Properties of solutions of quaternionic Riccati equations
aa r X i v : . [ m a t h . C A ] F e b MSC 34C99
Properties of solutions of quaternionicRiccati equationsG. A. Grigorian
Institute of Mathematics NAS of ArmeniaE -mail: [email protected]
Abstract. In this paper we study properties of regular solutions of quaternionic Riccatiequations. The obtained results we use for study of the asymptotic behavior of solutionsof two first-order linear quaternionic ordinary differential equations.Key words: quaternions, the matrix representation of quaternions, quaternionic Riccatiequations, regular, normal and extremal solutions of Riccati equations, normal, irreconci-lable, sub extremal and super extremal systems, principal and non principal solutions.
1. Introduction . Let a ( t ) , b ( t ) , c ( t ) and d ( t ) be quaternionic-valued continuousfunctions on [ t , + ∞ ) , i.e.: a ( t ) ≡ a ( t ) + ia ( t ) + ja ( t ) + ka ( t ) , b ( t ) ≡ b ( t ) + ib ( t ) + jb ( t ) + kb ( t ) , c ( t ) ≡ c ( t ) + ic ( t ) + jc ( t ) + kc ( t ) , d ( t ) ≡ d ( t ) + id ( t ) + jd ( t ) + kd ( t ) , where a n ( t ) , b n ( t ) , c n ( t ) , d n ( t ) ( n = 0 , are real-valued continuous functionson [ t , + ∞ ) , i, j, k are the imaginary unities satisfying the conditions i = j = k = ijk = − , ij = − ji = k. (1 . Consider the quaternionic Riccati equation q ′ + qa ( t ) q + b ( t ) q + qc ( t ) + d ( t ) = 0 , t ≥ t . (1 . Particular cases of this equation appear in various problems of mathematics, in particularin problems of mathematical physics (e. g., in the Euler’s vorticity dynamics [1], in theEuler’s fluid dynamics [2], in the problem of classification of diffeomorphisms of S [3], andin the other ones [4, 5]). A quaternionic-valued function q = q ( t ) , defined on [ t , t ) ( t ≤ t < t ≤ + ∞ ) is called a solution of Eq. (1.2) on [ t , t ) , if it is continuously differentiableon [ t , t ) and satisfies (1.2) on [ t , t ) . It follows from the general theory of ordinarydifferential equations that for every t ≥ t and γ ∈ H (here and after H denotes thealgebra of quaternions) there exists t > t ( t ≤ + ∞ ) such that Eq. (1.2) has the uniquesolution q ( t ) on [ t , t ) , satisfying the initial condition q ( t ) = γ . Thus for every t ≥ t γ ∈ H a solution q ( t ) of Eq. (1.2) with q ( t ) = γ exists or else on some finite interval [ t , t ) or else on [ t , + ∞ ) . In the last case the solution q ( t ) we will call a t -regular (orsimply regular) solution of Eq. (1.2). Notice that some sufficient conditions for existenceof regular solutions are obtained in the works [1], [6], [7]. In the real case properties ofregular solutions of Eq. (1.2) are studied in [8] and have found several applications (see [9- 13]). In this paper we study the properties of regular solutions of Eq. (1.2). We use theobtained result to study the asymptotic behavior of solutions of systems of two first-orderlinear quaternionic differential equations.
2. Auxiliary propositions . It is not difficult to verify that there exists a one to onecorrespondence q ↔ Q between the quaternions q = q + iq + jq + kq , q k ∈ R , k = 0 , and the skew symmetric matrices Q ≡ q q q − q − q q − q − q − q q q q q q − q q , keeping the arithmetic operations: q m ↔ Q m , m = 1 , ⇒ q + q ↔ Q + Q ,q q ↔ Q Q , q − ↔ Q − ( q = 0) . The matrix Q we will call the symbol of q and will denote by b q . By | q | we denote the euclidian norm of the vector q : | q | ≡ p q + q + q + q . We also denote Re q ≡ q - the real part of q and Im q ≡ iq + jq + kq - the imaginary part of q . Finally by tr b q we denote the trace of b q . Lemma 2.1.
For every quaternion q the equalities det b q = | q | , tr b q = 4 Re q are valid.
Proof. By direct checking.Let A ( t ) , B ( t ) , C ( t ) and D ( t ) be the symbols of a ( t ) , b ( t ) , c ( t ) and d ( t ) respectively.Consider the matrix Riccati equation Y ′ + Y A ( t ) Y + B ( t ) Y + Y C ( t ) + D ( t ) = 0 , t ≥ t . (2 . Obviously the solutions q ( t ) of Eq. (1.2), existing on an interval [ t , t ) ( t ≤ t < t ≤ + ∞ ) are connected with solutions Y ( t ) of Eq. (2.1) by relation d q ( t ) = Y ( t ) , t ∈ [ t , t ) . (2 . Y ( t ) be a solution of Eq. (2.1) on [ t , t ) Then every solution Y ( t ) of Eq. (2.1) on [ t , t ) is connected with Y ( t ) by the formula (see [14], pp. 139, 140, 158, 159, Theorem 6.2) Y ( t ) = Y ( t ) + [Φ Y ( t )Λ − ( t )( I + Λ( t ) M Y ( t , t ))Ψ Y ( t )] − , t ∈ [ t , t ) , where Φ Y ( t ) and Ψ Y ( t ) are the solutions of the linear matrix equations Φ ′ = [ A ( t ) Y ( t ) + C ( t )]Φ , t ∈ [ t , t ) , Ψ ′ = Ψ[ B ( t ) + Y ( t ) A ( t )] , t ∈ [ t , t ) respectively with Φ Y ( t ) = Ψ Y ( t ) = I, I is the identity matrix of dimension × , M Y ( t , t ) ≡ t Z t Φ − Y ( τ ) A ( τ )Ψ − Y ( τ ) dτ, t ∈ [ t , t ) , Λ( t ) ≡ Y ( t ) − Y ( t ) , provided det Λ( t ) = 0 . From here we obtain Y ( t ) = Y ( t ) + Ψ − Y ( t )[ I + Λ( t ) M Y ( t , t )] − Λ( t )Φ − Y ( t ) , t ∈ [ t , t ) . (2 . By