UDC 517.934

MATHEMATICS

I. T. Kiguradze

ON MONOTONE SOLUTIONS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS OF THE \(n\)-TH ORDER

(Presented by Academician I. N. Vekua, 11 XII 1967)

In the present paper we study the problem of the existence of a solution \(u(t)\) of the differential equation

\[ u^{(n)}=f(t,u,u',\ldots,u^{(n-1)}), \tag{1} \]

defined on the interval \((0,+\infty)\) and satisfying the conditions

\[ \lim_{t\to 0} u(t)=u_0,\qquad (-1)^k u^{(k)}(t)\ge 0\quad \text{for } t>0\ (k=0,1,\ldots,n-1). \tag{2} \]

Previously this problem had been investigated only for the case \(n=2\) \((^{1-5})\).

By a solution of problem (1)—(2) we mean a function \(u(t)\), absolutely continuous together with its derivatives up to order \(n-1\) inclusive on every segment contained in the interval \((0,+\infty)\), and satisfying equation (1) and conditions (2) on this interval.

Before proceeding to the formulation of the existence theorem, we give some definitions and one auxiliary proposition.

Definition 1. \(\omega(t,x)\in B_n(r;a)\) if \(\omega(t,x)\) is defined and nonnegative in the domain \(t\in(0,a]\), \(x\in[0,+\infty)\), and there exist a number \(a_0\in(0,a)\) and a function \(b(t)\), continuous on the half-interval \((0,a]\), with \(t^{n-2}b(t)\in L(0,a)\), such that, for any \(\alpha\in(0,a_0]\), for every function \(u(t)\) absolutely continuous together with its derivatives up to order \(n-1\) inclusive on the interval \([\alpha,a]\) and satisfying on this interval the inequalities
\[ (-1)^k u^{(k)}(t)\ge 0\quad (k=0,1,\ldots,n),\qquad u(t)\le r,\qquad |u^{(n)}(t)|\le \omega(t,|u^{(n-1)}(t)|), \]
we have \(|u^{(n-1)}(t)|\le b(t)\) for \(t\in[\alpha,a]\).

Definition 2. \(\psi(t,x_1,\ldots,x_m)\in K_t(\alpha,\beta)\) if \(m\ge 1\), \(\psi(t,x_1,\ldots,x_m)\) is measurable in \(t\) on the interval \((\alpha,\beta)\) for every \((x_1,\ldots,x_m)\in R_m\), continuous in \((x_1,\ldots,x_m)\) in the space \(R_m\) for almost all \(t\in(\alpha,\beta)\), and for every \(r>0\) there exists a function \(\psi(t;r)\in L(\alpha,\beta)\) such that
\[ |\psi(t,x_1,\ldots,x_m)|\le \psi(t;r) \]
for \(t\in(\alpha,\beta)\), \(|x_k|\le r\) \((k=1,2,\ldots,m)\) (local Carathéodory conditions).

In what follows it is everywhere assumed that \(n\ge 2\) and \(f(t,x_1,\ldots,x_n)\in K_t(\alpha,\beta)\) for arbitrary \(\alpha,\beta\in(0,+\infty)\); here the case is not excluded when the function \(f(t,x_1,\ldots,x_n)\), having a singularity at \(t=0\), does not belong to the set \(K_t(0,\beta)\).

Lemma. If \(n\ge 3\), the function \(u(t)\) is absolutely continuous together with its derivatives up to order \(n-1\) inclusive on the interval \([\alpha,\beta]\), \(u^{(k)}(\beta)=0\) \((k=1,\ldots,n-2)\), and
\[ (-1)^k u^{(k)}(t)\ge 0 \]
for \(t\in[\alpha,\beta]\) \((k=0,1,\ldots,n)\), then for \(t\in[\alpha,\beta]\) the inequalities
\[ |u^{(k-1)}(t)|\le \frac{[(n-1)!]^{(n-k)/(n-1)}}{(n-k)!} \,|u(t)|^{(n-k)/(n-1)} \,|u^{(n-1)}(t)|^{(k-1)/(n-1)} \]

\[ (k=1,2,\ldots,n). \]

