UDC 517.941.9

ON A QUESTION OF UNIQUENESS OF SOLUTIONS OF LINEAR BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

V. V. OSTROUMOV

Consider the problem

\[ L[y]\equiv y^{(n)}-\sum_{k=0}^{n-1}g_k y^{(k)}=f, \]

\[ l_i[y;\,a,b]=A_i \quad (i=0,\ldots,n-1), \tag{1} \]

where \(g_k(t)\) are continuous on \([0,T]\), and \(l_i[y;\,a,b]\) are linear functionals defined on the space of functions \(n\) times continuously differentiable on \([0,T]\) and depending continuously on the parameters \(a,b\in[0,T]\).

As was shown in unpublished reports by the leader of the Izhevsk seminar, N. V. Azbelev, the consideration of the boundary value problem (1) as a function of the parameters proved fruitful for constructing criteria for the existence and preservation of the sign of the Green’s function of problem (1) for fixed values of the parameters \(a,b\). Using some of the ideas of the reports mentioned and of works [1—4], we propose below a general theorem on the existence of the Green’s function and illustrate the effectiveness of this theorem by several corollaries for the Vallée-Poussin problem.

1. Consider in the square \(t,s\in[a,b]\) a function \(W(t,s)\), assuming that \(L[W(t,s)]\) (where \(W^{(k)}(t,s)=\dfrac{\partial^k}{\partial t^k}W(t,s)\)) is continuous in \(t\) for \(t\in[a,b]\), \(t\ne s\), and any fixed \(s\in[a,b]\).

We shall say that \(W(t,s)\) satisfies condition \(G\) if

\[ u(t)=\int_a^b W(t,s)\xi(s)\,ds \]

for every function \(\xi(t)\) continuous on \([a,b]\) satisfies the boundary conditions of problem (1). We note that if \(\xi(t)\) is a solution of the equation

\[ \xi(t)+\int_a^b L[W(t,s)]\xi(s)\,ds=f(t), \tag{2} \]

then, as is easily verified by direct substitution, \(L[u]=f\). Thus, \(W(t,s)\) satisfies the following conditions:

a) for fixed \(s\), the function \(w(t)=W(t,s)\) satisfies the boundary conditions of problem (1), is continuous on \([a,b]\) together with \(w^{(n-2)}(t)\), and for \(t\in[a,s)\), \(t\in(s,b]\) has an absolutely continuous derivative \(w^{(n-1)}(t)\);

b)

\[ \left\{\frac{\partial^{\,n-1}}{\partial t^{\,n-1}}W(t,s)\right\}_{t=s+0} - \left\{\frac{\partial^{\,n-1}}{\partial t^{\,n-1}}W(t,s)\right\}_{t=s-0} =1. \]

If (2) has a unique solution, then (1) also has a unique solution \(y(t)\), the Green’s function \(G(t,s)\) of problem (1) exists, and moreover

\[ y(t)=\int_a^b G(t,s) f(s)\,ds. \]

It is now easy to prove the following assertion.

Theorem 1. For the existence of the Green’s function of problem (1) it is necessary and sufficient that, in the square \(t,s\in [a,b]\), there exist a function \(W(t,s)\) satisfying condition \(G\), and a continuous function \(v(t)\) such that on \([a,b]\)

\[ v(t)\geqslant 0,\qquad \psi(t)=v(t)-\int_a^b |L[W(t,s)]|v(s)\,ds\geqslant 0, \tag{3} \]

and the set of zeros of the residual \(\psi(t)\) is nowhere dense on \([a,b]\).

Sufficiency follows from what was said above and from Urison’s theorem [5] (see also Theorem 4 of [1]), which, in view of (3), guarantees that the first characteristic number of the kernel \(L[W(t,s)]\) is greater than one.

Necessity is obtained by putting \(G\equiv W\).

Remark. It is not difficult to see that, by supplementing the condition of the theorem with the inequality \(L[W(t,s)]\leqslant 0\) in the square \(t,s\in [a,b]\) and with the condition of preservation of sign of \(W(t,s)\) in the rectangles \(s\in [a,b]\), \(t\in [\alpha_j,\alpha_{j+1}]\), \(\alpha_j\in [a,b]\), we obtain not only existence, but also the assertion that the sign of the Green’s function in the indicated rectangles coincides with the sign of \(W(t,s)\).

