A generalization of Sturm's comparison theorem
aa r X i v : . [ m a t h . C A ] M a y MSC 34C10
A generalization of Sturm’s comparison theoremG. A. Grigorian
Institute of Mathematics NAS of ArmeniaE -mail: [email protected]
Abstract. The Riccati equation method is used to establish a new comparison theorem forsystems of two linear first order ordinary differential equation. This result is based on a,so called, concept of ”null-classes”, and is a generalization of Sturm’s comparison theorem.Key words: Sturm’s comparison theorem, Sturm’s majorant (minorant), linear systems,Riccati equation, null-elements, null-classes.
1. Introduction.
Let p j ( t ) , q j ( t ) , j = 1 , be real-valued continuous functions onthe interval [ a, b ] . Consider the second order linear ordinary differential equations ( p j ( t ) φ ′ ) ′ + q j ( t ) φ = 0 , j = 1 , . (1 . j ) Definition 1.1.
Eq. (1 . ) is called a Sturm majorant for Eq. (1 . ) on [ a, b ] if p ( t ) ≥ p ( t ) > and q ( t ) ≤ q ( t ) , t ∈ [ a, b ] . (1 . If in addition q ( t ) < q ( t ) or p ( t ) > p ( t ) > and q ( t ) = 0 for some t ∈ [ a, b ] , then Eq. (1 . ) is called a strict Sturm majorant for Eq. (1 . ) on [ a, b ] . The Sturm’s famous comparison theorem states (see [1], p. 334)
Theorem 1.1 (Sturm).
Let (1 . ) be a Sturm majorant for (1 . ) and let φ = φ ( t ) = 0 be a solution of Eq. (1 . ) having exactly n ( n ≥ zeroes t = t < t < ... < t n on ( t , t ] . Let φ = φ ( t ) = 0 be a solution of Eq. (1 . ) satisfying p ( t ) φ ′ ( t ) φ ( t ) ≥ p ( t ) φ ′ ( t ) φ ( t ) (1 . at t = t (the expression on the right [or left] of (1.3) is considered to be + ∞ if φ ( t ) = 0 [or φ ( t ) = 0 ]; in particular, (1.3) holds at t = t if φ ( t ) = 0 ). Then φ ( t ) has at least n eroes on ( t , t n ] . Furthermore φ ( t ) has at least n zeroes on ( t , t ) if either the inequality(1.3) holds at t = t or (1 . ) is a strict Sturm majorant for Eq. (1 . ) on [ t , t n ] . Note that if p j ( t ) = 0 , j = 1 , , t ∈ [ a, b ] , then the equations (1 . j ) , j = 1 , areequivalent (reducible) to the systems φ ′ = p j ( t ) ψ ′ ψ ′ = − q j ( t ) φ, t ∈ [ a, b ] . (1 . j ) j = 1 , . They are particular cases of more general systems φ ′ = f j ( t ) ψ,ψ ′ = − g j ( t ) φ, t ∈ [ a, b ] , (1 . j ) j = 1 , , where f j ( t ) , g j ( t ) , j = 1 , are real-valued continuous functions on [ a, b ] .Obviously the system (1 . j ) is reducible to the equation (1 . j ) if and only if f j ( t ) = 0 ,t ∈ [ a, b ] or g j ( t ) = 0 , t ∈ [ a, b ] ( j = 1 , . Therefore under this restriction Theorem 1.1can be reformulated for the systems (1 . j ) , j = 1 , , in particular the conditions (1.2)become f ( t ) ≥ f ( t ) > and g ( t ) ≥ g ( t ) , t ∈ [ a, b ] . (1 . Here arises the following question. Is it possible to obtain a comparison theorem forthe systems (1 . j ) , j = 1 , like of Theorem 1.1 (in the sense of mentioned abovereformulation) if we weaken the conditions (1.6) up to the following ones? f ( t ) ≥ f ( t ) ≥ and g ( t ) ≥ g ( t ) , t ∈ [ a, b ] . (1 . The answer is "No!’ via the following example
Example 1.1.