Theorem 1. If

\[ f(t,0,\ldots,0)\equiv 0,\qquad (-1)^n f(t,x_1,\ldots,x_n)\ge 0 \]

for \(t>0\), \((-1)^{k-1}x_k\ge 0\) \((k=1,2,\ldots,n)\),
\(a>0,\ r>0\), and in the domain

\[ t\in(0,a],\qquad 0\le x_1\le r,\qquad 0\le (-1)^{k-1}x_k\le \]

\[ \le \frac{[(n-1)!\,x_1]^{(n-k)/(n-1)}}{(n-k)!}\, |x_n|^{(k-1)/(n-1)} \]

\[ (k=2,\ldots,n-1),\qquad (-1)^{n-1}x_n\ge 0 \tag{4} \]

the inequality

\[ |f(t,x_1,\ldots,x_n)|\le \omega(t,|x_n|) \tag{5} \]

is satisfied, where \(\omega(t,x)\in B_n(r;a)\), then for every \(u_0\in[0,r]\) the problem (1)—(2) has at least one solution.

We indicate the scheme of the proof of Theorem 1. Let the number \(a_0\) and the function \(b(t)\) be chosen according to Definition 1. Put

\[ b_k(t)=(n-1)!\,r\,(a-a_0)^{-k} +\int_t^a \tau^{\,n-2-k}b(\tau)\,d\tau \quad (k=0,\ldots,n-2), \]

\[ b_{n-1}(t)=b(t); \tag{6} \]

\[ \rho(t)=\max_{t\le \tau\le a}\sum_{k=0}^{n-1} b_k(\tau); \tag{7} \]

\[ \chi(t;\tau)= \begin{cases} 1, & \text{for } 0\le t\le \tau,\\ 2-t/\tau, & \text{for } \tau<t<2\tau,\\ 0, & \text{for } t\ge 2\tau; \end{cases} \qquad \sigma_k(t)= \begin{cases} t, & \text{for } (-1)^{k-1}t>0,\\ 0, & \text{for } (-1)^{k-1}t\le 0; \end{cases} \tag{8} \]

\[ g_m(t,x_1,\ldots,x_n) = \chi\!\left(\sum_{i=1}^n |x_i|;\rho\!\left(\frac{a_0}{m}\right)\right) f(t,\sigma_1(x_1),\ldots,\sigma_n(x_n)). \tag{9} \]

Since \(t^{n-2}b(t)\in L(0,a)\), it is clear from (6) that

\[ b_1(t)\in L(0,a). \tag{10} \]

Taking into account conditions (3), with the aid of Schauder’s theorem it is easy to prove that, for every natural number \(m\), the differential equation

\[ u^{(n)}=g_m(t,u,u',\ldots,u^{(n-1)}) \tag{11} \]

has a solution \(u_m(t)\) satisfying the boundary conditions

\[ u_m(a_0/m)=u_0,\qquad u_m^{(k)}(a+m)=0 \quad (k=0,1,\ldots,n-2) \tag{12} \]

and the inequalities

\[ (-1)^k u_m^{(k)}(t)\ge 0 \quad \text{for } t\in\left[\frac{a_0}{m},\,a+m\right] \quad (k=0,1,\ldots,n). \tag{13} \]

Since \(u_0\in[0,r]\), according to (12) and (13), from the lemma cited above it is clear that, for \(t\in[a_0/m,a+m]\), the point
\((t,u(t),\ldots,u^{(n-1)}(t))\) belongs to the domain (4); therefore, by virtue of (5), (8), and (9), from (11) we have

\[ |u_m^{(n)}(t)|\le \omega(t,|u_m^{(n-1)}(t)|) \quad \text{for } t\in[a_0/m,a]. \]

