A Note on Superlinear Ambrosetti-Prodi Type Problem in a Ball

Size: px
Start display at page:

Download "A Note on Superlinear Ambrosetti-Prodi Type Problem in a Ball"

Transcription

1 A Noe on Superlinear Ambrosei-Prodi Type Problem in a Ball by P. N. Srikanh 1, Sanjiban Sanra 2 Absrac Using a careful analysis of he Morse Indices of he soluions obained by using he Mounain Pass Theorem applied o he associaed Euler Lagrange funcional acing boh on he full space H(Ω) 1 and on is subspace H,r(Ω) 1 of radially symmeric funcions we prove he exisence of non-radially symmeric soluions of a problem of Ambrosei Prodi ype in a Ball. Keywords: Mounain Pass Theorem, Non-radial soluions, Morse Index, Concenraion. AMS Subjec Classificaions (2): 35J65, 35J2 1 Inroducion We consider he following semi-linear ellipic problem (P ) { u = (u + ) p ϕ 1 in Ω u = on Ω where Ω = {x IR n : x < 1}. ϕ 1 is he eigenfuncion corresponding o he firs eigenvalue λ 1 of ( ) wih zero Dirichle daa on he boundary and normalized by aking ϕ 1 () = 1. is a real parameer. Due o requiremens of he variaional mehod used here, we have o resric he values of he dimension n o 1 < n < 6. In [2],i has been proved ha (S ) wih g(u) = u 2 admis a non-radially symmeric soluion for large > even if Ω is a ball.i is also easy o see ha he resul in [2] follows if g(u) = u p 1 TIFR Cenre, P.B. 1234, IISc Campus, Bangalore , India, srikanh@mah.ifrbng.res.in 2 Deparmen of Mahemaics, Indian Insiue of Science, Bangalore , India, sanjiban@mah.iisc.erne.in 1

2 where p (1, n+2 ) if n 3 and p (1, ) if n = 2.Moivaed by he resuls in [2] i has n 2 been proved in [1] ha (S ) has a non-symmeric soluion in Ω where Ω is a smooh bounded domain in R n where p (1, n+2 ) if n 3 and p (1, ) if n = 2. n 2 (S ) { u = g(u) ϕ1 in Ω u = on Ω where g(u) = u p.in [1] he mehod works for nonlineariy of he form g(u) = a u + p +b u p for a >,b >. I is no clear in wheher he mehod [1] works for a nonlineariy of he form g(u) = (u + ) p because of he esimae one needs o conrol he negaive soluion.in his noe we conclude ha he resul in [2] holds for g(u) = (u + ) p. So our main resuls are he following Theorem 1. For > sufficienly large,all he non-negaive radial soluions of (P ) have Morse Index larger han one. As a consequence of his heorem, we have Corollary. For > sufficienly large a soluion of (P ) in H 1,r(Ω) wih Morse Index 1 has o change sign. Theorem 2. For > sufficienly large he radial soluion of (P ) has Morse Index one in he space H 1,r(Ω) has Morse Index a leas wo on he whole space H 1 (Ω). Theorem 3. For > sufficienly large (P ) has soluions which are no radially symmeric. 2 Proof of he Theorems. Since we are in he case n 5, regulariy resuls imply ha he H(Ω)-soluions 1 obained here are indeed classical soluions, and his fac will be used hroughou he paper.also noe ha if u is a posiive soluion of (P ) hen u + = u. Remark 1.To sar wih, using his fac we obain he following properies of he radial 2

3 soluions of (P ). We recall ha a radial soluion u of (P ) saisfies he following equaion: u rr (n 1)u r r and boundary condiions u r () = u(1) = = ( u + ) p ϕ 1, 1) Each non-negaive radial soluion of (P ) has a mos a finie number of poins of maxima and minima. 2) A a poin of maximum a of a non-negaive radial soluion u we have u p (a ) ϕ 1 (a ). Also, here is an open inerval (a, d ) where u p (r) ϕ 1 (r) >. 3) A a poin of minimum m of a non-negaive radial soluion u we have u p (m ) ϕ 1 (m ). Also, here is an open inerval (e, m ) where u p (r) ϕ 1 (r) <. 4)The proofs of Theorem 1 and Corollary goes in he same way as in [2] so we will no prove his in our noe. 5)If u is a negaive soluion o (P ) hen we have { u = ϕ1 in Ω u = on Ω Hence he soluion o his problem is u = ϕ 1 λ 1 which is unique. 6) When he nonlineariy is u 2, hen one can easily show ha u + L C which is a consequence of he negaive par saisfying u L C. However in he conex of he problem we are discussing he negaive par behaves like u L which calls for a subsle argumen o obain he main resul of he Paper.Infac his seem o be he main reason why he resuls of [1] do no seem o work for he nonlineariies like (u + ) p. 3

4 The sraegy used o prove Theorem 1 consiss in showing ha any posiive soluion u of (P ) has Morse index a leas wo, for large. This can be done by jus considering hree ypes of posiive soluions: Type I: u has a unique maximum a. Type II: u has a leas wo maxima. Type III: u has a unique maximum a a poin a (, 1). Lemma 1: Suppose u is a posiive radial soluion of (P ) having unique maximum a he origin, hen for large such a soluion will have a high Morse index. Proof. Similar as in [2]. Noe, by definiion, if u is a soluion of (P ), is Morse Index is given by he dimension of he space where he form below is negaive definie, ha is w p u p 1 w, w <, w H 1 (Ω). Since he linearized problem is given by ( ) = (p u p 1 )( ) i follows ha, if he weigh ges very large on a inerval of fixed lengh, hen he Morse Index also ges large. Remark 2. We have made no assumpion ( u p ϕ 1 )(). All we wan in Lemma is ha is he unique maximum. In he case ( u p ϕ 1 )() =, u decreases near can be seen by differeniaing equaion u rr r (n 1) u r = ( u p ϕ 1 )r o obain u rrrr () < and u rrr () =. Noe u rr () = follows from ( u p ϕ 1 )() =. This implies u p ϕ 1 > in an inerval (, d ). Lemma 2. If u is a Radial soluion of (P ) for some > wih more han one maximum, hen is Morse Index is a leas wo on he Radial space. Proof. There are wo possibiliies. A) The soluion has precisely wo maxima: and a > ; B) The soluion has wo maxima < a 1 < a 2 < 1, and evenually more. We assume ha hose are he wo larges ones. 4

