Abstract
Full Text
MATHEMATICS
G. Ya. LYUBARSKII
ON A BOUNDARY-VALUE PROBLEM ON THE AXIS FOR A NONLINEAR EQUATION OF THE \(n\)-TH ORDER
(Presented by Academician S. L. Sobolev on 15 X 1962)
1. We shall be interested in those solutions \(x(t)\) of the autonomous nonlinear equation
\[ x^{(n)}+f\left(x,\dot{x},\ldots,x^{(n-1)}\right)=0, \tag{1} \]
which satisfy the following boundary conditions:
\[ x(-\infty)=q_1\ (q_1<0),\qquad x(+\infty)=q_2\ (q_2>0), \]
\[ x^{(s)}(\pm\infty)=0\quad (s=1,2,\ldots,n). \]
We shall call such solutions transition solutions.
As regards the function \(f\), we shall assume that it is continuous in the domain
\(q_1\leq x\leq 0,\ 0\leq x\leq q_2,\ -\infty<x^{(s)}<\infty\)
\((s=1,2,\ldots,n-1)\), and satisfies the relations
\[ f(q_i,0,\ldots,0)=0\ (i=1,2),\qquad f(x,0,\ldots,0)\neq 0\ (q_1<x<q_2). \tag{2} \]
Yu. A. Klokov \({}^{1}\) considered an analogous problem for the case \(n=3\), but under the opposite assumption \(f(x,0,0)\equiv 0\).
Below we give two sufficient conditions for the existence of transition solutions of equation (1) and note some properties of these solutions.
2. We introduce a number of auxiliary notations which will help formulate the results obtained. Consider a pair of polynomials
\[ P_{\pm}(\nu)=a_n\nu^n+a_{n-1}\nu^{n-1}+\cdots+a_1\nu+a_{\pm}, \]
possessing the following two properties: 1) they differ only in their constant terms, with \(a_->0\), \(a_+<0\), and \(a_1<0\); 2) the polynomial \(P_-(\nu)\) has \(l\) negative zeros
\(\lambda_1^->\lambda_2^->\cdots>\lambda_l^-\) and \(m\) positive zeros
\(\mu_1^-<\mu_2^-<\cdots<\mu_m^-\) \((l+m=n)\); the polynomial \(P_+(\nu)\) has \((l+1)\) negative zeros
\(\lambda_1^+>\lambda_2^+>\cdots>\lambda_{l+1}^+\) and \((m-1)\) positive zeros
\(\mu_1^+<\mu_2^+<\cdots<\mu_{m-1}^+\).
We shall denote the totality of all such pairs of polynomials by \(\Pi\).
Denote by \(K_-(P_{\pm})\) the cone consisting of all polynomials \(Q(\nu)\) of the form
\[ \begin{aligned} Q(\nu)={}&c_0+c_1\left(1-\frac{\nu}{\mu_1^+}\right) +c_2\left(1-\frac{\nu}{\mu_1^+}\right)\left(1-\frac{\nu}{\mu_2^+}\right)+\cdots \\ &+\cdots+c_{m-1}\left(1-\frac{\nu}{\mu_1^+}\right)\left(1-\frac{\nu}{\mu_2^+}\right)\cdots\left(1-\frac{\nu}{\mu_{m-1}^+}\right) \\ &+\prod_{1}^{m}\left(1-\frac{\nu}{\mu_j^-}\right) \left\{c_m+c_{m+1}\left(1-\frac{\nu}{\lambda_1^-}\right) +c_{m+2}\left(1-\frac{\nu}{\lambda_1^-}\right)\left(1-\frac{\nu}{\lambda_2^-}\right)\right.\\ &\left.\qquad\qquad+\cdots+c_{n-1}\left(1-\frac{\nu}{\lambda_1^-}\right)\cdots\left(1-\frac{\nu}{\lambda_{l-1}^-}\right)\right\}, \end{aligned} \]
where all coefficients \(c_j\) \((j=0,1,\ldots,n-1)\) are nonnegative.
In a similar way, denote by \(K_+(P_{\pm})\) the cone consisting of all polynomials \(Q(\nu)\) of the form
\[ Q(\nu)=c_0+c_1\left(1-\frac{\nu}{\lambda_1^-}\right) +c_2\left(1-\frac{\nu}{\lambda_1^-}\right)\left(1-\frac{\nu}{\lambda_2^-}\right)+\cdots \]
\[ \begin{aligned} &\cdots+c_l\left(1-\frac{\nu}{\lambda_1^-}\right)\left(1-\frac{\nu}{\lambda_2^-}\right)\cdots\left(1-\frac{\nu}{\lambda_l^-}\right) +\prod_{1}^{l+1}\left(1-\frac{\nu}{\lambda_j^+}\right)\times\\ &\times\left\{c_{l+1}+c_{l+2}\left(1-\frac{\nu}{\mu_1^+}\right) +\cdots+c_{n-1}\left(1-\frac{\nu}{\mu_1^+}\right)\left(1-\frac{\nu}{\mu_2^+}\right)\cdots\left(1-\frac{\nu}{\mu_{m-2}^+}\right)\right\}, \end{aligned} \]
where all coefficients \(c_j\) \((j=0,1,\ldots,n-1)\) are nonnegative.
