UDC 517.948.32

COMPARISON THEOREMS FOR VOLTERRA EQUATIONS WITH RETARDED ARGUMENT

A. I. LOGUNOV, Z. B. TSALYUK

Comparison theorems for solutions, as is well known, are widely used in the investigation of various questions in the qualitative theory of equations, for example questions of uniqueness, stability, and the asymptotic behavior of solutions. As a rule, in the study of the questions mentioned, sufficiently well-known methods are used (see, for example, [1, 2, 3, 6, 8, 12, 13, 15]). By analogous methods, on the basis of comparison theorems for solutions, it is expedient to investigate uniqueness, stability, and the asymptotic behavior of solutions of equations with retarded argument (see, for example, [4, 5, 7, 9, 10, 11, 14, 16]).

The present note is devoted to comparison theorems for solutions of equations with retarded argument and is a further development and generalization of the results of [1] and [9]. Below, on the basis of the results of [1], conditions are established for the existence of upper and lower solutions of a system of Volterra integral equations with retarded argument, and a theorem is given on comparison of solutions of two equations with different retardations. This theorem generalizes, in particular, the theorems on integral inequalities in [1, 5, 7, 14].

For brevity of exposition we shall restrict ourselves to the consideration of an equation with one retardation, although the corresponding assertions are valid for equations with any finite number of retardations (cf. [5, 7, 9]).

1. Consider the system

\[ x(t)=\int_a^t K(t,s,x(g(s)))\,ds+f(t)\qquad (t\ge a), \]

\[ x(t)=\varphi(t)\qquad (t\le a). \tag{1} \]

Suppose that:

1) the vector-function
\[ K(t,s,x)=\{K^1(t,s,x^1,\ldots,x^n),\ldots,K^n(t,s,x^1,\ldots,x^n)\} \]
satisfies in the domain \(a\le s\le t<b\), \(\|x\|<c\) \((b,c<\infty)\) the Carathéodory conditions [1];

2) for all \(t\) and almost all \(s\) the vector-function \(K(t,s,x)\) is nondecreasing in \(x^*\);

3) \(g(t)\) is continuous and \(g(t)\le t\) for \(t\in [a,b)\);

*) As in [1], the inequality \(x_1\ge x_2\) \((x_1>x_2)\) between vectors \(x_i=\{x_i^1,\ldots,x_i^n\}\) means that \(x_1^j\ge x_2^j\) \((x_1^j>x_2^j)\) \((j=1,2,\ldots,n)\).

4) \(f(t)=\{f^1(t),\ldots,f^n(t)\}\) and \(\varphi(t)=\{\varphi^1(t),\ldots,\varphi^n(t)\}\) are continuous vector functions. We consider only continuous solutions of system (1). Accordingly, we assume

\[ f(a)=\varphi(a)\quad \text{and}\quad \|\varphi(g(a))\|<c. \]

A solution \(x(t)\) of system (1), defined for \(t\leq t_0\), is called noncontinuable if for every \(\varepsilon>0\) there does not exist a solution \(y(t)\), defined for \(t\leq t_0+\varepsilon\), such that \(y(t)=x(t)\), \(t\leq t_0\).

A solution \(x(t)\), defined for \(t<t_0\), is called an upper (lower) solution if \(y(t)\leq x(t)\) \((y(t)\geq x(t))\) for \(t<\min\{t_0,t_1\}\), whatever solution \(y(t)\) defined for \(t<t_1\) may be (cf. [1]).

Theorem 1. There exists a noncontinuable solution \(u(t)\) of system (1) such that \(u(t)\) is an upper (lower) solution.

Proof. Put

\[ f_0(t)= \begin{cases} f(g(t)), & \text{if } g(t)\geq a,\\ \varphi(g(t)), & \text{if } g(t)\leq a; \end{cases} \tag{2} \]

\[ Q(t,s,x)= \begin{cases} 0, & \text{if } \max\{a,g(t)\}<s\leq t<b,\ \|x\|<c,\\ K(g(t),s,x), & \text{if } g(t)\geq a \text{ and } a\leq s\leq g(t)\leq\\ & \leq t<b,\ \|x\|<c. \end{cases} \]

By virtue of Theorem 1 of [1], the system

\[ y(t)=\int_a^t Q(t,s,y(s))\,ds+f_0(t) \tag{3} \]

has a noncontinuable solution \(y(t)\), defined on \([a,d)\), such that \(y(t)\) is an upper (lower) solution on \([a,d)\).