the Liouville formula we have: det Φ Y ( t ) = exp (cid:26) t Z t tr [ A ( τ ) Y ( τ ) + C ( τ )] dτ (cid:27) , t ∈ [ t , t ) , (2 . det Ψ Y ( t ) = exp (cid:26) t Z t tr [ A ( τ ) Y ( τ ) + B ( τ )] dτ (cid:27) , t ∈ [ t , t ) , (2 . Let q ( t ) be a solution of Eq. (1.2) on [ t , t ) . Then due to (2.2) from (2.3) it follows thatfor every solotion q ( t ) of Eq. (1.2) on [ t , t ) the equality q ( t ) = q ( t ) + ψ − q ( t )[1 + λ ( t ) µ q ( t , t )] − λ ( t ) φ − q ( t ) , t ∈ [ t , t ) (2 . is valid, where φ q ( t ) and ψ q ( t ) are the solutions of the linear equations φ ′ = [ a ( t ) q ( t ) + c ( t )] φ, t ∈ [ t , t ) ,ψ ′ = ψ [ b ( t ) + q ( t ) a ( t )] , t ∈ [ t , t ) φ q ( t ) = ψ q ( t ) = 1 , λ ( t ) ≡ q ( t ) − q ( t ) ,µ q ( t , t ) ≡ t Z t φ − q ( τ ) a ( τ ) ψ − q ( τ ) dτ, t ∈ [ t , t ) . By (2.3) and Lemma 2.1 from (2.5) and (2.6) we obtain | φ q ( t ) | = exp (cid:26) t Z t Re [ a ( τ ) q ( τ ) + c ( τ )] dτ (cid:27) , t ∈ [ t , t ) , (2 . | ψ q ( t ) | = exp (cid:26) t Z t Re [ a ( τ ) q ( τ ) + b ( τ )] dτ (cid:27) , t ∈ [ t , t ) . (2 . Let q m ( t ) , m = 1 , be solutions of Eq. (1.2) on [ t , t ) . Set: λ m,s ( t ) ≡ q m ( t ) − q s ( t ) ,m, s = 1 , . By (2.4) we have a ( t )[ q m ( t ) − q s ( t )] = a ( t ) ψ − q s ( t )[1 + λ m,s ( t ) µ q s ( t ; t )] − φ − q s ( t ) , t ∈ [ t , t ) . Hence, [1 + λ m,s ( t ) µ q s ( t ; t )] ′ = A q m ,q s ( t ; t )[1 + λ m,s ( t ) µ q s ( t ; t )] , t ∈ [ t , t ) , where A q m ,q s ( t ; t ) ≡ λ m,s ( t ) ψ − q s ( t )[ q m ( t ) − q s ( t )] φ − q s ( t ) λ − m,s ( t ) , t ∈ [ t , t ) , m = 1 , . From here it follows [ I + \ λ m,s ( t ) \ µ q s ( t ; t )] ′ = \ A q m ,q s ( t ; t )[ I + \ λ m,s ( t ) \ µ q s ( t ; t )] , t ∈ [ t , t ) , m = 1 , . By Lemma 2.1 and the Liouville’s formula from here we obtain | λ m,s ( t ) µ q s ( t ; t ) | = exp (cid:26) t Z t Re [ a ( τ )( q m ( τ ) − q s ( τ ))] dτ (cid:27) , t ∈ [ t , t ) , (2 . m, s = 1 , . From here we immediately get: | λ m,s ( t ) µ q s ( t ; t ) || λ s,m ( t ) µ q m ( t ; t ) | ≡ , t ∈ [ t , t ) , m, s = 1 , . (2 . . Properties of regular solutions of Eq. (1.2).Definition 3.1. A t -regular solution q ( t ) of Eq. (1.2) is called t -normal if thereexists a neighborhood U ( q ( t )) of q ( t ) such that every solution e q ( t ) of Eq. (1.2) with e q ( t ) ∈ U ( q ( t )) is also t -regular, otherwise q ( t ) is called t -extremal. Definition 3.2.
Eq. (1.2) is called regular if it has at least one regular solution.
Remark 3.1.
Since the solutions of Eq. (1.2) are continuously dependent on theirinitial values every t -normal ( t -extremal) solution of Eq. (1.2) is also a t -normal ( t -extremal) solution of Eq. (1.2) for all t > t . Due to this a t -normal ( t -extremal)solution of Eq. (1.2) we will just call a normal (a extremal) solution of Eq. (1.2). Notethat a t -normal ( t -extremal) solution of Eq. (1.2) may not be a t -normal ( t -extremal)solution of Eq.(1.2) if t < t , because a t -regular solution of Eq. (1.2) may not be t -regular for t < t . Theorem 3.1.
If Eq. (1.2) has a t -regular solution q ( t ) for some t ≥ t , then it hasalso another (different from q ( t ) ) t -regular solution. Proof. Let q ( t ) be a t -regular solution for some t ≥ t . Since µ q ( t ; t ) is continuouslydifferentiable by t there exists γ ∈ H \{ } such that µ q ( t ; t ) = γ for all t ≥ t ( µ q ( t ; t ) = 0 and the curve f ( t ) ≡ µ q ( t ; t ) , t ≥ t is not space filling). Therefore by (2.7) the solution q ( t ) of Eq. (1.2) with q ( t ) = q ( t ) − γ is a t -regular solution of Eq. (1.2), different from q ( t ) . The theorem is proved.Denote by Q ( t ; t ; λ ) the general solution of Eq. (1.2) in the region G t ≡ { ( t ; q ) : t ∈ I t ( λ ) , q, λ ∈ H } , where I t is the maximum existence interval for the solution q ( t ) of Eq. (1.2) with q ( t ) = λ . Example 3.1.
Consider the equation q ′ + qa ( t ) q = 0 , t ≥ − . (3 . The general solution of this equation in the region G ∩ [ − , + ∞ ) × H is given by formula Q ( t ; 0; λ ) = 11 + λ t R t a ( τ ) dτ λ, λ ∈ H , λ t Z t a ( τ ) dτ = 0 , t ≥ t . (3 . Assume a ( t ) has a bounded support. Then from (3.2) is seen that Eq. (3.1) has no -extremal solution, and all its solutions Q ( t, ; 0; λ ) with enough small | λ | are -normal. If a ( t ) is a non negative function with an unbounded support and I ≡ + ∞ R a ( τ ) dτ < + ∞ then from (3.2) is seen that the solution q ( t ) = Q ( t ; 0; − I ) is -extremal; all the solutions Q ( t ; 0; λ ) with λ ∈ H \ ( −∞ , − I ) are -normal and all the solutions Q ( t ; 0; λ ) with λ ∈ −∞ , − I ) are not -regular. Assume now t R a ( τ ) dτ = arctan(cos t + i sin t + j cos πt + k sin πt ) , t ≥ . Then from (3.2) is seen that all the solutions Q ( t ; 0; λ ) with | λ | = √ π are -extremal (since the set { √ (cos t + i sin t + j cos πt + k sin πt ) : t ≥ } is everywhere densein the unite sphere { q : | q | = 1 } ) and all solutions Q ( t ; 0; λ ) with | λ | < √ π are -normal. Example 3.2.
