c Pleiades Publishing, Ltd., 2017. ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 4, pp. 450–456. c G.A. Grigoryan, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 4, pp. 459–464. Original Russian Text
ORDINARY DIFFERENTIAL EQUATIONS
Global Solvability Tests for a Scalar Riccati Equation with Complex Coefficients G. A. Grigoryan Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan, 375019 Armenia e-mail:
[email protected] Received May 11, 2016
Abstract—We obtain two global solvability tests for a scalar Riccati equation with complex coefficients. One of them is used to prove a test for the existence of a solution of the Redheffer system, which arises when studying a physical model of electromagnetic wave distribution in a transmission line and in a physical model of diffraction of particles along a rod. DOI: 10.1134/S0012266117040048
1. INTRODUCTION Let a(t), b(t), and c(t) be continuous complex-valued functions on [t0 , +∞). Consider the Riccati equation t ≥ t0 . (1.1) z + a(t)z 2 + b(t)z + c(t) = 0, The existence of global solutions of this equation (i.e., solutions defined for all t ≥ t0 ), together with the well-known relationship of its solutions with solutions of two-dimensional linear differential systems and with second-order linear differential equations, implies the nonoscillation of the corresponding solutions of the latter. Numerous papers (see the monograph [1] and the papers cited therein as well as [2–9]) deal with the nonoscillation of solutions of differential equations and systems. This necessitates finding global solvability tests for Eq. (1.1). Such global solvability tests for Eq. (1.1) with real coefficients a(t), b(t), and c(t) were obtained in [8, 10] based on two comparison tests for scalar Riccati equations. In the present paper, we prove two global solvability tests for Eq. (1.1) based on one of the above-mentioned tests. One of the tests proved below is used to derive a test for the existence of a solution of the Redheffer system arising when studying a physical model of electromagnetic wave distribution in a transmission line and in a physical model of diffraction of particles along a rod. (In the case of real coefficients a(t), b(t), and c(t), two tests for the existence of a solution of the Redheffer system were proved in [8].) 2. AUXILIARY ASSERTIONS The solutions z(t) of Eq. (1.1) existing on some interval [t1 , t2 ) are related to the solutions (ϕ(t), ψ(t)) of the system ϕ = a(t)ψ,
ψ = −c(t)ϕ − b(t)ψ,
t ≥ t0 ,
(2.1)
by the formulas (see [12, pp. 153–154])
t ϕ(t) = ϕ(t1 ) exp
a(τ )z(τ ) dτ ,
ϕ(t1 ) = 0,
t1
Let z0 (t) be a solution of Eq. (1.1) on the interval [t1 , t2 ). 450
ψ(t) = z(t)ϕ(t).
(2.2)
GLOBAL SOLVABILITY TESTS FOR A SCALAR RICCATI EQUATION
451
Definition 2.1. We say that [t1 , t2 ) is the maximal interval of existence of the solution z0 (t) if z0 (t) cannot be extended to the right of t2 as a solution of Eq. (1.1).
t Lemma 2.1. If the function Ft1 (t) ≡ t1 Re [a(τ )z0 (τ )] dτ, t ∈ [t1 , t2 ), is bounded below on the interval [t1 , t2 ) and t2 < +∞, then [t1 , t2 ) is not the maximal interval of existence of the solution z0 (t). Proof. Let (ϕ0 (t), ψ0 (t)) be the solution of system (2.1) with ϕ0 (t1 ) = 1 and ψ0 (t1 ) = z0 (t1 ). Then, by relations (2.2), we have t ϕ0 (t) = exp
a(τ )z0 (τ ) dτ ,
t ∈ [t1 , t2 ).
ψ0 (t) = z0 (t)ϕ0 (t),
t1
This, together with the continuity of the function ϕ0 (t) and the assumptions of the lemma, implies that ϕ0 (t) = 0, t ∈ [t1 , t3 ), for some t3 > t2 . Therefore, z0 (t) ≡ ψ0 (t)/ϕ0 (t) is a solution of Eq. (1.1) on the interval [t1 , t3 ). Obviously, z0 (t) coincides with z0 (t) on [t1 , t2 ). Consequently, [t1 , t2 ) is not the maximal interval of existence of the solution z0 (t). The proof of the lemma is complete. We substitute z(t) = x(t) − iy(t), where x(t) ≡ Re z(t) and y(t) ≡ − Im z(t), into Eq. (1.1) and separate the real and imaginary parts; then we obtain the nonlinear system x + a1 (t)x2 + [b1 (t) + 2a2 (t)y]x + Py (t) = 0, y + a2 (t)y 2 + [b1 (t) + 2a1 (t)x]y + Qx (t) = 0,
(2.3)
t ≥ t0 ,
where Py (t) ≡ −a1 (t)y 2 (t)+b2 (t)y(t)+c1 (t), Qx (t) ≡ −a2 (t)x2 (t)−b2 (t)x(t)−c2 (t), a1 (t) ≡ Re a(t), a2 (t) ≡ Im a(t), b1 (t) ≡ Re b(t), b2 (t) ≡ Im b(t), c1 (t) ≡ Re c(t), and c2 (t) ≡ Im c(t), t ≥ t0 . Let f (t), g(t), h(t), f1 (t), g1 (t), and h1 (t) be continuous real-valued functions on the interval [t0 , +∞). Consider the Riccati equations y + f (t)y 2 + g(t)y + h(t) = 0, y + f1 (t)y 2 + g1 (t)y + h1 (t) = 0,
t ≥ t0 , t ≥ t0 ,
(2.4) (2.5)
t ≥ t0 , t ≥ t0 .
(2.6) (2.7)
and the differential inequalities y + f (t)y 2 + g(t)y + h(t) ≥ 0, y + f1 (t)y 2 + g1 (t)y + h1 (t) ≥ 0,
Remark 2.1. If f (t) ≥ 0 for t ≥ t0 , then every solution of the linear equation y + g(t)y + h(t) = 0 on an interval [t0 , τ0 ) (t0 < τ0 ≤ +∞) is a solution of inequality (2.6) on [t0 , τ0 ). Remark 2.2. Every solution of Eq. (2.5) on the interval [t0 , τ0 ) (t0 < τ0 ≤ +∞) is also a solution of inequality (2.7) on [t0 , τ0 ). Theorem 2.1. Let Eq. (2.5) have a real solution y1 (t) on an interval [t0 , τ0 ) (τ0 ≤ +∞), and let the following conditions be satisfied : f (t) ≥ 0 and τ t exp t0
[f (s)(η0 (s) + η1 (s)) + g(s)] ds t0
× [(f1 (τ ) − f (τ ))y12 (τ ) + (g1 (τ ) − g(τ ))y1 (τ ) + h1 (τ ) − h(τ )] dτ ≥ 0,
t ∈ [t0 , τ0 ),
where η0 (t) and η1 (t) are solutions of inequalities (2.6) and (2.7), respectively, on [t0 , τ0 ) such that ηj (t0 ) ≥ y1 (t0 ), j = 0, 1. Then for each γ0 ≥ y1 (t0 ) Eq. (2.4) has a solution y0 (t) on [t0 , τ0 ) satisfying the initial condition y0 (t0 ) = γ0 ; in addition, y0 (t) ≥ y1 (t), t ∈ [t0 , τ0 ). Proof. The proof of the theorem can be found in [11]. DIFFERENTIAL EQUATIONS
Vol. 53
No. 4
2017
