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On the Question of the Čaplygin Method for the Cauchy Problem
Yu. V. Komlenko
In works [1—3] and a number of continuations of these investigations (see, for example, [4, 5]), methods of monotonically convergent successive approximations of the type of the well-known Čaplygin method were proposed for the Cauchy problem in the case when the theorem on a differential inequality does not hold. Using the ideas of N. V. Azbelev’s theorems on the funnel [4, 6], we propose a new method of successive approximations, which includes as special cases the results of the aforementioned works. Since the Cauchy problem for an ordinary differential equation reduces to a Volterra integral equation, below we consider the more general problem of the Čaplygin method for a Volterra equation without the assumption that the theorem on an integral inequality is valid.
Consider the system of Volterra integral equations
\[ x(t)=f(t)+\int_{t_0}^{t} K(t,s,x(s))\,ds . \tag{1} \]
Here \(x(t)=\{x^1(t),\ldots,x^n(t)\}\), \(f(t)=\{f^1(t),\ldots,f^n(t)\}\), \(K(t,s,x)=\{K^1(t,s,x),\ldots,K^n(t,s,x)\}\) are \(n\)-dimensional vectors.
The continuous vector-functions \(f(t)\) and \(K(t,s,x)\) are defined in the domain
\[ G=\{t_0\le s\le t<t_1\le +\infty,\;+\infty\ge \xi_1(t)>x(t)>\xi_2(t)\ge -\infty\}. \]
The inequality \(u(t)\ge v(t)\) \((u(t)>v(t))\), \(t\in [t_0,t_1)\), between two vectors
\(u(t)=\{u^1(t),\ldots,u^n(t)\}\) and \(v(t)=\{v^1(t),\ldots,v^n(t)\}\) means the inequalities
\(u^i(t)\ge v^i(t)\) \((u^i(t)>v^i(t))\), \(i=1,\ldots,n\), \(t\in [t_0,t_1)\).
Let \(K(t,s,x)=l(t,s,x)+m(t,s,-x)\) and \(-K(t,s,x)=p(t,s,x)+q(t,s,-x)\), where \(l(t,s,y)\), \(m(t,s,y)\), \(p(t,s,y)\), \(q(t,s,y)\) are continuous in \(G\), nondecreasing with respect to \(y\), and satisfy a Lipschitz condition with respect to \(y\).
Theorem 1. Let the continuous vector-functions \(v_0(t)\) and \(w_0(t)\) satisfy, for \(t\in [t_0,t_1)\), the inclusion \(v_0,w_0\in(\xi_1,\xi_2)\) and the inequalities
\[ v_0(t)\ge f(t)+\int_{t_0}^{t} l(t,s,v_0(s))\,ds+\int_{t_0}^{t} m(t,s,-w_0(s))\,ds, \]
\[ w_0(t)\le f(t)-\int_{t_0}^{t} p(t,s,v_0(s))\,ds-\int_{t_0}^{t} q(t,s,-w_0(s))\,ds. \tag{2} \]
Then the sequences \(\{v_k(t)\}\) and \(\{w_k(t)\}\), formed according to the law
\[ v_{k+1}(t)=f(t)+\int_{t_0}^{t} l(t,s,v_k(s))\,ds+\int_{t_0}^{t} m(t,s,-w_k(s))\,ds, \]
\[ w_{k+1}(t)=f(t)-\int_{t_0}^{t} p(t,s,v_k(s))\,ds-\int_{t_0}^{t} q(t,s,-w_k(s))\,ds, \tag{3} \]
uniformly converge to the solution \(x(t)\) of system (1), and
\[ v_k(t)\gg v_{k+1}(t)\gg x(t)\gg w_{k+1}(t)\gg w_k(t),\qquad t\in [t_0,t_1). \tag{4} \]
Proof. Let us form a \(2n\)-th order system of equations
\[ y^1(t)=f(t)+\int_{t_0}^{t} l(t,s,y^1(s))\,ds+\int_{t_0}^{t} m(t,s,y^2(s))\,ds, \]
\[ y^2(t)=-f(t)+\int_{t_0}^{t} p(t,s,y^1(s))\,ds+\int_{t_0}^{t} q(t,s,y^2(s))\,ds. \tag{5} \]
Here \(y^1=\{y^{11},\ldots,y^{1n}\}\), \(y^2=\{y^{21},\ldots,y^{2n}\}\). Defining the \(2n\)-dimensional vectors
\[ y=\{y^1,y^2\}=\{y^{11},\ldots,y^{1n},y^{21},\ldots,y^{2n}\}, \]
\[ H(t,s,y)=\{l(t,s,y^1)+m(t,s,y^2),\ p(t,s,y^1)+q(t,s,y^2)\}, \]
\[ F(t)=\{f(t),-f(t)\}, \]
we write system (5) in the form
\[ y(t)=F(t)+\int_{t_0}^{t} H(t,s,y(s))\,ds. \tag{5′} \]
By direct substitution we verify that the \(2n\)-dimensional vector \(y=\{x,-x\}\), where \(x(t)\) is a solution of system (1), satisfies system (5′).