For u ∈ H and < r < R < + ∞ denote K r,R ( u ) ≡ { q ∈ H : r < | q − u | < R } - an annulus in H with a center u and radiuses r and R . For any ε > denote K ε,r,R ( u ) ≡ { ξ , ..., ξ m ∈ K r,R ( u ) : if u ∈ K r,R ( u ) then there exists s ∈ { , ..., m } such that | u − ξ s | < ε } - a finite ε -net for K r,R ( u ) (here m depends on ε ). Consider thesequence of n -nets: { K n , n ,n ( u ) } + ∞ n =1 . Let the function f ( t ) ≡ t R a ( τ ) dτ, t ≥ has thefollowing properties: f ( t ) = u , t ∈ [0 , ; when t varies from n to n + 1 ( n = 1 , , ... ) thecurve f ( t ) crosses all points of K n , n ,n ( u ) (i. e. for every v ∈ K n , n ,n ( u ) there exists ζ v ∈ [ n, n + 1] such that f ( ζ v ) = v ); f ( t ) ∈ K n , + ∞ ( u ) n = 1 , , ...., t ≥ . From theseproperties it follows that for every T ≥ the set { f ( t ) : t ≥ T } is everywhere dense in H and f ( t ) = u , t ≥ . Hence from (3.2) it follows that Eq. (3.1) has no t -normalsolutions for all t ≥ and has at least two extremal solutions: q ( t ) ≡ and q ( t ) with q (0) = − u . By analogy using n -nets K n , n ,n ( u ; ...u l ) ≡ { ξ , ..., ξ m ∈ l T k =0 K n ,n ( u k ) : u ∈ l T k =0 K n ,n ( u k ) ⇒ ∃ s ∈ { , ..., m } : | u − ξ s | < n } of the intersections l T k =1 K n ,n ( u k ) inplace of K n , n ,n ( u ) , n = 1 , , .... one can show that there exists a Riccati equation whichhas no t -normal solutions and has at least l + 2 t -extremal solutions for all t ≥ . Theorem 3.2. A t -regular solution q ( t ) of Eq. (1.2) is t -normal if and only if µ q ( t ; t ) is bounded by t . Proof. Sufficiency. Set M ≡ sup t ≥ t | µ q ( t ; t ) | . Let q ( t ) be a solution of Eq. (1.2) with | q ( t ) − q ( t ) | < M . Then obviously q ( t ) − q ( t )) µ q ( t ; t ) = 0 , t ≥ t . By (2.7) from here it follows that q ( t ) is t -normal.Necessity. Suppose µ q ( t ; t ) is unbounded by t on [ t , + ∞ ) . Let then t < t < ...t m , ... be an infinitely large sequence such that | µ q ( t ; t n ) | ≥ n, n = 2 , , ... (3 . Let q n ( t ) , n = 2 , , ... be the solutions of Eq. (1.2) with q n ( t ) − q ( t ) = − µ q ( t ; t n ) − , n = 2 , , ... (3 . q ( t ) is t -normal there exists δ > such that every solution e q ( t ) of Eq. (1.2) with | e q ( t ) − q ( t ) | < δ is t -regular. Hence from (3.3) and (3.4) it follows that for enough large n the solutions q n ( t ) are t -regular. On the other hand by (2.7) from (3.4) it follows thatfor enough large n every solution q n ( t ) is unbounded in the neighborhood of t n . It meansthat for enough large n the solutions q n ( t ) are not t -regular. The obtained contradictioncompletes the proof of the theorem.By (2.10) from Theorem 3.2 we immediately obtain Corollary 3.1.
The following statements are valid:1) any two t -regular solutions q ( t ) and q ( t ) of Eq. (1.2) are t -normal if and only if thefunction I q ,q ( t ) ≡ t Z t Re [ a ( τ )( q ( τ ) − q ( τ ))] dτ, t ≥ t is bounded;2) if q N ( t ) and q ∗ ( t ) are t -normal and t -extremal solutions of Eq. (1.2) respectively then lim sup t → + ∞ t Z t Re [ a ( τ )( q ∗ ( τ ) − q N ( τ ))] dτ < + ∞ , lim inf t → + ∞ t Z t Re [ a ( τ )( q ∗ ( τ ) − q N ( τ ))] dτ = −∞ ;
3) if q ∗ ( t ) and q ∗ ( t ) are t -extremal solutions of Eq. (1.2) then lim sup t → + ∞ t Z t Re [ a ( τ )( q ∗ ( τ ) − q ∗ ( τ ))] dτ = + ∞ , lim inf t → + ∞ t Z t Re [ a ( τ )( q ∗ ( τ ) − q ∗ ( τ ))] dτ = −∞ . (cid:4) Definition 3.3.
A regular Eq. (1.2) is called normal if it has no extremal solutions.
Definition 3.4.
A regular Eq. (1.2) is called irreconcilable if its every regular solutionis extremal.
Definition 3.5 . A regular Eq. (1.2) is called sub extremal if it has only one extremalsolution. efinition 3.6. A regular Eq. (1.2) is called super extremal if it has at least twoextremal solutions and normal solutions.
From Definitions 3.3 - 3.6 is seen that every regular Eq. (1.2) is or else normal orelse irreconcilable or else sub extremal or else super extremal. The examples, illustratedabove, show that all these types of equations exist.For any t -regular solution q ( t ) of Eq. (1.2) set ν q ( t ) ≡ + ∞ Z t φ − q ( τ ) a ( τ ) ψ − q ( τ ) dτ, t ≥ t , where φ q ( t ) and ψ q ( t ) are the solutions of the linear equations φ ′ = [ a ( t ) q ( t ) + c ( t )] φ, t ≥ t .ψ ′ = ψ [ b ( t ) + q ( t ) a ( t )] , t ≥ t respectively with φ q ( t ) = ψ q ( t ) = 1 . Theorem 3.3.