The question of whether an arbitrary polynomial belongs to the cone \(K_-(P_{\pm})\) or \(K_+(P_{\pm})\) is decided in an elementary way.
Let \(H_n\) be the \(n\)-dimensional space of vectors \(\{f_i\}_0^{n-1}\). In it we single out the set \(F_-(P_{\pm})\) of vectors distinguished by the following property:
\[ P_-(v)-a_n\bigl(v^n+f_{n-1}v^{n-1}+\cdots+f_1v+f_0\bigr)\in K_-(P_{\pm}). \]
In the same way, by means of the condition
\[ P_+(v)-a_n\bigl(v^n+f_{n+1}v^{n-1}+\cdots+f_1v+f_0\bigr)\in K_+(P_{\pm}) \]
we single out the set \(F_+(P_{\pm})\).
Let us introduce the differential operators
\[ M_j^-=-\frac{1}{\mu_j^-}\frac{d}{dt}+1\quad (j=1,2,\ldots,m); \]
\[ M_j^+=-\frac{1}{\mu_j^+}\frac{d}{dt}+1\quad (j=1,2,\ldots,m-1); \]
\[ L_j^-=-\frac{1}{\lambda_j^-}\frac{d}{dt}+1\quad (j=1,2,\ldots,l); \]
\[ L_j^+=-\frac{1}{\lambda_j^+}\frac{d}{dt}+1\quad (j=1,2,\ldots,l+1). \]
With the aid of these operators we associate with the function \(x(t)\) \(n\) functions \(y_j^-(t)\) \((j=0,1,\ldots,m-1)\), \(z_k^-(t)\) \((k=0,1,\ldots,l-1)\), defined on the negative half-axis in the following way:
\[ y_0^-(t)=x(t);\quad z_0^-(t)=M_1^-M_2^-\cdots M_m^-x(t);\quad y_j^-(t)=M_j^+y_{j-1}^-(t); \tag{3} \]
\[ z_k^-(t)=L_k^-z_{k-1}^-(t)\quad (t<0;\ j=1,2,\ldots,m-1;\ k=1,2,\ldots,l-1). \]
On the positive half-axis we define the functions \(y_j^+(t)\) \((j=0,1,\ldots,l)\) and \(z_k^+(t)\) \((k=0,1,\ldots,m-2)\)
\[ y_0^+(t)=x(t);\quad z_0^+(t)=L_1^+L_2^+\cdots L_{l+1}^+x(t);\quad y_j^+(t)=L_j^-y_{j-1}^+(t); \tag{4} \]
\[ z_k^+(t)=M_k^+z_{k-1}^+(t)\quad (t>0;\ j=1,2,\ldots,l;\ k=1,2,\ldots,m-2). \]
Denote by \(D_-\) \((D_+)\) that domain of values in the space of the variables \(x,\dot x,\ldots,x^{(n-1)}\) which corresponds to the cube \(q_1\leq y_j^-, z_k^-\leq 0\) \((0\leq y_j^+, z_k^+\leq q_2)\) in the space of the variables \(y^-,z^-\) \((y^+,z^+)\).
We shall say that the function \(f(x,\dot x,\ldots,x^{(n-1)})\) is majorized by the pair \(P_{\pm}\in\Pi\) if: 1) the function \(f(x,\dot x,\ldots,x^{(n-1)})\) is differentiable at every point of the domains \(D_-\) and \(D_+\); 2) the function \(f\) satisfies condition (2), and the sign of the function \(f(x,0,\ldots,0)\) in the interval \(q_1<x<q_2\) coincides with the sign of the coefficient \(a_n\); 3) \(a_-q_1=a_+q_2\); 4) the gradient of the function \(f\) (i.e. the vector with coordinates
\[ f_j=\frac{\partial f(x,\dot x,\ldots,x^{(n-1)})}{\partial x^{(j)}}) \]
at every point of the domain \(D_-\) belongs to the set \(F_-(P_{\pm})\), and at every point of the domain \(D_+\), to the set \(F_+(P_{\pm})\).
- Let us formulate the results obtained.