5 The proof in eiher case follows in he same way as in [2]. The basic fac is o use he funcion F (r) := u p ϕ 1, and noing ha F (r) = (p u p 1 )F (r) p(p 1) u 2 r u p 2 λ 1 ϕ 1 and observing also ha p(p 1) u p 1 u 2 r + λ 1 ϕ 1 >, r [, 1). Lemma 3. Suppose ha for all large here exiss a radial posiive soluion u in (, 1) of (P ), wih Morse Index one on he Radial space H 1,r(Ω) wih ( u p ϕ 1 )(), hen a 1 as, where a is he unique maximum in (, 1). Proof of Lemma 3. From he fac ha here is a unique maximum, we have u r > in (, a ) and u r in (a, 1). We denoe by b he unique poin in (, a ) where ( u p ϕ 1 ) =. i.e ( u p ϕ 1 )(b ) =. The uniqueness of b follows from he fac u is increasing in ([, a ) where as ϕ 1 is decreasing in (, 1). The proof of his lemma is essenially conained in Lemma 1, and similar o [2]. Remark 3. Le b be he poin in (, a ) as specified in he proof of Lemma 3. If b =, large, we see easily ha he Morse index canno be one. I is also clear from Lemma 3, since a 1, one should have ha b 1 as well in order o mainain Morse Index one in he conex of he posiive soluions we are discussing. We will now prove ha b 1 will lead o a conradicion, hus compleing he proof of Theorem 1. Lemma 4. Suppose ha for all large here exiss a radial posiive soluion u in (, 1) of (P ), wih Morse Index one on he Radial space wih ( u p ϕ 1 )() <. Then b canno converge o 1 as, where b is he unique poin in (, a ) such ha ( u p ϕ 1 )(b ) =, a being he unique maximum in (, 1). Proof of Lemma 4.See [2]. Proof of Theorem 1. Theorem 1 now follows direcly from he above lemmas. Proof of Theorem 2.By Theorem 1, we know ha he soluion obained by Mounain Pass in H 1,r(Ω) has o change sign. Noe ha here canno be more han one posiive par if Morse Index is one.noe ha each posiive par conribues o he Morse Index by one in 5

6 he radial space follows from, < u (pu p 1 bi )u, u > H 1,r (a i,b i )= (u p + ϕ 1 )u < a i in each (a i, b i ), where (a i, b i ) are disjoin inervals of posiiviy of u. u (ai,b i ) H 1,r(a i, b i ). In his conex we are in, due o he above facs we need o look a wo ypes of soluions. Type IV: he radial soluion u has a (unique) posiive maximum a and a unique negaive minimum a a (, 1). Type V: he radial soluion u has a (unique) negaive minimum a and a unique posiive maximum a a (, 1). All oher possibiliies are covered by reasonings similar o he ones used o handle Types I hrough V. We proceed now wih he proof of Theorem 2 by showing Lemmas 5, Proposiion 1 and Lemma 6. Lemma 5. Suppose u is a radial soluion of (P ) wih u() > and wih a negaive par hen u has Morse Index a leas wo on he space H 1 (Ω). Proof of Lemma 5. Noe ha, under he assumpions of his lemma, a ypical soluion is of Type 4. Dropping he prefix for he soluion we have ha i saisfies he equaion i.e. u = (u + ) p ϕ 1 in Ω u = on Ω (1) In course of our proof, we will discuss he case Ω IR 2 (See Remark 4 for he general case), jus o make he idea more explici. Denoing he coordinaes by (x, y) = (r cos θ, r sin θ), and differeniaing (1) wih respec o x and wriing w = u r cos θ, we see w saisfies w = (pu p 1 )w ϕ 1,r cos θ in B(, d ) w = ϕ 1,r cos θ in B(, a ) B(, d ) w = on B(, a ) Noe ha u r in (, a ), where a is he firs minima of u and u(d ) =. Since ϕ 1r, we have w (pu p 1 )w, w < H 1 (Ω) where w is defined by w in B(, a ) and zero elsewhere. Noe ha u + conribues o he Morse Index and w also conribues o he Morse Index and w and u + are orhogonal. In 6

7 fac he Morse Index of u is a leas wo. In fac he Morse Index of u is a leas hree since w 1 = u r sin θ will play he same role as w. Hence he lemma. Remark 4. I is clear u xi (i = 1,..., n) will work exacly like u r cos θ if we are in higher dimensions, leading o Morse index being (n + 1). Remark 5. In view of Theorem 1 and Lemma 5 i is clear ha Theorem 2 will follow if we can show ha Radial soluions wih u() < and having a unique zero in (, 1) has Morse Index a leas wo on he whole space for large. A ypical soluion we need o consider is of Type V. Noe ha if u() < and has wo nodal zeros, hen argumens as he ones used in Lemma 5 would lead o a high Morse Index on he whole space. Le d denoes he unique zero of u in (, 1), a he poin of maximum and b is he unique poin in (d, a ) such ha u p (b ) ϕ 1 (b ) =. Remark 6. If u denoes a soluion as discussed in Remark 5, we have ha u L if u r in [, a ], where u = max( u, ). I is easy o show ha u L.We now proceed o show ha u + L C 2 p+1 for C independen of, for all soluions under consideraion which is a major cause of difficuly. Proposiion 1. Suppose u is a radial soluion of (P ) for > wih u() < and wih u changing sign a a unique poin say d in (, 1) wih u > in (d, 1), having a unique maximum a and u r in (, a ). Then here exiss C > independen of such ha u + L C 2 p+1. (2) Remark 7. In he conex of Theorem 2, which is our main goal, i is clear, from earlier discussions, ha now we need jus o consider soluions saisfying hypohesis of Proposiion 1. Tha is, soluions saring wih u() < and having a unique posiive maximum a some poin a wih u r in [, a ]. Noe ha argumens similar o Lemma 5 would imply higher Morse Index if u r < in some subinerval of [, a ]. Also as observed earlier wo posiive maxima would make he soluions o have higher Morse Index on Radial space 7