Set: f ( t ) ≡ , t ∈ [0 , π ) , sin t, t ∈ [ π, π ] , g ( t ) ≡ , t ∈ [0 , π ] . Consider thesystem φ ′ = f ( t ) ψ,ψ ′ = g ( t ) φ, t ∈ [0 , π ] . It is not difficult to verify that for the non trivial solution ( φ ( t ) , ψ ( t )) of this system with φ (0) = 0 , ψ (0) = 1 the function φ ( t ) is identically zero on [0 , π ] . This example shows that under the conditions (1.7) the system (1 . j ) ( j = 1 , mayhave infinitely many zeroes (even continuum) on a finite interval. Hence it is impossible2o obtain a generalization of Theorem 1.1 (in the sense of mentioned above reformulation)with the conditions (1.7). However it is possible to obtain a generalization (in some othersense) of Theorem 1.1 for the systems (1 . j ) , j = 1 , with the conditions (1.7). The ideaof a generalization of Theorem 1.1 for the systems (1 . j ) , j = 1 , with the conditions(1.7) is based on the concept of so called ’null-classes’ of zeroes of components φ ( t ) , ψ ( t ) of solutions ( φ ( t ) , ψ ( t )) of two dimensional linear systems (this concept is introduced in[2]).In this paper we use the Riccati equation method to obtain a generalization of Sturm’scomparison theorem (Theorem 1.1), which is based on the concept of "null-classes".
2. Auxiliary propositions . Along with the systems (1 . j ) , j = 1 , consider theRiccati equations y ′ + f j ( t ) y + g j ( t ) = 0 , t ∈ [ a, b ] , j = 1 , , (2 . j ) and the differential inequalities η ′ + f j ( t ) η + g j ( t ) ≥ , t ∈ [ a, b ] , j = 1 , . (2 . j ) Remark 2.1.
Every solution of Eq. (2 . on an interval [ t , t ) ( ⊂ [ a, b ]) is also asolution of the inequality (2 . ) on [ t , t ) , Remark 2.2. If f ( t ) ≥ , t ∈ [ t , t ) , then for every λ ∈ ( −∞ , + ∞ ) the function η λ ( t ) ≡ λ − t R t g ( τ ) dτ, t ∈ [ t , r ) is a solution to the inequality (2 . ) on [ t , t ) . The following comparison theorem plays a crucial role in the proof of the main result.
Theorem 2.1.
Let y ( t ) be a solution of Eq. (2 . ) on [ t , τ ) ( ⊂ [ a, b ]) and let η ( t ) and η ( t ) be solutions of the inequalities (2 . ) and (2 . ) respectively on [ t , τ ) , moreoversuppose that y ( t ) ≤ η j ( t ) , j = 1 , . In addition let the following conditions be satisfied f ( t ) ≥ , t ∈ [ t , τ ); y (0) − y ( t ) + t R t exp (cid:26) τ R t f ( s )( η ( s ) + η ( s )) ds (cid:27)(cid:20) ( f ( τ ) − f ( τ )) y ( τ ) + g ( τ ) − g ( τ ) (cid:21) dτ ≥ , t ∈ [ t , τ ) , for some y (0) ∈ [ y ( t ) , η ( t )] . ThenEq. (2 . ) has a solution y ( t ) with the initial condition y ( t ) ≥ y (0) , on [ t , τ ) ; moreover y ( t ) ≥ y ( t ) , t ∈ [ t , τ ) . See the proof in [3].Besides of Theorem 2.1 for the proof of the main result we need also in the followingthree lemmas.
Lemma 2.1.
Let f ( t ) ≥ , t ∈ [ t , τ ) , and let ( t , t ) be the maximum existenceinterval for a solution y ( t ) of Eq. (2 . ) , where t < t < t < τ . , Then lim t → t − y ( t ) = −∞ , lim t → t +0 = + ∞ . (2 . Remark 2.3 . The first equality of (2.3) remains valid also in the case when ( t , t ) is not the maximum existence interval, but y ( t ) is not continuable to the right from thepoint t . Lemma 2.2.
Let f ( t ) ≥ , t ∈ [ t , τ ) , and let ( t k , t k )( ⊂ [ t , τ )) be the the maximumexistence interval for a solution y k ( t ) of Eq. (2 . ) , k = 1 , . In addition let y ( t ) > y ( t ) for some t ∈ ( t , t ) ∩ ( t , t ) . Then t > t and t > t . See the proof in [3].
Lemma 2.3.
