UDC 517.911

ON DIFFERENTIAL EQUATIONS AND INEQUALITIES WITH DISCONTINUOUS RIGHT-HAND SIDES. II

V. M. MATROSOV

In the present paper we study differential inequalities of the Chaplygin type [1, 2]. For strong right-sided solutions, \(K\)-solutions, and quasi-solutions with discontinuous comparison functions, assertions are proved which generalize the theorems of Vazhevsky [3], Szarski [4], Olech and Opial [5], and others ([6, 7]). For generalized solutions, corresponding results are obtained which modify the theorems on inequalities of V. M. Alekseev [8], A. I. Perov [9], Li Moon Su and A. B. Samarov [10].

The paper is a continuation of [11]; here all the notation, definitions, and assumptions introduced in [11] are used.

Consider the Cauchy problem

\[ \frac{dy}{dt}=f(y,t),\qquad y(t_0)=y_0 \tag{1.1} \]

on some interval \([t_0,\tau)\), \(y=\{y^1,\ldots,y^k\}\in R_k\). The vector-function \(f(y,t)\) is measurable in the domain \(A\) and bounded in every closed bounded domain \(\bar B\subset A\) by a number \(\varphi_B\) (in norm); \(\{y_0,t_0\}\in A\) are fixed.

In what follows it is also assumed that \(f(y,t)\) is superpositionally measurable ([14], p. 361) and satisfies the following condition of Vazhevsky [3]: each component \(f^s(y^1,\ldots,y^{s-1},y^s,y^{s+1},\ldots,y^k,t)\) is nondecreasing in the off-diagonal variables \(y^1,\ldots,y^{s-1},y^{s+1},\ldots,y^k\), i.e. \(f^s(u,t)\le f^s(y,t)\) when \(u^s=y^s\), \(u^\nu\le y^\nu\) \((\nu\ne s;\ \nu,s=1,\ldots,k)\) (quasimonotonicity in the terminology of [7]). The functions \(m_y f(y,t)\), \(M_y f(y,t)\), \(\underline f_y(y,t)\), \(\bar f_y(y,t)\), \(\underline f(y,t)\), \(\bar f(y,t)\), \(\underline f^{*}(y,t)\), \(\bar f^{*}(y,t)\), \(f^\nu(y,t)\) also satisfy these conditions.

In [11] various definitions of solutions of problem (1.1) were introduced, and the conditions for their existence and their classification were studied*). In addition, in the present paper the concept of an upper solution of problem (1.1) (of some type) on the interval \([t_0,\tau)\) is used (cf. [2–10]): a solution \(\bar y(t)\) of problem (1.1) on \([t_0,\tau)\) (of some type) such that, for every solution \(y(t)\) of this problem (of the same type) on any interval \([t_0,\tau_1]\subseteq [t_0,\tau)\), one has \(y(t)\le \bar y(t)\).

DIFFERENTIAL INEQUALITIES FOR COMPARISON WITH SOLUTIONS AND QUASI-SOLUTIONS

A measurable vector-function \(u(t)\) is called a comparison function if \(\{u(t),t\}\in A\), it is upper right-semicontinuous and lower left-semicontinuous on \([t_0,\tau)\), i.e.

\[ \bar u_+(t)=\lim_{\delta\to 0}\sup_{0<t'-t<\delta} u(t')\le u(t)\le \underline u_-(t)= \]

\[ =\lim_{\delta\to 0}\inf_{0<t-t'<\delta} u(t'),\qquad t,t'\in [t_0,\tau), \tag{2.1} \]

*) In the figure in [11] (p. 406) errors were made: it should be \(0\mathrm{IV}\subseteq 0\mathrm{III}\), \(\mathrm{II}0\mathrm{IV}\subseteq \mathrm{II}0\mathrm{III}\) (whereas inclusions of the opposite sense were printed).

and satisfies the differential inequality

\[ \overline{D}_{+}u(t)\leqslant f(u(t),t) \tag{2.2} \]

at least for almost all \(t\in [t_0,\tau)\), and also the initial condition
\(u(t_0)\leqslant y_0\).

The comparison function \(u(t)\) is called, following [8], absolutely upper semicontinuous on the interval \([t_0,\tau']\), if for every \(\varepsilon\) there is a \(\delta\) such that for any finite collection of nonintersecting intervals \([t_i,\tau_i]\)
\((t_0\leqslant t_i<\tau_i\leqslant \tau')\) with total length
\(\sum_i(\tau_i-t_i)<\delta\), one has
\(\sum_i (u^s(\tau_i)-u^s(t_i))<\varepsilon\) \((s=1,\ldots,k)\).

Lemma 2.1. Let \(u(t)\) be a comparison function, and let \(y^s(t)\) be continuous on \([t_0,\tau)\). If, for some \(\tau_1,\tau_2\in [t_0,\tau)\), \(\tau_1\leqslant \tau_2\), the relations
\(u^s(\tau_1)\leqslant y^s(\tau_1)\), \(y^s(\tau_2)\leqslant u^s(\tau_2)\) hold, then there exists \(\tau_3\in[\tau_1,\tau_2]\) such that
\(u^s(\tau_3)=y^s(\tau_3)\). If
\(\overline{D}_{+}[u^s(t)-y^s(t)]\leqslant 0\) for
\(t\in[\tau',\tau'']\subset [t_0,\tau)\), then
\(u^s(\tau'')-y^s(\tau'')\leqslant u^s(\tau')-y^s(\tau')\).

