On the oscillatory integration of some ordinary differential equations
aa r X i v : . [ m a t h . C A ] J a n On the oscillatory integration of some ordinarydifferential equations
Octavian G. Mustafa ∗ Faculty of Mathematics, D.A.L.,University of Craiova, Romaniae-mail: [email protected]
Abstract
Conditions are given for a class of nonlinear ordinary differentialequations x ′′ + a ( t ) w ( x ) = 0, t ≥ t ≥
1, which includes the linear equationto possess solutions x ( t ) with prescribed oblique asymptote that have anoscillatory pseudo-wronskian x ′ ( t ) − x ( t ) t . Keywords:
Ordinary differential equation; Asymptotic integration; Pre-scribed asymptote; Non-oscillation of solutions
A certain interest has been shown recently in studying the existence ofbounded and positive solutions to a large class of elliptic partial differentialequations which can be displayed as∆ u + f ( x, u ) + g ( | x | ) x · ∇ u = 0 , x ∈ G R , (1)where G R = { x ∈ R n : | x | > R } for any R ≥ n ≥
2. We would like tomention the contributions [1], [3], [8] – [11], [13, 14], [18] and their referencesin this respect.It has been established, see [8, 9], that it is sufficient for the functions f , g to be H¨older continuous, respectively continuously differentiable in orderto analyze the asymptotic behavior of the solutions to (1) by the comparison ∗ Correspondence address: Str. Tudor Vladimirescu, Nr. 26, 200534 Craiova, Dolj,Romania scillatory integration ζ >
0, let us assume that there exist a contin-uous function A : [ R, + ∞ ) → [0 , + ∞ ) and a nondecreasing, continuouslydifferentiable function W : [0 , ζ ] → [0 , + ∞ ) such that0 ≤ f ( x, u ) ≤ A ( | x | ) W ( u ) for all x ∈ G R , u ∈ [0 , ζ ]and W ( u ) > u >
0. Then we are interested in the positive solutions U = U ( | x | ) of the elliptic partial differential equation∆ U + A ( | x | ) W ( U ) = 0 , x ∈ G R , for the rˆole of super-solutions to (1).M. Ehrnstr¨om [13] noticed that, by imposing the restriction x · ∇ U ( x ) ≤ , x ∈ G R , upon the super-solutions U , an improvement of the conclusions from theliterature is achieved for the special subclass of equations (1) where g takesonly nonnegative values. Further developments of Ehrnstr¨om’s idea are givenin [1, 3, 11, 14].Translated into the language of ordinary differential equations, the re-search about U reads as follows: given c , c ≥
0, find (if any) a positivesolution x ( t ) of the nonlinear differential equation x ′′ + a ( t ) w ( x ) = 0 , t ≥ t ≥ , (2)where the coefficient a : [ t , + ∞ ) → R and the nonlinearity w : R → R arecontinuous and given by means of A , W , such that x ( t ) = c t + c + o (1) when t → + ∞ (3)and W ( x, t ) = 1 t (cid:12)(cid:12)(cid:12)(cid:12) x ′ ( t ) 1 x ( t ) t (cid:12)(cid:12)(cid:12)(cid:12) = x ′ ( t ) − x ( t ) t < , t > t . (4)The symbol o ( f ) for a given functional quantity f has here its standardmeaning. In particular, by o (1) we refer to a function of t that decreases to0 as t increases to + ∞ .The papers [2, 3, 20, 22, 23] present various properties of the functionalquantity W , which shall be called pseudo-wronskian in the sequel. Our aim inthis note is to complete their conclusions by giving some sufficient conditionsupon a and w which lead to the existence of a solution x to (2) that verifies(3) while having an oscillatory pseudo-wronskian (this means that there existthe unbounded from above sequences ( t ± n ) n ≥ and ( t n ) n ≥ such that t n −
Lemma 1
Given x ∈ C ([ t , + ∞ ) , R ) , suppose that x ′′ ( t ) ≤ for all t ≥ t .Then W ( x, · ) can change from being nonnegative-valued to being negative-valued at most once in [ t , + ∞ ) . In fact, its set of zeros is an interval (pos-sibly degenerate). Proof.