8 iself. Also u() > has already been aken care hrough Lemma 5. Proof of Proposiion 1. Similar o [2]. Remark 8. In he conex of soluions discussed in Proposiion 1 i is clear ha argumens of Lemma 3 are applicable and ha a 1 as as long as we assume u has Morse Index one on he Radial space. Lemma 6. Le u be a soluion of (P ) wih Morse Index one on he Radial space and saisfying he hypohesis of Proposiion 1, hen d 1 as. Proof. Similar as [2]. Proof of Theorem 2 (compleed).in order o finish he proof of Theorem 2, we have jus o use he previous lemmas o show ha a radial soluion of Type V, has Morse index higher han 1 in he full space H 1 (Ω). This is done nex. We will discuss he proof Ω IR 2, and Ω IR n, n 3 separaely. Recall we are working in a siuaion where he soluion is like in Type V.Le us firs discuss he case when Ω IR n, when n 3 Hence Le w = u(r)(x 1 + x x n ) hen w = (x 1 + x x n ) u 2 n u xi i=1 w = (x 1 + x x n )( u p ϕ 1 ) 2 Then w saisfies on (d, 1) he equaion n u xi i=1 w = p( u p 1 ) w (p 1) u p (x 1 + x x n ) (x 1 + x x n )ϕ 1 2 n i=1 u xi in Ω d w = on Ω d where Ω d = {r : d < r < 1}. Thus w p( u p 1 )w, w = (p 1) (x 1 + x x n ) 2 u p+1 dx (x 1 + x x n ) 2 uϕ 1 dx Ω Ω n 2 (x 1 + x x n ) u u xi dx Ω i=1 8

9 i.e. w p( u p 1 ) w, w = (p 1) r 2 u p+1 r 2 uϕ 1 + n Ω Ω Ω u 2 Now ha u(r) > in Ω d.in order o prove ha he soluion has Morse Index higher han one on he whole space,i is enough o show ha he funcion F (r) = (p 1) u p r 2 ϕ 1 r 2 + n u(r) is negaive in (d, 1).We will show his o be he case for large. By Lemma 6,we know ha d 1.Hence aking > large enough we can assume r 2 3 in he region of ineres.observe ha 4 F (r) < if u p 1 (r) 4n.Hence 3(p 1) F (r) can occur only if < u p 1 (r) < 4n.Now noe ha 3(p 1) F (d ) <. and F (1) =. Firs we claim ha λ 1 u r (r) ϕ 1,r (r) r (d, a ) (3) Proof of he claim:we have ( u r r n 1 ) r = (( u + ) p ϕ 1 )r n 1 and inegraion from o s, where s a,we have Hence we have, s s s ( u r r n 1 ) r dr = ( u + ) p r n 1 dr ϕ 1 r n 1 dr. Thus we have, s n 1 u r (s) = s n 1 u r (s) = s d u p r n 1 dr s ϕ 1 r n 1 dr s n 1 u r (s) λ 1 s n 1 ϕ 1,r (s) s s ϕ 1 r n 1 dr d u p r n 1 dr λ 1 u r (r) ϕ 1,r (r) as u r (r) in (d, a ). 9

10 Noe ha in his case p < n+2 n 2 and λ 1 > nπ2 4. For r (d, a ) and < u p 1 (r) < 4n 3(p 1) we have F r (r) F r (r) = p(p 1) u r (r) u p 1 (r)r 2 2 u p (r)r ϕ 1,r r 2 2ϕ 1 r + n u r (r) = p(p 1) u r (r) u p 1 (r)r 2 + ϕ 1,r (r) r 2 + n u r (r) 2 u p (r)r 2ϕ 1 r (4) Now le us look o wha happens o F r (1). We have F r (1) = n u r (1) + ϕ 1r (1). Hence we have F r (1) = { n ur(1) + ϕ 1r (1) }. Dropping he prefix for he ime being, we have in (, a ). 1 2 (r n 1 u r (r)) r = ((u + ) p ϕ 1 )r n 1 Muliplying boh sides by r n 1 u r and inegraing beween o a we have a (r n 1 u r (r)) r (r n 1 a u r (r))dr = u p u r r 2n 2 a dr u r ϕ 1 r 2n 2 dr d a (r n 1 u r ) r (r n 1 u r )dr = 1 a (u p+1 ) p + 1 r r 2n 2 a dr u r ϕ 1 r 2n 2 dr d a ((r n 1 u r ) 2 ) r dr = 1 p + 1 up+1 (a )a 2n 2 2n 2 a u p+1 r 2n 3 a dr u r ϕ 1 r 2n 2 dr p + 1 d = 1 p + 1 up+1 (a )a 2n 2 + 2n 2 a u p+1 r 2n 3 a dr + u r ϕ 1 r 2n 2 dr (5) p + 1 d Again we have in (a, 1) (r n 1 u r (r)) r = (u p ϕ 1 )r n 1 Muliplying boh sides by r n 1 u r and inegraing beween a o 1 we have 1 (r n 1 u r ) r (r n 1 u r )dr = (u p+1 ) r r 2n 2 dr u r ϕ 1 r 2n 2 dr a p + 1 a a 1

11 1 2 1 ((r n 1 u r ) 2 ) r dr = 1 a p + 1 up+1 (a )a 2n 2 2n 2 p a u p+1 r 2n 3 dr 1 a u r ϕ 1 r 2n 2 dr u2 r(1) 2 = 1 p + 1 up+1 (a )a 2n 2 2n u p+1 r 2n 3 dr u r ϕ 1 r 2n 2 dr p + 1 a a u2 r(1) 2 = 1 p + 1 up+1 (a )a 2n 2 + 2n u p+1 r 2n 3 dr + u r ϕ 1 r 2n 2 dr (6) p + 1 a a Adding (5) and (6) we have u 2 r(1) 2 = 2n 2 p d u p+1 r 2n 3 dr + 1 a u r ϕ 1 r 2n 2 dr + a u r ϕ 1 r 2n 2 dr Hence we have, u 2 r(1) 2 2n 2 p d u p+1 r 2n 3 dr + a u r ϕ 1 r 2n 2 dr (7) Noe ha we have r n 1 ϕ 1r (r) is an increasing funcion and 1 rn 1 ϕ 1 dr = ϕ 1r(1) λ 1. Hence for large, (7) yeilds u 2 r(1) 2 2 ϕ 1r(1) λ 1 a r n 1 ϕ 1 dr λ 1 u r (1) 2 ϕ 1r(1) π 2 n 4 u r (1) 2 ϕ 1r(1) Thus we have u r (1) n < ϕ 1r(1) So we have F r (1) >.Hence F (r) < in a neighborhood of 1. Now we are required o prove ha F r (r) > for whenever < u p 1 (r) and r (a, 1). 11 4n 3(p 1)

12 Dropping he prefix for he ime being (for he sake of simpliciy). In (a, 1) we have, (r n 1 u r ) r = (u p ϕ 1 )r n 1 have Muliplying he above equaion by r n 1 u r and inegraing from a o r,(r > a ) we r (s n 1 u s ) s (s n 1 u s )ds = 1 r r (u p+1 ) s s 2n 2 ds u s ϕ 1 s 2n 2 ds a p + 1 a a 1 2 r ((s n 1 u s ) 2 ) s ds = a 1 p + 1 up+1 (r)r 2n 2 1 p + 1 up+1 (a )a 2n 2 2n 2 p + 1 r a u s ϕ 1 s 2n 2 ds r a u p+1 s 2n 3 ds 1 2 (rn 1 u r ) 2 (r) = 1 p + 1 up+1 (r)r 2n 2 1 p + 1 up+1 (a )a 2n 2 2n 2 p + 1 r a u s ϕ 1 s 2n 2 ds r a u p+1 s 2n 3 ds 1 2 r2n 2 u 2 r(r) = 1 p + 1 up+1 (r)r 2n p + 1 up+1 (a )a 2n 2 + 2n 2 r r u p+1 s 2n 3 ds + u s ϕ 1 s 2n 2 ds (8) p + 1 a a Adding (5) and (8) we have Hence we have i.e. for large we have 1 2 r2n 2 u 2 r(r) = 1 p + 1 up+1 (r)r 2n 2 + 2n 2 p r a u s ϕ 1 s 2n 2 ds + a r u s ϕ 1 s 2n 2 ds d u p+1 s 2n 3 ds 1 2 r2n 2 u 2 r(r) 2n 2 r u p+1 s 2n 3 a ds + u s ϕ 1 s 2n 2 ds p + 1 d 1 2 r2n 2 u 2 a r(r) u s ϕ 1 s 2n 2 ds 12

13 1 2 r2n 2 u 2 r(r) 2 r a n 1 ϕ 1 (a ) s n 1 ϕ 1 ds λ r2n 2 u 2 r(r) 2 r 2n 2 ϕ λ 2 1r (r) 2 1 λ 1 u r (r) 2 ϕ 1r(r) 4n 3 u r (r) < ϕ 1r(r) Now we have for r (a, 1) F r (r) = p(p 1) u r (r) u p 1 (r)r 2 p u p (r)r ϕ 1,r r 2 2ϕ 1 r + n u r (r) For large we have F r (r) u r (r) p(p 1) u p 1 (r)r 2 > Hence F r (r) >. Hence F (r) < r (a, 1) for large. Now we have o show ha F (r) < r (d, a ). Noe ha F r (d ) > and F r (1) > which implies F (r) is increasing in a neighborhood of d and 1. Hence if F has a sric zero in (d, 1), hen i mus have a leas wo zeros in (d, 1).We claim ha his canno happen. If possible, le here exis r (d, a ) such ha F (r ) =. Then we claim ha (1 r ) C. Since < u p 1 (r) < 4n 3(p 1) we have (p 1) u p (r )r 2 ϕ 1 (r )r 2 + u(r ) = (p 1) u p (r )r 2 + ϕ 1 (r )r 2 = u(r ) 13

14 ϕ 1 (r ) C Hence we have by mean value heorem ϕ 1 (1) ϕ 1 (r ) = ϕ 1,r (ξ )(1 r ) for some ξ (r, 1, ) i.e we have (1 r ) C. Now we claim ha ur(r) If r (d, a ) we have Inegraing beween from r o a we have can be made as small as we wish. (r n 1 u r ) r = ( u p ϕ 1 )r n 1 a (r n 1 a u r ) r dr = u p r n 1 a dr ϕ 1 r n 1 dr r r r r n 1 a u r (r ) = u p r n 1 a dr ϕ 1 r n 1 dr r r u r (r ) as. Hence we have F r (r ) > for large which is a conradicion. Hence F (r) <, r (d, 1). Now we discuss he case when Ω IR 2. Le w(x, y) be defined by Then w saisfies on (d, 1) he equaion w(x, y) = u(r) cos θ. { w = (p u p 1 ) w (p 1) u p cos θ ϕ 1 cos θ + u cos θ in Ω r 2 d w = on Ω d where Ω d = {r : d < r < 1} he annular domain. 14