Proof. Denote

\[ \tau_3=\inf [t\in[\tau_1,\tau_2]:u^s(t)\geqslant y^s(t)]. \]

It is clear that this number exists on the interval \([\tau_1,\tau_2]\), and, if one assumes
\(u^s(\tau_3)\ne y^s(\tau_3)\), then we obtain
\(\max [u^s(\tau_3),\overline{u}^{\,s}_{+}(\tau_3)]\geqslant y^s(\tau_3)\),
\(\underline{u}^{\,s}_{-}(\tau_3)\leqslant \underline{u}^{\,s}_{-}(\tau_3)\leqslant y^s(\tau_3)\)
(for \(\tau_3=\tau_1\) only the first inequality), which contradicts (2.1) in this case.

The contradiction proves the first part of the lemma. The second part of the lemma in the case of continuous \(u(t)\) is known [12].

Consider \(u(t)\) satisfying conditions (2.1). Suppose, to the contrary, that
\(v^s(\tau'')>v^s(\tau')\), where \(v^s(t)\equiv u^s(t)-y^s(t)\) satisfies (2.1). Then
\(v^s(\tau'')-v^s(\tau')-\varepsilon(\tau''-\tau')>0\) for sufficiently small \(\varepsilon\). But
\(\overline{D}_{+}v^s(\tau')\leqslant 0\), and therefore the function
\(v^s(t)-v^s(\tau')-\varepsilon(t-\tau')<0\) for \(t\) sufficiently close to \(\tau'\); and, by the preceding, there exists \(\tau'''\in(\tau',\tau'')\) such that
\(v^s(\tau''')-v^s(\tau')=\varepsilon(\tau'''-\tau')\). Denote

\[ \tau^*=\sup[\tau''':v^s(\tau''')-v^s(\tau')=\varepsilon(\tau'''-\tau')]; \quad \tau^*<\tau'' \quad\text{(by (2.1))}. \]

Then
\(v^s(t)-v^s(\tau')-\varepsilon(t-\tau')>0\) for
\(t\in(\tau^*,\tau'']\), and, consequently,
\(\overline{D}_{+}v^s(\tau^*)\geqslant \varepsilon>0\), which contradicts the condition.

The lemma is proved.

Lemma 2.2. Let the Cauchy problem (1.1) have on \([t_0,\tau)\) a strong right-hand solution \(y(t)\), or a \(K\)-solution \(y(t)\), for which
\(\underline{D}_{+}y(t)\geqslant f(y(t),t)\) for \(t\in[t_0,\tau)\), and let the comparison function \(u(t)\) satisfy on \([t_0,\tau)\) the differential inequality

\[ \overline{D}_{+}u(t)<f(u(t),t). \tag{2.3} \]

Then \(u(t)\leqslant y(t)\) for \(t\in[t_0,\tau)\).

Proof. For sufficiently small \(\Delta\tau_0>0\) we have
\(u(t)<y(t)\) for \(t\in(t_0,t_0+\Delta\tau_0)\). Indeed, for those \(s\) for which
\(u^s(t_0)<y_0^s\), this follows from the upper-right semicontinuity of the function
\(u(t)-y(t)\), and for those \(s\) for which \(u^s(t_0)=y^s(t_0)\), it follows from
(2.3), the Wazewski condition, and \(u(t_0)\leqslant y_0\), whence

\[ \overline D_{+}u^{s}(t_0)-D_{-}y^{s}(t_0) =\overline D_{+}\bigl[u^{s}(t_0)-y^{s}(t_0)\bigr] <f^{s}(u(t_0),t_0)-f^{s}(y_0,t_0)\leq 0 \]

and, consequently, there exists no sequence \(\tau_i\to t_0+0\) for which it would be that \(u^{s}(\tau_i)>y^{s}(\tau_i)\).

Suppose, to the contrary, that at some moments \(t\in[t_0+\Delta\tau_0,\tau)\) the condition \(u(t)\leq y(t)\) is violated. Denote by \(\tau_*\) the lower bound of such time moments \((\tau_*\in[t_0+\Delta\tau_0,\tau))\). Then \(u(\tau_i)\leq y(\tau_i)\) for \(\tau_i\in[t_0,\tau_*)\), and, taking (2.1) into account,
\(u(\tau_*)\leq u_{-}(\tau_*)\leq \bar u_{-}(\tau_*)\leq y(\tau_*)\), while for some \(s\) there exists a sequence \(t_i\to\tau_*+0\), as \(i\to\infty\), for which \(u^{s}(t_i)>y^{s}(t_i)\). Taking the moment \(\tau_*\) as the initial one, one can carry out the preceding argument and prove the existence of such a \(\Delta\tau_*>0\) that \(u(t)<y(t)\) for \(t\in(\tau_*,\tau_*+\Delta\tau_*)\). This contradicts the convergence of the sequence \(\{t_i\}\) to \(\tau_*+0\).