Notice that d dt [ x ( t )] = 1 t · ddt [ t W ( x, t )] , t ≥ t . The function t t W ( x, t ) being nonincreasing, it is clear that, if it haszeros, it has either a unique zero or an interval of zeros. (cid:3) The result has an obvious counterpart.
Lemma 2
Given x ∈ C ([ t , + ∞ ) , R ) , suppose that x ′′ ( t ) ≥ for all t ≥ t .Then, W ( x, · ) can change from being nonpositive-valued to being positive-valued at most once in [ t , + ∞ ) . Again, its set of zeros is an interval (possiblyreduced to one point). Consider that x is a positive solution of equation (2) in the case where a ( t ) ≥ t , + ∞ ) and w ( u ) > u >
0. Then, we have d W dt = − W t − a ( t ) w ( x ( t )) , t ≥ t , which leads to W ( x, t ) = 1 t (cid:20) t W − Z tt sa ( s ) w ( x ( s )) ds (cid:21) , W = W ( x, t ) , (5)throughout [ t , + ∞ ) by means of Lagrange’s variation of constants formula.The integrand in (5) being nonnegative-valued, we regain the conclusionof Lemma 1. In fact, if T ∈ [ t , + ∞ ) is a zero of W ( x, · ) then it is a solutionof the equation t W = Z Tt sa ( s ) w ( x ( s )) ds. (6) scillatory integration x is positive-valued throughout[ t , + ∞ ) then it is necessary to have( t W ≥ ) Z + ∞ t sa ( s ) w ( x ( s )) ds < + ∞ . (7)It has become clear at this point that whenever the equation (2) has a pos-itive solution x such that W ≤
0, the functional coefficient a is nonnegative-valued and has at most isolated zeros and w ( u ) > u >
0, the pseudo-wronskian W satisfies the restriction (4). Now, returning to the problemstated in the Introduction, we can evaluate the main difficulty of the inves-tigation: if the positive solution x has prescribed asymptotic behavior, seeformula (3) or a similar development, then we cannot decide upfront whetheror not W ≤
0. The formula (6) shows that there are also certain difficultiesto estimate the zeros of the pseudo-wronskian. W Let us survey in this section some of the recent results regarding thepseudo-wronskian.It has been established that its presence in the structure of a nonlineardifferential equation x ′′ + f ( t, x, x ′ ) = 0 , t ≥ t ≥ , (8)where the nonlinearity f : [ t , + ∞ ) × R → R is continuous, allows for aremarkable flexibility of the hypotheses when searching for solutions withthe asymptotic development (3) (or similar). Theorem 1 ([20, p. 177])
Assume that there exist the nonnegative-valued,continuous functions a ( t ) and g ( s ) such that g ( s ) > for all s > and xg ( s ) ≤ g ( x − α s ) , where x ≥ t and s ≥ , for a certain α ∈ (0 , . Supposefurther that | f ( t, x, x ′ ) | ≤ a ( t ) g (cid:16)(cid:12)(cid:12)(cid:12) x ′ − xt (cid:12)(cid:12)(cid:12)(cid:17) and Z + ∞ t a ( s ) s α ds < Z + ∞ c + |W | t − α dug ( u ) . Then the solution of equation (8) given by (5) exists throughout [ t , + ∞ ) andhas the asymptotic behavior x ( t ) = c · t + o ( t ) , x ′ ( t ) = c + o (1) when t → + ∞ (9) for some c = c ( x ) ∈ R . scillatory integration L (( t , + ∞ ) , R ). Theorem 2 ([3, p. 371])
Assume that f does not depend explicitly of x ′ and there exists the continuous function F : [ t , + ∞ ) × [0 , + ∞ ) → [0 , + ∞ ) ,which is nondecreasing with respect to the second variable, such that | f ( t, x ) | ≤ F (cid:18) t, | x | t (cid:19) and Z + ∞ t t (cid:20) (cid:18) tt (cid:19)(cid:21) F (cid:18) t, | c | + εt (cid:19) dt < ε for certain numbers c = 0 and ε > . Then there exists a solution x ( t ) ofequation (8) defined in [ t , + ∞ ) such that x ( t ) = c · t + o (1) when t → + ∞ and W ( x, · ) ∈ L . The effect of perturbations upon the pseudo-wronskian is investigated inthe papers [2, 20, 22].