15 Noe ha u(r) > in Ω d. In order o prove ha he soluion has a Morse Index higher han one on he whole space, i is enough o prove ha he funcion F (r) defined by F (r) = (p 1) u p (r)r 2 ϕ 1 (r)r 2 + u(r) is negaive in (d, 1). We will show his o be he case for large. Similar as above F (r) can only occur if < u(r) <, r (d, 1). 4n. We claim ha 3(p 1) F (r) < Noe ha from (4), F r (d ) > and F r (1) <.Hence F (r) is increasing in a neighborhood of d and 1,i now follows ha if F has a sric zero in (d, 1),hen i mus have a leas wo zeros in (d, 1). If possible, le here exis r (d, 1) such ha F (r ) =. Then we claim ha (1 r ) C. Since < u p 1 (r) < 4n 3(p 1) we have (p 1) u p (r )r 2 ϕ 1 (r )r 2 + u(r ) = (p 1) u p (r )r 2 + ϕ 1 (r )r 2 = u(r ) Hence we have by mean value heorem ϕ 1 (r ) C ϕ 1 (1) ϕ 1 (r ) = ϕ 1r (ξ )(1 r ) for some ξ (r, 1) i.e we have (1 r ) C. Now we claim ha ur(r) If r (d, 1) we have can be made as small as we wish. (r u r ) r = ( u p ϕ 1 )r Inegraing beween from r o a we have a a (r u r ) r dr = u p a rdr ϕ 1 rdr r r r 15

16 a r u r (r ) = u p a rdr ϕ 1 rdr r r u r (r ) as. Hence we have F r (r ) > for large, which is a conradicion. Similar is he case when r (a, 1). Hence Theorem 2 follows. Proof of Theorem 3. The funcional associaed o (P ) is I(u) = 1 2 Ω u 2 1 (u + ) p+1 + ϕ 1 u p + 1 Ω Ω on H 1 (Ω).From Remark 1, i follows ha he negaive soluion u is unique and is a sric local minimum for he funcional I.Also noe ha he funional is unbounded below and hence i saisfies all he condiions of Mounain Pass Theorem and if I(u ) = c and hence here exis a criical level c > c as a consequence of Mounain Pass heorem.also i is clear ha I saisfies all he hypohesis of Theorem 1.2 of [5](See page 222 and Theorem 5.1 of [3]) and hence a a level c here exis a soluion u of (P ) wih Morse Index less han or equal o one.if his soluion is non-radial hen we are done.on he oher hand if his soluion is radial,since I(u ) = c > c, u u and u canno be negaive. Hence i has a posiive par.this implies u has Morse Index one on he radial space and hence he soluion we are analysizing has o mainain Morse Index one on he whole space.however, Theorem 2 implies such a soluion has Morse Index a leas wo on he whole space.then i implies ha u has o be non-radial.hence he Theorem. Acknowledgemen. The second auhor acknowledges he parial suppor received from CSIR, India during he course of his work. References [1] Dancer E.N. and Yan Shusen, Mulipliciy and Profile of he changing sign soluions for an Ellipic Problem of Ambrosei-Prodi ype, (Pre-Prin). 16

17 [2] defigueiredo, D.G.,Srikanh P.N.,Sanra Sanjiban,Non-radially Symmeric Soluions for a Superlinear Ambrosei-Prodi Type Problem in a Ball, Communicaions in Conemporarary Mahemaics,(o appear). [3] Ekeland,I.,Ghoussoub,N.,Seleced new aspecs of he calculus of variaion in he large,bull.of AMS,39(22) [4] Fang,G.,Ghoussoub,N.,Morse-Type Informaion on Palais-Smale sequences obained by Min-Max Principles,Communicaions in Pure and Applied Mahemaics,47 (1994) [5] Ghoussoub,N.,Dualiy and Perurbaion Mehods in Criical Poin Theory, Cambridge Tracs in Mahemaics,17,Cambridge Universiy Press. 17

Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson

Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,

More information

EXERCISES FOR SECTION 1.5

EXERCISES FOR SECTION 1.5 1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler

More information

Hamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:

Hamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation: M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno

More information

arxiv:math/ v1 [math.nt] 3 Nov 2005

arxiv:math/ v1 [math.nt] 3 Nov 2005 arxiv:mah/0511092v1 [mah.nt] 3 Nov 2005 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON AND S. M. GONEK Absrac. Le πs denoe he argumen of he Riemann zea-funcion a he poin 1 + i. Assuming

More information

Differential Equations

Differential Equations Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding

More information

MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE

MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS

More information

The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales

The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions

More information

Convergence of the Neumann series in higher norms

Convergence of the Neumann series in higher norms Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann

More information

An Introduction to Malliavin calculus and its applications

An Introduction to Malliavin calculus and its applications An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214

More information

Chapter 6. Systems of First Order Linear Differential Equations

Chapter 6. Systems of First Order Linear Differential Equations Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh

More information

Lecture 10: The Poincaré Inequality in Euclidean space

Lecture 10: The Poincaré Inequality in Euclidean space Deparmens of Mahemaics Monana Sae Universiy Fall 215 Prof. Kevin Wildrick n inroducion o non-smooh analysis and geomery Lecure 1: The Poincaré Inequaliy in Euclidean space 1. Wha is he Poincaré inequaliy?