Example 3 [11] with the comparison function

\[ u(t)= \begin{cases} \dfrac{1}{2}(t-1), & \text{for } t\in[0,1),\\[6pt] -(t-1), & \text{for } t\in[1,+\infty), \end{cases} \]

satisfying condition (2.3) on \([0,+\infty)\) and the initial condition \(u(t_0)=u(0)=-\dfrac12<y_0=0\), but with \(\tau_*=1\), for which \(u(\tau_*)=u(1)=0=y(1)\), shows that the relation \(u(t)\leq y(t)\) in the assertion of Lemma 2.2 cannot in general be replaced by the strict inequality \(u(t)<y(t)\).

Theorem 2.1. Let \(f(y,t)\) be the limit, converging everywhere in \(A\) uniformly with respect to \(y\) for almost all \(t\in[t_0,\tau)\), from above strictly, of a sequence \(\{f_\nu(y,t)\}\) \((f_\nu(y,t)\to f(y,t)+0,\ f(y,t)<f_\nu(y,t)\) for \(\{y,t\}\in A)\) of functions measurable in \(A\) and uniformly bounded in every bounded closed domain \(\overline B\subset A\), satisfying the existence conditions on \([t_0,\tau)\) for \(K\)-solutions \(y_\nu(t)\) of the Cauchy problems

\[ \frac{dy_\nu}{dt}=f_\nu(y_\nu,t),\qquad y_\nu(t_0)=y_{\nu0}, \tag{2.4} \]

for which \(y_0\leq y_{\nu0}\), \(y_{\nu0}\to y_0\), \(f(y_\nu(t),t)\leq D_{+}y_\nu(t)\) for all \(t\in[t_0,\tau)\) (for example, strong right-hand solutions), and moreover, for any \(\tau_1\in(t_0,\tau)\) there exists such a \(\overline B\) that \(\{y_\nu(t),t\}\in\overline B\) for \(t\in[t_0,\tau_1]\) \((\nu=1,2,\ldots)\). Then there exists a quasi-solution \(\tilde y(t)\) of problem (1.1) on \([t_0,\tau)\) such that, if the comparison function \(u(t)\) satisfies (2.2) everywhere on \([t_0,\tau)\), then \(u(t)\leq \tilde y(t)\) for \(t\in[t_0,\tau)\). If, in addition, one of the following conditions is fulfilled: a) \(f_{\nu+1}(y,t)<f_\nu(y,t)\) for \(\{y,t\}\in A\) \((\nu=1,2,\ldots)\), the function \(f(y,t)\) is continuous in \(y\) from the right for almost all \(t\in[t_0,\tau)\); b) \(f(y,t)\) satisfies the Carathéodory conditions, then \(\tilde y(t)\) is a \(K\)-solution (an upper \(K\)-solution in the case of upper semicontinuity of \(f(y,t)\) from above with respect to \(\{y,t\}\in A\), from the right with respect to \(t\)).

Proof. According to Theorem 1.2 [11], there exists a quasi-solution \(y(t)\) of problem (1.1) on \([t_0,\tau)\), which, recalling the proof of this theorem, can be represented as the limit of some subsequence \(\{y_i(t)\}\subset\{y_\nu(t)\}\) of \(K\)-solutions of problems (2.4) on \([t_0,\tau)\), converging

*) One-sided Carathéodory condition: \(f(y,t)=\lim\limits_{y'\to y+0} f(y',t)\).

uniformly on any segment \([t_0,\tau_1]\subset [t_0,\tau)\). The comparison function \(u(t)\) satisfies the conditions

\[ \overline{D}_{+}u(t)\leq f(u(t),t)< f_i(u(t),t) \quad \text{for } \quad t\in [t_0,\tau), \]

therefore, by Lemma 2.2, \(u(t)\leq y_i(t)\) for \(t\in [t_0,\tau)\) \((i=1,2,\ldots)\). Hence, passing to the limit, we obtain \(u(t)\leq \lim_{i\to\infty} y_i(t)=\tilde y(t)\) for \(t\in [t_0,\tau)\).

Under condition a), by Lemma 2.2, \(y_{i+1}(t)\leq y_i(t)\), so that \(y_i(t)\downarrow \tilde y(t)\) on \([t_0,\tau)\), and by the right-continuity of \(f(y,t)\) with respect to \(y\) we have \(f(y_i(t),t)\to f(\tilde y(t),t)\) almost everywhere in \([t_0,\tau)\). Under condition b), the latter relation also obviously holds. Applying now Lemma 1.1 [11], we verify that \(\tilde y(t)\) is a \(K\)-solution.

