Properties of solutions of Riccati equation - Springer Link

c Allerton Press, Inc., 2007. ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2007, Vol. 42, No. 4, pp. 184–197. c G. A. Grigoryan, 2007, published in Izvestiya NAN Armenii. Matematika, 2007, No. 4, pp. 13–31. Original Russian Text

DIFFERENTIAL EQUATIONS

Properties of Solutions of Riccati Equation* G. A. Grigoryan1* 1

Institute of Mathematics, National Academy of Sciences of Armenia Received February 13 2007

Abstract—The paper studies some properties of solutions of the Riccati equation y (t) + a(t)y 2 (t) + b(t)y(t) + c(t) = 0 on a semiaxis [t0 , +∞) for different types of initial value sets. Two types of solutions are singled out: normal, that are in a sense stable, and extremal, that are non-stable in the Lyapunov sense. Relations expressing the extremal solutions by means of a given normal solution in quadratures and elementary functions are obtained and some relations between solutions the extendable to [t0 , +∞) are derived. MSC2000 numbers : 34A12, 34D20 DOI: 10.3103/S1068362307040024 Key words: Regular solution; normal solution; regular initial value; characteristic function; acceptable parameter; asymptotic closeness of solutions; Vieta formulas.

1. INTRODUCTION Consider the Riccati equation y (t) + a(t)y 2 (t) + b(t)y(t) + c(t) = 0,

(1.1)

where t ∈ [t0 , +∞) ⊂ (l, +∞) while a(t), b(t) and c(t) are continuous functions defined in (l, +∞). Formulas t a(τ )y(τ )dτ and ψ(t) = y(t)φ(t) (1.2) φ(t) = λ0 exp t0

(see [1], pp. 153-154), where λ0 is a constant, connect the equation (1.1) and the system ⎧ ⎨ φ (t) = a(t)ψ(t), ⎩ ψ (t) = −c(t)φ(t) − b(t)φ(t).

(1.3)

For a(t) ≡ 1 the latter system is equivalent to the equation φ (t) + b(t)φ (t) + c(t)φ(t) = 0,

t ∈ [t0 , +∞).

(1.4)

A solution y(t) of the equation (1.1), that satisfies the initial condition y(t0 ) = y(0) , has either a continuation to [t0 , +∞) or a vertical asymptote at a point t1 > t0 . In the first case, y(t) is called t0 regular solution and y(0) is called a regular initial value. I. M. Sobol (see [2], [3]) has successfully applied the properties of t0 -regular solutions of the equation y (t) + y 2 (t) + c(t) = 0,

t ∈ [t0 , +∞),

(1.5)

to the study of asymptotical properties of solutions of the corresponding non-oscillating equation φ (t) + c(t)φ(t) = 0, *

E-mail: [email protected]

184

t ∈ [t0 , +∞).

(1.6)

PROPERTIES OF SOLUTIONS OF RICCATI EQUATION

185

It is known [3] that the set of t0 -regular initial values of a solution of (1.5) if nonempty, is of the form [y(0) , +∞). Below we show, that in the general case the set of t0 -regular initial values of the equation (1.1) if nonempty, is of one of the following types: (α, β), [α, β), (α, β], [α, β], and if α(β) is not a t0 regular initial value, then it can become −∞(+∞), and the interval [α, β] can reduce to a point. A t0 -regular solution with initial value from the interior (or the boundary) of the set of t0 -regular initial values we call normal (or extremal) solution. Theorem 2 of [3] implies that to any two different normal solutions y1 (t) and y2 (t) of the equation (1.5) correspond two solutions φ1 (t) and φ2 (t) of the equation (1.6) that are asymptotically equivalent in the sense that the ratios φ1 (t) φ2 (t)

and

φ2 (t) φ1 (t)

are bounded in [t0 , +∞). To the extremal value y (t) corresponds the “minimal" solution φ∗ (t), whose asymptotics at +∞ differs from that of φ1 (t) and φ2 (t) (they are linearly independent of the “minimal" solution). We will see that these properties in some sense remain in force in a more general case. Namely, to any two different normal solutions of the equation (1.1) correspond via (1.2) some solutions of the system (1.3), that are asymptotically equivalent in the above mentioned sense. To each extremal solution of (1.1) corresponds a solution of the system, the asymptotics of which at +∞ differs from that of other, linearly independent of it solutions. Thus, if for any t1 ≥ t0 the equation (1.1) has no t1 -extremal solutions (for instance, the equation y (t) + cos t y 2 (t) = 0), then all solutions of the system (1.1) are asymptotically equivalent. This case differs from the others by the scarcity of asymptotical behavior patterns at +∞ of the solutions of the system (1.3). In this case the system (1.3) can not have more than one finite Lyapunov characteristic value if for the normal solutions y(t) = o(t) as t → +∞ (such are all normal solutions of the equation y (t) + cos ty 2 (t) = 0). If the equation (1.1) has normal solutions and two extremal solutions (another extremal case), then we have a system (1.3) yielding a “rich" variety of asymptotical behavior patterns at +∞. It is known (see [1], pp. 143-146 and [4]) that in the general case the equation (1.1) is not solvable in quadratures and elementary functions. To solve (1.1) numerical methods are used, for instance, those of Runge–Kutt, see [5], pp. 387-395. If the desired solution is not stable, then the numerical solution becomes difficult. If the solution is normal, then in many cases it is stable in Lyapunov sense, and in some cases all normal solutions can even asymptotically come close together (see [3] and [6], p. 95). Thus, if there is a formula which explicitly expresses the extremal solution by the normal one, then in many cases that formula can be used to avoid the mentioned difficulties. Below we find some formulas which explicitly express any extremal solution by normal ones in quadratures and elementary functions. Also, relations between any two regular solutions are obtained, which generalize the Vieta formulas for the case of real, different roots of the quadratic equation. One of these relations is used to obtain an asymptotical closeness criterion for the normal solutions of the equation (1.1). 2. SOME DEFINITIONS AND PRELIMINARY RELATIONS Below always t1 ≥ t0 . Definition 1. A solution of the equation (1.1) is called t1 -regular if it exists on the whole semiaxis [t1 , +∞). Let Y (t, t1 , λ) be the general solution of the equation (1.1) in the domain Gt1 = {(t, y) : t ∈ It1 (λ), y ∈ R} , where It1 (λ) is the largest interval where the solution yλ (t) of (1.1) exists with yλ (t1 ) = λ (λ ∈ R). JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