More information

Expert Advice for Amateurs

Expert Advice for Amateurs Exper Advice for Amaeurs Ernes K. Lai Online Appendix - Exisence of Equilibria The analysis in his secion is performed under more general payoff funcions. Wihou aking an explici form, he payoffs of he

More information

23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes

23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion

More information

THE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant).

THE WAVE EQUATION. part hand-in for week 9 b. Any dilation v(x, t) = u(λx, λt) of u(x, t) is also a solution (where λ is constant). THE WAVE EQUATION 43. (S) Le u(x, ) be a soluion of he wave equaion u u xx = 0. Show ha Q43(a) (c) is a. Any ranslaion v(x, ) = u(x + x 0, + 0 ) of u(x, ) is also a soluion (where x 0, 0 are consans).

More information

Existence of positive solutions for second order m-point boundary value problems

Existence of positive solutions for second order m-point boundary value problems ANNALES POLONICI MATHEMATICI LXXIX.3 (22 Exisence of posiive soluions for second order m-poin boundary value problems by Ruyun Ma (Lanzhou Absrac. Le α, β, γ, δ and ϱ := γβ + αγ + αδ >. Le ψ( = β + α,

More information

Predator - Prey Model Trajectories and the nonlinear conservation law

Predator - Prey Model Trajectories and the nonlinear conservation law Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories

More information

Properties Of Solutions To A Generalized Liénard Equation With Forcing Term

Properties Of Solutions To A Generalized Liénard Equation With Forcing Term Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick

More information

Ann. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:

Ann. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL: Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS

More information

Lecture 20: Riccati Equations and Least Squares Feedback Control

Lecture 20: Riccati Equations and Least Squares Feedback Control 34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he

More information

Optimality Conditions for Unconstrained Problems

Optimality Conditions for Unconstrained Problems 62 CHAPTER 6 Opimaliy Condiions for Unconsrained Problems 1 Unconsrained Opimizaion 11 Exisence Consider he problem of minimizing he funcion f : R n R where f is coninuous on all of R n : P min f(x) x

More information

Hamilton Jacobi equations

Hamilton Jacobi equations Hamilon Jacobi equaions Inoducion o PDE The rigorous suff from Evans, mosly. We discuss firs u + H( u = 0, (1 where H(p is convex, and superlinear a infiniy, H(p lim p p = + This by comes by inegraion

More information

Homework sheet Exercises done during the lecture of March 12, 2014

Homework sheet Exercises done during the lecture of March 12, 2014 EXERCISE SESSION 2A FOR THE COURSE GÉOMÉTRIE EUCLIDIENNE, NON EUCLIDIENNE ET PROJECTIVE MATTEO TOMMASINI Homework shee 3-4 - Exercises done during he lecure of March 2, 204 Exercise 2 Is i rue ha he parameerized

More information

Stability and Bifurcation in a Neural Network Model with Two Delays

Stability and Bifurcation in a Neural Network Model with Two Delays Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy

More information

Hamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t

Hamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n

More information

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Solutions from Chapter 9.1 and 9.2

Solutions from Chapter 9.1 and 9.2 Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is

More information

Question 1: Question 2: Topology Exercise Sheet 3

Question 1: Question 2: Topology Exercise Sheet 3 Topology Exercise Shee 3 Prof. Dr. Alessandro Siso Due o 14 March Quesions 1 and 6 are more concepual and should have prioriy. Quesions 4 and 5 admi a relaively shor soluion. Quesion 7 is harder, and you

More information

A NOTE ON S(t) AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION

A NOTE ON S(t) AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION Bull. London Mah. Soc. 39 2007 482 486 C 2007 London Mahemaical Sociey doi:10.1112/blms/bdm032 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON and S. M. GONEK Absrac Le πs denoe he

More information

Asymptotic instability of nonlinear differential equations

Asymptotic instability of nonlinear differential equations Elecronic Journal of Differenial Equaions, Vol. 1997(1997), No. 16, pp. 1 7. ISSN: 172-6691. URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp (login: fp) 147.26.13.11 or 129.12.3.113 Asympoic insabiliy

More information

KEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow

KEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering

More information

Existence Theory of Second Order Random Differential Equations

Existence Theory of Second Order Random Differential Equations Global Journal of Mahemaical Sciences: Theory and Pracical. ISSN 974-32 Volume 4, Number 3 (22), pp. 33-3 Inernaional Research Publicaion House hp://www.irphouse.com Exisence Theory of Second Order Random

More information

Matlab and Python programming: how to get started

Matlab and Python programming: how to get started Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,

More information

Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k

Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,

More information

t + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that

t + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so

More information

Existence of positive solution for a third-order three-point BVP with sign-changing Green s function

Existence of positive solution for a third-order three-point BVP with sign-changing Green s function Elecronic Journal of Qualiaive Theory of Differenial Equaions 13, No. 3, 1-11; hp://www.mah.u-szeged.hu/ejqde/ Exisence of posiive soluion for a hird-order hree-poin BVP wih sign-changing Green s funcion

More information

Solutions to Assignment 1

Solutions to Assignment 1 MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we

More information

SPECTRAL EVOLUTION OF A ONE PARAMETER EXTENSION OF A REAL SYMMETRIC TOEPLITZ MATRIX* William F. Trench. SIAM J. Matrix Anal. Appl. 11 (1990),

SPECTRAL EVOLUTION OF A ONE PARAMETER EXTENSION OF A REAL SYMMETRIC TOEPLITZ MATRIX* William F. Trench. SIAM J. Matrix Anal. Appl. 11 (1990), SPECTRAL EVOLUTION OF A ONE PARAMETER EXTENSION OF A REAL SYMMETRIC TOEPLITZ MATRIX* William F Trench SIAM J Marix Anal Appl 11 (1990), 601-611 Absrac Le T n = ( i j ) n i,j=1 (n 3) be a real symmeric