If \(f(y,t)\) is upper semicontinuous (from the right with respect to \(t\)), then for any other \(K\)-solution \(y(t)\) on \([t_0,\tau_1)\subset [t_0,\tau)\), \(f(y(t),t)\) will be upper semicontinuous from the upper right for \(t\in [t_0,\tau_1)\). Arguing as in the proof of necessity in Lemma 1.2 [11], we obtain

\[ \overline{D}_{+}y(t)\leq f(y(t),t)< f_i(y(t),t) \quad \text{for } \quad t\in [t_0,\tau_1). \]

Hence, applying Lemma 2.2, we have \(y(t)\leq y_i(t)\) for \(t\in [t_0,\tau_1)\), and therefore, passing to the limit, \(y(t)\leq \lim y_i(t)=\tilde y(t)\) as \(i\to\infty\), i.e. \(\tilde y(t)\) is an upper \(K\)-solution.

The theorem proved substantially generalizes one assertion from [6]. It is easy to see from an example that, in the hypotheses of Lemma 2.2 and of the first part of Theorem 2.1, the requirement of the existence of strong right-hand solutions or \(K\)-solutions under the additional condition \(f(y(t),t)\leq \underline{D}_{+}y(t)\) \((f_\nu(y_\nu(t),t)\leq \underline{D}_{+}y_\nu(t))\) on \([t_0,\tau)\) is essential and cannot be replaced by the requirement of the existence simply of \(K\)-solutions, or even of weak right-hand solutions (with the other conditions retained in the same form).

Example 1.

\[ k=1,\qquad f(y)= \begin{cases} -3 & \text{if } y>0,\\ +1 & \text{if } y=0,\\ -2 & \text{if } y<0, \end{cases} \qquad y_0=0,\quad t_0=0. \]

There exists a unique right-hand (even weakened) solution \(y=-2t\) for \(t\in [0,+\infty)\), which is not strong. The comparison function \(u(t)=0\) satisfies a condition of the form (2.3):

\[ \frac{du(t)}{dt}=0<f(u(t))=f(0)=+1 \quad \text{for } \quad t\in [0,+\infty),\quad u(0)=y_0=0. \]

But there does not exist even a quasi-solution \(\tilde y(t)\geq u(t)\). For \(f_\nu(y)=f(y)+\dfrac{1}{\nu}\) \((\nu=1,2,\ldots)\), the conditions of the first part of Theorem 2.1 are satisfied, except for the requirement of the existence of strong right-hand solutions of the systems

\[ \frac{dy_\nu}{dt}=f_\nu(y), \]

which is replaced by the existence of weakened solutions.

If \(f(y)=\{0 \text{ if } y\geq 0;\ -2 \text{ if } y<0\}\), then for \(f_\nu(y)=f(y)+1/\nu\) the conditions of Theorem 2.1 are satisfied and there exists an upper classical solution \(y(t)=0=u(t)\) for \(t\in [t_0,+\infty)\).

Theorem 2.2. Let \(f(y,t)\) be continuous in \(\{y,t\}\in A\) from the right with respect to \(t\). Then there exists an upper strong right-hand solution \(\bar y(t)\) of problem (1.1) on the interval \([t_0,\tau_0)\). If, moreover, the comparison function \(u(t)\) satisfies condition (2.2) for all \(t\in [t_0,\tau_0)\), then \(u(t)\leq \bar y(t)\) for \(t\in [t_0,\tau_0)\).

Proof. Let \(f_\nu(y,t)=f(y,t)+1/\nu\). Each function \(f_\nu\) satisfies the conditions of Theorem 1.1 [11], according to which there exists a strong right-hand solution \(y_\nu(t)\) of the problem

\[ \frac{dy_\nu}{dt}=f_\nu(y_\nu,t),\qquad y_\nu(t_0)=y_0 \]

on some interval, and

\[ \|y_\nu(t)-y_0\|\leq (\varphi_B+1/\nu)(t-t_0) \]

and each solution \(y_\nu(t)\) can be extended up to the boundary \(\bar B\setminus B\) \((K\subset B)\); therefore one may take \(\tau_\nu\geq \tau_0\) for sufficiently large \(\nu\). The convergence \(f_\nu(y,t)\to f(y,t)+0\) is uniform in \(\{y,t\}\in A\); in any \(\bar B\), the \(f_\nu(y,t)\) are uniformly bounded by the number \(\varphi_B+1\). Thus all the conditions of Theorem 2.1 are satisfied, according to which there exists an upper \(K\)-solution \(\bar y(t)\) of problem (1.1) on \([t_0,\tau_0)\), such that \(u(t)\leq \bar y(t)\) for \(t\in [t_0,\tau_0)\). But when \(f(y,t)\) is continuous in \(\{y,t\}\in A\) from the right with respect to \(t\), every \(K\)-solution is a strong right-hand solution of problem (1.1).

The theorem is proved.

Corollary 2.1. Let \(f(y,t)\) be continuous in \(A\). Then on the interval \([t_0,\tau_0)\) there exists an upper classical solution \(\bar y(t)\) of problem (1.1). If, moreover, the comparison function \(u(t)\) satisfies condition (2.2) for all \(t\in [t_0,\tau_0)\), then \(u(t)\leq \bar y(t)\) for \(t\in [t_0,\tau_0)\).