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Lemma 1. If y0 (t) is a t0 -regular solution of the equation (1.1), then the general solution Y (t, t1 , λ) can be written in [t0 , +∞) × R ∩ Gt1 as Y (t, t1 , λ) = y0 (t) + λ where

ey0 (t1 , t) , 1 + λµy0 (t1 , t)

(2.1)

t ey0 (t1 , t) = exp − [2a(s)y0 (s) + b(s)]ds ,

t1 t

µy0 (t1 , t) =

a(τ )ey0 (t1 , τ )dτ,

t1 , t ≥ t0 ,

λ ∈ R.

t1

The proof of this lemma is elementary. Let yi (t) (i = 1, 2) be two t0 -regular solutions of the equation (1.1). Then by Lemma 1 for any t1 , t ∈ [t0 , +∞) yi (t) = yj (t) + λij

eyj (t1 , t) , 1 + λij (t1 )µyj (t1 , t)

(2.2)

where λij (t1 ) = yi (t1 ) − yj (t1 ). Hence yi (t) − yj (t) = λij

eyj (t1 , t) , 1 + λij (t1 )µyj (t1 , t)

t1 , t ∈ [t0 , +∞),

i = 1, 2.

(2.3)

The left-hand sides of these equalities are independent of t1 , and hence this is true also for the right-hand sides. So we can denote 1 + λij (t1 )µyj (t1 , t) , t ≥ t0 , i, j = 1, 2 (i = j), νyi ,yj (t) = − λij (t1 )eyj (t1 , t) and (2.2) can be written as yi (t) = yj (t) −

1 νyi ,yj (t)

,

t ≥ t0 ,

i = 1, 2

(i = j).

(2.4)

It obviously follows that νy1 ,y2 (t) = −νy2 ,y1 (t),

t ≥ t0 .

(2.5)

Theorem 1. For any two t0 -regular solutions y1 (t) and y2 (t) νy 1 ,y2 (t) − b(t), t ∈ [t0 , +∞), νy1 ,y2 (t) νy ,y (t) + c(t), t ∈ [t0 , +∞). a(t)y1 (t)y2 (t) = y1 (t) + y1 (t) 1 2 νy1 ,y2 (t) a(t)[y1 (t) + y2 (t)] =

(2.6) (2.7)

Proof: It is not difficult to verify that νy 1 ,y2 (t) = −a(t) + [2a(t)y2 (t) + b(t)]νy1 ,y2 (t),

t ≥ t0 .

Besides, νy1 ,y2 (t) = 0, t ∈ [t0 , +∞), and therefore νy 1 ,y2 (t) a(t) − a(t)y2 (t) − b(t) = − + 2a(t)y2 (t) + b(t) νy1 ,y2 (t) νy1 ,y2 (t) 1 , t ∈ [t0 , +∞). − a(t)y2 (t) − b(t) = a(t) y2 (t) − νy1 ,y2 (t) JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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Hence (2.6) follows by (2.4). Further, multiplying (2.6) by y1 (t) we get a(t)y12 (t) + a(t)y1 (t)y2 (t) = y1 (t)

νy 1 ,y2 (t) − b(t)y1 (t), νy1 ,y2 (t)

t ≥ t0 .

By (1.1) we get (2.7), and the proof is complete. From (2.2) we derive −2a(t)yi (t) − b(t) = −2a(t)yj (t) − b(t) − 2λij

a(t)eyj (t1 , t) , 1 + λij (t1 )µyj (t1 , t)

i = 1, 2.