1. Consider the two-point Vallée-Poussin problem

\[ L[y]\equiv y^{(n)}-g(t)y=f(t), \]

\[ y^{(j)}(a)=0\quad (j=0,\ldots,n-k-1), \tag{4} \]

\[ y^{(j)}(b)=0\quad (j=0,\ldots,k-1). \]

Denote by \(r_{n-k,k}(\alpha)=\sup \beta\), where \(\beta\) is such that for \(a,b\in [\alpha,\beta)\) problem (4) has a unique solution. Following the terminology of the Izhevsk seminar [2, 4, 6], we shall call \([\alpha,r_{n-k,k}(\alpha))\) the subcritical interval of problem (4). As is known (see, for example, [2, 3]), if \(g(t)\geqslant g_1(t)\), then \(r_{n-k,k}(\alpha)\geqslant r^1_{n-k,k}(\alpha)\) \((\leqslant)\) for \(k\) odd (even), where \(r^1_{n-k,k}(\alpha)\) corresponds to the equation \(y^{(n)}-g_1y=0\). Hence, and from the fact that \(r_{n-k,k}(\alpha)=\infty\) for the equation \(y^{(n)}=0\), the following assertion follows: if \(g\geqslant 0\) \((\leqslant)\), then \(r_{n-k,k}(\alpha)=\infty\) for \(k\) odd (even).

Let \(a=0\), \(h=b\). Denote

\[ g_s(t)=\frac{1}{2}\left(|g(t)|+(-1)^k g(t)\right). \]

Then \(r_{n-k,k}(\alpha)\geqslant r^s_{n-k,k}(\alpha)\), where \(r^s_{n-k,k}(\alpha)\) corresponds to the equation \(y^{(n)}-(-1)^k g_s y=0\). The following is true.

Theorem 2. If

\[ g_s(t)\leqslant \frac{n!}{t^{\,n-k}(h-t)^k}, \]

then \(h<r_{n-k,k}(0)\).

Proof. Denote by \(G^0(t,s)\) the Green’s function of the problem \(y^{(n)}=0\);

\[ y^{(j)}(0)=0\quad (j=0,\ldots,n-k-1),\qquad y^{(j)}(h)=0\quad (j=0,\ldots,k-1). \]

Putting \(W(t,s)=G^0(t,s)\), we obtain equation (2) in the form

\[ \xi(t)=\int_0^h G^0(t,s)(-1)^k g_s(t)\xi(s)\,ds+f(t). \]

Let us note that the kernel \(G^0(t,s)(-1)^k g_s(t)\) is nonnegative by virtue of the definition of \(g_s(t)\) and the sign of \(G^0(t,s)\) [4]. As the function \(v(t)\) entering into the condition

of Theorem 1, take

\[ v(t)=\int_0^h |G^0(t,s)|\,ds=\frac{t^{\,n-k}(h-t)^k}{n!}. \]

Then, by virtue of Theorem 1, the inequality

\[ \psi(t)=\int_0^h |G^0(t,s)|\,ds -\int_0^h \left\{G^0(t,s)(-1)^k g_s(t)\int_0^h |G^0(s,\tau)|\,d\tau\right\}\,ds \geqslant 0 \]

guarantees the uniqueness of the solution of problem (4). It is easy to see that \(\psi(t)\geqslant 0\) if

\[ g_s(t)\leqslant \frac{n!}{t^{\,n-k}(h-t)^k}. \tag{5} \]

It is obvious that inequality (5), and hence also the inequality \(\psi(t)\geqslant 0\), is satisfied if \(h^1=b-a\leqslant h\). This means that the Green’s function of problem (4) exists for any \(a,b\subset[0,h]\), which is the assertion of our theorem.

Remark. Inequality (5) is satisfied if

\[ \max_{t\in[0,h]} g_s(t)\leqslant \frac{n!\,n^n}{(n-k)^{\,n-k}k^k h^n}. \tag{6} \]

Let now

\[ v_1(t)=\int_0^h |H(t,s)|\,ds, \]

where \(H(t,s)\) is the Green’s function of the problem

\[ y^{(n-1)}=0;\qquad y^{(j)}(0)=0\quad (j=0,\ldots,n-k-1);\qquad y^{(j)}(h)=0\quad (j=0,\ldots,k-2). \]

As in the proof of the preceding theorem, consider the residual

\[ \psi_1(t)=v_1(t)-\int_0^h G^0(t,s)(-1)^k g_s(t)v_1(s)\,ds, \]

and, using Lemma 1 of [4], we obtain the following assertion.