Theorem 3 ([20, p. 183])
Consider the nonlinear differential equation x ′′ + f ( t, x, x ′ ) = p ( t ) , t ≥ t ≥ , (10) where the functions f : [ t , + ∞ ) × R → R and p : [ t , + ∞ ) → R arecontinuous and verify the hypotheses | f ( t, x, x ′ ) | ≤ a ( t ) (cid:12)(cid:12)(cid:12) x ′ − xt (cid:12)(cid:12)(cid:12) , Z + ∞ t ta ( t ) dt < + ∞ and lim t → + ∞ t Z tt sp ( s ) ds = C ∈ R − { } . Then, given x ∈ R , there exists a solution x ( t ) of equation (10) defined in [ t , + ∞ ) such that x ( t ) = x and lim t → + ∞ W ( x, t ) = C. In particular, lim t → + ∞ x ( t ) t ln t = C. scillatory integration
6A slight modification of the discussion in [22, Remark 3], see [2, p. 47],leads to the next result.
Theorem 4
Assume that f in (10) does not depend explicitly of x ′ and thereexists the continuous function F : [ t , + ∞ ) × [0 , + ∞ ) → [0 , + ∞ ) , which isnondecreasing with respect to the second variable, such that | f ( t, x ) | ≤ F ( t, | x | ) and Z + ∞ t sF (cid:18) s, | P ( s ) | + sup τ ≥ s { q ( τ ) } (cid:19) ds ≤ q ( t ) , t ≥ t , for a certain positive-valued, continuous function q ( t ) possibly decaying to as t → + ∞ . Here, P is the twice continuously differentiable antiderivativeof p , that is P ′′ ( t ) = p ( t ) for all t ≥ t . Suppose further that lim sup t → + ∞ (cid:20) t W ( P, t ) q ( t ) (cid:21) > and lim inf t → + ∞ (cid:20) t W ( P, t ) q ( t ) (cid:21) < − . Then equation (10) has a solution x ( t ) throughout [ t , + ∞ ) such that x ( t ) = P ( t ) + o (1) when t → + ∞ and W ( x, · ) oscillates. Finally, the presence of the pseudo-wronskian in the structure of a nonlin-ear differential equation can lead to multiplicity when searching for solutionswith the asymptotic development (3).
Theorem 5 ([23, Theorem 1])
Given the numbers x , x , c ∈ R , with c = 0 ,and t ≥ such that t x − x = c , consider the Cauchy problem (cid:26) x ′′ = t g ( tx ′ − x ) , t ≥ t ≥ ,x ( t ) = x , x ′ ( t ) = x , (11) where the function g : R × R → R is continuous, g ( c ) = g (3 c ) = 0 and g ( u ) > for all u = c . Assume further that Z cc + dug ( u ) < + ∞ and Z (3 c ) − c dug ( u ) = + ∞ . Then problem (11) has an infinity of solutions x ( t ) defined in [ t , + ∞ ) anddevelopable as x ( t ) = c t + c + o (1) when t → + ∞ for some c = c ( x ) and c = c ( x ) ∈ R . scillatory integration K ( x )( t ) = x ( t ) x ′ ( t ) , t ≥ t , employed in the theory of Kneser-solutions , see the papers [6, 7] for the linearand respectively the nonlinear case and the monograph [19], and HW ( x ) = Z + ∞ t x ( s ) w ( x ( s )) ds. The latter quantity is the core of the nonlinear version of
