CRITERIA AND ESTIMATES FOR DECAYING OSCILLATORY SOLUTIONS FOR SOME SECOND-ORDER QUASILINEAR ODES

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1 Electronic Journal of Differential Equations, Vol ), No 28, pp 1 10 ISSN: URL: or CRITERIA AND ESTIMATES FOR DECAYING OSCILLATORY SOLUTIONS FOR SOME SECOND-ORDER QUASILINEAR ODES TADIE Dedicated to the family Teku Kuate Kamguem Ebenizer Abstract Oscillation criteria for the solutions of quasilinear second order ODE are revisited In our early works 6, 7], we obtained basic oscillation criteria for φαu t)) + αct)φβ ut)) = 0 by estimating of the diameters of the nodal sets of the solutions The focus of this work is to estimate the decay of the oscillatory solutions Let u be a strongly oscillatory solution, t m) the increasing sequence of zeros of u, and D m the nodal set of u that contains t m We estimate ut m) := max t Dm ut) and the diameter of D m as m 1 Introduction For some constants b, β, q, c 0, α > 0 and φ γ S) := S γ 1 S with γ > 0), we consider problems of the type φα u t)) } + αct)φβ ut)) = 0, t > 0; 11) u0) = 0, u 0) = b > 0, where c C 1 R +, c 0, )), with c > 0 and ct) = Ot q ) as t We will review some oscillation criteria for such equations and establish estimates of the decay of oscillatory solutions of 11) Definition 11 A function u is said to be oscillatory if it has a zeros in every exterior domain Ω T := T, ) with T 0 A function u is said to be strongly oscillatory if its zeros are isolated, or if it has nodal sets in every Ω T A nodal set of a function v is an interval Dv) = t, s] such that vt) = vs) = 0 and v 0 in t, s) For the function v + t) = max0, vt)}, the nodal set Dv + ) = t, s] is such that vt) = vs) = 0 with v > 0 in t, s) An equation or a problem) is said to be oscillatory in Ω T if its bounded and non-trivial solutions belong to C 2 Ω T ) and are strongly oscillatory For a strongly oscillatory function u, Du)] will denote the set of nodal sets of u In this case there are two increasing sequences x k ) and t k ) such that x k < t k < x k+1, ux k ) = 0, and u t k ) = 0 We denote D k := D k u) = x k, x k+1 ] as 2010 Mathematics Subject Classification 34C10, 34K15 Key words and phrases Oscillation criteria for ODE and Estimates of decaying solutions c 2017 Texas State University Submitted September 14, 2016 Published January 24,

2 2 TADIE EJDE-2017/28 a nodal set of u We denote ut k ) := max Dk u) ut) For a, b R, we define a b := mina, b} Our main result for problem 11) reads as follows Theorem 12 For all b, α, β, c 0 > 0, any non-trivial and bounded solution of 11) is strongly oscillatory in 0, ) With the corresponding elements as defined above as m, with β := α β, we have π α cx m+1 )] 1/β +1) < x m+1 x m < where π α, 12) cx m )] 1/β +1) ut m ) constct m )] 1/β +1) = constt m ] q/β +1), 13) π α := 2π α + 1) sinπ/α + 1)] The result in 4] is limited to estimates 12) of the diameters of the nodal sets, for the case α = β > 0 Now we present some Picone-type formulae which will be used throughout this article To start, for w, y C 1 R, R) and γ > 0 we define see eg 1, 2]) ζ γ w, y) := w γ+1 γ + 1)w φ γ w y y ) + γ w y y γ+1 14) which is strictly positive and is null only if there exists µ R such that w µy Let C, C 1, α, β > 0 and w, z, u C 1 R), respectively, be solutions in R +, for φα u t)) } + ct)αφβ ut)) = 0; Using that for γ > 0 and S, T R, φα z ) ) + Cαφα z) = 0, φα w ) ) + C1 αφ β w) = 0 Sφ γs) = γφ γ S), Sφ γ S) = S γ+1, φ γ ST ) = φ γ S)φ γ T ), wherever u 0, we have zφα z ) zφ α z u u ) ] = ζα z, u) + α z α+1 ct) u β α C } ; wφα w ) wφ α w u u ) ] = ζα w, u) + α w β+1 ct) u w β α C 1 } 15) = ζ α w, u) + α w α+1 ct) u β α C1 w β α} Note that: 1) For µ > 0, if the function Zt) := µzt) is used in 15)i), φ α Z ) ) + Cαφ α Z) = 0 and 15)i) remains the same with Z replacing z 2) But if W t) := µwt) then φα W ) ) + µ α β C 1 αφ β W ) = 0 and 15)ii) with W holds with C 1 replaced by µ α β C 1 3) The Picone-type formulae in 15) will be the main tools in this work In fact as the formulae make sense only wherever u 0 w 0), if the righthand side of the formula happens to be strictly positive in a set D then the integration over D would give 0 at the left and a strictly positive value at the right if u 0 w 0) in D and u D = 0 Therefore if the right-hand

3 EJDE-2017/28 DECAYING OSCILLATORY SOLUTIONS 3 side of 15) is strictly positive on a set D, we cannot have u w) 0 inside D and u D = 0, implying that u has to have a zero inside such a D Now we study equations with positive constant coefficients Theorem 13 For each k, θ, β > 0, any bounded and non-trivial solution u of the problem φθ u ) } + kθφβ u) = 0, t > 0; u0) = 0; u 0) = A > 0 16) is oscillatory and β + 1) u t) θ+1 + kθ + 1) ut) β+1 = β + 1)A θ+1 t > 0, ut ) = 0 u T ) = A T > 0 u S) = 0 us) = β + 1)A θ+1 ] 1 β+1 S > 0, kθ + 1) which implies max u = β + 1)A θ+1 ] 1 β+1 and max R + kθ + 1) R u = A + When β = θ > 0, 17)iv) reads max u = 1 ] 1 β+1 A and max u = A R + k R + 17) Proof That this problem is oscillatory has been established in 6, 7] but for selfcontained purpose we show it using a method relevant to the present work Let u C 2 R + ) be a non-trivial and bounded solution of 14) Then φθ u ) ) = u 2 ] θ 1)/2 u ) and Similarly = u u 2 ] θ 1)/2 ) + u u 2 ) θ 1)/2) = u u θ 1) + θ 1)u u θ 1 = θu u θ 1 u φ θ u ) ) θ = 2 u 2 ) u 2) θ 1)/2 θ = u θ+1) θ + 1) θku φ β u) = θku u u β 1 = θ 2 ku2 ) u 2) β 1)/2 = θ β + 1) k u β+1) The two inequalities above lead to β + 1) u θ+1 + kθ + 1) u β+1} = 0 18) and 17)i) follows 17)ii) to 17)v) follow immediately Assume that u > ν > 0 in some Ω T Then with k replacing ct), in 15)i), k u β α C kν β α C > 0 if we take C small enough With this, the integration over Dz + ) Ω T would lead to a contradiction as the left hand side would be zero and the right strictly positive If in such an Ω T u > 0 and u 0 then i) or ii) would be violated Therefore u has to have a zero in any Ω T

4 4 TADIE EJDE-2017/28 Corollary 14 Let A 1, A 2, θ, β > 0 θ β Let u 1 and u 2, respectively, be oscillatory solutions for φθ u i) } + ki θφ β u i ) = 0, t > 0; u0) = 0; u 0) = A i > 0 19) with A θ+1 1 k 1 < Aθ+1 2 k 2 Let Du + i ) denote a nodal set of u+ i, and assume that Du+ 1 ) Du+ 2 ) R D := Du + 1 ) Du+ 2 ) with u 1R) = u 2R) = 0, then If max Du + 1 ) u 1 := u 1 R) > max Du + 2 ) u 2 := u 2 R) 110) Let u 1, u 2, u 3, respectively, be non-trivial oscillatory solutions for φθ u i) } + ki θφ β u i ) = 0, t > 0; u0) = 0; u 0) = A > 0, 111) where k 1 > k 2 > k 3 > 0 Then if there is S > 0 such that for some Du + 1 ), Du+ 2 ) and Du + 3 ), then S Du + 1 ) Du+ 2 ) Du+ 3 ), Du is) = 0, for i = 1, 2, 3, max u + Du + 3 ) 3 t) = u 3S) max u + Du + 2 ) 2 t) = u 2S) max u + Du + 1 ) 1 t) = u 1S) The proof of the above corollary follows straight from 17)iv) Remark 15 1) It is easy to show that when the coefficient of φ β is a positive constant, the solutions are periodic 2) There are