Theorem 3. If

\[ \max_{t\in[0,h]} g_s(t)\leqslant \frac{(2n-1)!(n-k)!(k-1)!}{(n-1)!(n-1)!h^n}, \tag{7} \]

then \(h<\min[r_{n-k,k}(0);\,r_{k,n-k}(0)]\).

Remark. Comparison of the right-hand sides of inequalities (6) and (7) leads us to the following. If \(k_n=E(x)\), where \(x\) is such a root of the equation

\[ \ln\frac{n-k}{k}=\frac{n-2k+1}{n-1}, \]

that \(1\leqslant x\leqslant E\left(\frac n2\right)\), then estimate (6) ((7)) gives the better result for \(k>k_n\) (\(<\)).

Using the remark to Theorem 1 and the arguments in the proof of Theorems 2 and 3, it is easy to verify the validity of the following assertion.

Theorem 4. Let \(g\geqslant 0\) (\(\leqslant\)) and let \(k\) be even (odd). Suppose, further, that inequality (6) or (7) is satisfied. Then the Green’s function of problem (4) is nonnegative (nonpositive) for \(t,s\in[0,h]\).

3. Denote

\[ r(a)=\min_{k=1,\ldots,n-1}[r_{n-k,k}(a)]. \tag{8} \]

As is known (see, for example, [2, 4]), on the interval \([a,r(a))\) any nontrivial solution of the equation \(L[y]\equiv0\) has no more than \(n-1\) zeros, counting a multiple zero as many times as its multiplicity.

Let \(B_2=8,\quad B_3=60,\quad B_4=560,\quad B_5=3780\)

\[ B_n=\min\left[ \frac{(2n-1)!(n-k)!(k-1)!}{(n-1)!(n-1)!}; \right. \]

\[ \left. \frac{n!\,n^n}{(n-k-1)^{\,n-k-1}(k+1)^{\,k+1}} \right], \quad \text{if } n\ge 6. \]

From (8), from the estimates for \(r_{n-k,k}(0)\) obtained above, and from the remark to Theorem 3, it follows that

Theorem 5. If

\[ \ln\frac{n-k}{k} \ge \frac{n-2k+1}{n-1}, \]

but

\[ \ln\frac{n-k-1}{k+1}\le \frac{n-2k-1}{n-1}, \]

then

\[ r(0)>\sqrt[n]{\frac{B_n}{\max_{t\in[0,h]}|g(t)|}}. \]

Remark. In the work of A. Yu. Levin [7] the estimate

\[ r(0)>\sqrt[n]{\frac{C_n}{\max_{t\in[0,h]}|g(t)|}}, \quad \text{where } C_n=2^n n\left[\frac{n-1}{2}\right]!\left[\frac{n}{2}\right]!. \]

was proposed.

We note that

\[ \frac{B_n}{C_n}\ge \frac{1}{2}\cdot \frac{n+1}{2}\cdot \frac{n+2}{4}\cdots \frac{2n-1}{2n-2}. \]

The author takes this opportunity to thank the staff of the Izhevsk seminar for their attention to this work.

References

1. Azbelev N. V., Tsalyuk Z. B. DAN SSSR, 156, No. 2, 239—242, 1964.
2. Azbelev N. V., Tsalyuk Z. B. Matem. sb., new series, 51, no. 4, 475—485, 1960.
3. Azbelev N. V., Khokhryakov A. Ya., Tsalyuk Z. B. Matem. sb., 59 (101), supplement, 1962, pp. 125—144.
4. Ostroumov V. V. Differential Equations, 1, No. 5, 625—630, 1965.
5. Uryson P. S. Works on Topology and Other Areas of Mathematics, 1, Moscow—Leningrad, GITTL, 1951, pp. 82—86.
6. Aliev R. G., Inozemtseva I. N. Proceedings of the Izhevsk Seminar, issue 1, 1963, pp. 36—38.
7. Levin A. Yu. Matem. sb., new series, 64, issue 3, 1963, pp. 396—409.
8. Agamä O. RZh Matem., 1965, 4B588.

Received by the editors
January 12, 1966

Izhevsk Mechanical Institute