By the local existence theorem, the solution \(y(t)\) of system (5′) is defined in some neighborhood of the initial point. Let \(\tau\leq t_1\), and let \([t_0,\tau)\) be the maximal interval on which \(y\) is defined. In \([t_0,\tau)\) construct a sequence \(\{z_k(t)\}\) according to the rule
\[ z_{k+1}(t)=F(t)+\int_{t_0}^{t} H(t,s,z_k(s))\,ds, \]
where
\[ z_0(t)\gg F(t)+\int_{t_0}^{t} H(t,s,z_0(s))\,ds,\qquad z_0\in(\zeta_1,\zeta_2), \]
\[ \zeta_1=\{\xi_1,-\xi_2\},\qquad \zeta_2=\{\xi_2,-\xi_1\}. \]
This sequence satisfies the inequalities
\[ z_0\gg z_1\gg\cdots\gg z_k\gg z_{k+1}\gg\cdots\gg y. \tag{6} \]
Indeed, if \(z_k\ll z_{k-1}\), then \(z_{k+1}\ll z_k\), since
\[ z_{k+1}-z_k=\int_{t_0}^{t}\bigl[H(t,s,z_k(s))-H(t,s,z_{k-1}(s))\bigr]\,ds\ll 0 \]
by virtue of the monotonicity of \(H\). But
\[ z_1-z_0\ll \int_{t_0}^{t}\bigl[H(t,s,z_0(s))-H(t,s,z_0(s))\bigr]\,ds=0 \]
by the definition of \(z_0\). The inequalities \(z_k\gg y\) follow from the theorem on integral inequalities (see [16]).
Next we shall show that the sequence \(\{z_k(t)\}\) converges uniformly to the solution of system (5′).
Let \(z_k-z_{k+1}=\varphi_k\). If the series \(\sum_{k=0}^{\infty}\|\varphi_k(t)\|\) converges uniformly, then the sequence \(\{z_k(t)\}\) also converges uniformly. Here \(\|\varphi\|=\max_{i=11,\ldots,1n,\,21,\ldots,2n}|\varphi^i|\). Since
\[ \varphi_k(t)=z_k(t)-z_{k+1}(t)=\int_{t_0}^{t}[H(t,s,z_{k-1}(s))-H(t,s,z_k(s))]\,dt, \]
\[ \|\varphi_k(t)\|\leq \int_{t_0}^{t}\|H(t,s,z_{k-1}(s))-H(t,s,z_k(s))\|\,ds\leq \]
\[ \leq L\int_{t_0}^{t}\|z_{k-1}(s)-z_k(s)\|\,ds = L\int_{t_0}^{t}\|\varphi_{k-1}(s)\|\,ds, \]
where \(L\) is a constant Lipschitz matrix, then
\[ \|\varphi_k(t)\|\leq L^k\int_{t_0}^{t}\frac{(t-s)^{k-1}}{(k-1)!}\|\varphi_0(s)\|\,ds = \frac{ML^k(\tau-t_0)^k}{k!},\qquad M=\|\varphi_0(t)\|. \]
Let us show that \(\lim_{k\to\infty}z_k(t)=z(t)\) satisfies system \((5')\). Passing in the inequality
\[ \left\|z(t)-F(t)-\int_{t_0}^{t}H(t,s,z(s))\,ds\right\| = \|z(t)-F(t)- \]
\[ -\int_{t_0}^{t}H(t,s,z(s))\,ds-z_{k+1}(t)+F(t)+ \]
\[ +\int_{t_0}^{t}H(t,s,z_k(s))\,ds\| = \left\|z(t)-z_{k+1}(t)+\int_{t_0}^{t}[H(t,s,z_k(s))-\right. \]
\[ \left.-H(t,s,z(s))]\,ds\right\| \leq \|z(t)-z_{k+1}(t)\|+ L\int_{t_0}^{t}\|z_k(s)-z(s)\|\,ds \]
to the limit as \(k\to\infty\), we obtain
\[ \left\|z(t)-F(t)-\int_{t_0}^{t}H(t,s,z(s))\,ds\right\|\leq 0, \]
i.e.
\[ z(t)=F(t)+\int_{t_0}^{t}H(t,s,z(s))\,ds. \]
Since the solution of system \((5')\) is unique, it follows that \(z(t)=y(t)=\{x(t),-x(t)\}\). Now putting
\(z_0=\{v_0^1,\ldots,v_0^n,-w_0^1,\ldots,-w_0^n\}\), we obtain inequalities (4).
Let us show that \(\tau=t_1\). Suppose that \(\tau<t_1\). Then, by the continuation theorem for a solution (see [6]), it follows that for some component either
\(\lim_{t\to\tau}x^i(t)=\xi_1^i(\tau)\), or
\(\lim_{t\to\tau}x^i(t)=\xi_2^i(\tau)\). But this contradicts the inequalities
\[ \xi_1^i(\tau)>\lim_{t\to\tau}v_k^i(t)\geq \lim_{t\to\tau}x^i(t)\geq \lim_{t\to\tau}w_k^i(t)>\xi_2^i(\tau). \]
The theorem is proved.
