UDC 517.934

MATHEMATICS

A. Ya. LEPIN, A. D. MYSHKIS

EXISTENCE OF A SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR AN ORDINARY NONLINEAR DIFFERENTIAL EQUATION OF ORDER \(n\)

(Presented by Academician I. G. Petrovsky on 28 X 1965)

The two-point boundary-value problem for nonlinear ordinary differential equations of the second order has been studied comparatively well, beginning with the classical work of S. N. Bernstein \((^1)\). The analogous problem for equations of order higher than the second has, in the general case, been little studied (see \((^{2,3})\), where further bibliography is given).

Here we shall consider the boundary-value problem

\[ y^{(n)}=f(x,y,y',\ldots,y^{(n-1)}); \tag{1} \]

\[ y(a)=\alpha_0,\quad y'(a)=\alpha_1,\ldots,\quad y^{(n-2)}(a)=\alpha_{n-2}; \tag{2} \]

\[ y^{(k)}(b)=\beta, \tag{3} \]

where the function \(f\) is given and continuous for \(x\in [a,b]\) \((a<b)\), \(-\infty<y,y',\ldots,y^{(n-1)}<\infty\); \(a,b,\alpha_0,\ldots,\alpha_{n-2},k,\beta\) are given; \(0\le k\le n-1\); all quantities are finite and real. Analogous methods can be applied to boundary conditions of a more general form.

Theorem 1. Let equation (1) be such that the totality of all its solutions \(y(x)\) \((x\in [a_y,b_y]\subseteq [a,b])\) has the following properties:

A. For every \(M>0\) there exists an \(N>0\) such that if
\[ |y(a_y)|<M,\quad |y'(a_y)|<M,\ldots,\quad |y^{(n-1)}(a_y)|<M,\quad |y^{(k)}(b_y)|<M, \]
then
\[ |y^{(k)}(x)|<N \quad (a_y\le x\le b_y). \]

B. For every \(M>0\) there exists an \(N>0\) such that if
\[ |y(a_y)|<M,\quad |y'(a_y)|<M,\ldots,\quad |y^{(n-1)}(a_y)|<M,\quad |y^{(k)}(x)|<M, \]
then \((a_y\le x\le b_y)\),
\[ |y^{(n-1)}(x)|<N \quad (a_y\le x\le b_y). \]

C. For any \(a_y<b\), \(M_1>0\), \(M_2>0\) there exists an \(N>0\) such that if
\[ |y(a_y)|<M_1,\ldots,\quad |y^{(n-2)}(a_y)|<M_1,\quad |y^{(n-1)}(a_y)|>N, \]
then there exists a continuation \(\bar y(x)\) \((a_y\le x\le b_y^-;\ b_y\le b_y^-\le b)\) of the solution \(y(x)\) and a point \(c\in [a_y,b_y^-]\) such that
\[ |\bar y^{(i)}(c)|>M_2,\quad \operatorname{sign}\bar y^{(i)}(c)=\operatorname{sign} y^{(n-1)}(a_y) \]
\[ (i=k,\ldots,n-1). \]

D. For every \(M\) there exists an \(N>0\) such that if
\[ |y^{(k)}(a_y)|>N,\ldots,\quad |y^{(n-1)}(a_y)|>N, \]
and all \(y^{(k)}(a_y),\ldots,y^{(n-1)}(a_y)\) have the same sign, then
\[ |y^{(k)}(b_y)|>M. \]

Then problem (1)—(3) has at least one solution.

For the proof of this theorem one must add to (2) the condition
\[ y^{(n-1)}(a)=\alpha \]
and consider the continuation of the resulting Cauchy problem as \(\alpha\) varies continuously.

Remark. In conditions A—C it is sufficient to restrict oneself to solutions for which \(a_y=a\) and the equalities (2) hold.

Conditions A—D are not sufficiently effective, and therefore we shall give a number of sufficient criteria for their fulfillment, expressed directly in terms of the properties of the function \(f\).