Integrating this equality from t1 to t, we obtain that for t1 , t ≥ t0 , i = 1, 2, t t

2

2a(t)yi (τ ) + b(τ ) dτ = 2a(τ )yj (τ ) + b(τ ) dτ − ln 1 + λij (t1 )µyj (t1 , t) . t1

t1

Hence eyi (t1 , t) =

eyj (t1 , t) , (1 + λij (t1 )µyj (t1 , t))2

t1 , t ≥ t0 ,

i, j = 1, 2,

and consequently for t1 , t ≥ t0 ey1 (t1 , t)ey2 (t1 , t) =

ey1 (t1 , t)ey2 (t1 , t)

2

2 . 1 + λ1,2 (t1 )µy2 (t1 , t) 1 + λ2,1 (t1 )µy1 (t1 , t)

(2.8)

Note that 1 + λij (t1 )µyj (t1 , t1 ) = 1 > 0 (i, j = 1, 2), and the solutions yi (t) (i = 1, 2) are t0 -regular. Therefore, by (2.2) (1 + λij (t1 )µyj (t1 , t)) > 0 for t1 , t ≥ t0 , i, j = 1, 2. Consequently, by (2.8)

1 + λ1,2 (t1 )µy2 (t1 , t) 1 + λ2,1 (t1 )µy1 (t1 , t) ≡ 1. We come to the following relations: µyi (t1 , t) =

µyj (t1 , t) , 1 + λij (t1 )µyj (t1 , t1 )

t1 , t ≥ t0 ,

i, j = 1, 2.

(2.9)

Definition 2. A t1 -regular solution y0 (t) is called t1 -normal if there is a δ-neighborhood Uδ (y0 (t1 )) = (y0 (t1 ) − δ, y0 (t1 ) + δ) of the point y0 (t1 ), such that any solution y(t) of the equation (1.1) with y(t1 ) ∈ Uδ (y0 (t1 )) is t1 -regular. Otherwise, the solution y0 (t) is called t1 -extremal or t1 -bounding. Definition 3. A t1 -extremal solution y0 (t) is called lower (or upper) t1 -extremal if no t1 -regular solutions y(t) of the equation (1.1) with y(t1 ) < y0 (t1 ) (y(t1 ) > y0 (t1 )) exist. Definition 4. A number λ is called t1 -acceptable parameter of the function µy0 (t1 , t) if 1 + λµy0 (t1 , t) = 0 for t ∈ [t1 , +∞). The set of t1 -acceptable parameters of the function µy0 (t1 , t) will be denoted by Dy0 (t1 ). Henceforth, we shall assume that the equation (1.1) has at least one t0 -regular solution. From the representation (2.1) of the general solution of (1.1), one can see that the solution y0 (t) is t1 normal if and only if there exists a δ-neighborhood (−δ, δ) of the origin, such that for any λ ∈ (−δ, δ) the solution Y (t, t1 , λ) is t1 -regular. Besides, the solution y0 (t) is lower (or upper) t1 -extremal if and only if for any λ < 0 (or > 0) Y (t, t1 , λ) is not t1 -regular. Hence, by (2.1) we come to the following theorem. Theorem 2. The following statements are true: A) A t1 -regular solution y0 (t) of the equation (1.1) is t1 -normal if and only if µy0 (t1 , t) is bounded by t ∈ [t1 , +∞), JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

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B) a t1 -regular solution y0 (t) of the equation (1.1) is a lower but not upper (or upper but not lower) t1 -extremal solution if and only if µy0 (t1 , t) is bounded from below but not from above (or bounded from above but not from below) by t ∈ [t1 , +∞), C) a t1 -regular solution y0 (t) of the equation (1.1) is simultaneously lower and upper extremal if and only if µy0 (t1 , t) is bounded neither from below nor from above for t ∈ [t1 , +∞). Henceforth, we call µy0 (t1 , t) a characteristic function of the solution y0 (t) of the equation (1.1). Let y0 (t) be a t0 -regular solution of (1.1). Then for any t1 > t0 this solution is t1 -regular. Besides, µy0 (t0 , t) = µy0 (t0 , t1 ) + ey0 (t0 , t1 )µy0 (t1 , t),

t1 , t ≥ t0 ,

(2.10)

and hence the below corollary immediately follows from Theorem 2. Corollary 1. If t1 > t0 , then the following statements are true: A ) a t0 -regular solution of the equation (1.1) is t1 -normal if and only if it is t0 -normal, B ) a t0 -regular solution of the equation (1.1) is lower but not upper (or upper but not lower) t1 -extremal if and only if it is a lower but not upper (or upper but not lower) t0 -extremal, C ) a t0 -regular solution of the equation (1.1) is simultaneously lower and upper t1 -extremal if and only if it simultaneously is lower and upper t0 -extremal. We see that the type of a t0 -regular solution (normal, lower, upper or extremal) does not depend on the set [t1 , +∞) ⊂ [t0 , +∞), where we consider it. Henceforth we shall call a t0 -regular (or t0 -normal, t0 -extremal) solution regular (normal or extremal) solution of the equation (1.1), independent of the set [t1 , +∞) (t1 ≥ t0 ) where we consider this solution. Now assuming that t1 ≥ t0 , we introduce the following quantities for a regular solution y0 (t): µ(y0 , t1 ) =

inf t∈[t1 ,+∞)

µy0 (t1 , t),

µ(y0 , t1 ) =

sup

µy0 (t1 , t).

t∈[t1 ,+∞)