Remark 1. It is not difficult to see that, along the way, we have proved the following nonlocal assertion on the existence and estimate of a solution of equation (1), generalizing a number of assertions in works [6; 7].
Theorem 1′. If on \([t_0,t_1)\) there are defined continuous vector functions \(v(t)\) and \(w(t)\) satisfying, on \([t_0,t_1)\), the integral inequalities
\[ v(t) \geqslant f(t)+\int_{t_0}^{t} l(t,s,v(s))\,ds+\int_{t_0}^{t} m(t,s,-w(s))\,ds, \]
\[ w(t) \leqslant f(t)-\int_{t_0}^{t} p(t,s,v(s))\,ds-\int_{t_0}^{t} q(t,s,-w(s))\,ds, \]
and the inclusions \(v,w\in(\xi_1,\xi_2)\), then on \([t_0,t_1)\) there exists a solution \(x(t)\) of equation (1), and the estimate
\(v(t)\geqslant x(t)\geqslant w(t)\), \(t\in[t_0,t_1)\), is valid.
Remark 2. For the construction of the initial pair \(v_0(t)\) and \(w_0(t)\) one can use the following lemma.
Lemma. If the continuous vector function \(v(t)\) satisfies, for \(t\in[t_0,t_1)\), the inclusions
\[ v(t)\in(\xi_1,\xi_2),\qquad -v(t)\in(\xi_1,\xi_2) \]
and the integral inequality
\[ v(t)\geqslant 3A\int_{t_0}^{t} v(s)\,ds+\varphi(t), \]
where \(A\) is the constant Lipschitz matrix of the vector function \(K(t,s,x)\),
\[ \varphi(t)\geqslant \left|\int_{t_0}^{t} K(t,s,0)\,ds+f(t)\right|, \]
then \(v_0=v\), \(w_0=-v\) satisfy the integral inequalities (2).
Proof. It is sufficient to show that the inequalities (2) are satisfied:
\[ \begin{aligned} &v(t)-f(t)-\int_{t_0}^{t} l(t,s,v(s))\,ds-\int_{t_0}^{t} m(t,s,v(s))\,ds \geqslant 3A\int_{t_0}^{t} v(s)\,ds+ \\ &\quad+\left|\int_{t_0}^{t} K(t,s,0)\,ds+f(t)\right|-f(t) -\int_{t_0}^{t} l(t,s,v(s))\,ds-\int_{t_0}^{t} m(t,s,v(s))\,ds \geqslant \\ &\quad\geqslant 2A\int_{t_0}^{t} v(s)\,ds+\int_{t_0}^{t} A|v(s)-0|\,ds+\int_{t_0}^{t} K(t,s,0)\,ds \\ &\qquad-\int_{t_0}^{t} l(t,s,v(s))\,ds-\int_{t_0}^{t} m(t,s,v(s))\,ds \geqslant 2A\int_{t_0}^{t} v(s)\,ds+ \\ &\quad+\int_{t_0}^{t} |K(t,s,v(s))-K(t,s,0)|\,ds+\int_{t_0}^{t} K(t,s,0)\,ds \\ &\qquad-\int_{t_0}^{t} l(t,s,v(s))\,ds-\int_{t_0}^{t} m(t,s,v(s))\,ds \geqslant \\ &\quad\geqslant 2A\int_{t_0}^{t} v(s)\,ds-\int_{t_0}^{t} [m(t,s,v(s))-m(t,s,-v(s))]\,ds = \\ &\quad=\int_{t_0}^{t} \{A|v(s)-(-v(s))|-[m(t,s,v(s))-m(t,s,-v(s))]\}\,ds \geqslant 0. \end{aligned} \]
The second inequality is also fulfilled, since \(w=-v\), and
\[ p(t,s,v)+q(t,s,v)=-\,l(t,s,v)-m(t,s,v). \]
References
- N. V. Azbelev, DAN SSSR, 83, No. 4, 1952, pp. 517–519.
- N. V. Azbelev, DAN SSSR, 99, No. 4, 1954, pp. 493–494.
- S. N. Slugin, DAN SSSR, 110, No. 6, 1956, pp. 936–939.
- N. V. Azbelev, Scientific Reports of Higher Education Institutions, Physics and Mathematics, No. 6, 1958, pp. 30–35.
- S. A. Pak, Siberian Mathematical Journal, 3, No. 4, 1962, pp. 569–574.
- N. V. Azbelev, Z. B. Tsalyuk, Mathematical Collection, 56 (98), No. 3, 1962, pp. 325–342.
- S. M. Lozinskii, DAN SSSR, 92, No. 2, 1953, pp. 225–228.
Received by the editors
February 20, 1965
Izhevsk Mechanical Institute