two transformations which could be used in some proofs: i) For any oscillatory function u, and λ > 0, the associated function u λ t) := λut) is also oscillatory, having exactly the same zeros as u but with u λ = λ u and u λ = λ u ii) For ξ R, the translated function U ξ t) := ut+ξ) would be also oscillatory as u and the curve t, U ξ t)) would be that of u, slit alongside the t-axis forward if ξ < 0) or backward if ξ > 0) 3) Let u and v, respectively, be oscillatory solutions of φα u ) ) + ct)φβ u) = 0; t > 0; φα v ) ) + Cφβ v) = 0, t > 0; v0) = 0, v 0) = b > 0 If some of their nodal sets satisfy Du + ) Dv + ), and R Du + ) satisfies u R) = 0, then ξ can be chosen such that the transformed W t) := vt + ξ) has the same singularity R in the resulted DW + ) ie W R) = u R) = 0 In summary if a) Du + ) Dv + ) and b) u has a zero inside Dv + ), then there is ξ, λ) R R + such that for some R Du + ), then the function V t) := λvt + ξ) satisfies V R) = u R) = 0 and V = λ v

5 EJDE-2017/28 DECAYING OSCILLATORY SOLUTIONS 5 2 Equations with increasing and unbounded coefficients It is known that if ct) is increasing and unbounded then if x n ) n N denotes the increasing successive zeros of the oscillatory solution z of φα z ) ) + αct)φα z) = 0, 21) then x n+1 x n = O π α cx n )] 1/α+1)) for large n In fact as for large m N, cx m ) ct) cx m+1 ), inside D m := x m, x m+1 ]; from 4], with Cx) := cx)] 1/α+1), we have π α Cx m+1 ) x m+1 x m π α Cx m ) 22) Lemma 21 For some c 0 > 0 let c C 1 R, c 0, )) be increasing, and let α, β > 0 Then any non-trivial and bounded solution u of φα u t)) } + ct)αφβ ut)) = 0, t > 0; is oscillatory u0) = 0, u 0) = b > 0 Proof The oscillatory character of the equations have been established in our early papers 4, 7] but for later use purpose, we provide some slightly different proofs using Picone-type formulae 1) Assume that α β > 0 Let u be such a solution and with some C > 0 Let z be an oscillatory solution of φα z ) ) + Cαφα z) = 0; t > 0 If we suppose that u > µ > 0 in some Ω S then ct) ut) β α > ct)µ β α for t > S and the right-hand side of 15)i) is eventually strictly positive in Ω S Assume that u > 0 in some Ω T for some T > 0 and u 0 as t Still because 0 < β α, the function ct) ut) β α is unbounded in Ω T and the righthand side of 15)i) is eventually strictly positive in Ω S for large S > T In those cases, the right-hand side of 15)i) is strictly positive in any such a Dz + ) Ω T Thus the assumption cannot stand; u has a zero in any Ω T 2) Assume that β > α > 0 For a constant C > 0 and an oscillatory solution z of φα w ) ) + αcφβ w) = 0, t > 0; w0) = 0, w 0) = b > 0 wherever u 0 in some interval D, 15)ii) holds with C instead of C 1 ) As C is constant, from 17), w + has a constant maximum value in any nodal set Dw + ) which is w := w CDw + )) = max w = β + 1)b α+1 ] 1 β+1) Dw + ) α + 1)C We see that the smaller b := w 0) is, the smaller w will be If there exists ν > 0 such that u > ν in Ω R then as c is unbounded, the righthand side of 15)ii) is eventually strictly positive in any nodal set Dw + ) Ω S for large enough S > R as we would have ct) C u w β α 1 } > ct) C u β α } 1 w

