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UDC 517.934.9
ON THE QUESTION OF ESTIMATING THE INTERVAL OF APPLICABILITY OF CHAPLYGIN’S THEOREM
V. S. BEZDOMNIKOV, Yu. V. KOMLENKO
Consider the problem
\[ L[y]\equiv y^{(n)}+\sum_{k=0}^{n-1} g_k(t)y^{(k)}=0, \tag{1} \]
\[ y^{(i)}(a)=y^{(m)}(b)=0 \quad (i=0,1,\ldots,n-2;\; m\leq n-1). \tag{2} \]
We shall assume the coefficients of the operation \(L[y]\) to be continuous. We shall call \([0,R)\) a subcritical interval of the problem (1), (2) if, for any \(a,b\in[0,R)\) (\(a<b\)), the problem (1), (2) has a unique solution. An interval \([0,T)\) on which every solution of equation (1) has no more than \(n-1\) zeros, counting a multiple zero as many times as its multiplicity, will be called an interval of nonoscillation of equation (1).
Below we propose methods, based on the ideas of [1, 2], for obtaining estimates of the subcritical interval of the problem (1), (2).
- Let \(u_0(t),u_1(t),\ldots,u_{n-1}(t)\) be a fundamental system of equation (1), and let \(W(t)\) be the Wronskian of this system,
\[ C(t,s)=\frac{1}{W(s)} \begin{vmatrix} u_0(s) & u_1(s) & \cdots & u_{n-1}(s)\\ u'_0(s) & u'_1(s) & \cdots & u'_{n-1}(s)\\ \cdots & \cdots & \cdots & \cdots\\ u_0^{(n-2)}(s) & u_1^{(n-2)}(s) & \cdots & u_{n-1}^{(n-2)}(s)\\ u_0(t) & u_1(t) & \cdots & u_{n-1}(t) \end{vmatrix} \]
is called the Cauchy function of equation (1). It is not difficult to verify that
\[ K(t,s)=\frac{(t-s)^{\,n-1}}{(n-1)!} \]
is the Cauchy function of the equation \(y^{(n)}=f(t)\), and the Green’s function for this equation with boundary conditions (2) has the form
\[ G(t,s)= \begin{cases} K(t,s)-\dfrac{K^{(m)}(b,s)K(t,a)}{K^{(m)}(b,a)}, & (t>s),\\[1.2em] -\dfrac{K^{(m)}(b,s)K(t,a)}{K^{(m)}(b,a)}, & (t\leq s). \end{cases} \]
Let \(H_k\) be numbers such that ...
\[ H_k \geq \sup_{t,s\in [a,b]}\left|\frac{K^{(k)}(s,a)}{K^{(i)}(t,a)}\,G^{(i)}(t,s)\right| \tag{3} \]
\[ (i,\ k=0,\ 1,\ \ldots,\ n-1) \]
for any \(a,\ b\in [0,R)\). A direct calculation shows that
\[ H_k=\frac{(n-m-1)^{\,n-m-1}(n-k-1)^{\,n-k-1}} {(n-k-1)!(2n-m-k-2)^{\,2n-m-k-2}}\,R^{\,n-k-1} \tag{4} \]
satisfy inequality (3) for any \(a,\ b\in [0,R)\).
Consider the operation
\[ L_1[y]\equiv y^{(n)}+\sum_{k=0}^{n-1}p_k(t)y^{(k)}, \]
where \(p_k(t)=g_k(t)\) \((k=1,\ 2,\ldots,\ n-1)\), \(p_0(t)=\max_{t\in[a,b]}[0,g_0(t)]\). By the comparison theorem (see [5]), the subcritical interval for the equation \(L_1[y]=0\) with boundary conditions (2) is contained in the subcritical interval of problem (1), (2). Taking this into account, we shall prove the following assertion.
Theorem 1. If
\[ \sum_{k=0}^{n-1} H_k \int_0^R |p_k(t)|\,dt <1, \]
then \([0,R)\) is a subcritical interval of problem (1), (2).
