UDC 517.9+532

MATHEMATICS

G. V. SHCHERBINA

ON A BOUNDARY-VALUE PROBLEM, ARISING IN APPLICATIONS, FOR A SECOND-ORDER DIFFERENTIAL EQUATION ON THE HALF-LINE

(Presented by Academician L. I. Sedov on March 22, 1967)

In mechanics the following boundary-value problem is known:
\[ x''' + 2xx'' + 2\beta(1 - x'^2)=0,\qquad x(0)=x'(0)=0,\qquad x'(+\infty)=1, \tag{1'} \]
arising in the study of self-similar solutions of boundary-layer equations \((^1)\). The change of variables \(x'(t)=1-y(x)\) brings this problem to the form
\[ y''=F(x,y,y'),\qquad y(0)=a,\qquad y(+\infty)=0. \tag{1} \]

In the investigation of the question of mixing of two gas jets, the boundary-value problem \((^2)\) was obtained:
\[ x''' + xx''=0,\qquad x'(-\infty)=a,\qquad x(0)=0,\qquad x'(+\infty)=0, \tag{2'} \]
which is similarly reduced to the form
\[ y''=F(x,y,y'),\qquad y(-\infty)=a,\qquad y(+\infty)=0. \tag{2} \]

From physical considerations one can often indicate inequalities that the desired solution must satisfy. Thus, in problem \((1')\) we have \(0\le x'(t)\le 1\), while in problem \((2')\), \(0\le x'(t)\le a\).

Boundary-value problems of the form (1), (2) in various special cases have been studied by many authors \((^{3-9})\). In the present note an existence theorem will be proved for a solution of problem (1) or (2) for a sufficiently broad class of functions \(F\). In particular, the existence theorems in papers \((^{3-9})\) can be obtained from Theorem 1. We shall show how Theorem 1 can be applied in some new cases.*

Assume that there exist functions \(f_+(x)\) and \(f_-(x)\), continuous for \(x\ge 0\), such that the function \(F(x,y,z)\) is defined in the strip
\[ \pi:\ \{x\ge 0,\ f_-(x)\le y\le f_+(x),\ |z|<\infty\}, \]
is continuous in \(\pi\) with respect to \(x\), and satisfies a Lipschitz condition with respect to \(y\) and \(z\). Suppose, moreover, that the inequalities
\[ f_+''(x)\le F(x,f_+(x),f_+'(x)),\qquad f_-''(x)\ge F(x,f_-(x),f_-'(x)), \]
\[ f_+'(x-0)\ge f_+'(x+0),\qquad f_-'(x-0)\le f_-'(x+0), \]
\[ |f_+(x)|+|f_-(x)|\le M, \]
\[ f_+(0)\ge a\ge f_-(0),\qquad \lim_{x\to\infty} f_-(x)\le 0\le \lim_{x\to\infty} f_+(x), \]
\[ |F(x,y,z)|\le C(x,y)\varphi(z), \tag{A} \]
are satisfied, where \(C(x,y)\) is continuous,
\[ \int_0^\infty z[\varphi(z)]^{-1}\,dz=\infty. \]
We shall assume that

* The present work was reported at the World Congress of Mathematicians in Moscow in 1966.

the discontinuities \(f_+'\) and \(f_-'\) are isolated, of the first kind, and, where \(f_-'(f_+')\) exists, \(f_+''(f_-'')\) exists.

In what follows, for simplicity we shall consider problem (1) under the assumption

\[ \lim_{x\to\infty} f_-(x)=0 . \]

The considerations in the general case and for problem (2) are entirely analogous.