6 6 TADIE EJDE-2017/28 with an unbounded ct) Assume that u > 0 and u decreases to zero at in some Ω T, with T > 0 Then for any R > T and J R := R, 2R], we define ν := u2r) := min JR u + ] We take C := cr) := C 1 and R > T so big that wr) = OR q/β+1) ) With such a large cr), w + has many nodal sets Dw + ) in J R and with b small enough, w < ν and ct) ν β α CR) w } > 0 in many of them The integration over such a Dw + ) of 15)ii) would lead to a contradiction as the left hand side would give 0 and the right strictly positive Thus u > 0 cannot hold in any Ω T This, as above, completes the oscillatory character of u Theorem 22 1) Let u and z, respectively, be oscillatory solutions of φα u t)) } + ct)αφβ ut)) = 0, φα z ) ) + mαφα z) = 0, t > 0 where for some c 0 > 0, c C 1 R, c 0, )) is an increasing and unbounded function and α β > 0 Assume that there are two overlapping nodal sets Dz + ) and Du + ) such that i) thee exists R Dz + ) Du + ) such that z R) = u R) = 0; ii) u has a zero inside Dz + ) and ct) u β α m } > 0 in Dz + ) Then Du + ) Dz + ) whence diam Du + ) ] diam Dz + ) ] = O 1 ] 1/α+1) ) 23) m 2) Also if 0 < α < β instead of z the solution w of φα w ) ) + mαφβ w) = 0, t > 0 is used, then under the conditions i) and ii) the results hold with w replacing z with the following changes: ct) u β α m } > 0 in Dw + ) and we have w diam Du + ) ] diam Dw + ) ] = O 1 ] 1/β+1) ) 24) m Proof Let Dz + ) := t 1, t 2 ] and Du + ) := x 1, x 2 ] with t 1 < x 1 < R < t 2 We claim that R < x 2 < t 2 Otherwise if u > 0 in R, t 2 ) the integration of 15)i) where m = C) over R, t 2 ) leads to an absurdity as unlike the right-hand side, the left would be zero Thus x 2 has to be between R and t 2 and using 22), it leads to 23) For the case of w we just use 15)ii) instead of 15)i) As a prelude for the next results we have the following Lemma; Lemma 23 For the strongly oscillatory solution u of φα u ) ) + αct)φβ u) = 0, t > 0; u0) = 0, u 0) = b > 0 25) define the increasing sequences T k ) and S k ) such that 1) for all n N, T n, T n+1 ] := D n Du + ) ], S n D n ; u S n ) = 0; 2) c n t) = ct) for t 0, T n ] and c n t) = ct n ) for t T n For any n, let u n and z n, respectively, be the solutions of φα u ) ) + αcn t)φ β u) = 0, φα z ) ) + αctn )φ β z) = 0; z0) = 0, z 0) = u T n )

7 EJDE-2017/28 DECAYING OSCILLATORY SOLUTIONS 7 Then u n z n in Ω Tn and with β := maxα, β}, as n, β + 1)u u n Du + n ) = z T n ) θ+1 ] 1 β +1 ns n ) = = O T n ] q/β +1)) 26) ct n )θ + 1) Proof The identity u n z n in Ω Tn is due to the fact that the two satisfy the same initial values at T n In fact if w and v are two C 2 Ω T ) solutions for φα u ) ) + αct)φβ u) = 0; ut ) = 0, u T ) = b > 0 then without loss of generality we assume that u > v > 0 in some T, τ) From φ α w ) = α w w φ α w ) as Sφ αs) = αφ α S)), and from their equations v u u v = ct) u v 1 α v β u α 1 u β v α 1] := ct) u v 1 α Γu, v), Γu, v) = 0 at T and remains strictly positive as long as v > 0 Therefore as long as v > 0, u v is increasing as v u u v = v ) 2 ) u v But from these formulae, v should not be zero while u > 0 Thus v and u have the same first zero after T which is a contradiction 26) follows from 23) and 24) 3 Estimates for some decaying oscillatory solutions Now we take for oscillatory functions z := z R which will a fortiori depend upon the function u through their bounded coefficients Namely we will use z, a solution of φα z ) } + αcφβ z) = 0; t > 0; z0) = 0; u 0) = b > 0 where C will be the value of c at some point R > 0 Theorem 31 Let R, c 0, β, α > 0 and c C 1 R +, c 0, )) be unbounded and increasing Then if u and z := z R are, respectively, two non-trivial oscillatory solutions of φα u t)) } + ct)αφβ ut)) = 0, φα z ) ) + cr)αφβ z) = 0, t > 0; z0) = 0; z 0) = b > 0 31) Then