Let f ( t ) ≥ , t ∈ [ t , τ ) , let y ( t ) be a solution of Eq. (2 . ) on [ t , τ ) ,and let η ( t ) be a solution of the inequality (2 . ) on [ t , τ ) ; moreover let y ( t ) ≤ η ( t ) .Then y ( t ) ≤ η ( t ) , t ∈ [ t , τ ) . See the proof in [3].
3. Main result . On the set R of subsets of real numbers R define the order relation ≺ , assuming x ≺ y if and only if for every t x ∈ x ∈ R , t y ∈ y ∈ R the inequality t x < t y is valid. Let ( φ ( t ) , ψ ( t )) be a nontrivial solution of the system (1 . ) . Since φ ( t ) isa continuous function on [ a, b ] thats zeroes form a closed set. Definition 3.1.
A connected component of zeroes of the function φ ( t ) of a solution ( φ ( t ) , ψ ( t )) of the system (1 . ) is called a null-element of φ ( t ) . Let z ( t ) be a solution of Eq. (2 . ) with z ( a ) = i . Then z ( t ) exists on [ a, b ] and y ( t ) ≡ Im z ( t ) > , t ∈ [ a, b ] and φ ( t ) = µ p y ( t ) sin (cid:18) t Z a f ( τ ) y ( τ ) dτ + θ (cid:19) , t ∈ [ a, b ] , (3 . where µ and θ are some real constants (see [2]). Let n ( φ ) be a null-element of the function φ ( t ) . By (3.1) t Z a f ( τ ) y ( τ ) dτ + θ = πk , t ∈ N ( φ ) , k ∈ Z . (3 . Hereafter by [ t , t ] we mean the set of all points of R lying between t and t , includingthemselves. Definition 3.2.
Null-elements N ( φ ) and N ( φ ) of the function φ ( t ) are called congene-rous, if for every t j ∈ N ( φ ) , j = 1 , the inequality (cid:12)(cid:12)(cid:12)(cid:12) t Z t f ( τ ) y ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) < π, t ∈ [ t , t ] s valid. It was shown in [2] that the congeniality relation between null-elements N ( φ ) and N ( φ ) is an equivalence relation. Definition 3.3.
An equivalence class of congenerous null-elements N ( φ ) of the functionof a solution ( φ ( t ) , ψ ( t )) of the system (1 . ) is called a null-class of φ ( t ) . Remark 3.1.
It follows from (3.1) and (3.2) that if f ( t ) ≥ , t ∈ [ a, b ] , then everynull-class consists of only one null-element. It was shown in [2] that for every solution ( φ ( t ) , ψ ( t )) of the system (1 . ) the function φ ( t ) has a finite number of null-classes, and these null-classes are linearly ordered by ≺ .Let ( φ j ( t ) , ψ j ( t )) be a solution of the system (1 . j ) , j = 1 , , and let N ( φ ) = [ t , τ ] ≺ N ( φ ) = [ t , τ ] ≺ ... ≺ N n ( φ ) = [ t n , τ n ] be all of the null-classes of the function φ ( t ) on [ t , t ] , t ∈ ( t , τ ) . Definition 3.4.