Hermann Weyl’slimit-point/limit-circle classification designed for equation (2), see the well-documented monograph [5] and the paper [21]. W We shall assume in the sequel that the nonlinearity w of equation (2)verifies some of the hypotheses listed below: | w ( x ) − w ( y ) | ≤ k | x − y | , where k > , (12)and w (0) = 0 , w ( x ) > x > , | w ( xy ) | ≤ w ( | x | ) w ( | y | ) (13)for all x , y ∈ R . We notice that restriction (13) implies the existence of amajorizing function F , as in Theorem 2, given by the estimates | f ( t, x ) | = | a ( t ) w ( x ) | ≤ | a ( t ) | · w ( t ) w (cid:18) | x | t (cid:19) = F (cid:18) t, | x | t (cid:19) . We can now use the paper [24] to recall the main conclusions of an asymp-totic integration of equation (2). It has been established that whenever R + ∞ t tw ( t ) | a ( t ) | dt < + ∞ , all the solutions of (2) have asymptotes (3) andtheir first derivatives are developable as x ′ ( t ) = c + o (cid:0) t − (cid:1) when t → + ∞ . (14)Consequently, W ( x, t ) = − c t − + o ( t − ) for all large t ’s. In this case (thefunctional coefficient a has varying sign), when dealing with the sign of the scillatory integration c = 0. Here, theasymptotic development does not even ensure that W is eventually negative.Enlarging the family of coefficients to the ones subjected to the restriction R + ∞ t t ε w ( t ) | a ( t ) | dt < + ∞ , where ε ∈ [0 , x ( t ) = ct + o (cid:0) t − ε (cid:1) , x ′ ( t ) = c + o (cid:0) t − ε (cid:1) , c ∈ R , (15)yielding the less precise estimate W ( x, t ) = o ( t − ε ) when t → + ∞ . Wehave again a lack of precision in the asymptotic development of W ( x, · ) withrespect to the sign issue. We also deduce on the basis of (3), (15) that someof the coefficients a in these classes verify (7), a fact that complicates thediscussion.The next result establishes the existence of a positive solution to (2)subjected to (4), (15) for the largest class of functional coefficients: ε = 0.By taking into account Lemmas 1, 2 and the non-oscillatory character ofequation (2) when the nonlinearity w verifies (13), we conclude that for aninvestigation within this class of coefficients a of the solutions with oscillatorypseudo-wronskian it is necessary that a itself oscillates . Also, when a is non-negative valued we recall that the condition Z + ∞ t a ( t ) dt < + ∞ is necessary for the linear case of equation (2) to be non-oscillatory, see [16],while in the case given by w ( x ) = x λ , x ∈ R , with λ > Emden-Fowler equation , see the monograph [19]) thecondition Z + ∞ t ta ( t ) dt = + ∞ (16)is necessary and sufficient for oscillation, see [4]. In the case of Emden-Fowlerequations with λ ∈ (0 ,
1) and a continuously differentiable coefficient a suchthat a ( t ) ≥ a ′ ( t ) ≤ t , + ∞ ), another result establishesthat equation (2) has no oscillatory solutions provided that condition (16)fails, see [17].Regardless of the oscillation of a , it is known [3, p. 360] that the linear caseof equation (2) has bounded and positive solutions with eventually negativepseudo-wronskian. Theorem 6
Assume that the nonlinearity w verifies hypothesis (13) and isnondecreasing. Given c , d > , suppose that the functional coefficient a is scillatory integration nonnegative-valued, with eventual isolated zeros, and Z + ∞ t w ( t ) a ( t ) dt ≤ dw ( c + d ) . Then, the equation (2) has a solution x such that W = 0 , c − d ≤ x ′ ( t ) < x ( t ) t ≤ c + d for all t > t (17) and lim t → + ∞ x ′ ( t ) = lim t → + ∞ x ( t ) t = c. (18) Proof.