Theorem 1. If there exists a pair of polynomials \(P_{\pm}\in\Pi\) which majorizes the function \(f(x,\dot x,\ldots,x^{(n-1)})\), then equation (1) has a transitional solution \(x(t)\) satisfying the conditions
\[ q_1\leq x(t)\leq 0\quad (t\leq 0),\qquad 0\leq x(t)\leq q_2\quad (t\geq 0). \tag{5} \]
Theorem 1 can be made somewhat more precise. To this end, denote by \(S\) the totality of all \(n\)-times differentiable functions \(x(t)\) possessing
with the following property: for every \(t<0\) (\(t>0\)) the set of \(n\) numbers \(\{x(t),\dot x(t),\ldots,x^{(n-1)}(t)\}\) belongs to the domain \(D_-\) (\(D_+\)). If \(x_1(t)-x_2(t)\in S\), then we shall say that the function \(x_1(t)\) is steeper than the function \(x_2(t)\), and write \(x_1\gg x_2\).
Theorem 2. Suppose that all the conditions of Theorem 1 are fulfilled. Then equation (1) has a solution \(x(t)\in S\). If there are several such solutions, then among them there exist two “extreme” solutions: the steepest \(x_1(t)\) and the least steep \(x_2(t)\).
The following theorem shows how transition solutions change when the function \(f\) is varied.
Theorem 3. Let two functions \(f_i(x,\dot x,\ldots,x^{(n-1)})\) \((i=1,2)\) be majorized by one and the same pair of polynomials \(P_\pm\in\Pi\). Suppose that in the domain \(D_-+D_+\) the inequality \(a_n f_1\gg a_n f_2\) holds. Then the extreme solutions \(x_1\) and \(x_2\) of equation (1) for \(f=f_1\) are steeper than those for \(f=f_2\).
Theorem 1 can be considerably strengthened if one uses Schauder’s fixed-point theorem \((^2)\). This gives
Theorem 4. Let \(f_i(x,\dot x,\ldots,x^{(n-1)})\) \((i=1,2)\) be any two functions majorized by some pair \(P_\pm\in\Pi\). If the function \(f(x,\dot x,\ldots,x^{(n-1)})\) is continuous in the domains \(D_-\) and \(D_+\), satisfies condition (2), and if in the domain \(D_-+D_+\) the inequality \(a_n f_1\ll a_n f\ll a_n f_2\) holds, then equation (1) has at least one solution \(x(t)\in S\). Moreover, if \(x_i(t)\) is some transition solution of equation (1) for \(f=f_i\) \((i=1,2)\), and the function \(x_1(t)\) is less steep than \(x_2(t)\), then among the functions \(x(t)\) such that \(x_1\ll x\ll x_2\) there exists at least one transition solution of equation (1).
4. Let us outline the proof of the theorems of the preceding section. Equation (1) is evidently equivalent to the equation
\[ P\left(\frac{d}{dt}\right)x+a(x)x= \]
\[ =-a_n f(x,\dot x,\ldots,x^{(n-1)})+a_{n-1}x^{(n-1)}+\cdots+a_1\dot x+a(x)x \equiv \]
\[ \equiv g(x,\dot x,\ldots,x^{(n-1)}), \tag{6} \]
where \(P(\nu)=a_n\nu^n+a_{n-1}\nu^{n-1}+\cdots+a_1\nu\); \(a(x)=a_-\), if \(x<0\), and \(a(x)=a_+\), if \(x>0\). We shall seek solutions of this equation among functions \(x(t)\) satisfying condition (5). For such functions the relation \(a(x(t))=a(t)\) is valid. Therefore the nonlinear operator \(P(d/dt)+a(x)\), standing on the left-hand side of (6), can be replaced by the linear operator \(L=P(d/dt)+a(t)\). The operator \(L\) has a Green’s function \(K_0(t,s)\) satisfying the conditions \(K_0(\pm\infty,s)=K_0(0,s)=0\). With its aid, equation (6) can be replaced by the equivalent integro-differential equation
\[ x(t)=\int_{-\infty}^{\infty} K_0(t,s)\,g(x(s),\dot x(s),\ldots,x^{(n-1)}(s))\,ds. \tag{7} \]
Using definitions (3) and (4), it is easy to obtain from this
\[ y_j^-(t)=\int_{-\infty}^{\infty} K_j^-(t,s)\,g(x(s),\dot x(s),\ldots,x^{(n-1)}(s))\,ds \]
\[ (t<0;\ j=0,1,2,\ldots,m-1), \]