Put

\[ u(t)= \begin{cases} \varphi(t), & \text{if } t<a,\\ \displaystyle \int_a^t K(t,s,y(s))\,ds+f(t), & \text{if } t\in[a,d). \end{cases} \tag{4} \]

We shall show that \(u(g(t))=y(t)\) for \(t\in[a,d)\). Hence, according to (4), it will follow that the continuous vector function \(u(t)\) is a solution of system (1). For \(g(t)\geq a\) we have

\[ u(g(t))=\int_a^{g(t)} K(g(t),s,y(s))\,ds+f_0(t)=\int_a^t Q(t,s,y(s))\,ds+ \]

\[ +f_0(t)=y(t). \]

If \(g(t)<a\), then \(Q(t,s,x)\equiv0\) for \(a\leq s\leq t\) and, consequently, \(y(t)=f_0(t)=\varphi(g(t))=u(g(t))\). Let us show that \(u(t)\) is an upper solution of system (1). Indeed, let \(x(t)\) be a solution of system (1) defined for \(t\leq d_1\). Then, as is not difficult to see, the vector function \(x(g(t))\) satisfies system (3) on \([a,d_1)\), and, consequently, \(x(g(t))\leq y(t)\) for \(t\in[a,\min\{d_0,d_1\}]\). By the monotonicity of \(K(t,s,x)\), it follows from this that, for \(t\in[a,\min\{d_0,d\}]\),

\[ u(t)=\int_a^t K(t,s,y(s))\,ds+f(t)\ge \]

\[ \ge \int_a^t K(t,s,x(g(s)))\,ds+f(t)=x(t). \]

Finally, let us show that \(u(t)\) is a noncontinuable solution. Indeed, if the solution \(x(t)\) of system (1), defined for \(t<d+\varepsilon\), \(\varepsilon>0\), coincides for \(t<d\) with \(u(t)\), then \(y(t)\) satisfies system (3) on \([a,d+\varepsilon)\) and coincides with \(y(t)=u(g(t))\) for \(t\in [a,d)\). This contradicts the noncontinuability of \(y(t)\).

1. Consider the system

\[ x(t)=\int_a^t G(t,s,x(h(s)))\,ds+r(t)\quad (t\ge a), \tag{5} \]

\[ x(t)=\rho(t)\quad (t\le a). \]

Assume that this system satisfies conditions 1, 3, and 4 (monotonicity of \(G\) is not assumed). Without loss of generality, we shall suppose that the solutions of systems (1) and (5) considered below are defined on \([a,b)\).

Put

\[ r_0(t)= \begin{cases} r(h(t)), & \text{if } h(t)\ge a,\\ \rho(h(t)), & \text{if } h(t)\le a \end{cases} \]

and define \(f_0(t)\) by equality (2).

Theorem 2. Let, for any continuous vector-function \(x(t)\) such that \(\|x(h(t))\|<c\) for \(t\in [a,b)\), the following inequalities hold for \(t\in [a,b)\):

\[ \int_a^{\max\{a,g(t)\}} K(g(t),s,x(h(s)))\,ds - \int_a^{\max\{a,h(t)\}} G(h(t),s,x(h(s)))\,ds + f_0(t)-r_0(t)\ge 0\quad (\le 0), \]

\[ \int_a^t \{K(t,s,x(h(s)))-G(t,s,x(h(s)))\}\,ds+f(t)-r(t)\ge 0\quad (\le 0). \]

Then, whatever the solution \(v(t)\) of system (5) on \([a,b)\) may be, the estimates
\[ v(t)\le u(t),\qquad v(h(t))\le u(g(t)), \]
\[ \bigl(v(t)\ge u(t),\qquad v(h(t))\ge u(g(t))\bigr) \]
are valid, where \(u(t)\) is the upper (lower) solution of system (1).