This corollary generalizes the classical theorem on differential inequalities of Wazewski [3] to the case of a discontinuous comparison function satisfying the condition
\[ \underline u_+(t)\leq u(t)\leq \underline u_-(t). \]

From the given functions \(f(y,t)\) and \(u(t)\) we construct, following [9], an auxiliary function \(f_*(y,t)=f(\max[u(t),y],t)\), which will obviously satisfy Wazewski’s condition and be superpositionally measurable in some neighborhood \(A\) of the graph of the comparison function.

Theorem 2.3. Let the auxiliary Cauchy problem

\[ \frac{dx}{dt}=f_*(x,t),\qquad x(t_0)=y_0 \tag{2.5} \]

have on \([t_0,\tau)\) a \(K\)-solution \(x(t)\), with \(\{\max[u(t),x(t)],t\}\in A\) for \(t\in [t_0,\tau)\), and let one of the following conditions be fulfilled: 1) the comparison function \(u(t)\) satisfies the differential inequality (2.2) for all \(t\in [t_0,\tau)\), and the \(K\)-solution \(x(t)\) the differential inequality

\[ f_*(x(t),t)\leq D_+x(t)\quad\text{for } t\in [t_0,\tau) \]

(for example, it is a strong right-hand solution); 2) the comparison function \(u(t)\) is absolutely upper semicontinuous on every segment \([t_0,\tau_1]\subset [t_0,\tau)\). Then \(x(t)\) is a \(K\)-solution of problem (1.1) on \([t_0,\tau)\), and \(u(t)\leq x(t)\) for \(t\in [t_0,\tau)\).

Proof. We first show that \(u(t)\leq x(t)\) for \(t\in [t_0,\tau)\). Suppose the contrary, that for some \(s\) and \(\tau_1\in [t_0,\tau)\) one has

\(x^s(\tau_1)<u^s(\tau_1)\). According to Lemma 2.1, there exist \(\tau'\in[t_0,\tau_1)\) such that \(x^s(\tau')=u^s(\tau')\).

Denote
\[ \tau_2=\sup[\tau'\in[t_0,\tau_1): x^s(\tau')=u^s(\tau')]. \]
In view of (2.1), \(\tau_2<\tau_1\) and \(x^s(\tau_2)=u^s(\tau_2)\). It is also obvious that \(x^s(t)<u^s(t)\) on \((\tau_2,\tau_1]\), since otherwise, by Lemma 2.1, there would be a \(\tau_3\in(\tau_2,\tau_1)\) such that \(x^s(\tau_3)=u^s(\tau_3)\), but this would contradict the definition of \(\tau_2\). Taking all this into account, on \([\tau_2,\tau_1]\) we obtain
\[ f^s(u(t),t)\leq f^s(\max[u^1(t),x^1(t)],\ldots,\max[u^{s-1}(t),x^{s-1}(t)], \]
\[ u^s(t),\max[u^{s+1}(t),x^{s+1}(t)],\ldots,\max[u^k(t),x^k(t)],t)= \]
\[ =f^s(\max[u(t),x(t)],t)=f_*^s(x(t),t). \]

But then, under condition 1), on \([\tau_2,\tau_1]\)
\[ \overline D_+u^s(t)\leq f^s(u(t),t)\leq f_*^s(x(t),t). \tag{2.6} \]
Hence \(\overline D_+u^s(t)\leq D_+x^s(t)\), i.e.
\[ \overline D_+[u^s(t)-x^s(t)]\leq 0 \]
for \(t\in[\tau_2,\tau_1]\), and, according to Lemma 2.1,
\[ u^s(\tau_1)-x^s(\tau_1)\leq u^s(\tau_2)-x^s(\tau_2). \]

Under condition 2), (2.6) holds almost everywhere on \([\tau_2,\tau_1]\), and, applying Lemma 6 from [8], we obtain, taking into account that \(x(t)\) is a \(K\)-solution of problem (2.5), the same result:
\[ u^s(\tau_1)-u^s(\tau_2)\leq \int_{\tau_2}^{\tau_1} f_*^s(x(t),t)\,dt = x^s(\tau_1)-x^s(\tau_2). \]

But the condition thus obtained,
\[ u^s(\tau_1)-x^s(\tau_1)\leq u^s(\tau_2)-x^s(\tau_2)=0, \]
contradicts the assumption made.

Thus, \(u(t)\leq x(t)\) for \(t\in[t_0,\tau)\), and therefore
\[ f_*(x(t),t)=f(x(t),t), \]
i.e. \(x(t)\) is a \(K\)-solution of problem (1.1) on \([t_0,\tau)\), for which \(u(t)\leq x(t)\).

Corollary 2.2. Let the comparison function \(u(t)\) be absolutely upper semicontinuous on any interval \([t_0,\tau_1]\subset[t_0,\tau)\), and let \(f(y,t)\) satisfy the Carathéodory condition and let all \(K\)-solutions of problem (1.1) be continuable to \([t_0,\tau)\). Then there exists an upper \(K\)-solution \(\overline y(t)\) of this problem such that
\[ u(t)\leq \overline y(t)\quad\text{for }t\in[t_0,\tau). \]

This corollary generalizes the theorems of Szarski [4], Olech and Opial [5], and others (see [7]) (proved by the authors for absolutely continuous comparison functions or for discontinuous ones of bounded variation and with nonincreasing singular part).