If y0 (t) is normal, then by Theorem 2 A) the characteristic function of y0 (t) satisfies one of the following conditions: 1) −∞ < µ(y0 , t1 ) < µy0 (t1 , t) < µ(y0 , t1 ) < +∞, t ∈ [t1 , +∞), 2) −∞ < µ(y0 , t1 ) ≤ µy0 (t1 , t) < µ(y0 , t1 ) < +∞, t ∈ [t1 , +∞), µy0 (t1 , t2 ) = µ(y0 , t1 ) for some t2 ≥ t1 , 3) −∞ < µ(y0 , t1 ) < µy0 (t1 , t) ≤ µ(y0 , t1 ) < +∞, t ∈ [t1 , +∞), µy0 (t1 , t2 ) = µ(y0 , t1 ) for some t2 ≥ t1 , 4) −∞ < µ(y0 , t1 ) ≤ µy0 (t1 , t) ≤ µ(y0 , t1 ) < +∞, t ∈ [t1 , +∞), µy0 (t1 , t2 ) = µ(y0 , t1 ) and µy0 (t1 , t3 ) = µ(y0 , t1 ) for some t2 , t3 ≥ t1 . If y0 (t) is a lower but not an upper extremal solution, then by Theorem 2 B) µ(y0 , t1 ) > −∞,

µ(y0 , t1 ) = +∞,

and the following cases are possible: 5) −∞ < µ(y0 , t1 ) < µy0 (t1 , t) < µ(y0 , t1 ) = +∞, t ∈ [t1 , +∞), 6) −∞ < µ(y0 , t1 ) ≤ µy0 (t1 , t) < µ(y0 , t1 ) = +∞, t ∈ [t1 , +∞), µy0 (t1 , t2 ) = µ(y0 , t1 ) for some t2 ≥ t1 . JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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Similarly, if y0 (t) is an upper but not a lower extremal solution, then the following cases are possible: 7) −∞ = µ(y0 , t1 ) < µy0 (t1 , t) < µ(y0 , t1 ) < +∞, t ∈ [t1 , +∞), 8) −∞ = µ(y0 , t1 ) < µy0 (t1 , t) ≤ µ(y0 , t1 ) < +∞, t ∈ [t1 , +∞), µy0 (t1 , t2 ) = µ(y0 , t1 ) for some t2 ≥ t1 . At last, if a solution y0 (t) is simultaneously lower and upper extremal, then: 9) −∞ = µ(y0 , t1 ) < µy0 (t1 , t) < µ(y0 , t1 ) = +∞, t ∈ [t1 , +∞). One can verify that in the cases 1)-9) the set of t1 -acceptable parameters happens to be: 1 1 ,− (it is clear that in this case µ(y0 , t1 ) < 0, µ(y0 , t1 ) > 0), 1) Dy0 (t1 ) = − µ(y0 , t1 ) µ(y0 , t1 )

1 1 ,− if µ(y0 , t1 ) = 0 and 2) Dy0 (t1 ) = − µ(y0 , t1 ) µ(y0 , t1 ) 1 , +∞ if µ(y0 , t1 ) = 0 (obviously µ(y0 , t1 ) > 0)), Dy0 (t1 ) = − µ(y0 , t1 )

1 1 ,− if µ(y0 , t1 ) = 0 and 3) Dy0 (t1 ) = − µ(y0 , t1 ) µ(y0 , t1 ) 1 if µ(y0 , t1 ) = 0, Dy0 (t1 ) = −∞, − µ(y0 , t1 )

1 1 ,− if µ(y0 , t1 ) < 0 and µ(y0 , t1 ) > 0, 4) Dy0 (t1 ) = − µ(y0 , t1 ) µ(y0 , t1 ) 1 if µ(y0 , t1 ) = 0 and besides Dy0 (t1 ) = −∞, − µ(y0 , t1 ) 1 , +∞ if µ(y0 , t1 ) > 0 and µ(y0 , t1 ) < 0, Dy0 (t1 ) = − µ(y0 , t1 ) µ(y0 , t1 ) = 0, Dy0 (t1 ) = (−∞, +∞) if µ(y0 , t1 ) = µ(y0 , t1 ) = 0,

1 , 5) Dy0 (t1 ) = 0, − µ(y0 , t1 )

1 , 6) Dy0 (t1 ) = 0, − µ(y0 , t1 )

1 ,0 , 7) Dy0 (t1 ) = − µ(y0 , t1 )

8) Dy0 (t1 ) =

1 ,0 , − µ(y0 , t1 )