there is R 1 > 0 such that u has a zero inside any nodal set Dz + ) Ω R for all R > R 1 Proof Let u and z be such oscillatory solutions We saw that any multiplication of z by a positive λ > 0 would not affect any Dz) but only that λz = λ z Also for all T > 0, there are a multitude of Dz + ) and Du + ) inside Ω T 1) Suppose that β > α > 0 Let T 1 > 0 be such that ct) > 1 for all t > T 1 Assume that there exists T > T 1 such that for all R > T there is a nodal set Dz + R ) := D 1z + ) Ω R such that u > 0 in D 1 z + ) We take T 1 big enough for J R := R, 2R] to contain many nodal sets of z + including D 1 z + ) which is guaranteed by the fact that bigger R is, the smaller diamdz + R )) is If for some ν > 0, u β α > ν β α > 0 in D 1 z + ), then, in D 1 z + ) := DZ + ), the function Zt) =: νzt) satisfies φα Z ) ) + ν β α cr)αφ β Z) = 0, t > 0, Zφα Z ) Zφ α Z u u ) ] = ζα Z, u) + α Z α+1 ct) u β α ν β α cr) } > 0 32)

8 8 TADIE EJDE-2017/28 The integration over DZ + ) of 32) provides a contradiction Therefore the assumption cannot be true and u has to have a zero in D 1 z + ) 2) Assume that α β > 0 For this case 15)i) is used instead of 32), and the same conclusion is obtained Corollary 32 1) Let u and z be the two solutions in 31) where C > 0 is arbitrary Let two of their nodal sets, Let Du + ) and Dz + ), be such that u has a zero in Dz + ) and S Du + ) is the singularity of u + therein Then there is ξ R such that the translated function Zt) := zt + ξ) satisfies Z S) = u S) = 0, Du + ) DZ + ), diam Du + ) diam Dz + ) 2) Moreover, for t large enough, max u + := u Du + ) max Z + := Z + DZ + ) = z + Dz + ) 33) Du + ) DZ + ) Proof 1) This follows from Theorem 22 and Theorem 31 2) follows from Lemma 23 Proof of the Theorem 12 Any such a solution of 11) is strongly oscillatory by Lemma 21 and 4, 7] The estimates follow from Theorem 22, Theorem 31 and Corollary 32 4 An application For a restoring h CR) ie y R \ 0}, yhy) > 0) consider the problem φα u ) } + αct)hu) = 0, t > 0; u0) = 0, u 0) = b > 0, 41) where α, β, q > 0 and c being as before and for small S > 0, hs) = O S β) For the strongly oscillatory solution z of φ α z ) } + αcφα z) = 0, t > 0, and w of φ α w ) } + αcφβ w) = 0, wherever u 0, we have zφα z ) zφ α z u u ) ] = ζα z, u) + αc z α+1 ct)hu) Cφ α u) 1}, wφα w ) wφ α w u u ) ] = ζα w, u) + α w α+1 ct)hu) Cφ α u) w β α} 42) As h is a restoring function, we can define the function h 1 CR +, R + ) by hs) := h 1 S 2 )S for all S R and define H 1 t) := t 0 sh 1s 2 )ds such that equation 41) reads φα u ) ) + αct)h1 u 2 )u = 0, t > 0; u0) = 0, u 0) = b > 0 43) Thus, similar to Theorem 13, we have the following result Lemma 41 With h 1 defined in 43), C, α, b, β > 0 the problem φu ) ) + αch1 u 2 )u = 0, t > 0; u0) = 0, u 0) = b is strongly oscillatory furthermore and for its solution u, and all t > 0, we have 2 u t) α+1 + α + 1)CH 1 u 2 t)) = 2b α+1, us) = 0 and u T ) = 0 = u S) = b and ut ) = H 1 1 2b α+1 α + 1)C )] 1/2 44)