The quarter ( φ , ψ , f , g ) is called a majorant of the quarter ( φ , ψ , f , g ) on [ t , t ] if the following conditions are satisfied1. ψ ( t ) /φ ( t ) ≤ ψ ( t ) /φ ( t ) (The expression on the left hand [respectively on theright hand] side of the inequality is considered to be equal + ∞ if φ ( t ) = 0 [respectively,if φ ( t ) = 0 ] in particular this is in the case if φ ( t ) = 0 );2. f ( t ) ≥ f ( t ) ≥ , t ∈ [ t , t ];
3. there exists ξ k ∈ ( τ k , t k +1 ) ( k = 0 , ..., n − such that( ) g ( t ) ≥ g ( t ) , t ∈ [ τ k , ξ k ] ( k = 0 , ..., n − ( ) any solutions η jk ( t ) of the inequalities (2 . j ) ( j = 1 , on [ ξ k , t k +1 ] such that η jk ( ξ k ) > ψ ( ξ k ) /φ ( ξ k ) (such solutions always exist by condition 2 and Remark 2.2)satisfy the inequalities t Z ξ k exp (cid:26) τ Z ξ k f ( s ) (cid:2) η ,k ( s ) + η ,k ( s ) (cid:3) ds (cid:27)(cid:20) g ( τ ) − g ( τ ) (cid:21) dτ ≥ , t ∈ [ ξ k , t k +1 ] , k = 0 , ..., n − are satisfied. In addition, suppose that f ( t ) > , t ∈ [ t , t ] and either the strict inequalitytakes place in condition 1 or at least one of the following conditions is satisfied:1’. f ( t ′ ) > f ( t ′ ) and g ( t ′ ) = 0 for some t ′ ∈ [ t , t ] , g ( t ′ ) > g ( t ′ ) for some t ′ ∈ n − S k =1 [ τ k , ξ k ] , t k +1 R ξ k exp (cid:26) τ R ξ k f ( s ) (cid:2) η k ( s )+ η k ( s ) (cid:3) ds (cid:27)(cid:2) g ( τ ) − g ( τ ) (cid:3) dτ > for some k ∈ { , ..., n − } .Then the quarter ( φ , ψ , f , g ) is called a strict majorant of the quarter ( φ , ψ , f , g ) on [ t , t ] . heorem 3.1 (main result). Let ( φ ( t ) , ψ ( t )) be a solution of the system (1 . ) ,let N ( φ ) = [ t , τ ] ≺ ... ≺ N n ( φ ) = [ t n , τ n ] be all of the null-classes of φ ( t ) on [ t , t ]( t ∈ ( t , τ )) and let ( φ ( t ) , ψ ( t )) be a solution of the system (1 . ) . If ( φ , ψ , f , g ) isa majorant for ( φ , ψ , f , g ) on [ t , t ] , then the function φ ( t ) has at least one null-classin ( τ k , τ k +1 ] for each k = 0 , ..., n − . In addition, if ( φ , ψ , f , g ) is a strict majorant for ( φ , ψ , f , g ) on [ t , t ] , then the function φ ( t ) has at least n null-classes on ( t , t ) . Proof. Suppose that the function φ ( t ) has no zeroes on [ τ k , t k +1 ] for some k .Then the function ψ ( t ) /φ ( t ) exists on ( τ k , e τ k +1 ] , foe some e τ k +1 > t k +1 and is asolution of Eq. (2 . ) there. At first consider the case when k = 0 . In this case ( τ k , t k +1 ) is the maximal existence interval for the solution y ( t ) ≡ ψ ( t ) /φ ( t ) of Eq. (2 . ) . Let η j,k ( t ) ( j = 1 , be solutions of the respective inequalities (2 . j ) on [ ξ k , t k +1 ] with theinitial conditions η j,k ( ξ k ) > y ( ξ k ) , j = 1 , (in virtue of condition 1 and Remark 2.2these solutions exist always). Let e y ( t ) be a solution of Eq. (2 . ) such that y ( ξ k ) < e y ( ξ k ) ≤ min j =1 , { η j,k ( ξ k ) } and let ( e t k , e t k +1 ) be the maximum existence interval for e y ( t ) .By Lemma 2.2 it follows from the inequality y ( ξ k ) < e y ( ξ k ) that τ k < e t k and t k +1 < e t k +1 . We assume that e y ( ξ k ) is close enough to y ( ξ k ) to ensure that e t k ∈ ( t k , ξ k ) and e t k +1 ∈ ( t k , e τ k +1 ) . Since e t k is the left endpoint of the maximum existence interval of e y ( t ) by Lemma 2.1 we have e y ( e t k + 0) = + ∞ and since e y ( e t k ) < + ∞ [because of theinclusion e t k ∈ ( τ k , ξ k ]] we have y ( ζ k ) ≤ e y ( ζ k ) , (3 . for some ζ k ∈ ( e