Proof. Rewrite the equation \(L_1[y]=0\) in the form
\[ y^{(n)}=-\sum_{k=0}^{n-1}p_k(t)y^{(k)}. \tag{5} \]
Then problem (5), (2) is equivalent to the equation
\[ y(t)=-\sum_{k=0}^{n-1}\int_a^b G(t,s)p_k(s)y^{(k)}(s)\,ds \]
or, if we put \(y^{(k)}(t)=x_k(t)\), to the system
\[ x_i(t)=-\sum_{k=0}^{n-1}\int_a^b G^{(i)}(t,s)p_k(s)x_k(s)\,ds \tag{6} \]
\[ (i=0,\ 1,\ldots,\ n-1). \]
By virtue of Theorem 4 of [1], this system has a unique solution if there exist continuous functions \(z_i(t)>0\) such that
\[ z_i(t)>\sum_{k=0}^{n-1}\int_a^b |G^{(i)}(t,s)p_k(s)|z_k(s)\,ds \]
\[ (i=0,\ 1,\ldots,\ n-1). \]
We shall show that such functions exist for any \(a, b \in [0, R)\). From the condition of the theorem and (4) it follows that the root of the equation
\[ \left| \begin{array}{cccc} H_0\displaystyle\int_0^R |p_0(s)|\,ds-\lambda & \cdots & H_{n-1}\displaystyle\int_0^R |p_{n-1}(s)|\,ds \\ \cdot & \cdot & \cdot \\ H_0\displaystyle\int_0^R |p_0(s)|\,ds & \cdots & H_{n-1}\displaystyle\int_0^R |p_{n-1}(s)|\,ds-\lambda \end{array} \right| = \]
\[ =(-\lambda)^{n-1} \left\{ \lambda-\sum_{k=0}^{n-1} H_k\int_0^R |p_k(s)|\,ds \right\} =0 \]
is less than unity. And this means that, for the system
\[ \xi_i=\sum_{k=0}^{n-1} a_k \xi_k+\eta_i \qquad (i=0,1,\ldots,n-1), \tag{7} \]
where
\[ a_k=H_k\int_0^R |p_k(s)|\,ds, \tag{8} \]
Chaplygin’s theorem is valid, i.e., for \(\eta_i>0\) we have \(\xi_i>0\)\(^*\). Let \(c_i>0\) be a solution of system (7); then
\[ c_i>\sum_{k=0}^{n-1} a_k c_k \qquad (i=0,1,\ldots,n-1). \]
Using (3) and (8), we strengthen the last inequalities:
\[ c_i> \sum_{k=0}^{n-1}\int_a^b |G^{(i)}(t,s)p_k(s)| \frac{c_k K^{(k)}(s,a)}{K^{(i)}(t,a)}\,ds. \]
It follows from this that \(z_k(t)=c_k K^{(k)}(t,a)\) satisfy inequalities (6) for any \(a,b\in[0,R)\).
The theorem is proved.
Remark. It is not difficult to see that \([0,R)\) is a subcritical interval of problem (1), (2) if and only if in the triangle \(0\le s<t<R\) the inequality \(K^{(m)}(t,s)>0\) holds. Thus, the subcritical interval of problem (1), (2) is the interval of applicability of Chaplygin’s theorem for the equation \(L[y]=f(t)\) (see [4, 5]).
We note that the estimate of the interval of applicability of Chaplygin’s theorem [6, 8] is obtained as a special case of our theorem when \(m=r\), \(p_k(t)\equiv 0\) \((k=r+1,r+2,\ldots,n-1)\).
- In the case \(p_k(t)\ge 0\) \((k=1,2,\ldots,n-1)\), one may propose the following supplement to the results of [4].
\(^*\) This follows directly from the fact that, under the indicated conditions, the method of successive approximations for system (7) converges.
Theorem 2. If
\[ \sum_{k=0}^{n-1}\frac{(n-m-1)!\,R^{\,n-k}}{(2n-m-k-1)!}\,P_k<1, \]
where
\[ P_k=\max_{[a,b]\in[0,R)}[p_k(t)], \]
then \([0,R)\) is a precritical interval for problem (1), (2).