9 EJDE-2017/28 DECAYING OSCILLATORY SOLUTIONS 9 Proof From φu ) ) + αch1 u 2 )u = 0, u u φ αu ) + αch 1 u 2 )uu = αu φ α u ) + α C 2 u2 ) h 1 u 2 ) = 0 thus 1 2 u 2 ) u 2) α C 2 u2 ) h 1 u 2 ) = 1 α + 1 u α+1 + C 2 H 1u 2 ) ] = 0 leading to 44)i) Then 44)ii) follows as well The oscillation of the solution is obtained as for the Theorem 13 Theorem 42 For c 0, α, β, q > 0, let h 1 CR, 0, )) with h 1 S 2 )S = Os β ) for small S > 0 and c C 1 R, c 0, )) with c > 0 and ct) = Ot q ) as t Then any non-trivial and bounded solution of φα u ) ) + αct)h1 u 2 )u = 0, t > 0; u0) = 0; u 0) = b > 0 45) is strongly oscillatory 1) Moreover for any R > 0 let z := z R be a non-trivial oscillatory solution of φα z ) ) + cr)αφα z) = 0; t > 0 Then for S > 0 large enough, the oscillatory solution u of 45) has a zero in any nodal set Dz + R ) Ω S for R > S 2) Consequently as t, for β := α β the solution in 45) has the estimates ut) constt] q 1 ] 1/β +1) β+1 := const, ct) diamdu + 1 ] ) 46) 1/β +1) )) = O ct) Proof 1) For some C > 0 let z be a strongly oscillatory solution to φα z ) } + αcφα z) = 0 Then 42)i) with hu) replaced by h 1 u 2 )u becomes zφα z ) zφ α z u u ) ] = ζα z, u) + αc z α+1 ct)h 1 u 2 )u Cφ α u) If we assume that u > ν > 0 in some Ω R, then with ζ α z, u) + αc z α+1 ct)h 1 u 2 )u Cφ α u) Gν) := inf u ν 1 } 1 } > ζ α z, u) + αc z α+1 ct)gν) 1 } ct)h 1 u 2 )u Cφ α u) Because ct) is unbounded, ct)gν) 1} is eventually strictly positive Assume that that u > 0 and decreases to zero in some Ω S a) Case where α > β > 0 For very large R > S, as u 0, ct)h 1 u 2 )u Cφ α u) 1 } > const ct) C u β α 1 ] > 0 eventually and the integration over Dz) of 42)i) leads to a contradiction b) Let β α > 0 and w the oscillatory solution in 42)ii) Assume that u > 0 in some Ω S We use h 1 u 2 )u instead of hu) there For T > 0, We define h J T := T, 2T ) and ν := νt ) = inf 1u 2 )u JT Cφ αu) We take R > S so large that w+

10 10 TADIE EJDE-2017/28 has many nodal sets in J R and ct) > C there We choose b = w 0) such that w + ) β α < νr) Then in J R, ct)h 1 u 2 )u Cφ α u) w β α} > 0, and integration over Dw) of 42)ii) leads to a contradiction Therefore u cannot remain positive throughout any Ω T Assume that there is T > 0 such that for all R > T, there is a nodal set Dz + R ) := D R J R such that for some µ > 0, u > µ on D R We remind that ct) cr) for all t > R Then similar to a) and b) above, we see that as we can make z + R arbitrary small in J R, we cannot find T and µ > 0 such that the assumption holds 2) The estimates are obtained through the Corollary 32, keeping in mind that as h 1 τ 2 ) constτ β, we have H 1 τ) constτ β+1 References 1] J Jaros, T Kusano; A Picone-type identity for second order half-linear differential equations Acta Math Univ Comenianae, 1999) vol LXVIII, ] J Jaros, T Kusano, N Yosida; Picone-type Inequalities for Nonlinear Elliptic Equations and their Applications J of Inequal & Appl, 2001), vol 6, ] Patricia Pucci, James Serrin; The strong maximum principle revisited, Journal of Differential Equations, ), ] Tadié; Semilinear Second-Order Ordinary Differential Equations: Distances Between Consecutive Zeros of Oscillatory Solutions, Constanda, Christian ed) et al, Integral methods in science and engineering Theoretical and computational advances Papers based on the presentations at the international conference, Zbl , Springer International Publishing Switzerland 2015) 5] Tadié; Oscillation criteria for damped quasilinear second-order elliptic equations, Electronic J of Differential Equations, vol ), No 151, pp ] Tadié; On strong oscillation criteria for bounded solutions for some quasilinear second order elliptic equations, Communications in Mathematical Analysis, vol 13, No ), pp ] Tadié; Oscillation Criteria for Some Emden-Fowler Type Half-Linear Elliptic Equations, Int Journal of Math Analysis, vol ), Tadie Mathematics Institut, Universitetsparken 5, 2100 Copenhagen, Denmark address: tadietadie@yahoocom