We introduce the set D given by D = { u ∈ C ([ t , + ∞ ) , R ) : ct ≤ u ( t ) ≤ ( c + d ) t for every t ≥ t } . A partial order on D is provided by the usual pointwise order ” ≤ ”, thatis, we say that v ≤ v if and only if v ( t ) ≤ v ( t ) for all t ≥ t , where v , v ∈ D . It is not hard to see that ( D, ≤ ) is a complete lattice.For the operator V : D → C ([ t , + ∞ ) , R ) with the formula V ( u )( t ) = t (cid:26) c + Z + ∞ t s Z st τ a ( τ ) w ( u ( τ )) dτ ds (cid:27) , u ∈ D, t ≥ t , the next estimates hold c ≤ V ( u )( t ) t = c + Z + ∞ t s Z st τ a ( τ ) · w ( τ ) w (cid:18) u ( τ ) τ (cid:19) dτ ds ≤ c + sup ξ ∈ [0 ,c + d ] { w ( ξ ) } · Z + ∞ t s Z st τ w ( τ ) a ( τ ) dτ ds = c + w ( c + d ) (cid:20) t Z tt τ w ( τ ) a ( τ ) dτ + Z + ∞ t w ( τ ) a ( τ ) dτ (cid:21) ≤ c + w ( c + d ) Z + ∞ t w ( τ ) a ( τ ) dτ ≤ c + d by means of (13). These imply that V ( D ) ⊆ D .Since c · t ≤ V ( c · t ) for all t ≥ t , by applying the Knaster-Tarski fixedpoint theorem [12, p. 14], we deduce that the operator V has a fixed point u in D . This is the pointwise limit of the sequence of functions ( V n ( c · Id I )) n ≥ ,where V = V , V n +1 = V n ◦ V and I = [ t , + ∞ ).We deduce that u ′ ( t ) = [ V ( u )] ′ ( t ) = u ( t ) t − t Z tt τ a ( τ ) w ( u ( τ )) dτ < u ( t ) t , when t > t , and thus (17), (18) hold true.The proof is complete. (cid:3) scillatory integration Let the continuous functional coefficient a with varying sign satisfy therestriction Z + ∞ t t | a ( t ) | dt < + ∞ . We call the problem studied in the sequel an oscillatory (asymptotic)integration of equation (2).
Theorem 7
Assume that w verifies (12), w (0) = 0 and there exists c > such that L c + > > L c − , (19) where L c + = lim sup t → + ∞ t R + ∞ t sw ( cs ) a ( s ) ds R + ∞ t s | a ( s ) | ds , L c − = lim inf t → + ∞ t R + ∞ t sw ( cs ) a ( s ) ds R + ∞ t s | a ( s ) | ds . Then the equation (2) has a solution x ( t ) with oscillatory pseudo-wronskiansuch that x ( t ) = c · t + o (1) when t → + ∞ . (20) Proof.
There exist η > L c + > η , L c − < − η and two increasing,unbounded from above sequences ( t n ) n ≥ , ( t n ) n ≥ of numbers from ( t , + ∞ )such that t n ∈ ( t n , t n +1 ) and t n Z + ∞ t n sw ( cs ) a ( s ) ds + kη Z + ∞ t n s | a ( s ) | ds < t n Z + ∞ t n sw ( cs ) a ( s ) ds − kη Z + ∞ t n s | a ( s ) | ds > n ≥ Z + ∞ t τ | a ( τ ) | dτ ≤ ηk ( c + η ) scillatory integration S = ( D, δ ) given by D = { y ∈ C ([ t , + ∞ ) , R ) : t | y ( t ) | ≤ η for every t ≥ t } and δ ( y , y ) = sup t ≥ t { t | y ( t ) − y ( t ) |} , y , y ∈ D. For the operator V : D → C ([ t , + ∞ ) , R ) with the formula V ( y )( t ) = 1 t Z + ∞ t sa ( s ) w (cid:18) s (cid:20) c − Z + ∞ s y ( τ ) τ dτ (cid:21)(cid:19) ds, y ∈ D, t ≥ t , the next estimates hold (notice that | w ( x ) | ≤ k | x | for all x ∈ R ) t | V ( y )( t ) | ≤ k Z + ∞ t s | a ( s ) | (cid:20) c + η Z + ∞ s dττ (cid:21) ds ≤ η (23)and