Let q ( t ) be a t -regular solution of Eq. (1.2) such that the integral ν q ( t ) is convergent. Then in order that Eq. (1.2) has a t -extremal solution it is necessaryand sufficient that ν q ( t ) = 0 , t ≥ t . If this condition is satisfied then:1) the unique t -extremal solution q ∗ ( t ) of Eq. (1.2) is given by the formula q ∗ ( t ) = q ( t ) − ν q ( t ) , t ≥ t ; (3 .
2) for all t -normal solutions q ( t ) of Eq. (1.2) and only for them the integrals ν q ( t ) convergefor all t ≥ t and ν q ( t ) = 0 , t ≥ t ;
3) for all t ≥ t ν q ∗ ( t ) = ∞ ; (3 .
4) for two arbitrary t -normal solutions q ( t ) and q ( t ) the integral + ∞ Z t Re [ a ( τ )( q ( τ ) − q ( τ ))] dτ converges;5) for every t -normal solution q N ( t ) of Eq. (1.2) the equality + ∞ Z t Re [ a ( τ )( q ∗ ( τ ) − q N ( τ ))] dτ = −∞ (3 . s valid. Proof. Let q ( t ) be a t -regular solution of Eq. (1.2) for which ν q ( t ) converges and ν q ( t ) = 0 t ≥ t . Then − ν q ( t ) µ q ( t ; t ) = 0 , t ≥ t . (3 . Indeed otherwise if for some t > t ν q = µ q ( t ; t ) then from the equality ν q ( t ) = µ q ( t ; t ) + ν q ( t ) it follows that ν q ( t ) = 0 , which contradicts our assumption. Let q ∗ ( t ) be the solution of Eq. (1.2) with q ∗ ( t ) = q ( t ) − ν q ( t ) . Then by (2.7) from (3.8) it followsthat q ∗ ( t ) is t -regular and according to (2.11) we have (cid:12)(cid:12)(cid:12)(cid:12) ν q ∗ ( t ) µ q ∗ ( t ; t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ν q ( t ) µ q ( t ; t ) (cid:12)(cid:12)(cid:12)(cid:12) ≡ , t ≥ t . From here it follows ν q ∗ ( t ) = lim t → + ∞ µ q ∗ ( t ; t ) = ∞ . Then by virtue of Theorem 3.2 q ∗ ( t ) is t -extremal and (3.6) is valid. Assume now Eq. (1.2) has a t -extremal solution q ∗ ( t ) .Show that ν q ( t ) = 0 , t ≥ t . Suppose for some t ≥ t ν q ( t ) = 0 . Then obviously lim t → + ∞ [1 + ( q ∗ ( t ) − q ( t )) µ q ( t ; t )] = 1 . (3 . By (2.11) we have | q ( t ) − q ∗ ( t )) µ q ∗ ( t ; t ) || q ∗ ( t ) − q ( t )) µ q ( t ; t ) | ≡ , t ≥ t . This together wit (3.9) implies that µ q ∗ ( t ; t ) is bounded by t on [ t , + ∞ ) . Therefore µ q ∗ ( t ; t ) is bounded by t on [ t , + ∞ ) , and according to Theorem 3.2 q ∗ ( t ) is t -normal,which contradicts our assumption. The obtained contradiction shows that ν q ( t ) = 0 ,t ≥ t . Let us prove (3.5). Suppose for some t ≥ t q ∗ ( t ) = q ( t ) − ν q ( t ) . Then there exists a finite limit lim t → + ∞ [1 + ( q ∗ ( t ) − q ( t )) µ q ( t ; t )] = 0 . (3 . By (2.11) we have | q ( t ) − q ∗ ( t )) µ q ∗ ( t ; t ) || q ∗ ( t ) − q ( t )) µ q ( t ; t ) | ≡ , t ≥ t . µ q ∗ ( t ; t ) is bounded by t on [ t , + ∞ ) . Therefore µ q ∗ ( t ; t ) is bounded by t on [ t , + ∞ ) . By virtue of Theorem 3.2 from here it follows that q ∗ ( t ) is t -normal, which contradicts our assumption. The obtained contradiction proves(3.5).Let q ( t ) be a t -normal solution of Eq. (1.2). By (2.11) we have | q ( t ) − q ∗ ( t )) µ q ∗ ( t ; t ) || q ∗ ( t ) − q ( t )) µ q ( t ; t ) | ≡ , t ≥ t . This together with (3.6) implies lim t → + ∞ [1 + ( q ∗ ( t ) − q ( t )) µ q ( t ; t )] = 0 . Therefore the integrals ν q ( t ) converge for all t ≥ t . The inequality ν q ( t ) = 0 , t ≥ t follows immediately from the already proven necessary condition of existence of a t -extremal solution of Eq. (1.2).Let q ( t ) and q ( t ) be t -normal solutions of Eq. (1.2). By (2.10) we have | q ( t ) − q ( t )) µ q ( t ; t ) | = exp (cid:26) t Z t Re [ a ( τ )( q ( τ ) − q ( τ ))] dτ (cid:27) , t ≥ t . From here and from the convergence of ν q ( t ) it follows the convergence of the integral + ∞ Z t Re [ a ( τ )( q ( τ ) − q ( τ ))] dτ. Let q N ( t ) be a t -normal solution of Eq. (1.2). By (2.10) we have | q ( t N ) − q ∗ ( t )) µ q ∗ ( t ; t ) | = exp (cid:26) t Z t Re [ a ( τ )( q ∗ ( τ ) − q N ( τ ))] dτ (cid:27) , t ≥ t . This together with (3.6) implies (3.7). The theorem is proved.
Corollary 3.2.
Let Eq. (1.2) have a t -regular solution q ∗ ( t ) such that ν q ∗ ( t ) = ∞ . Then the statements 1) - 5) of Theorem 3.3 are valid.