Theorem 2. If for every \(M>0\) there exists a function nondecreasing in all variables \(\varphi_M(y_{k+1},\ldots,y_{n-1})\) \((y_{k+1},\ldots,y_{n-1}\ge 0)\), for which

second,

\[ \left| f(x, y_0, \ldots, y_{n-1}) \right| \leq \varphi_M\left(\left|y_{k+1}\right|, \ldots, \left|y_{n-1}\right|\right) \tag{4} \]

for \(|y_i| < M\) \((i=0,\ldots,k)\), \(-\infty < y_{k+1}, \ldots, y_{n-1} < \infty\),
\[ \lim_{y\to\infty}\varphi_M(y, y^2,\ldots,y^{n-k-1})y^{k-n}=0, \]
then conditions B and C are satisfied.

The proof is based on estimating \(y^{(k+1)}, \ldots, y^{(n-1)}\) in terms of \(y^{(k)}\) and \(y^{(n)}\) by means of the Kolmogorov–Gorny inequality, similarly to the proof of the main theorem of the paper (4).

Remark. For \(k=n-1\), conditions B and C are satisfied automatically. For \(k=n-2\), inequality (4) may be weakened by replacing the right-hand side by \(\varphi_M(|y_{n-1}|)\), where \(\varphi_M(s)\) \((s_0<s<\infty)\) is an arbitrary nondecreasing function for which
\[ \int^\infty s[\varphi_M(s)]^{-1}\,ds=\infty \]
(this is Nagumo’s condition (5)); in particular, one may take \(\varphi_M(s)=A_Ms^2\) (this is Bernstein’s condition).

Theorem 3. Suppose equation (1) satisfies condition A for some \(k\); then this condition is also fulfilled for any \(k'<k\). The same is true for condition D.

The proof is easily carried out for \(k'=k-1\), whence the assertion for any \(k'\) follows.

We shall say that the equation

\[ y^{(p)}=\varphi(x,y,y',\ldots,y^{(p-1)};\eta_1,\ldots,\eta_r) \tag{5} \]

with continuous right-hand side containing the parameters \(\eta_1,\ldots,\eta_r\)
\((a\leq x\leq b; -\infty<y,\ldots,y^{(p-1)};\eta_1,\ldots,\eta_r<\infty)\), satisfies condition \(\overline{A}\) if for every \(M>0\) there exists an \(N>0\) such that if a function \(y(x)\in C_p[a_y,b_y]\) and continuous functions \(\eta_1(x),\ldots,\eta_r(x)\) \((a_y\leq x\leq b_y)\) satisfy the relations

\[ y^{(p)}(x)=\varphi(x,y(x),\ldots,y^{(p-1)}(x);\eta_1(x),\ldots,\eta_r(x))\quad (a_y\leq x\leq b_y), \]

\[ |y(a_y)|<M,\ldots,|y^{(p-1)}(a_y)|<M,\quad |y(b_y)|<M, \]

then \(|y(x)|<N\) \((a_y\leq x\leq b_y)\). Analogously, from condition D one obtains condition \(\overline{D}\) for equation (5). (In condition \(\overline{A}\) it is sufficient to take \(a_y=a\).) From Theorem 3 it immediately follows:

Theorem 4. In order that equation (1) satisfy conditions A and D, it is sufficient that, for some natural number \(p\leq n-k\), the equation

\[ y^{(p)}=f(x,\eta_1,\ldots,\eta_{n-p},y,y',\ldots,y^{(n-p)}) \]

satisfy conditions \(\overline{A}\) and \(\overline{D}\).

Thus, for an equation of high order it becomes possible to apply criteria obtained for equations of lower orders. We indicate some criteria sufficient for satisfying conditions \(\overline{A}\) and \(\overline{D}\) for equations of the first three orders, denoting, for brevity, \(\eta=(\eta_1,\ldots,\eta_r)\).

Theorem 5. Suppose the right-hand side of the equation

\[ y'=\varphi(x,y;\eta) \tag{6} \]

is such that there exist two sequences \(u_i(x), v_i(x)\) \((a\leq x\leq b;\ i=1,2,\ldots)\) of continuously differentiable functions for which

\[ u_i'=\psi_{1i}(x,u_i)<\varphi(x,u_i;\eta),\qquad \min_{[a,b]}u_i(x)\to\infty \quad \text{as } i\to\infty, \]

\[ v_i'=\psi_{2i}(x,v_i)>\varphi(x,v_i;\eta),\qquad \max_{[a,b]}v_i(x)\to-\infty \quad \text{as } i\to\infty. \]

Then equation (6) satisfies conditions \(\overline{A}\) and \(\overline{D}\).

The proof is obtained by comparing \(y\) with \(u_i\) and \(v_i\). We note that in the formulation of the theorem the functions \(\psi_i\) are completely arbitrary; by choosing specific functions \(\psi_i\), one can obtain various classes of functions \(\varphi\) for which conditions \(\overline{\mathrm A}\) and \(\overline{\mathrm D}\) are satisfied. Thus, one may set \(\psi_{1i}=-c_1u_i-c_2,\ \psi_{2i}=-c_1v_i+c_2\) \((c_{1,2}\ge 0)\).

Theorem 6. Let the right-hand side of the equation

\[ y''=\varphi(x,y,y';\eta) \tag{7} \]

be such that there exist two sequences \(u_i(x), v_i(x)\) \((a\le x\le b;\ i=1,2,\ldots)\) of twice continuously differentiable functions for which

\[ u_i''=\psi_{1i}(x,u_i,u_i')<\varphi(x,y,u_i';\eta)\quad (y\ge u_i),\qquad \min_{[a,b]} u_i(x)\to\infty \quad \text{as } i\to\infty, \]

\[ v_i''=\psi_{2i}(x,v_i,v_i')>\varphi(x,y,v_i';\eta)\quad (y\le v_i),\qquad \max_{[a,b]} v_i(x)\to-\infty \quad \text{as } i\to\infty . \]

Then equation (7) satisfies conditions \(\overline{\mathrm A}\) and \(\overline{\mathrm D}\).

The proof is obtained by comparing \(y\) with \(u_i\) and \(v_i\). In particular, one may set \(\psi_{1i}=c_1u_i'-c_2,\ \psi_{2i}=c_1v_i'+c_2\) \((c_{1,2}\ge0)\).

Theorem 7. Let the right-hand side of equation (7) be such that there exist two sequences \(u_i(x), v_i(x)\) \((a\le x\le b;\ i=1,2,\ldots)\) of continuously differentiable functions for which \(u_i'=\psi_{1i}(x,u_i)\), \(v_i'=\psi_{2i}(x,v_i)\); \(\psi_{1i}, \psi_{2i}\) are continuously differentiable in both variables, \(\min_{[a,b]}u_i(x)\to\infty\) as \(i\to\infty\), \(\max_{[a,b]}v_i(x)\to-\infty\) as \(i\to\infty\), and

\[ \psi'_{1ix}(x,u_i)+\psi'_{1iu}(x,u_i)\psi_{1i}(x,u_i) <\varphi(x,y,\psi_{1i}(x,u_i);\eta)\quad (y\ge u_i), \]

\[ \psi'_{2ix}(x,v_i)+\psi'_{2iv}(x,v_i)\psi_{2i}(x,v_i) >\varphi(x,y,\psi_{2i}(x,v_i);\eta)\quad (y\le v_i). \tag{8} \]

Then equation (7) satisfies conditions \(\overline{\mathrm A}\) and \(\overline{\mathrm D}\).

The proof is obtained by comparing \(y\) with \(u_i\) and \(v_i\). Condition (8) is simplified if \(\psi_i\) does not depend on \(x\) or \(y\).

Theorem 8. If the right-hand side of the equation

\[ y'''=\varphi(x,y,y',y'';\eta) \]

for all values of the arguments satisfies the inequality \(y\varphi(x,y,y_1,y_2;\eta)\ge0\), then this equation satisfies conditions \(\overline{\mathrm A}\) and \(\overline{\mathrm D}\).