More information

Matrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality

Matrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]

More information

t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...

t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t... Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger

More information

Some Basic Information about M-S-D Systems

Some Basic Information about M-S-D Systems Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,

More information

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous

More information

CHAPTER 2 Signals And Spectra

CHAPTER 2 Signals And Spectra CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par

More information

On a Fractional Stochastic Landau-Ginzburg Equation

On a Fractional Stochastic Landau-Ginzburg Equation Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn

More information

A Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero

A Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero JOURNAL OF MAEMAICAL ANALYSIS AND APPLICAIONS 24, 7887 1997 ARICLE NO. AY965143 A Necessary and Sufficien Condiion for he Soluions of a Funcional Differenial Equaion o Be Oscillaory or end o Zero Piambar

More information

CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS

CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22 (2014, 67 76 DOI: 10.5644/SJM.10.1.09 CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ Absrac. This paper presens sufficien

More information

System of Linear Differential Equations

System of Linear Differential Equations Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y

More information

Notes for Lecture 17-18

Notes for Lecture 17-18 U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up

More information

Endpoint Strichartz estimates

Endpoint Strichartz estimates Endpoin Sricharz esimaes Markus Keel and Terence Tao (Amer. J. Mah. 10 (1998) 955 980) Presener : Nobu Kishimoo (Kyoo Universiy) 013 Paricipaing School in Analysis of PDE 013/8/6 30, Jeju 1 Absrac of he

More information

THE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI

THE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI THE 2-BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2-body problem. INTRODUCTION. Solving he 2-body problem from scrach

More information

dy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page

dy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,

More information

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities: Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial

More information

MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR RESONANT ELLIPTIC PROBLEMS. Francisco O. V. de Paiva. 1. Introduction. Let us consider the problem (1.

MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR RESONANT ELLIPTIC PROBLEMS. Francisco O. V. de Paiva. 1. Introduction. Let us consider the problem (1. Topological Mehods in Nonlinear Analysis Journal of he Juliusz Schauder Cener Volume 1, 003, 7 47 MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR RESONANT ELLIPTIC PROBLEMS Francisco O. V. de Paiva Absrac.

More information

On Two Integrability Methods of Improper Integrals

On Two Integrability Methods of Improper Integrals Inernaional Journal of Mahemaics and Compuer Science, 13(218), no. 1, 45 5 M CS On Two Inegrabiliy Mehods of Improper Inegrals H. N. ÖZGEN Mahemaics Deparmen Faculy of Educaion Mersin Universiy, TR-33169

More information

Math 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.

Math 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11. 1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be

More information

POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER

POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER GUANG ZHANG AND SUI SUN CHENG Received 5 November 21 This aricle invesigaes he exisence of posiive

More information

Class Meeting # 10: Introduction to the Wave Equation

Class Meeting # 10: Introduction to the Wave Equation MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion

More information

arxiv: v1 [math.ca] 15 Nov 2016

arxiv: v1 [math.ca] 15 Nov 2016 arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy

More information

MA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions

MA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by

More information

MATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence

MATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,

More information

2. Nonlinear Conservation Law Equations

2. Nonlinear Conservation Law Equations . Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear

More information

t 2 B F x,t n dsdt t u x,t dxdt

t 2 B F x,t n dsdt t u x,t dxdt Evoluion Equaions For 0, fixed, le U U0, where U denoes a bounded open se in R n.suppose ha U is filled wih a maerial in which a conaminan is being ranspored by various means including diffusion and convecion.

More information

11!Hí MATHEMATICS : ERDŐS AND ULAM PROC. N. A. S. of decomposiion, properly speaking) conradics he possibiliy of defining a counably addiive real-valu

11!Hí MATHEMATICS : ERDŐS AND ULAM PROC. N. A. S. of decomposiion, properly speaking) conradics he possibiliy of defining a counably addiive real-valu ON EQUATIONS WITH SETS AS UNKNOWNS BY PAUL ERDŐS AND S. ULAM DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO, BOULDER Communicaed May 27, 1968 We shall presen here a number of resuls in se heory concerning

More information

Attractors for a deconvolution model of turbulence

Attractors for a deconvolution model of turbulence Aracors for a deconvoluion model of urbulence Roger Lewandowski and Yves Preaux April 0, 2008 Absrac We consider a deconvoluion model for 3D periodic flows. We show he exisence of a global aracor for he

More information

POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION

POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Novi Sad J. Mah. Vol. 32, No. 2, 2002, 95-108 95 POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Hajnalka Péics 1, János Karsai 2 Absrac. We consider he scalar nonauonomous neural delay differenial

More information

A New Perturbative Approach in Nonlinear Singularity Analysis

A New Perturbative Approach in Nonlinear Singularity Analysis Journal of Mahemaics and Saisics 7 (: 49-54, ISSN 549-644 Science Publicaions A New Perurbaive Approach in Nonlinear Singulariy Analysis Ta-Leung Yee Deparmen of Mahemaics and Informaion Technology, The

More information

Example on p. 157

Example on p. 157 Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =

More information

Mapping Properties Of The General Integral Operator On The Classes R k (ρ, b) And V k (ρ, b)

Mapping Properties Of The General Integral Operator On The Classes R k (ρ, b) And V k (ρ, b) Applied Mahemaics E-Noes, 15(215), 14-21 c ISSN 167-251 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Mapping Properies Of The General Inegral Operaor On The Classes R k (ρ, b) And V k