Proof. By Theorem 3.3 it is enough to show that Eq. (1.2) has a t -regular solution q ( t ) such that ν q ( t ) converges and ν q ( t ) = 0 , t ≥ t . Let q ( t ) be a t -regular solutionof Eq. (1.2), different from q ∗ ( t ) . In virtue of (2.11) we have | q ( t ) − q ∗ ( t )) µ q ∗ ( t ; t ) || q ∗ ( t ) − q ( t )) µ q ( t ; t ) | ≡ , t ≥ t . (3 . lim t → + ∞ | q ( t ) − q ∗ ( t )) µ q ∗ ( t ; t ) | = + ∞ From here and from (3.11) it follows that q ( t ) is t -normal and the integral ν q ( t ) converges. Moreover by virtue of Theorem 3.2 from the condition of the corollary it followsthat q ∗ ( t ) is t -extremal. Since q ( t ) is an arbitrary t -regular solution of Eq. (1.2), differentfrom q ∗ ( t ) it follows that q ∗ ( t ) is the unique t -extremal solution of Eq. (1.2). Then byTheorem 3.3 ν q ( t ) = 0 , t ≥ t . The corollary is proved.Theorem 3.3 and Corollary 3.2 allow us to give the following equivalent definitions.
Definition 3.7.
Eq. (1.2) is called extremal if for some t ≥ t it has a t -regularsolution q ( t ) such that ν q ( t ) converges and ν q ( t ) = 0 , t ≥ t . Definition 3.8.
Eq. (1.2) is called extremal if for some t ≥ t it has a t -regularsolution q ( t ) such that ν q ( t ) = ∞ . Example 3.3.
Let λ ( t ) be a quaternionic valued continuously differentiable function on [ t , + ∞ ) , α ( t ) ≡ α ( t ) + iα ( t ) , β ( t ) ≡ β ( t ) + jβ ( t ) , t ≥ t , where α ( t ) , α ( t ) , β ( t ) and β ( t ) are some real-valued continuous functions on [ t , + ∞ ) . Consider the Riccatiequation q ′ + qa ( t ) q − [ λ ( t ) a ( t ) + α ( t )] q − q [ a ( t ) λ ( t ) + β ( t )] − λ ′ ( t ) ++ λ ( t ) a ( t ) λ ( t ) + α ( t ) λ ( t ) + λ ( t ) β ( t ) = 0 , t ≥ t . (3 . It is not difficult to verify that q = λ ( t ) is a t -regular solution of this equation and φ λ ( t ) = exp (cid:26) − t Z t β ( τ ) dτ (cid:27) , ψ λ ( t ) = exp (cid:26) − t Z t α ( τ ) dτ (cid:27) , t ≥ t . So ν λ ( t ) = + ∞ Z t exp (cid:26) τ Z t β ( s ) (cid:27) a ( τ ) exp (cid:26) τ Z t α ( s ) ds (cid:27) dτ, t ≥ t . Therefore if ν λ ( t ) converges and ν λ ( t ) = 0 , t ≥ t t for some t ≥ t or if ν λ ( t ) = ∞ ,then Eq. (3.12) is extremal. If ν λ ( t ) converges and ν λ ( t ) has arbitrary large zeroes, thenEq. (1.2) is normal. Obviously every extremal Eq. (1.2) is sub extremal. The next example shows that notall sub extremal equations are extremal. 11 xample 3.4.
Consider the Riccati equation q ′ + q ( t cos t ) q = 0 , t ≥ t , t sin t + cos t = 0 . (3 . For every λ ∈ H the solution q ( t ) of this equation with q ( t ) = λ has the form q ( t ) = 11 + λ t R t τ cos τ dτ λ = 11 + λ ( t sin t + cos t ) λ, λ ( t sin t + cos t ) = 0 . Hence every solution q ( t ) of this equation with q ( t ) ∈ H \ ( R \{ } ) is t -regular and for q ( t ) ∈ R \{ } q ( t ) is not t -regular. Therefore q ( t ) ≡ is a t -extremal solution of Eq.(3.13) and all its solutions q ( t ) with q ( t ) ∈ H \ R are t -normal. From here it follows thatEq. (3.13) is sub extremal. Obviously the integral ν q ( t ) = + ∞ Z t t cos tdt neither is convergent nor divergent to ∞ . Therefore Eq. (3.13) is not extremal.
4. The asymptotic behavior of solutions of systems of two first-order linearquaternionic ordinary differential equations . Let a ml ( t ) , m, l = 1 , be quaternionic-valued continuous functions on [ t , + ∞ ) . Consider the linear system φ ′ = a ( t ) φ + a ( t ) ψ,ψ ′ = a ( t ) φ + a ( t ) ψ, t ≥ t (4 . and the quaternionic Riccati equation q ′ + qa ( t ) q + qa ( t ) − a ( t ) q − a ( t ) = 0 , t ≥ t ) . (4 . It is not difficult to verify that the solutions q ( t ) of Eq. (4.2), existing on some interval [ t , t ) ( t ≤ t < t ≤ + ∞ ) are connected with solutions ( φ ( t ) , ψ ( t )) of the system (4.1)by relations φ ′ ( t ) = [ a ( t ) q ( t ) + a ( t )] φ ( t ) , ψ ( t ) = q ( t ) φ ( t ) , t ∈ [ t , t ) . (4 . From here it follows d φ ( t ) ′ = [ \ a ( t ) d q ( t ) + \ a ( t )] d φ ( t ) , t ∈ [ t , t ) .
12y Liouville’s formula from here we obtain det d φ ( t ) = det [ φ ( t ) exp (cid:26) t Z t tr [ \ a ( τ ) d q ( t ) + \ a ( τ )] dτ (cid:27) , t ∈ [ t ; t ) . ] By virtue of Lemma 2.1 from here it follows | φ ( t ) | = | φ ( t ) | exp (cid:26) t Z t Re [ a ( τ ) q ( τ ) + a ( τ )] dτ (cid:27) , t ∈ [ t , t ) . (4 . So if φ ( t ) = 0 , then φ ( t ) = 0 , t ∈ [ t , t ) . (4 . Remark 4.1.
It can be shown that if for a solution ( φ ( t ) , ψ ( t )) of the system (4.1) thefunction φ ( t ) does not vanish on [ t , t ) then q ( t ) = ψ ( t ) φ − ( t ) , t ∈ [ t , t ) is a solutionof Eq. (4.2) on [ t , t ) . Definition 4.1.
A solution ( φ ( t ) , ψ ( t )) of the system (4.1) is called t -regular ( t ≥ t )if φ ( t ) = 0 , t ≥ t . Definition 4.2. A t -regular ( t ≥ t ) solution ( φ ( t ) , ψ ( t )) of the system (4.1) is calledprincipal (non principal) if q ( t ) ≡ ψ ( t ) φ − ( t ) , t ≥ t is a t -extremal ( t -normal) solutionof Eq. (4.2). Definition 4.3.