The proof is based on the application of the elementary identity

\[ y(x_2)=y(x_1)+\frac12\,[y'(x_1)+y'(x_2)](x_2-x_1) +\frac12\int_{x_1}^{x_2}(x-x_1)(x-x_2)y'''(x)\,dx \]

Remark. From Theorems 5, 6, and 8, in particular, it follows that if for equation (5) the inequality

\[ y\varphi(x,y,y_1,\ldots,y_{p-1};\eta)\ge0, \]

is satisfied, then for \(p=1,2,3\) conditions \(\overline{\mathrm A}\) and \(\overline{\mathrm D}\) are satisfied. It is noteworthy that for \(p\ge5\) this is, generally speaking, not so (the case \(p=4\) remains open). Namely, it is verified directly that the function

\[ v(x)=(-x)^{-p+1}\sin[\beta\ln((-x))]\qquad (-1\le x\le0), \]

where \(\beta=\beta_p\) is chosen so that

\[ \sum_{i=1}^{p}\arccos\frac{p-2+i}{\sqrt{(p-2+i)^2+\beta^2}}=2\pi, \]

satisfies equations of the form

\[ y^{(p)}=\psi_1(x)\bigl(y^2+|y^{(p-1)}|\bigr)y \quad \text{or} \quad y^{(p)}=\psi_2(x)\bigl(y^2+|y^{(p-2)}|^{(2p-2)/(2p-3)}\bigr)y, \tag{9} \]

where \(\psi_i \geq 0,\ \psi_i \in C^{p-3}[-1,0]\). Here condition A is not satisfied. The boundary-value problem for the first of equations (9)

\[ (y-1)=v(-1),\ldots,\ y^{(p-2)}(-1)=v^{(p-2)}(-1),\ y^k(0)=\beta \]

has no solution for any \(k\) and \(\beta\), although the conditions of Theorem 2 are satisfied.

Generalization of the main problem. The following more general problem for equation (1) is studied analogously. Instead of conditions (2), it is required that, for a solution \(y(x)\), the line \((y(x),y'(x),\ldots,y^{(n-1)}(x))\) in the space \((x,y,y',\ldots,y^{(n-1)})\) contain at least one point of a given nonempty connected set \(\Phi\) (the solution \(y(x)\) passes through \(\Phi\)); instead of condition (3), it is required that this solution be extendable up to \(x=b\) and that \(F(y(b),y'(b),\ldots,y^{(n-1)}(b))=0\), where \(F(y,y_1,\ldots,y_{n-1})\) is a continuous function given for all \(y,y_1,\ldots,y_{n-1}\). For the existence of at least one solution of this problem it is sufficient that, for some \(k=0,\ldots,n-1\), conditions A and B be satisfied and that there exist two solutions \(y_1,y_2\), passing through \(\Phi\) and extendable up to \(x=b\), for which, at \(x=b\), \(F(y_1,\ldots,y_1^{(n-1)})\geq 0,\ F(y_2,\ldots,y_2^{(n-1)})\leq 0\). To verify conditions A and B one may use Theorems 2–8, and for the fulfillment of the last condition \(F\gtrless 0\) it is sufficient, under the hypotheses of Theorem 2, that condition D be fulfilled; for example, that
\(y_kF(y,y_1,\ldots,y_{n-1})>0\) for all sufficiently large \(|y_k|\), and that there exist a double sequence of points
\((x_p,y_p,y_{1p},\ldots,y_{n-1,p})\in \Phi\) \((p=\pm1,\pm2,\ldots)\), for which

\[ \sup_p x_p<b,\quad \sup_p |y_{mp}|<\infty \quad (m=0,\ldots,k), \]

\[ \sup_p |y_{mp}|^{\,n-k-1}|y_{n-1,p}|^{\,k-m}<\infty \quad (m=k+1,\ldots,n-2), \]

\[ \lim_{p\to-\infty} y_{n-1,p}=\infty,\qquad \lim_{p\to-\infty} y_{n-1,p}=-\infty . \]

Latvian State University
named after Pēteris Stučka

Received
27 X 1965

REFERENCES

1. S. N. Bernstein, Collected Works, 3, Publishing House of the Academy of Sciences of the USSR, 1960.
2. J. Sansone, Ordinary Differential Equations, 2, Moscow, 1954.
3. Yu. A. Klokov, Boundary-value problems with a condition at infinity for equations of mathematical physics, Riga, 1963.
4. A. Ya. Lepin, A. P. Myshkis, Differential Equations, No. 9 (1965).
5. M. Nagumo, Proc. Phys.-Math. Soc. Japan, 3, 19, 861 (1937).