Hence, according to Definition 1 and conditions (6), (7), (10), (12), and (13), we easily find that

\[ |u_m^{(k)}(t)|\le b_k(t) \quad \text{for } t\in\left[\frac{a_0}{m},a\right] \quad (k=0,1,\ldots,n-1), \]

\[ \sum_{k=0}^{n-1}|u_m^{(k)}(t)| \le \rho(a_0/m) \quad \text{for } t\in\left[\frac{a_0}{m},a+m\right]; \tag{14} \]

\[ |u_m(t)-u_0| \le \int_0^t b_1(\tau)\,d\tau \quad \text{for } t\in\left[\frac{a_0}{m},a+m\right]. \tag{15} \]

By virtue of (8), (9), and (14), it follows from (11) that \(u_m(t)\) is a solution of equation (1) on the interval \([a_0/m,a+m]\).

Relying on inequalities (13) and (14), one can show that the sequence \(\{u_m(t)\}\) contains a subsequence \(\{u_{m_i}(t)\}\) such that \(\{u_{m_i}^{(k)}(t)\}\) \((k=0,1,\ldots,n-1)\) converges uniformly on every segment contained in \((0,+\infty)\).

It is not hard to prove that \(u(t)=\lim\limits_{i\to\infty}u_{m_i}(t)\) is a solution of equation (1), defined on the interval \((0,+\infty)\), and, on the other hand, it follows directly from (13) and (15) that \(u(t)\) satisfies conditions (2).

Corollary 1. Suppose that conditions (3) are satisfied, and that in the domain (4) inequality (5) holds, where \(\omega(t,x)\ge 0\) for \(t\in(0,a)\), \(x\ge 0\), \(\omega(t,x)\in K_t(a,a)\) for any \(a\in(0,a)\), and there exists a positive number \(\rho_0\) such that the upper solution \(\rho(t)\) of the problem

\[ d\rho/dt=-\omega(t,\rho),\qquad \rho(a)=\rho_0 \]

is defined on the interval \((0,a]\) and

\[ (n-2)!\,r<\int_0^a t^{\,n-2}\rho(t)\,dt<+\infty . \]

Then problem (1)—(2) is solvable for any \(u_0\in[0,r]\).

The condition of Corollary 1 is satisfied, for example, by the function \(\omega(t,x)=\psi(t)\omega(x)\), if \(\psi(t)>0\) for \(t\in(0,a)\), \(\omega(x)\) is positive and continuous on the interval \((0,+\infty)\), and either

\[ \int_0^a \psi(t)\,dt<\int_0^{+\infty}\frac{dt}{\omega(t)}<+\infty,\qquad \int_0^a t^{n-2}\Omega^{-1}\!\left[\int_0^t \psi(\tau)\,d\tau\right]dt>(n-2)!\,r, \]

where \(\Omega^{-1}(t)\) is the function inverse to

\[ \Omega(t)=\int_t^{+\infty}\frac{d\tau}{\omega(\tau)}, \]

or

\[ \Omega_\delta(t)=\int_\delta^t \frac{d\tau}{\omega(\tau)}\to+\infty \quad\text{as }t\to+\infty,\qquad t^{n-2}\Omega_\delta^{-1}\!\left[\int_t^a \psi(\tau)\,d\tau\right]\in L(0,a), \]

where

\[ \delta>(n-1)!\,a^{1-n}r, \]

and \(\Omega_\delta^{-1}(t)\) is the function inverse to \(\Omega_\delta(t)\).

From Corollary 1 one easily obtains the following simple

Corollary 2. Suppose that conditions (2) are satisfied, \(a>0\), \(r>0\), and in the domain \(t\in(0,a)\), \((-1)^{k-1}x_k\ge 0\) \((k=1,2,\ldots,n)\), we have

\[ |f(t,x_1,\ldots,x_n)|\le A t^{(n-1)\lambda-n}|\ln t|^{\lambda-1-\varepsilon} \sum_{k=2}^{n}(1+|x_n|)^{\mu_k}(1+|x_k|)^{\frac{n-1}{k-1}\nu_k}, \]

where \(A\ge0\), \(\nu_k\ge0\), \(\mu_k+\nu_k\le\lambda\) \((k=2,\ldots,n)\), \(\varepsilon>0\) for \(\lambda\le1\) and \(\varepsilon=0\) for \(\lambda>1\). Then problem (1)—(2) is solvable for any \(u_0\in[0,r]\).