Suppose that for every \(\gamma\) \((0<\gamma\le d=\overline{\lim}_{x\to\infty} f_+(x))\) there is a function \(F_\gamma(x,y,z)\), nonincreasing in \(y\), such that \(F\ge F_\gamma\) in some strip

\[ \pi_\gamma:\quad \{|y-\gamma|\le \varepsilon_\gamma<\gamma,\ (x,y,z)\subset \pi\}. \]

Consider the solutions of the equation \(y''=F_\gamma\) satisfying the conditions

\[ y(x_0)=\gamma-\delta,\qquad y'(x_0)=\alpha . \]

The exact lower bound of the \(\alpha\)’s for which, from the fulfillment on the interval \(x_0\le t\le x\) of the inequality

\[ y(t)\le \gamma+\varepsilon_\gamma,\qquad (t,y,y')\subset \pi , \]

there follows the inequality \(y(x)\ge \gamma-\varepsilon_\gamma\), will be denoted by \(m(x_0,\gamma,\delta)\). If there are no such \(\alpha\), we put \(m(x_0,\gamma,\delta)=+\infty\). The solution of the equation \(y''=F_\gamma\) satisfying the conditions

\[ y(x_0)=\gamma-\delta,\qquad y'(x_0)=m(x_0,\gamma,\delta), \]

will be denoted by \(M(x,x_0,\gamma,\delta)\).

Just as in [8], it can be shown that, under conditions (A), the problem \(y(0)=a,\ y(x_n)=y_n\), where \(x_n>0,\ f_-(x_n)\le y_n\le f_+(x_n)\), has a solution \(y_n(x)\) satisfying, for \(0\le x\le x_n\), the inequality

\[ f_-(x)\le y_n(x)\le f_+(x), \]

and that there exist subsequences \(x_{n_k}\to\infty,\ y_{n_k}\to0\) such that \(y_{n_k}(x)\to Y(x)\), \(Y(0)=a\). Obviously,

\[ f_-(x)\le Y(x)\le f_+(x). \tag{3} \]

Theorem 1. If conditions (A) are fulfilled and for every \(\gamma\) \((0<\gamma\le d)\) there is a \(\delta\) \((0<\delta<\varepsilon_\gamma)\) such that

\[ \int^\infty m(x_0,\gamma,\delta)\,dx_0=-\infty,\qquad \int^\infty \{|m|+m\}\,dx_0<+\infty, \]

then

\[ \lim_{x\to\infty}Y(x)=0, \]

i.e. the boundary-value problem (1) has a solution satisfying inequality (3).

Proof. Suppose this is not so and

\[ \overline{\lim}_{x\to\infty}Y(x)=\gamma>0 . \]

Then there exist points \(\xi_i\) such that \(Y(\xi_i)\to\gamma,\ \xi_i\to\infty\). Choose \(A_0\) so large that for \(\xi_i>A_0\) the inequalities

\[ |Y(\xi_i)-\gamma|<\delta/3,\qquad \int_0^\infty \{|m|+m\}\,dx_0<\delta/3 \]

hold. We shall show that for some \(\xi_i\) \((\xi_i\to+\infty)\) the inequality

\[ Y'(\xi_i)>m(\xi_i,\gamma,\delta) \]

is valid. There are two possible cases: either \(|Y(x)-\gamma|<\delta\) for \(x>\xi_i\), or

\[ \underline{\lim}_{x\to\infty}Y(x)\le \gamma-\delta . \]

In the first case the inequality

\[ Y'(x)\le m(x,\gamma,\delta) \]

contradicts the fact that

\[ \int^\infty m\,dx_0=-\infty, \]

and in the second it contradicts the fact that

\[ \int_A^\infty \{|m|+m\}\,dx_0<\delta/3 . \]

It follows from what has been said that

\[ y_{n_i}(\xi_i)\ge \gamma-\delta,\qquad y_{n_i}'(\xi_i)\ge m(\xi_i,\gamma,\delta) \]

on some subsequence \(n_i\to\infty\). Applying the comparison theorem proved in [10], we would obtain that, if \(y_{n_i}(x)\le \gamma+\delta\) for \(x\ge \xi_i\), then

\[ y_{n_i}(x_{n_i})\ge M(x_{n_i},\xi_i,\gamma,\delta)\ge \gamma-\varepsilon_\gamma>0, \]

which is impossible. Hence \(y_{n_i}(x)\) attains at a point \(\omega_i>\xi_i\) a maximum \(\gamma_i\), and \(\gamma_i\ge\gamma+\delta\). Denoting \(\gamma_0=\lim_{i\to\infty}\gamma_i\) and arguing as before, we arrive at a contradiction. The theorem is proved.

