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International Workshop QUALITDE – 2018, December 1 – 3, 2018, Tbilisi, Georgia 67 Productivity of Riccati Differential Equations Jaroslav Jaroš Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842-48 Bratislava, Slovakia E-mail: jaros@fmph.uniba.sk Takaŝi Kusano Department of Mathematics, Faculty of Science, Hiroshima University, 739-8526 Higashi-Hiroshima, Japan E-mail: kusanot@zj8.so-net.ne.jp Department of Mathematical Sciences, Osaka Prefecture University, 599-8531 Osaka, Japan E-mail: ttanigawa@ms.osakafu-u.ac.jp Tomoyuki Tanigawa 1 Introduction Consider the second order half-linear differential equation (p(t)φ(x′))′ + q(t)φ(x) = 0; (E) where is a positive constant, p(t) and q(t) are positive continuous functions on [a; 1), a 0, and φ(u) = juj sgn u, u 2 R. We assume that equation (E) is nonoscillatory.
Given a solution x(t) of (E) we call the function p(t)φ(x′(t)) the quasi-derivative of x(t) and denote it by Dx(t).
If u(t) and v(t) are defined by u(t) = Dx(t) φ(x(t)) and v(t) = x(t) φ1=(Dx(t)) ; then they satisfy the first order nonlinear differential equations u′ = q(t) p(t) 1 juj1+ 1 ; v′ = p(t) 1 + 1 q(t)jvj1+; (R1) (R2) for all large t. Equations (R1) and (R2) are referred to as the first and the second Riccati equations associated with (E).

Note that (R2) has recently been discovered by Mirzov [3].
Conversely, suppose that (R1) and (R2) have solutions u(t) and v(t) defined for all large t, say on [T; 1).

Such solutions u(t) and v(t) are termed global solutions of (R1) and (R2), respectively.
Form the function x(t) on [T; 1) by one of the following formulas which are collectively called the reproducing formulas x(t) = exp ( t∫ T ) p(s) 1 φ 1 (u(s)) ds or x(t) = exp ( 1∫ t ) p(s) 1 φ 1 (u(s)) ds ; (1.1) 68 International Workshop QUALITDE – 2018, December 1 – 3, 2018, Tbilisi, Georgia x(t) = 1 (u(t)) exp φ 1 ( 1 t∫ T q(s) u(s) ) ds or x(t) = ( t∫ T ds 1 v(s) p(s) ) ) or x(t) = exp t ( x(t) = exp x(t) = v(t) exp ( 1 t∫ T 1 (u(t)) exp 1∫ ( 1 q(s) u(s) ) ds ; φ 1 ( 1∫ t ) ds 1 v(s) p(s) ; ) 1 1∫ t q(s)φ(v(s)) ds or x(t) = v(t) exp q(s)φ(v(s)) ds : (1.2) Then, x(t) gives a nonoscillatory solution of equation (E) on [T; 1).
This shows that equation (E) is nonoscillatory if and only if the Riccati equation (R1) (or (R2)) has a global solution.

We expect that the Riccati equations will be more productive in the sense that all nonoscillatory solutions of equation (E) can be reproduced from the global solutions of (R1) and/or (R2).

As a result of our efforts made in [2] it has turned out that a majority of solutions of (E) can really be reproduced by solving (R1) and (R2) by means of fixed point techniques.
Worthy of note is that both (R1) and (R2) are indispensable in the reproduction processes.

2 Main results We need the following notations: P(t) = t∫ a Ip = 1∫ a 1∫ p(t) 1 dt; Iq = q(t) dt; a 1∫ p(s) 1 ds if Ip = 1; (t) = p(s) 1 ds if Ip < 1; Q(t) = t∫ a t 1∫ q(s) ds if Iq = 1; (t) = q(s) ds if Iq < 1: t Of crucial importance is the following classification of nonoscillatory solutions of (E).
Let x(t) be a solution of (E) such that x(t)Dx(t) ̸= 0 on [T; 1).

Both x(t) and Dx(t) are monotone and have the limits x(1) = lim Dx(t) in the extended real number system.

t!1 The pair (x(1); Dx(1)), referred to as the terminal state of x(t), is a decisive indicator of the asymptotic behavior at infinity of a solution x(t) of (E).

All possible types of terminal states of solutions x(t) of (E) can be enumerated as follows.
x(t) and Dx(1) = lim t!1 (I) (The case where Ip = 1 ^ Iq < 1) (All solutions satisfy x(t)Dx(t) > 0) I(i): jx(1)j = 1, 0 < jDx(1)j < 1, I(ii): jx(1)j = 1, Dx(1) = 0, I(iii): 0 < jx(1)j < 1, Dx(1) = 0.

(II) (The case where Ip < 1 ^ Iq = 1) (All solutions satisfy x(t)Dx(t) < 0) II(i): 0 < jx(1)j < 1, jDx(1)j = 1, II(ii): x(1) = 0, jDx(1)j = 1, International Workshop QUALITDE – 2018, December 1 – 3, 2018, Tbilisi, Georgia 69 II(iii): x(1) = 0, 0 < jDx(1)j < 1.
(III) (The case where Ip < 1 ^ Iq < 1) III(i) = I(iii) (x(t)Dx(t) > 0), III(ii) = II(iii) (x(t)Dx(t) < 0), III(iii): 0 < jx(1)j < 1, 0 < jDx(1)j < 1) (x(t)Dx(t) > 0 or x(t)Dx(t) < 0).