where \(K_j^-(t,s)=M_j^+M_{j-1}^+\cdots M_1^+K_0(t,s)\) \((t<0)\). Analogous relations are also obtained for the functions \(z_k^-(t)\) for \(t<0\) and the functions \(y_j^+(t)\) and \(z_k^+(t)\) for \(t>0\). If in these relations the functions \(x(s),\dot x(s),\ldots,x^{(n-1)}(s)\) are replaced by the corresponding linear combinations of the functions \(y_j^-(s)\)
and \(z_k^-(s)\) for \(s<0\) and by combinations of the functions \(y_j^+(s)\) and \(z_k^+(s)\) for \(s>0\), then one obtains a system of integral equations which is equivalent on \(S\) to equation (1). Let \(K=K_j^-(t,s)\) \(\bigl(K=K_j^+(t,s)\bigr)\) be the kernel of one of the equations of this system. For \(t<0\) (\(t>0\)) it has the following properties: a) \(a(t)K(t,s)>0\); b)
\[ 0\leq a(t)\int_{-\infty}^{\infty} K(t,s)\,ds\leq 1; \]
c) the function
\[ \int_{-\infty}^{\infty} K(t,s)\,ds \]
decreases monotonically as \(t\) increases. The validity of these assertions is easily verified if one uses the following representation of the Green’s function \(K_0(t,s)\) \((^3)\):
\[ K_0(t,s)=\frac{a_- - a_+}{a_n^2}\int_0^t Y(\lambda^+,\mu^-,t-\sigma)\,Y(\lambda^-,\mu^+,\sigma-s)\,d\sigma, \tag{8} \]
where
\[ Y(\lambda^+,\mu^-,t)=\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{e^{\nu t}\,d\nu}{\Pi(\nu-\lambda^+)\Pi(\nu-\mu^-)}; \]
\[ Y(\lambda^-,\mu^+,t)=\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{e^{\nu t}\,d\nu}{\Pi(\nu-\lambda^-)\Pi(\nu-\mu^+)}. \]
On the other hand, the fact that the function \(f\) is majorized by the pair \(P_\pm\in\Pi\) makes it possible to derive the following properties of the function \(g\): a) if \(x_i\in S\) \((i=1,2)\) and \(x_1\gg x_2\), then
\[ g(x_1,\dot{x}_1,\ldots,x_1^{(n-1)})\leq g(x_2,\dot{x}_2,\ldots,x_2^{(n-1)}); \]
b) if \(x\in S\), then
\[ q_1a_- = q_2a_+ \leq g(x,\dot{x},\ldots,x^{(n-1)}) \leq 0. \]
It is now easy to construct two iterative sequences (one sequence with increasing steepness, the other with decreasing steepness), each of which converges to a transitional solution of equation (1). In this way theorems 1–3 are obtained naturally.
- Applying theorem 1 to the second-order equation
\[ \ddot{x}+f(x,\dot{x})=0, \tag{9} \]
we obtain the following assertion:
If the function \(f(x,0)\) has two adjacent zeros \(q_1<0\) and \(q_2>0\) and is positive inside the interval \((q_1,q_2)\), while the function \(f(x,\dot{x})\) is continuous and differentiable in the region \(q_1\leq x\leq 0\), \(0\leq x\leq q_2\), \(-\infty<\dot{x}<\infty\), then, for the existence of a transitional solution of equation (9), it is sufficient that the system of inequalities
\[ \max_{q_1\leq x,\,y\leq 0} \left\{\mu^2+\mu f'_{\dot{x}}\bigl(x,\mu(x-y)\bigr) +f'_x\bigl(x,\mu(x-y)\bigr)\right\} \leq \frac{q_2-q_1}{q_1}\lambda\mu, \]
\[ \min_{0\leq x,\,z\leq q_2} \left\{\lambda^2+\lambda f'_{\dot{x}}\bigl(x,\lambda(x-z)\bigr) +f'_x\bigl(x,\lambda(x-z)\bigr)\right\}\geq 0,\qquad \lambda<0, \]
\[ \min_{q_1\leq x,\,y\leq 0} \left\{f'_{\dot{x}}\bigl(x,\mu(x-y)\bigr)\right\} +\lambda+\mu\geq 0,\qquad \lambda+\mu>0, \]
\[ \min_{0\leq x,\,z\leq q_2} \left\{f'_{\dot{x}}\bigl(x,\lambda(x-z)\bigr)\right\} +\lambda+\mu\geq 0. \]
be consistent. With the aid of this assertion one can, for example, show that the equation
\[ \ddot{x}-2(1+x^2)\dot{x}+1-x^2=0 \]
has a transitional solution. When the function \(f(x,0)\) is negative on the interval \((q_1,q_2)\), an analogous criterion for the existence of a transitional solution can be formulated.
Physico-Technical Institute
Academy of Sciences of the Ukrainian SSR
Received
10 X 1962
CITED LITERATURE
- Yu. A. Klokov, DAN, 139, No. 4 (1961).
- E. L. Elsgolts, Qualitative Methods in Mathematical Analysis, Moscow, 1955, p. 96.
- G. Ya. Lyubarskii, DAN, 140, No. 6 (1961).