MATHEMATICS

CHAN CHAN KHUN

EXISTENCE AND UNIQUENESS OF THE SOLUTION OF BOUNDARY-VALUE PROBLEMS FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

(Presented by Academician S. L. Sobolev on 21 XI 1956)

In the works \((^3,^9,^{10})\), mainly two-point boundary-value problems for nonlinear differential equations of the second order, solved with respect to the highest derivatives, were considered. Functional methods in related problems were applied in the works \((^4,^5)\).

1. Consider the equation

\[ y^{(n)}=\varphi\left(x,y,y',\ldots,y^{(n-1)}\right). \tag{1} \]

We shall say that \(\varphi\) satisfies condition \(L^0\) if it is defined in
\(G:\ -\infty<x<+\infty,\ -\infty<y^{(k)}<+\infty\ (k=0,1,2,\ldots,n-1)\), is continuous in all variables, continuously differentiable with respect to \(y,y',\ldots,y^{(n-1)}\), and \(\varphi_{y^{(k)}}\ge 0\ (k=0,1,2,\ldots,n-3)\), \(\varphi_{y^{(n-2)}}>0\) in \(G\). If \(S\ge \varphi_{y^{(k)}}\ge 0\ (k=0,1,2,\ldots,n-3)\), \(\varphi_{y^{(n-2)}}<\mu>0\), where \(S\) and \(\mu\) are some constant numbers, then we shall say that \(\varphi\) satisfies condition \(L_S^\mu\).

Lemma 1. Suppose we have a solution \(y(x)\) of equation (1), whose function \(\varphi\) satisfies condition \(L^0\), with initial data

\[ y^{(k)}(a)=y_0^{(k)} \qquad (k=0,1,\ldots,n-1), \tag{*} \]

and an \(n\)-times continuously differentiable function \(z(x)\) on \([a,b]\) such that

\[ z^{(n)}(x)\ge \varphi\left(x,z(x),z'(x),\ldots,z^{(n-1)}(x)\right),\quad z^{(k)}(a)\ge y_0^{(k)} \quad (k=0,1,2,\ldots,n-2) \]

and

\[ z^{(n-1)}(a)>y_0^{(n-1)}. \]

Then the differential inequality holds:

\[ z^{(k)}(x)>y^{(k)}(x)\quad \text{on }(a,b]\qquad (k=0,1,2,\ldots,n-2). \]

The lemma is proved with the aid of the ideas of N. V. Azbelev \((^1)\) and B. N. Babkin \((^2)\) on Chaplygin differential inequalities. From this lemma it follows:

Theorem 1. If equation (1), where \(\varphi\) satisfies condition \(L^0\), has a solution with boundary conditions

\[ y^{(k)}(a)=y_0^{(k)} \qquad (k=0,1,2,\ldots,n-2) \]

and

\[ y^{(i)}(b)=y_1^{(i)},\quad \text{where } i \text{ is one of the numbers }0,1,2,\ldots,n-2, \]

then the solution is unique.

Theorem 2. Let \(\varphi\) satisfy condition \(L_S^\mu\) and the following condition:

I. At every point of \(G\) the relation

\[ |\varphi|\le A\left(x,y,y',\ldots,y^{(n-2)}\right)\left(y^{(n-1)}\right)^2 + B\left(x,y,y',\ldots,y^{(n-2)}\right), \]

holds, where \(A,B\) are continuous and nonnegative in every closed domain \(G\).

Then, for \(b-a<\ln(1+\mu/S)\), there exists a unique solution of equation (1) satisfying the boundary data \((*)\) and
\[ y^{(n-2)}(b)=y_1^{(n-2)}. \]

Theorem 2 is proved with the aid of estimates of the derivatives of the solution and Lemma 1 by the method of analytic continuation of S. N. Bernstein’s solution with respect to a parameter. This theorem is a generalization of S. N. Bernstein’s theorem \((^3)\).

Theorem 3. Suppose that \(\varphi\) satisfies the conditions \(L_S^\mu\) and condition I of Theorem 2, and that two functions \(y_1(x)\), \(y_2(x)\) are known, satisfying equation (1), the initial conditions \((*)\), and the inequalities
\[ y_1^{(k)}(x)>y_2^{(k)}(x)\quad \text{on }(a,b]\qquad (k=0,1,2,\ldots,n-2). \]
Then there exists a unique solution satisfying the boundary conditions \((*)\) and
\[ y^{(i)}(b)=y^{*(i)},\qquad y_2^{(i)}(b)<y^{*(i)}<y_1^{(i)}(b), \]
where \(b-a<\ln(1+\mu/S)\).

The existence of \(y_1(x)\) and \(y_2(x)\) follows from Lemma 1 and Theorem 2.

2. We now consider the boundary-value problem for a system of \(n\) second-order differential equations
\[ u''=F(t,u,u'), \tag{2} \]
\[ u(a)=\alpha,\qquad u(b)=\beta, \tag{3} \]
where \(u\) is a vector-function of the \(n\)-dimensional vector space \(E_n\), defined on \([a,b]\), and \(\alpha,\beta\) are new constant vectors.

Assume that \(F(t,u,u')\) has the following properties:

a) it is defined in \(D\):
\[ a\le t\le b,\qquad \|u\|_{E_n},\ \|u'\|_{E_n}<+\infty; \]

b) for the matrix \(F_u\), with arbitrary combinations of arguments, the condition
\[ (F_u w,w)\ge m\|w\|_{E_n}^2,\qquad w\in E_n \]
is satisfied, and for the matrix \(F_{u'}\), with arbitrary combinations of arguments, the relation
\[ |(F_{u'}v,w)|\le m_1\|v\|_{E_n}^2+m_2\|w\|_{E_n}^2 \]
holds, where \(1>m_1>0,\ m>m_2>0\), and these are constants;

c) for any points of \(D\),
\[ \|F(t,u,u')\|_{E_n}^2\le A_1(t,u)\|u'\|_{E_n}^2+B_1(t,u), \]
where \(A_1\) and \(B_1\) are continuous nonnegative functionals in any closed domain \(D\).