9) Dy0 (t1 ) = {0}. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 42

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3. TYPES OF SETS OF REGULAR INITIAL VALUES. RELATIONS BETWEEN NORMAL AND EXTREMAL SOLUTIONS Assuming that t1 ≥ t0 , we introduce the following definition. Definition 5. A number y(0) (∈ R) is called t1 -regular initial value of the equation (1.1) if its solution y0 (t) with y0 (t1 ) = y(0) is t1 -regular. The set of t1 -regular initial values of the equation (1.1) will be denoted by reg(t1 ). If y(0) ∈ reg(t1 ) and λ ∈ Dy0 (t1 ), then by (2.1) y(0) = y0 (t1 ) + λ. Obviously reg(t1 ) is independent of the choice of y0 (t)). Therefore, reg(t1 ) takes the following values in the cases 1)-9) correspondingly: 1 1 , y0 (t1 ) − , 1 ) reg(t1 ) = y0 (t1 ) − µ(y0 , t1 ) µ(y0 , t1 ) 1 1 , y0 (t1 ) − if µ(y0 , t1 ) = 0, besides 2 ) reg(t1 ) = y0 (t1 ) − µ(y0 , t1 ) µ(y0 , t1 ) reg(t1 ) = y0 (t1 ) − µ(y01,t1 ) , +∞ if µ(y0 , t1 ) = 0, 1 1 , y0 (t1 ) − if µ(y0 , t1 ) = 0, besides reg(t1 ) = y0 (t1 ) − µ(y0 , t1 ) µ(y0 , t1 ) 1 if µ(y0 , t1 ) = 0, reg(t1 ) = −∞, y0 (t) − µ(y0 , t1 ) 1 1 , y0 (t1 ) − if µ(y0 , t1 ) < 0 and µ(y0 , t1 ) > 0, reg(t1 ) = y0 (t1 ) − µ(y0 , t1 ) µ(y0 , t1 ) 1 if µ(y0 , t1 ) < 0 and µ(y0 , t1 ) = 0, reg(t1 ) = −∞, y0 (t1 ) − µ(y0 , t1 ) 1 , +∞ if µ(y0 , t1 ) = 0 and µ(y0 , t1 ) > 0, reg(t1 ) = y0 (t1 ) − µ(y0 , t1 ) reg(t1 ) = (−∞, +∞) if µ(y0 , t1 ) = µ(y0 , t1 ) = 0, 1 , reg(t1 ) = y0 (t1 ), y0 (t1 ) − µ(y0 , t1 ) 1 , reg(t1 ) = y0 (t1 ), y0 (t1 ) − µ(y0 , t1 ) 1 , y0 (t1 ) , reg(t1 ) = y0 (t1 ) − µ(y0 , t1 ) 1 ,0 , reg(t1 ) = y0 (t1 ) − µ(y0 , t1 )

3 )

4 )

5 ) 6 ) 7 ) 8 )

9 ) reg(t1 ) = {y0 (t1 )}. We need the notation A(t1 ) = inf{t : t ∈ reg(t1 )},

B(t1 ) = sup{t : t ∈ reg(t1 )}.

Then by 1)-9) the set reg(t1 ) can be of one of the following types: JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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i) reg(t1 ) = (A(t1 ), B(t1 )), ii) reg(t1 ) = [A(t1 ), B(t1 )), iii) reg(t1 ) = [A(t1 ), B(t1 )], iv) reg(t1 ) = [A(t1 ), B(t1 )), and if A(t1 ) ∈ / reg(t1 ) (or B(t1 ) ∈ / reg(t1 )), then A(t1 ) (or B(t1 )) can take the value −∞ (or +∞). It is obvious that if A(t0 ) ∈ reg(t0 ) (or B(t0 ) ∈ reg(t0 )), then by Corollary 1 A(t) ≡ y∗ (t) (or B(t) ≡ y ∗ (t)) is a lower (or upper) extremal solution of the equation (1.1). Therefore, by i)-iii) we may correspondingly write i ) reg(t) = [y∗ (t), y ∗ (t)], ii ) reg(t) = [y∗ (t), B(t)), iii ) reg(t) = (A(t), y ∗ (t)], t ∈ [t0 , +∞). A simple example shows that the case iii) can appear for any numbers A(t1 ) and B(t1 ). Similar examples can be given for the cases i), ii) and iv) as well. Example 1. Riccati equation y (t) + a(t)y 2 (t) = 0,

t ∈ [t0 , +∞),

(3.1)

has a regular solution y0 (t) ≡ 0 whose characteristic function is t def a(τ )dτ = A(t1 , t), t1 , t ≥ t0 . µy0 (t1 , t) = t1

For briefness, we denote A(t1 ) = µ(y0 , t1 ),

A(t1 ) = µ(y0 , t1 ),

where y0 (t) ≡ 0. Then we suppose that Sn , n = 1, 2, . . . is a bounded sequence, such that 0 < S1 < S3 < . . . < S2m+1 < . . . , 0 > S2 > S4 > . . . S2m . . . , 1 1 − = B(t1 ) − A(t1 ) > 0. lim S2m+1 lim S2m m→∞

(3.2)

m→∞

and besides, that t1 < t2 < . . . < tn < . . . and tn → +∞ as n → +∞. Let a(t) be such that (−1)k+1 a(t) > 0 for t ∈ (tk , tk+1 ), k = 1, 2, . . ., and t3 t4 t2 a(τ )dτ = S1 > 0, a(τ )dτ = S2 − S1 < 0, a(τ )dτ = S3 − S2 > 0, t1

t2 tk+1

...

a(τ )dτ = Sk − Sk−1 ,

t3

((Sk − Sk−1 )(−1)k−1 > 0), . . .

tk

Then, 0 > A(t1 ) = lim S2m > −∞ and m→∞

0 < A(t1 ) = lim S2m+1 < +∞

(3.3)

m→∞

and A(t1 ) < A(t1 , t) < A(t1 ),

t ≥ t1 .