It should be noted that the above-stated conditions of the form (5) are, in a certain sense, close to necessary conditions. As examples, let us consider the equations

\[ u^{(n)}=(-1)^n t^{(n-1)\lambda-n}(1+|\ln t|)^\nu(1+|u^{(n-1)}|)^\lambda u; \tag{16} \]

\[ u^{(n)}=(-1)^n t^{(n-1)\mu-n}(1+|\ln t|)^\eta(u+|u^{(n-1)}|)^\mu, \tag{17} \]

where \(\lambda\le1\), \(\mu>1\). It follows directly from Corollary 2 of Theorem 1 that, if \(\nu<\lambda-1\) and \(\eta\le\mu-1\), then, for any \(u_0\ge0\), equations (16) and (17) have solutions satisfying conditions (2); on the other hand, one can show that if \(\nu\ge\lambda-1\), then problem (16)—(2) has no solution for any \(u_0>0\), while if \(\eta>\mu-1\), then problem (17)—(2) has no solution for sufficiently large positive \(u_0\).

Theorem 2. If \(f(t,x_1,\ldots,x_n)\in K_t(0,a)\) for any \(a\in(0,+\infty)\) and conditions (3) are satisfied, then there exists a positive number \(r\) such that problem (1)—(2) is solvable for any \(u_0\in[0,r]\).

In conclusion we give two theorems on the behavior of solutions of problem (1)—(2) as \(t \to +\infty\).

Theorem 3. Suppose that conditions (3) hold, \(a>0\), \(r>0\), \(\varepsilon>0\), and in the domain \(t \in [a,+\infty)\), \(0 \le x_1 \le r\), \(0 \le (-1)^{k-1}x_k \le \varepsilon\) \((k=2,\ldots,n)\) we have
\[ \left| f(t,x_1,\ldots,x_n) \right| \ge \sigma_1(t,x_1)+\sigma_2(t,x_1)|x_k|, \]
where \(\sigma_k(t,x_1)\) \((k=1,2)\) are functions nondecreasing in \(x_1\), \(\sigma_k(t,x_1)\in K_t(0,\alpha)\) for every \(\alpha\in(a,+\infty)\) \((k=1,2)\), and for every \(x_1\in(0,r]\)
\[ \lim_{s\to+\infty} \int_a^s t^{\,n-2} \left\{ \int_t^s \sigma_1(\tau,x_1) \exp\left[ \int_t^\tau \sigma_2(\xi,x_1)\,d\xi \right]d\tau \right\}dt =+\infty. \]

Then, whatever \(u_0\in[0,r]\), every solution \(u(t)\) of problem (1)—(2) satisfies the conditions
\[ \lim_{t\to+\infty} t^k u^{(k)}(t)=0 \qquad (k=0,1,\ldots,n-1). \]

Theorem 4. Suppose that conditions (3) hold, \(r>0\), and, for any \(x>0\), \(a>0\), the function \(f(t,x,0,\ldots,0)\) is different from zero on a set of positive measure of the interval \([a,+\infty)\), and for sufficiently large \(t\), in the domain (4), we have
\[ \left| f(t,x_1,\ldots,x_n) \right| \ge \delta t^{(k-1)\lambda-n}|x_k|^\lambda, \]
where \(1\le k\le n\), \(\delta>0\), \(0<\lambda<1\). Then, whatever \(u_0\in[0,r]\), for every solution \(u(t)\) of problem (1)—(2) there exists a number \(t_0\) such that
\[ u(t)\equiv 0 \quad \text{for } t\ge t_0. \]

Tbilisi State University

Received
6 XII 1967

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