The existence of solutions of the types I(i), I(iii), II(i) and II(iii) can be completely characterized.

Theorem 2.1.

Assume that Ip = 1 ^ Iq < 1.

(i) (E) has a solution of type I(i) if and only if 1∫ a q(t)P(t) dt < 1.

(ii) (E) has a solution of type I(iii) if and only if Theorem 2.2.
Assume that Ip < 1 ^ Iq = 1.

(i) (E) has a solution of type II(i) if and only if 1∫ a p(t) 1 (t) 1 dt < 1.

1∫ a p(t) 1 Q(t) 1 dt < 1.

(ii) (E) has a solution of type II(iii) if and only if 1∫ a q(t)(t) dt < 1.

Only the proofs of the “if” parts of Theorem 2.1 are outlined.

Proof of the “if” part of Theorem 2.1-(i).
Choose T > a so that 1)2+1, define the set 1∫ T q(s)P(s) ds =( + V = { v 2 CP[T; 1) : P(t) v(t) 2P(t); t T } ; where CP[T; 1) denotes the Banach space of all continuous functions w(t) on [T; 1) such that jw(t)j=P(t) is bounded with the norm ∥w∥P = supfjw(t)j=P(t) : t T g, and show that the integral operator given by Gv(t) = P(t) + 1 t∫ T q(s)jv(s)j+1 ds; t T; is a contraction such that ∥Gv1 Gv2∥P ∥v1 v2∥P for any v1; v2 2 V. Therefore, G has a unique fixed point v 2 V which gives a solution v(t) of (R2) on [T; 1) such that v(t) P(t) as t !
1.

With this v(t) define x(t) by the second formula in (1.2).

Then, it is a solution of (E) satisfying x(t) P(t) and Dx(t) 1 as t !

1.

 1 2 Proof of the “if” part of Theorem 2.1-(ii).
Choose T > a so that 1=( + 1)21+ 1 and consider the set { U = v 2 C0[T; 1) : (t) u(t) 2(t); t T } ; 1∫ T p(s) 1 (s) 1 ds 70 International Workshop QUALITDE – 2018, December 1 – 3, 2018, Tbilisi, Georgia where C0[T; 1) denotes the set of all continuous functions w(t) on [T; 1) tending to zero as t !

1.
1.

It is a Banach space with the sup-norm ∥w∥0 = supfjw(t)j : t T g. Show that the integral operator given by 1∫ F u(t) = (t) + p(s) 1 ju(s)j1+ 1 ds; t T; t ∥u1 u2∥0 for any u1; u2 2 U .

Let u 2 U be a unique is a contraction such that ∥F u1 F u2∥0 1 2 fixed point of F .

Then, it is a solution u(t) of (R1) on [T; 1) such that u(t) (t) as t !

1.
1.

Using this u(t) define x(t) according to the second reproducing formula of (1.1).

Then, it is a positive solution of (E) satisfying x(t) 1 and Dx(t) (t) as t !

1.

Note that any solution of the type III(iii) of (E) in the case Ip < 1 ^ Iq < 1 can also be reproduced from a suitable solution of (R1) or (R2).
As for solutions of the types I(ii) and II(ii) of (E), often referred to as intermediate solutions, In [2] we have very little is known about their existence and asymptotic behavior at infinity.

indicated several nontrivial cases of (E) whose intermediate solutions can actually be reproduced with the aid of (R1) and (R2).

Theorem 2.3.

(i) Assume that Ip = 1 ^ Iq < 1.


Equation (E) has an intermediate solution of the type I(ii) if 1∫ a p(t) 1 (t) 1 dt = 1; 1∫ a q(t)P(t) dt < 1: (ii) Assume that Ip < 1 ^ Iq = 1.

Equation (E) has an intermediate solution of the type II(ii) if 1∫ a q(t)(t) dt = 1; 1∫ a p(t) 1 Q(t) 1 dt < 1: Outline of proof of (i).

Let any constant A > 1 be given.


Put r(t) = 1∫ q(s)P(s) ds and choose t T > a so that r(T ) (A 1)A1.

Define the integral operator F by F u(t) = (t) + 1∫ t p(s) 1 ju(s)j1+ 1 ds; t T; and let it act on the set U defined by { u 2 C[T; 1) : (t) u(t) Ar(t)P (t); t T } ; U = which is a closed convex subset of the locally convex space C[T; 1).
Then, it can be shown that F is a continuous self-map of U sending U into a relatively compact subset of C[T; 1).

Therefore, by the Schauder–Tychonoff fixed point theorem there exists a u in U such that u = F u, which means that u(t) is a global solution of (R1).

With this u(t) apply the first reproducing formula of (1.1) to construct a positive solution x(t) of (E) on [T; 1).

This is an intermediate solution of the type I(ii) since it is easily verified that x(1) = 1 and Dx(1) = 0.

Remark.
Remark.

Some of our results are already known; see e.g., [1].

However, our approach based on the Riccati equations makes the asymptotic analysis of equation (E) much easier and clearer.

International Workshop QUALITDE – 2018, December 1 – 3, 2018, Tbilisi, Georgia 71 References [1] Z. Došlá and I. Vrkoč, On an extension of the Fubini theorem and its applications in ODEs.

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57 (2004), no.

4, 531–548.


[2] J. Jaroš, T. Kusano and T. Tanigawa, Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach.

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(to appear).

[3] J. D. Mirzov, Asymptotic Properties of Solutions of Systems of Nonlinear Nonautonomous Ordinary Differential Equations.

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