Let us give a simple corollary of Theorem 1:

If conditions (A) are fulfilled and

\[ \lim f_+(x)=\lim f_-(x), \]

then the boundary-value problem (1) has a solution. If \(F\ge F_1\) for \(x\ge A,\ (x,y,z)\subset\pi\), and the boundary-value problem

\[ y(A)=f_+(A),\qquad y(\infty)=0 \]

for the equation \(y''=F_1(x,y,y')\) has

solution \(y_1(x)\) lying in \(\pi\), then the boundary-value problem (1) also has a solution \(Y(x)\), and \(f_-(x)\leq Y(x)\leq y_1(x)\) for \(x\geq A\).

Thus, in order to prove the existence of a solution of problem (1) for a given equation, it suffices to construct \(f_+(x)\) and \(f_-(x)\), choose \(F_\gamma\) (if \(\lim_{x\to\infty} f_-(x)=0\)), and show that \(m(x_0,\gamma,\delta)\) has the required properties. Under the restrictions imposed on \(F\) in papers \((^3\text{--}^9)\), this is not hard to do. Thus, in problem \((1')\), after the substitution \(x'=1-y\), we obtain
\(f_+(x)\equiv 1,\ f_-(x)\equiv 0,\ F\geq -2x\,dy/dx+4\beta y\), and for \(\beta\geq 0\) one may use the corollary of Theorem 1:
\(0<y\leq c x^{-(2\beta+1)}e^{-x^2}\times [1+o(1/x)]\).

Consider now problem (1) for \(F=\psi(x)\varphi(y)\). Suppose that \(\varphi\) and \(\psi\) are defined and continuous for \(x\geq 0,\ y\geq 0\), that \(\varphi\) satisfies a Lipschitz condition and, moreover,

\[ \varphi(0)=0,\quad \varphi(y)>0\ \text{for }y>0,\quad \int_0^\infty x\psi(x)\,dx=+\infty,\quad \int_0^\infty x[|\psi|-\psi]\,dx<\infty . \tag{4} \]

If \(\psi\geq 0\), this problem was solved in paper \((^5)\).

Under these assumptions there exists a solution of the equation \(y''=\varphi(y)\psi(x)\) tending to zero and positive for large \(x\). If \(\varphi(y)\) has a finite limit as \(y\to+\infty\), then in this case the existence of a solution of the boundary-value problem (1) follows from conditions (4). If

\[ \varphi(y)\equiv y,\qquad \psi_+=\frac12\{\psi+|\psi|\},\qquad \psi_-=\frac12\{\psi-|\psi|\}, \]

\[ \int_\xi^x dt\int_\xi^t \psi_-(s)\,ds +q\int_\xi^x dt\int_\xi^t \psi_+(s)\,ds \geq -1+q \tag{5} \]

for some \(q\) \((0<q<1)\), all \(\xi\) and \(x\,(x>\xi)\), and \(\psi(x)\) satisfies conditions (4), then the boundary-value problem (1) has a positive solution. We note that condition (5) is required only for the construction of \(f_+(x)\), and therefore it may be replaced by the requirement that a bounded positive solution exist.

In conclusion, the author expresses gratitude to M. A. Krasnosel’skii, G. Ya. Lyubarskii, A. Yu. Levin, and A. I. Perov for their interest in this work and for a number of useful comments.

Physico-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR

Received
18 III 1967

REFERENCES

\(^1\) V. M. Folkner, S. W. Skan, Phil. Mag., 12 (1931).
\(^2\) Napolitano, Quart. Appl. Math., 16, No. 4, 397 (1959).
\(^3\) H. Weyl, Proc. Nat. Acad. Sci., Washington, 27, 578 (1941).
\(^4\) R. Iglisch, ZAMM, 33 (1953).
\(^5\) B. Hartman, A. Wintner, Am. J. Math., 73, 390 (1951).
\(^6\) Yu. A. Klokov, Izv. vyssh. uchebn. zaved., ser. matem., No. 6 (13) (1959).
\(^7\) Yu. A. Klokov, DAN, 139, No. 4 (1961).
\(^8\) G. V. Shcherbina, DAN, 140, No. 2 (1961).
\(^9\) G. V. Shcherbina, Candidate’s dissertation, 1963.
\(^10\) E. Kamke, Acta Math., 58, 82 (1932).