In Example 1,
\[ f_*(x,t)=f(\max[u(t),x])= \begin{cases} -3, & \text{for } y>0,\\ +1, & \text{for } y\leq 0, \end{cases} \]
the auxiliary Cauchy problem (2.5) has no \(K\)-solution. Despite the fact that the comparison function \(u(t)\equiv0\) is absolutely continuous and everywhere in \([0,+\infty)\) satisfies condition (2.3), while the main Cauchy problem has a \(K\)-solution (even a weakened solution) \(y(t)=-2t\), we have
\[ u(t)=0>y(t)=-2t\quad\text{for }t\in[0,+\infty). \]

This example shows that the requirement of the existence of a \(K\)-solution of the auxiliary Cauchy problem (2.5) in the condition of Theorem 2.3 cannot in general be replaced by the same requirement for the main Cauchy problem (1.1).

DIFFERENTIAL INEQUALITIES FOR COMPARISON WITH GENERALIZED SOLUTIONS

A generalized comparison function of type I, or a 0I-comparison function (respectively, 0II-, 0III-, 0IV-comparison functions), is a measurable vector function \(u(t)\), whose graph lies in \(A\), is upper-right semicontinuous, lower-left semicontinuous on \([t_0,\tau)\), and satisfies the differential inequality

\[ \overline{D}_{+}u(t) \leq M_u f(u(t),t) \tag{2.7} \]

(respectively

\[ \overline{D}_{+}u(t) \leq \overline{f}_u(u(t),t), \tag{2.8} \]

\[ \overline{D}_{+}u(t) \leq \overline{f}(u(t),t), \tag{2.9} \]

\[ \overline{D}_{+}u(t) \leq \overline{f}^{*}(u(t),t)) \tag{2.10} \]

at least for almost all \(t\in [t_0,\tau)\), and also the initial condition
\(u(t_0)\leq y_0\).

Theorem 2.4. Let there exist an upper 0I (respectively, 0II)-solution \(y(t)\) of problem (1.1) on \([t_0,\tau_0)\). If the 0I (respectively, 0II) comparison function \(u(t)\) satisfies one of the conditions: 1) the differential inequality (2.7) (respectively, (2.8)) is fulfilled for all \(t\in [t_0,\tau_0)\); 2) \(u(t)\) is absolutely upper semicontinuous on every interval \([t_0,\tau_1]\subset [t_0,\tau_0)\), \(\tau=\tau_0\), and in case 1) \(M_y f(y,t)\) (respectively, \(\overline{f}_y(y,t)\)) is continuous in \(\{y,t\}\) on \(K\) from the right with respect to \(t\), then \(u(t)\leq y(t)\) for \(t\in [t_0,\tau_0)\).

Proof. We consider 0II-solutions; for 0I-solutions the proof is analogous.

The function \(\overline{f}_y(y,t)\) is upper semicontinuous in \(y\) for every fixed \(t\) in \(K\), satisfies Vazhëvskii’s condition, is measurable in \(t\), and is bounded in \(K\) by the number \(\varphi_B\) in norm \((K\subset B\subset \overline{B}\subset A)\). According to [9, 15, 13], for every \(t\in [t_0,\tau_0]\) there exists a decreasing sequence of functions \(f_i(y,t)\), continuous in \(y\) in \(K\), nondecreasing in each of the off-diagonal variables \(y^s\) inside \(K\), and hence [3] satisfying Vazhëvskii’s condition, measurable in \(t\) for fixed \(y\), and uniformly bounded in \(K\) by the number \(\varphi_B\),

\[ f_i(y,t)=\max_{\{y',t\}\in K} \left[ \overline{f}_y(y',t)-i\sum_{s=1}^{k}|y^{s'}-y^s| \right], \qquad \{y,t\}\in K. \]

The sequence \(\{f_i(y,t)\}\) converges to \(\overline{f}_y(y,t)\) on \(K\):

\[ f_i(y,t)\downarrow \overline{f}_y(y,t) \quad \left(\overline{f}_y(y,t)\leq f_{i+1}(y,t)\leq f_i(y,t)\right). \]

Each function \(f_i(y,t)\) satisfies the Carathéodory condition on \(K\); therefore the auxiliary Cauchy problems

\[ \frac{dy_i}{dt}=f_i(y_i,t),\qquad y_i(t_0)=y_0 \tag{2.11} \]

have \(K\)-solutions \(y_i(t)\) on \([t_0,\tau_0)\). Every \(K\)-solution \(y_i(t)\) of each problem (2.11) is extendable to \([t_0,\tau_0]\), and almost everywhere in \([t_0,\tau_0]\)

\[ \frac{dy_i(t)}{dt}=f_i(y_i(t),t)\leq f_{i-1}(y_i(t),t). \]

Therefore, according to Corollary 2.2, there exists an upper \(K\)-solution \(\bar y_{i-1}(t)\) such that for any \(K\)-solution \(y_i(t)\) one has \(y_i(t) \leq \bar y_{i-1}(t)\) for \(t \in [t_0,\tau_0)\). Among the \(y_i(t)\) there is an upper \(K\)-solution \(\bar y_i(t)\) such that for any \(K\)-solution \(y_{i+1}(t)\) one has \(y_{i+1}(t) \leq \bar y_i(t)\) for \(t \in [t_0,\tau_0)\), and so on.