Proof. From the remark to Theorem 1 it is clear that the question of estimating the precritical interval for problem (1), (2) is equivalent to the question of the conditions for positivity of \(C^{(m)}(t,s)\), where \(C(t,s)\) is the Cauchy function of equation (5). For a lower estimate of \(C^{(m)}(t,s)\), define the sequence \(\{W_i(t,s)\}\) as follows:
\[ W_0(t,s)=\frac{(t-s)^{n-1}}{(n-1)!}, \]
\[ W_{i+1}(t,s)=W_i(t,s)-\int_s^t W_i(t,\tau)L_1[W_i(\tau,s)]\,d\tau . \tag{9} \]
By Theorem 2 of [3], if \(L_1[W_0(\tau,s)]\ge 0\) in the triangle \(0\le s<t<R\), then \(W_i^{(m)}(t,s)\le W_{i+1}^{(m)}(t,s)\le C^{(m)}(t,s)\) in this triangle. Restricting ourselves to the first step of the iterative process (9), we have: if \(W_1^{(m)}(t,s)>0\) in the triangle \(0<s<t<R\), then also \(C^{(m)}(t,s)>0\) in this triangle. The inequality
\[ W_1^{(m)}(t,s)= \frac{(t-s)^{n-m-1}}{(n-m-1)!} - \int_s^t \frac{(t-\tau)^{n-m-1}}{(n-m-1)!} \times \]
\[ \times \left[ \sum_{k=0}^{n-1} p_k(\tau)\frac{(\tau-s)^{n-k-m-1}}{(n-k-m-1)!} \right]\,d\tau>0 \]
will hold if
\[ \sum_{k=0}^{n-1} \frac{(t-s)^{2n-m-k-1}(n-m-1)!}{(2n-m-k-1)!}\,p_k < (t-s)^{n-m-1}. \]
The last inequality is valid if
\[ \sum_{k=0}^{n-1} \frac{(n-m-1)!\,R^{n-k}}{(2n-m-k-1)!}\,p_k<1. \]
The theorem is proved.
3. Consider the equation
\[ y'''+p(t)y'+g(t)y=0. \tag{10} \]
We shall assume that \(p(t)\) has a continuous derivative. Denote
\[ h(t)=\frac{1}{2}\{|g(t)|+g(t)\}. \]
\[ h_1(t)=\frac{1}{2}\{|p'(t)-g(t)|+|p'(t)-g(t)|\}, \]
\[ H=\max_{t\in[0,T]}[h(t)],\qquad H_1=\max_{t\in[0,T]}[h_1(t)]. \]
Taking into account theorem 2 of [2], we obtain the following consequences of theorems 1 and 2.
Corollary 1. If the inequalities
\[ \frac{4T}{27}\int_0^T |p(t)|\,dt+\frac{T^2}{32}\int_0^T h(t)\,dt<1, \]
\[ \frac{4T}{27}\int_0^T |p(t)|\,dt+\frac{T^2}{32}\int_0^T h_1(t)\,dt<1, \]
are satisfied, then \([0,T)\) is an interval of nonoscillation of equation (10).
Corollary 2. Let \(P\geq p(t)\geq 0\) and let the inequalities
\[ \frac{T^2}{12}\,P+\frac{T^3}{60}\,H<1,\qquad \frac{T^2}{12}\,P+\frac{T^3}{60}\,H_1<1, \]
be satisfied; then \([0,T)\) is an interval of nonoscillation of equation (10).
- Let us consider some consequences of the above theorems for an equation of the special form
\[ y^{(n)}+g(t)y=0. \tag{11} \]
As is known (see, for example, [7]), when \(g(t)\geq 0\), the subcritical interval of problem (11), (2) for \(m=0\) is an interval of nonoscillation for equation (11). On the basis of what has been said and theorems 1 and 2, we have
Corollary 3. If \(g(t)\geq 0\) on \([0,T)\) and one of the inequalities
\[ \frac{T^{n-1}}{4^{\,n-1}(n-1)!}\int_0^T g(t)\,dt<1 \quad\text{or}\quad g(t)<\frac{(2n-1)!}{(n-1)!\,T^n}, \]
is satisfied, then \([0,T)\) is an interval of nonoscillation of equation (11).
For odd \(n\), the equations
\[ y^{(n)}+g(t)y=0 \quad\text{and}\quad y^{(n)}-g(t)y=0 \]
are adjoint. Since for adjoint equations the intervals of nonoscillation coincide, one may formulate
Corollary 4. If \(g(t)\leq 0\) on \([0,T)\) and one of the inequalities
\[ \frac{T^{n-1}}{4^{\,n-1}(n-1)!}\int_0^T |g(t)|\,dt<1 \quad\text{or}\quad |g(t)|<\frac{(2n-1)!}{(n-1)!\,T^n}, \]
is satisfied, then \([0,T)\) is an interval of nonoscillation of equation (11) for odd \(n\).
References
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- N. V. Azbelev, I. M. Smolin, Z. B. Tsalyuk. Dokl. Akad. Nauk SSSR, 135, No. 3, 511—514, 1960.
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Received by the editors
April 30, 1965
Izhevsk Mechanical
Institute