tau)) is continuous in all variables for
(x_1\leq x\leq x_2,\ a\leq t,\tau\leq b), is infinitely differentiable with respect to (x) for (x_1\leq x\leq x_2); (\mu\to 0) ((\mu>0)).

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

[
\mu y_i''+\mu \sum_{i=1}^{n} p_{ji}(x)y_i' = y_j
\qquad (j=1,2,\ldots,n),
]

where the functions (p_{ji}(x)) are continuous and infinitely differentiable for (x_1 \le x \le x_2); (\mu \to 0) ((\mu>0)).

Kazakh State University
named after S. M. Kirov

Received
9 VIII 1961

CITED LITERATURE

1. L. A. Lyusternik, V. I. Sobolev, Elements of Functional Analysis, Moscow–Leningrad, 1951.
2. L. A. Groza, DAN, 135, No. 5 (1960).