Proof. Since

\[ v(h(t))-\int_a^t Q(t,s,v(h(s)))\,ds-f_0(t)= \]

\[ = \int_a^{\max\{a,h(t)\}} G(h(t),s,v(h(s)))\,ds - \int_a^{\max\{a,g(t)\}} K(g(t),s,v(h(s)))\,ds + \]

\[ +r_0(t)-f_0(t)\le 0\quad (\ge 0), \]

then, by the theorem on integral inequalities [1], we obtain that \(v(h(t)) \leq y(t)\) \((v(h(t)) \geq y(t))\) for \(t \in [a,b)\), where \(y(t)\) is the upper (lower) solution of system (3). But in the proof of Theorem 1 it was shown that \(y(t)=u(g(t))\). Hence
\(v(h(t)) \leq u(g(t))\) \((v(h(t)) \geq u(g(t)))\) for \(t \in [a,b)\). Further, for \(t \in [a,b)\), by virtue of the monotonicity of \(K\), we have

\[ v(t)=\int_a^t G(t,s,v(h(s)))\,ds+r(t) \leq \int_a^t K(t,s,v(h(s)))\,ds+ \]

\[ +\,f(t)\leq \int_a^t K(t,s,u(g(s)))\,ds+f(t)=u(t). \]

The theorem is proved.

It is easy to see that from Theorem 2 there follows immediately the following generalization of the assertions of the works [1, 5, 7, 9, 14].

Corollary 1. If a continuous vector-function \(z(t)\) \((\|z(g(t))\|<c,\ t\in[a,b))\) satisfies the integral inequality

\[ z(t)-\int_a^t K(t,s,z(g(s)))\,ds-f(t)\leq 0 \quad (\geq 0),\quad t\in[a,b), \]

\[ z(t)\leq \varphi(t)\quad (z(t)\geq \varphi(t)),\quad t\leq a, \]

then \(z(t)\leq u(t)\) \((z(t)\geq u(t))\) for \(t\in[a,b)\), where \(u(t)\) is the upper (lower) solution of system (1).

Corollary 2. Suppose that for every continuous solution \(v(t)\) of system (5), for \(t\in[a,b)\) the inequalities hold:

\[ 1)\quad \int_a^t \{K(t,s,v(h(s)))-G(t,s,v(h(s)))\}\,ds+f(t)-r(t)\geq 0 \quad (\leq 0); \]

\[ 2)\quad \int_a^{\max\{a,g(t)\}} G(g(t),s,v(h(s)))\,ds - \int_a^{\max\{a,h(t)\}} G(h(t),s,v(h(s)))\,ds+ \]

\[ +\,r_1(t)-r_0(t)\geq 0 \quad (\leq 0), \]

where

\[ r_1(t)= \begin{cases} r(g(t)), & \text{if } g(t)\geq a,\\ \rho(g(t)), & \text{if } g(t)\leq a. \end{cases} \]

Then for \(t\in[a,b)\) the estimates \(u(t)\geq v(t)\) \((v(t)\geq u(t))\) hold, where \(u(t)\) is the upper (lower) solution of system (1).

Proof. By virtue of the second of the inequalities, for any solution \(v(t)\) of system (5), for \(t\in[a,b)\) we have

\[ v(g(t))= \int_a^{\max\{a,g(t)\}} G(g(t),s,v(h(s)))\,ds+r_1(t)\geq \]

\[ \geq \int_a^{\max\{a,h(t)\}} G(h(t),s,v(h(s)))\,ds+r_0(t)=v(h(t)). \]

Hence, by virtue of the first of the inequalities and the monotonicity of \(K\), it follows that for \(t\in[a,b)\) the integral inequality holds

\[ v(t)-\int_a^t K(t,s,v(g(s)))\,ds - f(t) \leq v(t) - \]

\[ -\int_a^t K(t,s,v(h(s)))\,ds - f(t) \leq v(t) - \int_a^t G(t,s,v(h(s)))\,ds - r(t)=0. \]

The validity of the assertion now follows from Corollary 1.

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Received by the editors
May 27, 1965

Udmurt State University