t | V ( y )( t ) − V ( y )( t ) | ≤ k Z + ∞ t s | a ( s ) | (cid:18)Z + ∞ s dττ (cid:19) ds · δ ( y , y ) ≤ kt Z + ∞ t s | a ( s ) | ds ≤ ηc + η · δ ( y , y ) . These imply that V ( D ) ⊆ D and thus V : S → S is a contraction.From the formula of operator V we notice also thatlim t → + ∞ tV ( y )( t ) = 0 for all y ∈ D. (24)Given y ∈ D the unique fixed point of V , one of the solutions to (2)has the formula x ( t ) = t h c − R + ∞ t y ( s ) s ds i for all t ≥ t . Via (24) andL’Hospital’s rule, we provide also an asymptotic development for this solu-tion, namelylim t → + ∞ [ x ( t ) − c · t ] = − lim t → + ∞ t Z + ∞ t y ( s ) s ds = − lim t → + ∞ ty ( t )= − lim t → + ∞ tV ( y )( t ) = 0 . The estimate (cid:12)(cid:12)(cid:12)(cid:12) ty ( t ) − Z + ∞ t sw ( cs ) a ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ k Z + ∞ t s | a ( s ) | (cid:20)Z + ∞ s | y ( τ ) | τ dτ (cid:21) ds ≤ kη · t Z + ∞ t s | a ( s ) | ds, t ≥ t , scillatory integration y ( t n ) = W ( x , t n ) < y ( t n ) = W ( x , t n ) > . (25)The proof is complete. (cid:3) Remark 1
When equation (2) is linear, that is w ( x ) = x for all x ∈ R , theformula (19) can be recast as L + = lim sup t → + ∞ t R + ∞ t s a ( s ) ds R + ∞ t s | a ( s ) | ds > > lim inf t → + ∞ t R + ∞ t s a ( s ) ds R + ∞ t s | a ( s ) | ds = L − . We claim that for all c = 0 there exists a solution x ( t ) with oscillatory pseudo-wronskian which verifies (20). In fact, replace c with c in the formulas (21),(22) for a certain c subjected to the inequality min { L + , − L − } > ηc . It isobvious that, when L + = − L − = + ∞ , formulas (21), (22) hold for all c , η >
0. Given c ∈ R − { } , there exists λ = 0 such that c = λc . Thesolution of equation (2) that we are looking for has the formula x = λ · x ,where x ( t ) = t h c − R + ∞ t y ( s ) s ds i for all t ≥ t and y is the fixed point ofoperator V in D . Its pseudo-wronskian oscillates as a consequence of theobvious identity λ · W ( x , t ) = W ( x, t ) , t ≥ t . Example 1
An immediate example of functional coefficient a for the prob-lem of linear oscillatory integration is given by a ( t ) = t − e − t cos t , where t ≥ Z + ∞ t s a ( s ) ds = 1 √ (cid:16) t + π (cid:17) e − t and Z + ∞ t s | a ( s ) | ds ≤ e − t throughout [1 , + ∞ ) which yields L + = + ∞ , L − = −∞ .Sufficient conditions are provided now for an oscillatory pseudo-wronskianto be in L p (( t , + ∞ ) , R ), where p >
0. Since lim t → + ∞ W ( x, t ) = 0 for anysolution x ( t ) of equation (2) with the asymptotic development (20), (14), weare interested in the case p ∈ (0 , Theorem 8
Assume that, in the hypotheses of Theorem 7, the coefficient a verifies the condition Z + ∞ t " t R + ∞ t s | a ( s ) | ds − p t | a ( t ) | dt < + ∞ for some p ∈ (0 , . (26) Then the equation (2) has a solution x ( t ) with an oscillatory pseudo-wronskianin L p and the asymptotic expansion (20). scillatory integration Proof.