The system (4.1) is called regular if it has at least one t -regularsolution for some t ≥ t . Remark 4.2.
It follows from (4.5) and Remark 4.1 that the system (4.1) has a t -regular solution for some t ≥ t if and only if Eq. (4.2) has a t -regular solution. Remark 4.3. If ( φ ( t ) , ψ ( t )) is a solution of the system (4.1) then for every λ ∈ H ( φ ( t ) λ, ψ ( t ) λ ) is also a solution of the system (4.1), but ( λφ ( t ) , λψ ( t )) may not be asolution of the system (4.1). For example ( e it , e kt ) , t ≥ t is a solution of the system φ ′ = iφ,ψ ′ = kψ, t ≥ t but ( je it , je kt ) , t ≥ t is not a solution of this system. Definition 4.4.
The solutions ( φ m ( t ) , ψ m ( t )) , m = 1 , are called linearly dependentif there exists λ ∈ H \{ } such that φ ( t ) = φ ( t ) λ, ψ ( t ) = ψ ( t ) λ , otherwise they arecalled linearly independent. emark 4.4. It follows from Theorem 3.1 and Remark 4.1 that if the system (4.1)has a t -regular solution ( φ ( t ) , ψ ( t )) , then it has also another t -regular solution, linearlyindependent of ( φ ( t ) , ψ ( t )) . Definition 4.5.
The regular system (4.1) is called normal (irreconcilable, sub extremal,super extremal, extremal) if Eq. (4.2) is normal (irreconcilable, sub extremal, superextremal, extremal).
Hereafter every t -regular solution of the system (4.1) we will just call a regular solutionof the system (4.1). On the basis of (4.4) from Corollary 3.1 we immediately get. Theorem 4.1.
The following statements are valid:I) if the system (4.1) is normal then for its two regular solutions ( φ m ( t ) , ψ m ( t )) , m = 1 , the inequalities lim sup t → + ∞ | φ ( t ) || φ ( t ) | < + ∞ , lim sup t → + ∞ | φ ( t ) || φ ( t ) | < + ∞ are valid;II) if the system (4.1) is irreconcilable then for its two arbitrary linearly independentregular solutions ( φ m ( t ) , ψ m ( t )) , m = 1 , the equalities lim sup t → + ∞ | φ ( t ) || φ ( t ) | = lim sup t → + ∞ | φ ( t ) || φ ( t ) | = + ∞ are valid;III) If the system (4.1) is sub extremal then there exists a regular solution ( φ ∗ ( t ) , ψ ∗ ( t )) of (4.1) such that for every regular solutions ( φ m ( t ) , ψ m ( t )) , m = 1 , of (4.1) linearlyindependent of ( φ ∗ ( t ) , ψ ∗ ( t )) the relations lim sup t → + ∞ | φ ∗ ( t ) || φ ( t ) | < + ∞ , lim inf t → + ∞ | φ ∗ ( t ) || φ ( t ) | = 0 , lim sup t → + ∞ | φ ( t ) || φ ( t ) | < + ∞ , lim sup t → + ∞ | φ ( t ) || φ ( t ) | < + ∞ are valid;IV) if the system (4.1) is super extremal then there exist two regular solutions ( φ ∗ ( t ) , ψ ∗ ( t )) and ( φ ∗ ( t ) , ψ ∗ ( t )) of (4.1) such that lim sup t → + ∞ | φ ∗ ( t ) || φ ∗ ( t ) | = lim sup t → + ∞ | φ ∗ ( t ) || φ ∗ ( t ) | = + ∞ and for all two arbitrary solutions ( φ m ( t ) , ψ m ( t )) , m = 1 , of (4.1) linearly independentof each ( φ ∗ ( t ) , ψ ∗ ( t )) and ( φ ∗ ( t ) , ψ ∗ ( t )) the following relations are valid lim sup t → + ∞ | φ ( t ) || φ ( t ) | < + ∞ , lim sup t → + ∞ | φ ( t ) || φ ( t ) | < + ∞ , im sup t → + ∞ | φ ∗ ( t ) || φ m ( t ) | < + ∞ , lim sup t → + ∞ | φ ∗ ( t ) || φ m ( t ) | < + ∞ , lim inf t → + ∞ | φ ∗ ( t ) || φ m ( t ) | = lim inf t → + ∞ | φ ∗ ( t ) || φ m ( t ) | = 0 , m = 1 , . (cid:4) Theorem 4.1 shows that in the normal case of the system (4.1) all regular solutionsof (4.1) are asymptotically equivalent. This case differs from the other cases by thescarcity of asymptotic behavior patterns at + ∞ of the solutions of the system (4.1).In the supercritical case of (4.1) we have "the richest"(among the other cases) variety ofasymptotic behavior pattern at + ∞ of regular solutions of the system (4.1)Let a ( t ) = a ( t ) + ia ( t ) + ja ( t ) + ka ( t ) , − a ( t ) = b ( t ) + ib ( t ) + jb ( t ) + kb ( t ) ,a ( t ) = c ( t ) + ic ( t ) + jc ( t ) + kc ( t ) , − a ( t ) = d ( t ) + id ( t ) + jd ( t ) + kd ( t ) . where a m ( t ) , b m ( t ) , c m ( t ) and d m ( t ) , m = 0 , are real-valued continuous functions on [ t , + ∞ ) . Set: p ,m ( t ) ≡ b m ( t ) + c m ( t ) , m = 1 , p ( t ) ≡ b ( t ) + c ( t ) , p ( t ) ≡ b ( t ) − c ( t ) ,p ( t ) ≡ b ( t ) − c ( t ) , p ( t ) ≡ b ( t ) − c ( t ) ,p ( t ) ≡ b ( t ) + c ( t ) , p ( t ) ≡ b ( t ) − c ( t ) ,p m ( t ) ≡ b m ( t ) − c m ( t ) , m = 1 , , t ≥ t ,D ( t ) ≡ P m =1 p m ( t ) + 4 a ( t ) d ( t ) , if a ( t ) = 0 , d ( t ) if a ( t ) = 0 ,D n ( t ) ≡ P m =1 p nm ( t ) − a n ( t ) d n ( t ) , if a n ( t ) = 0 , − d n ( t ) if a n ( t ) = 0 , n − , , t ≥ t . Let S be a non empty subset of the set { , , , } and let D be its complement i. e. D = { , , , }\ S . Theorem 4.2 . Let the conditions ) a n ( t ) ≥ , t ≥ t , n ∈ S and if a n ( t ) = 0 then p nm ( t ) = 0 , m ∈ S , a n ( t ) ≡ ,n ∈ D , D n ( t ) ≤ , t ≥ t , n = 0 , ; β ) + ∞ R t | a ( τ ) | exp (cid:26) t R t h Re a ( s ) − Re a ( s ) i ds (cid:27) dτ < + ∞ .be satisfied. Then the following statements are valid:1) the system (4.1) is or else normal or else extremal:2) for all T -regular ( T ≥ t ) non principal solutions ( φ ( t ) , ψ ( t )) of the system (4.1) theintegral + ∞ Z T | a ( τ ) || φ ( τ ) | exp (cid:26) τ Z T h Re a ( s ) + Re a ( s ) i ds (cid:27) dτ converges;3) if the system (4.1) is extremal, then: ) for its unique (up to arbitrary right multiplier) principal solution ( φ ∗ ( t ) , ψ ∗ ( t )) theequality + ∞ Z T ∗ | a ( τ ) || φ ∗ ( τ ) | exp (cid:26) τ Z T ∗ [ Re a ( s ) + Re a ( s )] ds (cid:27) dτ = + ∞ ; (4 . is valid, where T ∗ ≥ t such that φ ∗ ( t ) = 0 , t ≥ T ∗ ; ) for all non principal solutions ( φ ( t ) , ψ ( t )) of the system (4.1) the equality lim t → + ∞ | φ ∗ ( t ) || φ ( t ) || = 0 (4 . is valid; ) for two arbitrary non principal solutions ( φ m ( t ) , ψ m ( t )) , m = 1 , of the system (4.1)the relation lim t → + ∞ | φ ( t ) || φ ( t ) || = c = 0 (4 . is valid. To prove this theorem we need in the following result from [7] (see [7, Theorem 3.1])
Theorem 4.3 . Let the conditions α ) of Theorem 4.2 be satisfied. Then for all γ n ≥ ,n ∈ S , γ n ∈ ( −∞ , + ∞ ) , n ∈ D Eq. (4.2) has a solution q ( t ) = q , ( t ) − i q , ( t ) − j q , ( t ) − k q , ( t ) on [ t , + ∞ ) with q ,n ( t ) = γ n , n = 0 , and q ,n ( t ) ≥ , n ∈ S , t ≥ t . (cid:4) Proof of Theorem 4.2.
Let q ( t ) be the solution of Eq. (4.2) with q ( t ) = 0 . In virtueof Theorem 4.3 it follows from the conditions α ) of the theorem that q ( t ) is t -regular16nd Re [ a ( t ) q ( t )] ≥ , t ≥ t . (4 . Consider the integral e ν q ( t ) ≡ + ∞ Z t φ − q ( τ ) a ( τ ) ψ − q ( τ ) dτ, t ≥ t , where φ q ( t ) and ψ q ( t ) are the solutions of the linear equations φ ′ = [ a ( t ) q ( t ) + a ( t )] φ, t ≥ t , (4 . ψ ′ = ψ [ q ( t ) a ( t ) − a ( t )] , t ≥ t respectively with φ q ( t ) = ψ q ( t ) = 1 . By (2.7) and (2.8) we have respectively | φ q ( t ) | = exp (cid:26) t Z t Re [ a ( τ ) q ( τ ) + a ( τ )] (cid:27) , (4 . | ψ q ( t ) | = exp (cid:26) t Z t Re [ a ( τ ) q ( τ ) − a ( τ )] (cid:27) , t ≥ t . Hence, | e ν q ( t ) | ≤ + ∞ Z t | a ( τ ) || φ q ( τ ) || ψ q ( τ ) | dτ == + ∞ Z t | a ( τ ) | exp (cid:26) − τ Z t [2 Re a ( s ) q ( s ) + Re a ( s ) − Re a ( s )] ds (cid:27) dτ, t ≥ t . This together with (4.9) and β ) implies that | e ν q ( t ) | ≤ + ∞ Z t | a ( τ ) | exp (cid:26) τ Z t [ Re a ( s ) − Re a ( s )] ds (cid:27) dτ < + ∞ t ≥ t . (4 . It follows from here that the integrals e ν q ( t ) , t ≥ t converge. Two cases are possible:17 ) e ν q ( t ) has arbitrary large zeroes; b ) e ν q ( t ) = 0 , t ≥ T for some T ≥ t .Then by Theorem 3.3 the system (4.1) is or else normal (in the case a )) or else extremal (inthe case b )). The statement 1) of the theorem is proved. Let ( φ ( t ) , ψ ( t )) be the solutionof the system (4.1) with φ ( t ) = 1 , ψ ( t ) = 0 . Then by (4.3) φ ( t ) is a solution ofEq. (4.10). So φ ( t ) coincides with φ q ( t ) . Therefore from β ), (4.9) and (4.11) it follows + ∞ Z t | a ( τ ) || φ ( τ ) | exp (cid:26) τ Z t [ Re a ( s ) + Re a ( s )] ds (cid:27) dτ ≤≤ + ∞ Z t | a ( τ ) | exp (cid:26) τ Z t [ Re a ( s ) − Re a ( s )] ds (cid:27) dτ < + ∞ , t ≥ t . (4 . Let ( φ ( t ) , ψ ( t )) be a T -regular ( T ≥ t ) non principal solution of the system (4.1). Then q ( t ) ≡ ψ ( t ) φ − ( t ) , t ≥ T is a T -normal solution of Eq. (4.2). It follows from (4.12) that µ q ( T ; t ) is bounded on [ T, + ∞ ) . Hence, according to the statement 1) of Corollary 3.1we have sup t ≥ T (cid:12)(cid:12)(cid:12)(cid:12) t Z T Re [ a ( τ )( q ( τ ) − q ( τ ))] dτ (cid:12)(cid:12)(cid:12)(cid:12) < + ∞ . This together with (4.10) implies + ∞ Z T | a ( τ ) || φ ( τ ) | exp (cid:26) τ Z T [ Re a ( s ) + Re a ( s )] ds (cid:27) dτ == + ∞ Z T | a ( τ ) || φ ( τ ) | exp (cid:26) τ Z T [ Re a ( s )+ Re a ( s )] ds (cid:27) exp (cid:26) τ Z T Re [ a ( s )( q ( s ) − q ( s ))] ds (cid:27) dτ ≤≤ M + ∞ Z T | a ( τ ) || φ ( τ ) | exp (cid:26) τ Z t [ Re a ( s ) + Re a ( s )] ds (cid:27) dτ < + ∞ , where M ≡ exp (cid:26) − T Z t [ Re a ( s )+ Re a ( s )] ds (cid:27) exp (cid:26) t ≥ T (cid:12)(cid:12)(cid:12) τ Z t Re [ a ( s )( q ( s ) − q ( s ))] ds (cid:12)(cid:12)(cid:12)(cid:27) < + ∞ . q ∗ ( t ) . Let q ∗ ( t ) be T ∗ -regular for some T ∗ ≥ t and let ( φ ∗ ( t ) , ψ ∗ ( t )) be the solution of the system (4.1) with φ ∗ ( T ∗ ) = 1 , ψ ∗ ( T ∗ ) = q ∗ ( T ∗ ) . Thenby (4.3) ( φ ∗ ( t ) , ψ ∗ ( t )) is the unique (up to arbitrary right multiplier) principal solution ofthe system (4.1) and φ ∗ ( t ) is a solution of the linear equation φ ′ = [ a ( t ) q ∗ ( t ) + a ( t )] φ, t ≥ T ∗ . (4 . Consider the integral e ν q ∗ ( T ∗ ) ≡ + ∞ Z T ∗ φ − q ∗ ( τ ) a ( τ ) ψ − q ∗ ( τ ) dτ, where φ q ∗ ( t ) and ψ q ∗ ( t ) are the solutions of Eq. (4.14) and the equation ψ ′ = ψ [ q ∗ ( t ) a ( t ) − a ( t )] , t ≥ T ∗ respectively with φ q ∗ ( T ∗ ) = ψ q ∗ ( T ∗ ) = 1 . Since q ∗ ( t ) is extremal in virtue of Theorem 3.3we have e ν q ∗ ( T ∗ ) = ∞ . (4 . By (2.7) and (2.8) we have respectively | φ q ∗ ( t ) | = exp (cid:26) t Z T ∗ Re [ a ( τ ) q ∗ ( τ ) + a ( τ )] dτ (cid:27) , t ≥ T ∗ , | ψ q ∗ ( t ) | = exp (cid:26) t Z T ∗ Re [ a ( τ ) q ∗ ( τ ) − a ( τ )] dτ (cid:27) , t ≥ T ∗ . Therefore | ψ q ∗ ( t ) | = | φ q ∗ ( t ) | exp (cid:26) − t Z T ∗ Re [ a ( τ ) + a ( τ )] dτ (cid:27) , t ≥ T ∗ . (4 . Obviously φ ∗ ( t ) = φ q ∗ ( t ) , t ≥ T ∗ . This together with (4.16) implies | e ν q ∗ ( T ∗ ) | ≤ + ∞ Z T ∗ | a ( τ ) || φ ∗ ( τ ) | exp (cid:26) τ Z T ∗ Re [ a ( s ) + a ( s )] ds (cid:27) dτ. ( φ ( t ) , ψ ( t )) be a non principal solution ofthe system (4.1). Without loss of generality we may take that ( φ ( t ) , ψ ( t )) is T ∗ -regular.Then q ( t ) ≡ ψ ( t ) φ − ( t ) , t ≥ T ∗ is a T ∗ -normal solution of Eq. (4.2). By (3.7) from hereit follows + ∞ Z T ∗ Re [ a ( τ )( q ∗ ( τ ) − q ( τ ))] dτ = −∞ . By (2.7) from here we obtain (4.7): lim t → + ∞ | φ ∗ ( t ) || φ ( t ) | = lim t → + ∞ exp (cid:26) t Z T ∗ Re [ a ( τ )( q ∗ ( τ ) − q ( τ ))] dτ (cid:27) = 0 . Let ( φ m ( t ) , ψ m ( t )) , m = 1 , be non principal T -regular ( T ≥ t ) solutions of the system(4.1). By (4.3) q m ( t ) = ψ m ( t ) φ − m ( t ) , t ≥ T, m = 1 , are T -normal solutions of Eq.(4.2). Then according to the statement 4) of Theorem 3.3 the integral + ∞ Z T Re [ a ( τ )( q ( τ ) − q ( τ ))] dτ converges. By (2.7) from here it follows (4.8). The theorem is proved. Remark 4.1.
From the estimate (4.12) is seen that if supp a ( t ) is bounded, then e ν q ( t ) has arbitrary large zeroes. Hence in this case under the conditions of Theorem 4.2the system is normal. If supp a ( t ) is unbounded and the coefficients of the system (4.1)are real-valued, then it is not difficult to verify that under the conditions of Theorem 4.1 e ν q ( t ) = 0 , t ≥ t . So in this case (4.1) is extremal. References
1. P. Wilzinski, Quaternionic-valued differential equations. The Riccati equations.Journal of Differential Equations, vol. 247. pp. 2167 - 2187, 2009.2. J. D. Gibbon, D. D. Holm, R. M. Kerr and I. Roulstone. Quaternions and periodicdynamics in the Euler fluid equations. Nonlinearity, vol. 19, pp. 1962 - 1983, 2006.3. H. Zoladek, Classification of diffeomorphisms of S induced by quaternionic Riccatiequations with periodic coefficients. Topological methods in Nonlinear Analysis. Journalof the Juliusz Shauder Center, vol. 33. pp. 205 - 2015, 2009.20. V. Cristioano and F. Smaraadase. An Exact Mapping from Navier-Stocks Equation toSchrodinger Equation via Riccati equation. Progress in Physics, vol. 1. pp. 38, 39, 2008.5. K. Leschke and K. Morya. Application of Quaternionic Holomorphic Geometry tominimal surfaces. Complex manifolds, vol. 3, pp. 282 - 300, 2006.6. J. Campos, J. Mavhin. Periodic solutions of quaternionic-valued ordinary differentialequations. Annali di Mathematica, vol. 185, pp. 109 - 127, 2006.7. G. A. Grigorian, Global solvability criteria for quaternionic Riccati equations. ArchivumMathematicum. In print.8. G. A. Grigorian, On some properties of solutions of the Riccati equation. Izvestiya NASof Armenia, vol. 42, N ◦◦