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For the equation (3.1) we have

1 1 . reg(t1 ) = − ,− A(t1 ) A(t1 )

Replacing y(t) = z(t) −

1 − A(t1 ) A(t1 )

in (3.1), we come to an equation for which reg(t1 ) = A(t1 ), A(t1 ) +

1 1 − . A(t1 ) A(t1 )

By (3.2) and (3.3) we conclude that for the initial equation (3.1) reg(t1 ) = [A(t1 ), B(t1 )]. If A(t1 ) = B(t1 ), then it suffices to choose the sequence Sn , n = 1, 2, . . ., so that limm→∞ S2m+1 = +∞ and limm→∞ S2m = −∞. Now, suppose y1 (t) and y2 (t) are some regular solutions of the equation (1.1). Then multiplying by a(t) and integrating from t0 to t, from (2.3) we obtain that for t1 = t0 t a(τ )(yi (τ ) − yj (τ ))dτ = ln(1 + λıj (t0 )µyj (t0 , t)), (3.4) t0

where t ∈ [t0 , +∞) and i = 1, 2. If y1 (t) and y2 (t) are normal solutions of the equation (1.1), then by Theorem 2 A) the corresponding characteristic functions µy1 (t0 , t) and µy2 (t0 , t) are bounded by t ∈ [t0 , +∞). Therefore, by (3.4) t a(τ )(y1 (τ ) − y2 (τ ))dτ ≤ M, t ∈ [t0 , +∞), (3.5) t0

for M great enough. Besides, if y1 (t) ≡ y0 (t) is normal and y2 (t) ≡ ynp (t) is extremal, then by Theorem 2 µy0 (t0 , t) is bounded by t ∈ [t0 , +∞), and therefore, (3.4) implies t a(τ )(ynp (τ ) − y0 (τ ))dτ ≤ M1 , t ∈ [t0 , +∞), (3.6) t0

for M1 great enough. If y2 (t) ≡ ynp (t) is an extremal solution, then

lim inf 1 + λ2,1 (t0 )µy1 (t0 , t) = 0, t→+∞

hence by (3.4)

t

lim inf t→+∞

a(τ )(ynp (τ ) − y0 (τ ))dτ = −∞.

(3.7)

(3.8)

t0

The last equality particularly implies that if y∗ (t) is a lower and y ∗ (y) is an upper extremal solution, then t lim inf t→+∞ t0 a(τ )(y ∗ (τ ) − y∗ (τ ))dτ = −∞, (3.9) t lim supt→+∞ t0 a(τ )(y ∗ (τ ) − y∗ (τ ))dτ = +∞. We consider an equivalency relation ∼ in the set of solutions of the system (1.3): (φ1 (t), ψ1 (t)) ∼ φ2 (t) φ1 (t) and are bounded. One can see that (3.5) implies that all (φ2 (t), ψ2 (t)) if and only if φ2 (t) φ1 (t) those solutions of (1.3), which correspond to normal solutions of equation (1.1) by formulas (1.2), are equivalent. By (3.8) and (3.9) those solutions of the system (1.3), that correspond to extremal solutions of equation (1.1), are never equivalent to each other or to the solutions corresponding to normal solutions of (1.1). JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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Further, by (3.6) and (3.8) a solution of the system (1.3), corresponding to an extremal solution of the equation (1.1), possesses some “minimality" property. Note that (3.5) and (3.8) imply Theorem 2 of [3]. Let y0 (t) be a normal and y∗ (t) be a lower extremal solution of the equation (1.1). Then by (2.2) y∗ (t) = y0 (t) + λ∗

ey0 (t0 , t) , 1 + λ∗ µy0 (t0 , t)

(3.10)

where λ∗ = y∗ (t0 ) − y0 (t0 ). Besides, inf (1 + λ∗ µy0 (t0 , t)) = 0

t≥t0

(otherwise (3.10) would imply that y∗ (t) is normal). By this equality, λ∗ = −

1 . µ(y0 , t0 )

Hence by (3.10) y∗ (t) = y0 (t) −

1 , ν y0 (t)

t ∈ [t0 , +∞),

(3.11)

where ν y0 (t) =

µ(y0 , t0 ) − µy0 (t0 , t) > 0, ey0 (t0 , t)

t ∈ [t0 , +∞).

(3.12)

Comparing (3.11) and (2.4), we conclude that νy0 y∗ (t) = ν y0 (t),

t ∈ [t0 , +∞).

(3.13)

νy0 y∗ (t) = ν y0 (t),

t ∈ [t0 , +∞),

(3.14)

Similarly, where y ∗ (t) is an upper extremal solution and ν y0 (t) =

µ(y0 , t0 ) − µy0 (t0 , t) < 0, ey0 (t0 , t)

t ∈ [t0 , +∞).

(3.15)

By formulas (2.2), (2.4) and (3.11) - (3.15) we conclude that if ν y0 (t) = 0 (or ν y0 (t) = 0), t ≥ t0 , for some normal solution y0 (t), then the equation (1.1) has a lower (or upper) extremal solution. Obviously the converse statement is also true. Thus, the following theorem holds. Theorem 3. Let y0 (t) be a normal solution of the equation (1.1). Then (1.1) possesses a lower (or upper) extremal solution if and only if ν y0 (t) = 0 (ν y0 (t) = 0) as

t ≥ t0 .