First of all we prove one basic lemma.

Lemma 2. Given the linear equation
\[ u''=A(t)u+B(t)u'. \]
If for the matrices \(A(t)\) and \(B(t)\) the relations
\[ (A(t)u,u)\ge m\|u\|_{E_n}^2,\qquad |(B(t)v,u)|\le m_1\|v\|_{E_n}^2+m_2\|u\|_{E_n}^2 \]
hold for any elements \(u,v\) of \(E_n\), where \(m>m_2>0,\ 1\ge m_2>0\), then the solution of this equation satisfying the zero boundary data is trivial.

Theorem 4. If \(F(t,u,u')\) satisfies conditions a) and b) and there exists a solution of equation (2) with boundary conditions (3), then the solution is unique.

The proof follows from Lemma 2.

Theorem 5. If \(F\) satisfies properties a), b), and c), then equation (2) has a unique solution satisfying the boundary conditions (3).

For the proof of existence we introduce into equation (2) a numerical parameter \(\lambda\) in the following way:

\[ u_\lambda''=\lambda F(t,u_\lambda,u_\lambda'),\qquad 0\leqslant \lambda\leqslant 1,\qquad u_\lambda(a)=\alpha,\qquad u_\lambda(b)=\beta . \tag{4} \]

Passing to a system of integral equations, we obtain an integral operator equation equivalent to equation with parameter (4). By virtue of the compactness of the totality of solutions \(\{u_\lambda(t)\}_{0\leqslant\lambda\leqslant1}\), together with their first derivatives, and Lemma 2, to the integral operator equation there is applicable a theorem generalizing, in Banach space, the usual implicit-function theorem \({}^{(7)}\) as a whole. The solution of equation (4) is continued in \(\lambda\) from 0 to 1.

1. Consider an equation of the form

\[ y''=\varphi(x,y,y')+\alpha y . \tag{5} \]

Theorem 6. Suppose that the following conditions are satisfied:

1) \(\varphi\) is defined in the domain \(\Omega:\ 0\leqslant x\leqslant 1,\ |y|,\ |v|<+\infty\), and at every point of the domain \(0\leqslant x\leqslant 1,\ |y|<+\infty,\ |v|\leqslant L_0\) is continuously differentiable with respect to \(y\) and \(v\), and \(\varphi_y\geqslant 0\);

2) for arbitrary points in \(\Omega\),

\[ |\varphi(x,y,v)|\leqslant \sigma_1|y|+\sigma_2|v|+\psi(x), \]

where \(0\leqslant \psi(x)\leqslant \sigma_3 m,\ |\varphi(x,0,0)|<m\);

3)

\[ \sigma_1\frac{1}{\alpha}+\sigma_2\frac{1}{\sqrt{\alpha}}+\sigma_3<1,\qquad L_0=\frac{1}{\sigma_2}e^{\sigma_3\{m(\frac{\sigma_1}{\alpha}+\sigma_3+1)+1\}}, \]

where \(m,\alpha,\sigma_1,\sigma_2,\sigma_3\) are positive constants.

Then there exists a unique solution satisfying the conditions

\[ y(0)-y(1)=0,\qquad y'(0)-y'(1)=0. \]

For the proof of the theorem we transform equation (5), taking as the unknown \(y''-\alpha y=u\), into the equation

\[ u(x)=\varphi\left(x,\int_0^1 K(x,s)u(s)\,ds,\frac{d}{dx}\int_0^1 K(x,s)u(s)\,ds\right), \tag{6} \]

where \(K(x,s)\) is the Green’s function of the equation \(y'-\alpha y=u\) with the given boundary conditions. On the basis of the Leray–Schauder principle \({}^{(8)}\), the existence of a solution of equation (6) is proved inside the sphere of radius \(m\) in the space \(C(0,1)\). The uniqueness of the solution follows from \(\varphi_y\geqslant 1\).

1. We use the case to consider the behavior of the solution of a nonlinear parabolic equation of the type

\[ u_t=u_{xx}-f(x,u,u_x), \tag{7} \]

\[ u(0,t)=u(1,t)=u(x,0)=0. \tag{8} \]

as \(t\) tends to infinity.

Take the ordinary differential equation

\[ w''=f(x,w,w'), \tag{9} \]

\[ w(0)=w(1)=0. \tag{10} \]

On the function \(f\) we impose the conditions:

a) \(f(x,u,v)\) is defined in the domain \(0\leqslant x\leqslant 1,\ |u|,\ |v|<+\infty\), is continuously differentiable with respect to \(u,v\), and \(f_u\geqslant -c\), where \(c\) is a positive constant such that \(0<c<\pi^2\);

b) \(f_v=k+\alpha(x)\), where \(k\) is a constant and \(\alpha(x)\) is continuous in \(x\);

c)

\[ \frac{c}{\pi^2}+\frac{\max|\alpha(x)|}{2}\left\{1+\frac{1}{\pi^2}\right\}<1. \]

Theorem 7. If the function \(f(x,u,v)\) satisfies conditions a), b), c), then the solution of equation (9) with boundary conditions (10) is unique.

Theorem 8. Suppose that equation (7), under conditions a), b), c), has a solution \(u(x,t)\) satisfying the boundary conditions (8). Then, as \(t\) tends to infinity, \(u(x,t)\) converges in the mean to the solution \(w(x)\) of equation (9) with boundary conditions (10).

Received
14 XI 1956

REFERENCES

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