t k , ξ k ) . Let ( φ , ψ , f , g ) be a majorant for ( φ , ψ , f , g ) . Then byvirtue of Theorem 2.1 it follows from (3.5) that y ( ξ k ) ≤ e y ( ξ k ) (3 . and since e y ( ξ k ) ≤ min j =1 , { η j,k ( ξ k ) } , we have y ( ξ k ) ≤ η ,k ( ξ k ) , e y ( ξ k ) ≤ η ,k ( ξ k ) . By Lemma 2.3 from here it follows that y ( t ) ≤ η ,k ( t ) , e y ( t ) ≤ η ,k ( t ) t ∈ [ ξ k , e t k +1 ) . Therefore e y ( e t k +1 − ≥ y ( e t k +1 ) > −∞ . Then by virtue of Lemma 2.1 ( e t k , e t k +1 ) is notthe maximum existence interval for e y ( t ) . The obtained contradiction shows that φ ( t ) hasat least one zero l on ( τ k , t k =1 ] , which belongs to a null-element of the function φ ( t ) .Hence according to Remark 3.1 l belongs to a null-class N ( φ ) of the function φ ( t ) . By63.1) from condition 2 it follows that if N ( φ ) ∩ N k ( φ ) = ∅ . Then N ( φ ) ⊂ N k ( φ ) , and,therefore, N ( φ ) ⊂ ( τ k , τ k +1 ] . Now consider the case when k = 0 . If φ ( t ) = 0 , thenthe proof of a null-class of φ ( t ) in ( t , t ] can be proved by analogy with the proof in thepreceding case. Suppose φ ( t ) = 0 . Then by condition 1 we have also φ ( t ) = 0 . Showthat φ ( t ) has at least one zero on ( t , t ] . Suppose φ ( t ) = 0 , t ∈ ( t , t ] . Then y ( t ) ≡ ψ ( t ) /φ ( t ) is defined at least on [ t , t ] and is a solution of Eq. (2 . ) on that interval. Thefunction y ( t ) ≡ ψ ( t ) /φ ( t ) is a solution of Eq. (2 . ) on [ t , t ) , in virtue of Remark 2.3 y ( t −
0) = −∞ . But on the other hand since y ( t ) ≥ y ( t ) and ( φ , ψ , f , g ) is amajorant for ( φ , ψ , f , g ) it follows from Theorem 2.1 that y ( t − ≥ y ( t ) > −∞ .The obtained contradiction shows that φ ( t ) has at least one zero on ( t , t ] . Then φ ( t ) has at least one null-class in [ t , τ ] . The first part of the theorem is proved. The second(last) part of the theorem can be proved by analogy of the proof of the second part ofTheorem 4.1 from [3], as far as the strict majorant condition implies the reducibility ofthe systems (1 . j ) , j = 1 , to the second order linear ordinary differential equations like (1 . j ) , j = 1 , respectively. The theorem is proved. Definition 3.5.
The system (1 . ) is called a majorant of the system (1 . ) on [ t , t ]( ⊂ [ a, b ]) if the following conditions are satisfied1.) f ( t ) ≥ f ( t ) ≥ , t ∈ [ t , t ] , g ( t ) ≥ g ( t ) , t ∈ [ t , t ] .In addition suppose that f ( t ) > , t ∈ [ t , t ] and at least one of the followingconditions is satisfied1’) f ( t ′ ) > f ( t ′ ) and g ( t ′ ) = 0 for some t ′ ∈ [ t , t ] ;2’) g ( t ′ ) > g ( t ′ ) for some t ′ ∈ [ t , t ] .Then the system (1 . ) is called a strict majorant of the system (1 . ) . From Theorem 3.1 we immediately get.
Corollary 3.1.
Let the system (1 . ) be a majorant for the system (1 . ) . Let ( φ ( t ) , ψ ( t )) be a nontrivial solution of the system (1 . ) and let φ ( t ) have exactly n ( ≥ null-classes N ( φ ) ≺ ... ≺ N n ( φ ) on [ t , t ] . Let ( φ ( t ) , ψ ( t )) be a nontrivial solution ofthe system (1 . ) satisfying ψ ( t ) φ ( t ) ≥ ψ ( t ) φ ( t ) (3 . (The expression on the right [or left] of (3.7) is considered to be + ∞ , if φ ( t ) = 0 [or φ ( t ) = 0 ]; in particular, (3.7) holds if φ ( t ) = 0 ). Then φ ( t ) has at least n null-classesin ( t , t n ) , where N n ( φ ) = [ t n , τ n ] , if either the strict inequality (3 . holds or (1 . ) is astrict majorant for (1 . ) on [ t , t n ] . (cid:3) Remark 3.2.