Consider the decreasing sequence of absolutely continuous functions \(\{\bar y_i(t)\}\) on \([t_0,\tau_0]\). For a 0П-solution \(y(t)\) of problem (1.1) on \([t_0,\tau_0)\),

\[ \frac{dy(t)}{dt} \leq \bar f_y(y(t),t) \leq f_i(y(t),t) \]

for almost all \(t \in [t_0,\tau_0)\) \((i=1,2,\ldots)\); therefore, applying Corollary 2.2 once more, we obtain \(y(t) \leq \bar y_i(t)\) for \(t \in [t_0,\tau_0)\) \((i=1,2,\ldots)\), so that the decreasing sequence \(\{\bar y_i(t)\}\) is bounded below on \([t_0,\tau_0]\). Moreover, it is uniformly bounded and equicontinuous and, by the Arzelà–Ascoli and Dini theorems, converges uniformly to some continuous function \(\bar y(t)\), with \(\bar y(t_0)=y_0\), \(\{\bar y(t),t\}\in K\) for \(t \in [t_0,\tau_0]\), and for any solution \(y(t)\) of problem (1.1), \(y(t)\leq \bar y(t)\) for \(t \in [t_0,\tau_0)\).

Let us show that the limiting function \(\bar y(t)\) is a 0П-solution and, consequently, also an upper 0П-solution of problem (1.1) on \([t_0,\tau_0)\).

By the upper semicontinuity of \(\bar f_y(y,t)\) with respect to \(y\), for any \(t \in [t_0,\tau_0)\) and \(\varepsilon\) there exists a \(\delta\) such that

\[ \bar f_y(y,t)\leq \bar f_y(\bar y(t),t)+\varepsilon,\quad \{y,t\}\in K \quad \text{when}\quad \|y-\bar y(t)\|\leq \delta . \]

Since \(y_i(t)\to \bar y(t)\), for this \(\delta\) there is an index \(i_0>4\varphi_B/\delta\) such that \(\|y_i(t)-\bar y(t)\|\leq \delta/2\) for \(i>i_0\). In what follows we shall consider only \(i>i_0\). Since for \(\|y'-y_i(t)\|>\delta/2\), \(\{y',t\}\in K\), we have

\[ \bar f_y(y',t)-i\sum_{s=1}^{k}|y^s-y_i^s(t)| \leq \varphi_B-i\delta/2 \leq \varphi_B-i_0\delta/2 \leq \]

\[ \leq \varphi_B-2\varphi_B=-\varphi_B\leq \bar f_y(y_i(t),t), \]

then

\[ f_i(y_i(t),t)= \max_{\{y,t\}\in K} \left[ \bar f_y(y,t)-i\sum_{s=1}^{k}|y^s-y_i^s(t)| \right] = \max_{\|y-y_i(t)\|<\delta/2} \left[ \bar f_y(y,t)- \right. \]

\[ \left. -i\sum_{s=1}^{k}|y^s-y_i^s(t)| \right] \leq \max_{\|y-y_i(t)\|<\delta/2}\bar f_y(y,t) < \bar f_y(\bar y(t),t)+\varepsilon, \]

for here

\[ \|y-\bar y(t)\|\leq \|y-y_i(t)\|+\|y_i(t)-\bar y(t)\|<\frac{\delta}{2}+\frac{\delta}{2}=\delta . \]

Hence, also taking into account \(f(y_i(t),t)\leq f_i(y_i(t),t)\), the definition of \(\underline f_y\), and the convergence \(y_i(t)\to \bar y(t)\), we obtain

\[ \underline f_y(\bar y(t),t) \leq \lim_{i\to\infty} f_i(y_i(t),t) \leq \overline{\lim_{i\to\infty}} f_i(y_i(t),t) \leq \bar f_y(\bar y(t),t). \]

We now argue analogously to the proof of Theorem 1.3 [11]. By virtue of the measurability and boundedness of the functions \(f_i(y_i(t),t)\), \(\underline f_y(\bar y(t),t)\), \(\bar f_y(\bar y(t),t)\), they are summable, and for any \(\tau_*, t \in [t_0,\tau_0)\), \(\tau_*<t\), we have

\[ \overline y(t)-\overline y(\tau_*)=\lim_{i\to\infty}\,[y_i(t)-y_i(\tau_*)] =\lim_{i\to\infty}\int_{\tau_*}^{t} f_i(y_i(s),s)\,ds \leq \]

\[ \leq \int_{\tau_*}^{t}\overline{\lim_{i\to\infty}}\, f_i(y_i(s),s)\,ds \]

(and the analogous inequality for the lower limit). Therefore

\[ \int_{\tau_*}^{t} \underline f_y(\overline y(s),s)\,ds \leq \overline y(t)-\overline y(\tau_*) \leq \int_{\tau_*}^{t} \overline f_y(\overline y(s),s)\,ds, \]

whence follow the absolute continuity of \(\overline y(t)\), its differentiability almost everywhere on \([t_0,\tau_0)\), and the relations

\[ \underline f_y(\overline y(t),t)\leq \frac{d\overline y(t)}{dt} \leq \overline f_y(\overline y(t),t) \quad \text{for almost all } t\in [t_0,\tau_0). \]

Thus, \(\overline y(t)\) is an upper OII-solution.