Under this condition, the lower (or upper) extremal solution is given by 1 1 y ∗ (t) = y0 (t) − , t ∈ [t0 , +∞). y∗ (t) = y0 (t) − ν y0 (t) ν y0 (t) For a regular solution y0 (t), consider the integral τ +∞ a(τ ) exp − [2a(s)y0 (s) + b(s)]ds dτ, νy0 (t) ≡ t

t ∈ [t0 , +∞).

t

Evidently, this integral is convergent or divergent for any t ≥ t0 . Let νy0 (t0 ) be convergent (if νy0 (t) converges for some t = t1 , then it converges also for the remaining values of t). Then νy0 (t) = lim µy0 (t0 , t) ≡ µy0 (t0 , +∞). t→∞


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Hence, µy0 (t0 , t) is bounded by t, and by Theorem 2 A the solution y0 (t) is normal. It is not difficult to show that µy (t0 , +∞) − µy0 (t0 , t) νy0 (t) = 0 , t ≥ t0 . (3.16) ey0 (t0 , t) Consequently, if νy0 (t) = 0 for t ≥ t0 , then µy0 (t0 , +∞) coincides with µ(y0 , t0 ) when µy0 (t0 , +∞) is positive, and with µ(y0 , t0 ) when µy0 (t0 , +∞) is negative. Thus, by (3.12), (3.15) and (3.16) ⎧ ⎨ ν (t) when µ (t , +∞) = ν (t ) > 0, y0 y0 0 y0 0 νy0 (t) = ⎩ ν (t) when µ (t , +∞) = ν (t ) < 0. y0

y0

0

y0

0

Let y1 (t) be a normal solution different from y0 (t). Then by (2.9) lim µy1 (t0 , t) ≡ µy1 (t0 , +∞) =

t→+∞

µy1 (t0 , +∞) = ±∞ 1 + λ1,0 µy1 (t0 , +∞)

(since µy1 (t0 , t) is bounded by t), where λ1,0 = y1 (t0 ) − y0 (t0 ). This means that the integral νy1 (t0 ) = µy1 (t0 , +∞) is convergent for any t ≥ t0 , and hence also νy1 (t) is convergent. The following Theorem is true by Theorem 3 and the statements after it. Theorem 4. Let the integral νy0 (t0 ) be convergent for some regular solution y0 (t). Then: I) the integrals νy (t) are convergent for any t ≥ t0 if and only if y(t) is a normal solution. II) the equation (1.1) has an extremal solution if and only if νy0 (t) = 0,

t ∈ [t0 , +∞).

Under this condition the extremal solution ynp (t) is unique: ynp (t) = y0 (t) −

1 , νy0 (t)

t ∈ [t0 , +∞).

(3.17)

Besides, ynp (t) is a lower (or upper) extremal solution if and only if νy0 (t0 ) > 0 (or νy0 (t0 ) < 0). One can see that Theorem 1,[3] follows from the statement I) of Theorem 4. Now let yi (t) (i = 1, 2) be a normal and ynp (t) an extremal solution of the equation (1.1), and let t a(τ )y1 (τ )dτ , ψ1 (t) = y1 (t)φ1 (t), φ1 (t) = exp t0 t

φ2 (t) = exp

a(τ )ynp (τ )dτ

,

ψ2 (t) = ynp (t)φ2 (t).

t0

Then by (1.2), {φi (t), ψi (t)} (i = 1, 2) is a solution of the system (1.3). Consequently, for any λ ∈ R the function ψ(t) + λψ2 (t) Y (t, t0 , λ) = φ1 (t) + λφ2 (t) is a solution of (1.1) for those values of t, for which φ1 (t) + λφ(t) = 0. Dividing the numerator and the denominator by φ1 (t), we get t y1 (t) + λynp (t) exp a(τ )(ynp (τ ) − y1 (τ ))dτ t0 t . (3.18) Y (t, t0 , λ) = 1 + λ exp a(τ )(ynp (τ ) − y1 (τ ))dτ t0

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Since y1 (t0 ) + λynp (t0 ) , Y (t0 , t0 , λ) = 1+λ by (3.18)

t a(τ )(ynp (τ ) − y1 (τ ))dτ y1 (t) + λ2 ynp (t) exp y2 (t) = t t0 , 1 + λ2 exp a(τ )(ynp (τ ) − y1 (τ ))dτ t0

where λ2 =

y2 (t0 ) − y1 (t0 ) . ynp (t0 ) − y2 (t0 )

We derive:

a(t)(y2 (t) − y1 (t)) =

t

λ2 a(t)(ynp (t) − y1 (t)) exp a(τ )(ynp (τ ) − y1 (τ ))dτ t0 t . 1 + λ2 exp a(τ )(ynp (τ ) − y1 (τ ))dτ t0

Integration from t0 to t yields t ynp (t0 ) − y2 (t0 ) a(τ )(y2 (τ ) − y1 (τ ))dτ = ln ynp (t0 ) − y1 (t0 ) t0 t a(τ )(ynp (τ ) − y1 (τ ))dτ , + ln 1 + λ2 exp

t ∈ [t0 , +∞).

(3.19)

t0

Theorem 5. Let y1 (t) and y2 (t) be some normal solutions of the equation (1.1), νy1 (t0 ) be convergent and νy1 (t) = 0,

t ∈ [t0 , +∞).