Recall that y is the fixed point of operator V . Then, formula(23) implies that | y ( t ) | ≤ k ( c + η ) · t Z + ∞ t s | a ( s ) | ds, t ≥ t . Via an integration by parts, we have1[ k ( c + η )] p Z Tt | y ( s ) | p ds ≤ T − p − p (cid:20)Z + ∞ T s | a ( s ) | ds (cid:21) p + p − p Z Tt " s R + ∞ s τ | a ( τ ) | dτ − p s | a ( s ) | ds for all T ≥ t ≥ t .The estimates T − p − p (cid:20)Z + ∞ T s | a ( s ) | ds (cid:21) p = T − p − p Z + ∞ T " R + ∞ T τ | a ( τ ) | dτ − p s | a ( s ) | ds ≤ − p Z + ∞ T " s R + ∞ s τ | a ( τ ) | dτ − p s | a ( s ) | ds allow us to establish that1[ k ( c + η )] p Z Tt | y ( s ) | p ds ≤ p − p Z + ∞ t " s R + ∞ s τ | a ( τ ) | dτ − p s | a ( s ) | ds. The conclusion follows by letting T → + ∞ .The proof is complete. (cid:3) Example 2
An example of functional coefficient a in the linear case thatverifies the hypotheses of Theorem 8 is given by the formula t a ( t ) = b ( t ) = a k ( t − k ) , t ∈ [9 k, k + 1] ,a k (9 k + 2 − t ) , t ∈ [9 k + 1 , k + 3] ,a k ( t − k − , t ∈ [9 k + 3 , k + 4] ,a k (9 k + 4 − t ) , t ∈ [9 k + 4 , k + 5] ,a k ( t − k − , t ∈ [9 k + 5 , k + 7] ,a k (9 k + 8 − t ) , t ∈ [9 k + 7 , k + 8] , , t ∈ [9 k + 8 , k + 1)] , k ≥ . Here, we take a k = k − α − ( k + 1) − α for a certain integer α > − pp . scillatory integration k -th ”cell” of the function b can be visu-alized next. (cid:0)(cid:0)(cid:0) ❅❅❅ ❅❅❅ (cid:0)(cid:0)(cid:0) ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅ k + 1 9 k + 3 9 k + 5 9 k + 7It is easy to observe that Z k +49 k b ( t ) dt = Z k +89 k +4 b ( t ) dt = 0 for all k ≥ . We have Z + ∞ k +2 b ( t ) dt = Z k +49 k +2 b ( t ) dt = − a k , Z + ∞ k +6 b ( t ) dt = Z k +89 k +6 b ( t ) dt = a k and respectively Z + ∞ k +2 | b ( t ) | dt = 3 a k + 4 + ∞ X m = k +1 a m , Z + ∞ k +6 | b ( t ) | dt = a k + 4 + ∞ X m = k +1 a m . By noticing that L + = lim k → + ∞ (9 k + 6) R + ∞ k +6 b ( t ) dt R + ∞ k +6 | b ( t ) | dt , L − = lim k → + ∞ (9 k + 2) R + ∞ k +2 b ( t ) dt R + ∞ k +2 | b ( t ) | dt , we obtain L + = α and L − = − α .To verify the condition (26), notice first that I k = Z k +1)9 k " t R + ∞ t | b ( s ) | ds − p t | a ( t ) | dt ≤ Z k +1)9 k " k + 1) R + ∞ k +1) | b ( s ) | ds − p a k dt, k ≥ . The elementary inequality a k ≤ (2 α − k + 1) − α implies that I k ≤ c α ( k + 1) (1+ α ) p − , where c α = 9 (cid:18) (cid:19) − p (2 α − , and the conclusion follows from the convergence of the series P k ≥ ( k +1) − (1+ α ) p . scillatory integration Acknowledgement
The author is indebted to Professor Ondrej Doˇslyand to a referee for valuable comments leading to an improvement of the ini-tial version of the manuscript. The author was financed during this researchby the Romanian AT Grant 97GR/25.05.2007 with the CNCSIS code 100.
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