Under condition 1), the differential inequality (2.8) is satisfied everywhere on \([t_0,\tau_0)\), and, taking into account the continuity of \(\overline f_y(y,t)\) with respect to \(\{y,t\}\) on \(K\) from the right with respect to \(t\), by Theorem 2.2 we conclude that there exists an upper strong right-hand solution \(y(t)\) on \([t_0,\tau_0)\) of the Cauchy problem

\[ \frac{dy}{dt}=\overline f_y(y,t),\qquad y(t_0)=y_0, \]

which satisfies the condition \(u(t)\leq y(t)\) for \(t\in [t_0,\tau_0)\). It is clear that this solution \(y(t)\) coincides with the upper OII-solution \(\overline y(t)\) of problem (1.1), and hence \(u(t)\leq \overline y(t)\) for \(t\in [t_0,\tau_0)\).

Under condition 2) the function \(u(t)\) is absolutely upper semicontinuous and almost everywhere in \([t_0,\tau_0)\) satisfies the differential inequalities

\[ \overline D_{+}u(t)\leq \overline f_u(u(t),t)\leq f_i(u(t),t) \quad (i=i_0,\ i_0+1,\ldots) \]

and, by Corollary 2.2, \(u(t)\leq \overline y_i(t)\) for \(t\in [t_0,\tau_0)\). Passing to the limit, we also obtain \(u(t)\leq \overline y(t)\) for \(t\in [t_0,\tau_0)\).

In the case of OI-solutions, Theorem 2.4 generalizes the results of V. M. Alekseev [8], Li Moon Su and A. B. Samarskii [10].

Theorem 2.5. There exists an upper OIII (respectively OIV)-solution \(\overline y(t)\) of problem (1.1) on \([t_0,\tau_0)\). If the OIII (respectively OIV) comparison function \(u(t)\) satisfies one of the conditions: 1) the differential inequality (2.9) (respectively (2.10)) holds for all \(t\in [t_0,\tau_0)\); 2) \(u(t)\) is absolutely upper semicontinuous on every segment \([t_0,\tau_1]\subset [t_0,\tau_0)\), \(\tau=\tau_0\), then \(u(t)\leq \overline y(t)\) for \(t\in [t_0,\tau_0)\).

Proof. Consider the relations (1.17) (respectively (1.18)) as an equation in contingencies (or an equation with multivalued right-hand side in the sense of [9]). The functions \(\underline f(y,t)\), \(\overline f(y,t)\) (respectively \(f^{*}(y,t)\), \(\overline f^{*}(y,t)\)) are semicontinuous in \(A\), satisfy Wazewski’s conditions, and are bounded in any domain \(B\) \((B\subset A)\) containing the cylinder \(K\). By the corresponding theorem from [9], there exists an upper solution \(\overline y(t)\) of this equation on \([t_0,\tau_0]\). It will be an upper Nagumo solution (respectively Rosenthal solution), i.e., an upper OIII (respectively OIV)-solution

of problem (1.1) on \([t_0,\tau_0)\). In this case, according to [9], there exists a sequence of functions \(f_i(y,t)\), continuous on \(K\), satisfying the Wazewski condition, and \(f_i(y,t)\to \bar f(y,t)\) (respectively, \(f_i(y,t)\to \bar f^{*}(y,t)\)) for \(\{y,t\}\in K\), while the sequence \(\bar y_i(t)\) of upper classical solutions on \([t_0,\tau_0]\) of the Cauchy problems

\[ \frac{dy_i}{dt}=f_i(y_i,t),\qquad y_i(t_0)=y_0 \]

decreases and converges to \(\bar y(t)\) uniformly on \([t_0,\tau_0]\) (here it is taken into account that the functions \(f_i(y,t)\) can be extended to \(R_k\times [t_0,\tau_0]\) with preservation of their properties and boundedness).

It is clear that \(\bar f(u(t),t)\leq f_i(u(t),t)\) (respectively, \(\bar f^{*}(u(t),t)\leq f_i(u(t),t)\)) for \(t\in [t_0,\tau_0]\); therefore, at least almost everywhere in \([t_0,\tau_0)\) we have \(\overline D_{+}u(t)\leq f_i(u(t),t)\). Under condition 1) this differential inequality holds for all \(t\in [t_0,\tau_0)\), and, applying Corollary 2.1, we have \(u(t)\leq y_i(t)\) for \(t\in [t_0,\tau_0)\). Under condition 2) the analogous conclusion follows from Corollary 2.2. Passing to the limit in the obtained relation, we find

\[ u(t)\leq \lim_{i\to\infty}y_i(t)=\bar y(t). \]

Theorem 2.5 modifies the results of A. I. Perov [9].

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Received by the editors
June 7, 1966

Kazan Aviation Institute