Then for the unique extremal solution ynp (t) of the equation (1.1) we have +∞ ynp (t0 ) − y2 (t0 ) . a(τ )(y2 (τ ) − y1 (τ ))dτ = ln ynp (t0 ) − y1 (t0 ) t0

(3.20)

Proof: In virtue of Theorem 4, our conditions provide the existence and uniqueness of the extremal solution ynp (t) of (1.1). Besides, νy1 (t0 ) = lim µy1 (t0 , t), t→+∞

and hence by (3.7) lim (1 + λnp µy1 (t0 , t)) = 0,

t→+∞

where λnp = ynp (t0 ) − y1 (t0 ). Consequently, by (3.4) +∞ a(τ )(ynp (τ ) − y1 (τ ))dτ = −∞.

(3.21)

t0

Letting t → +∞ in (3.19), by (3.21) we come to (3.20). The proof is complete. Let us assume the existence of some regular solution y0 (t) for which νy0 (t0 ) is convergent. By (3.17) νy (t) ynp (t) − y2 (t) = 1 , ynp (t) − y1 (t) νy2 (t)

t ∈ [t0 , +∞),


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for any normal solutions y1 (t), y2 (t) and the extremal solution ynp (t). Letting t0 → +∞, by (3.20) we obtain lim

t→+∞

ynp (t) − y2 (t) = 1. ynp (t) − y1 (t)

(3.23)

Hence by (3.22) lim =

t→+∞

νy1 (t) = 1, νy2 (t)

(3.24)

and by (3.17) and (3.23) lim =

t→+∞

y1 (t) − y2 (t) = lim νy (t)(y1 (t) − y2 (t)) = 0. ynp (t) − y1 (t) t→+∞ 1

Observe that by (3.24) in the above relation the function νy1 (t) can be replaced by νy0 (t), where y0 (t) is any normal solution. Consequently, if νy0 (t) ≥ > 0, t ∈ [t0 , +∞), for some normal solution, then all normal solutions are asymptotically coming together, i.e. the following relation is true for any two normal solutions y1 (t) and y2 (t): lim (y2 (t) − y1 (t)) = 0.

t→+∞

Besides, by (2.5) and (2.7) for any solutions y1 (t) and y2 (t) y1 (t)

−

y2 (t)

νy 1 ,y2 (t) , = −(y1 (t) − y2 (t)) νy1 ,y2 (t)

t ∈ [t0 , +∞),

and hence νy ,y (t) [y1 (t) − y2 (t)] =− 1 2 , y1 (t) − y2 (t) νy1 ,y2 (t)

t ∈ [t0 , +∞).

Integrating this equality from t0 to t, we get ln

νy ,y (t) y1 (t) − y2 (t) = − ln 1 2 , y1 (t0 ) − y2 (t0 ) νy1 ,y2 (t0 )

t ∈ [t0 , +∞),

implying y1 (t) − y2 (t) νy1 ,y2 (t) = 1, y1 (t0 ) − y2 (t0 ) νy1 ,y2 (t0 ) Since y1 (t) and y2 (t) are normal,

t ∈ [t0 , +∞).

(3.25)

inf 1 + λ1,2 (t0 )µy2 (t0 , t) > 0.

t≥t0

Therefore, νy1 ,y2 (t) → +∞ as t → +∞ if and only if e−1 y2 (t0 , t) → +∞ as t → +∞. By (3.25) we come to a criterion: Theorem 6. All normal solutions of the equation (1.1) asymptotically come together as t → +∞ if and only if +∞ [2a(τ )y0 (τ ) + b(τ )]dτ = +∞ t0

for some normal solution y0 (t). The latter theorem is a generalization of Theorem 4 of [3]. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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Example 2. Let a(t) ≡ a0 = 0, let b(t) ≡ b0 and let c(t) ≡ c0 , b20 − 4a0 c0 > 0. Then −b0 ± b20 − 4a0 c0 y± (t) ≡ y± = 2a0 are regular solutions of the equation (1.1). It is not difficult to show that νy+ (t0 ) converges and νy− (t0 ) diverges. Therefore by Theorem 4, y+ (t) is a normal solution and y− (t) is the unique extremal solution of the equation (1.1), and this extremal solution is lower when a0 > 0 and upper when a0 < 0. Besides, y+ ≡ 0 and νy+ (t) ≡ 0. Applying (2.6) and (2.7) to the solutions y± we come to Vieta’s well-known formulas: ⎧ ⎨ a (y + y ) = −b , 0 + − 0 ⎩a y y =c . Further, by 2a0 y+ + b0 =

0 + −

0

b20 − 4a0 c0 > 0 and +∞ [2a0 y+ + b0 ]dτ = +∞, t0

hence by Theorem 6 all normal solutions of the equation (1.1) are asymptotically coming together. REFERENCES 1. A. I. Egorov, Riccati Equations (Fizmatlit, Moscow, 2001). 2. I. M. Sobol, “On Asymptotical Behavior of the Solutions of Linear Differential Equations”, Doklady AN SSSR [USSR Academy of Sciences Reports] LXI (2), (1948). 3. I. M. Sobol, “On Rikatti Equations and Reducible to Them Linear Equations of Second Order”, Doklady AN SSSR [USSR Academy of Sciences Reports] LXV (3), (1949). 4. E. Kamke, Gide for Ordinary Differential Equations (Moscow, 1961). 5. Yu. P. Boglaev, Computational Mathematics and Programming (Visshaya Shkola, Moscow,1990). 6. R. Bellman, Stability Theory of Solutions of Differential Equations (Izd. In. Lit., Moscow, 1954).


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2007