More information

556: MATHEMATICAL STATISTICS I

556: MATHEMATICAL STATISTICS I 556: MATHEMATICAL STATISTICS I INEQUALITIES 5.1 Concenraion and Tail Probabiliy Inequaliies Lemma (CHEBYCHEV S LEMMA) c > 0, If X is a random variable, hen for non-negaive funcion h, and P X [h(x) c] E

More information

Let us start with a two dimensional case. We consider a vector ( x,

Let us start with a two dimensional case. We consider a vector ( x, Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our

More information

3, so θ = arccos

3, so θ = arccos Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,

More information

4. Advanced Stability Theory

4. Advanced Stability Theory Applied Nonlinear Conrol Nguyen an ien - 4 4 Advanced Sabiliy heory he objecive of his chaper is o presen sabiliy analysis for non-auonomous sysems 41 Conceps of Sabiliy for Non-Auonomous Sysems Equilibrium

More information

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures. HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =

More information

Optimizing heat exchangers

Optimizing heat exchangers Opimizing hea echangers Jean-Luc Thiffeaul Deparmen of Mahemaics, Universiy of Wisconsin Madison, 48 Lincoln Dr., Madison, WI 5376, USA wih: Florence Marcoe, Charles R. Doering, William R. Young (Daed:

More information

A remark on the H -calculus

A remark on the H -calculus A remark on he H -calculus Nigel J. Kalon Absrac If A, B are secorial operaors on a Hilber space wih he same domain range, if Ax Bx A 1 x B 1 x, hen i is a resul of Auscher, McInosh Nahmod ha if A has

More information

CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR

CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR Annales Academiæ Scieniarum Fennicæ Mahemaica Volumen 31, 2006, 39 46 CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR Joaquim Marín and Javier

More information

Chapter Three Systems of Linear Differential Equations

Chapter Three Systems of Linear Differential Equations Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n

More information

ASYMPTOTIC FORMS OF WEAKLY INCREASING POSITIVE SOLUTIONS FOR QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS

ASYMPTOTIC FORMS OF WEAKLY INCREASING POSITIVE SOLUTIONS FOR QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS Elecronic Journal of Differenial Equaions, Vol. 2007(2007), No. 126, pp. 1 12. ISSN: 1072-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu (login: fp) ASYMPTOTIC FORMS OF

More information

Monochromatic Infinite Sumsets

Monochromatic Infinite Sumsets Monochromaic Infinie Sumses Imre Leader Paul A. Russell July 25, 2017 Absrac WeshowhahereisaraionalvecorspaceV suchha,whenever V is finiely coloured, here is an infinie se X whose sumse X+X is monochromaic.

More information

Monotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type

Monotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type In. J. Conemp. Mah. Sci., Vol. 2, 27, no. 2, 89-2 Monoonic Soluions of a Class of Quadraic Singular Inegral Equaions of Volerra ype Mahmoud M. El Borai Deparmen of Mahemaics, Faculy of Science, Alexandria

More information

International Journal of Scientific & Engineering Research, Volume 4, Issue 10, October ISSN

International Journal of Scientific & Engineering Research, Volume 4, Issue 10, October ISSN Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 900 FUZZY MEAN RESIDUAL LIFE ORDERING OF FUZZY RANDOM VARIABLES J. EARNEST LAZARUS PIRIYAKUMAR 1, A. YAMUNA 2 1.

More information

L p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

L p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity ANNALES POLONICI MATHEMATICI LIV.2 99) L p -L q -Time decay esimae for soluion of he Cauchy problem for hyperbolic parial differenial equaions of linear hermoelasiciy by Jerzy Gawinecki Warszawa) Absrac.

More information

4.6 One Dimensional Kinematics and Integration

4.6 One Dimensional Kinematics and Integration 4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of

More information

Finish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!

Finish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details! MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his

More information

Some Regularity Properties of Three Dimensional Incompressible Magnetohydrodynamic Flows

Some Regularity Properties of Three Dimensional Incompressible Magnetohydrodynamic Flows Global Journal of Pure and Applied Mahemaics. ISSN 973-78 Volume 3, Number 7 (7), pp. 339-335 Research India Publicaions hp://www.ripublicaion.com Some Regulariy Properies of Three Dimensional Incompressible

More information

(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4)

(1) (2) Differentiation of (1) and then substitution of (3) leads to. Therefore, we will simply consider the second-order linear system given by (4) Phase Plane Analysis of Linear Sysems Adaped from Applied Nonlinear Conrol by Sloine and Li The general form of a linear second-order sysem is a c b d From and b bc d a Differeniaion of and hen subsiuion

More information

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still. Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in

More information

Lie Derivatives operator vector field flow push back Lie derivative of

Lie Derivatives operator vector field flow push back Lie derivative of Lie Derivaives The Lie derivaive is a mehod of compuing he direcional derivaive of a vecor field wih respec o anoher vecor field We already know how o make sense of a direcional derivaive of real valued

More information

Operators related to the Jacobi setting, for all admissible parameter values

Operators related to the Jacobi setting, for all admissible parameter values Operaors relaed o he Jacobi seing, for all admissible parameer values Peer Sjögren Universiy of Gohenburg Join work wih A. Nowak and T. Szarek Alba, June 2013 () 1 / 18 Le Pn α,β be he classical Jacobi

More information

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS XUAN THINH DUONG, JI LI, AND ADAM SIKORA Absrac Le M be a manifold wih ends consruced in [2] and be he Laplace-Belrami operaor on M

More information

Chapter 2. First Order Scalar Equations

Chapter 2. First Order Scalar Equations Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.

More information

14 Autoregressive Moving Average Models

14 Autoregressive Moving Average Models 14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class

More information

15. Vector Valued Functions

15. Vector Valued Functions 1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,

More information

MATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018

MATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018 MATH 5720: Gradien Mehods Hung Phan, UMass Lowell Ocober 4, 208 Descen Direcion Mehods Consider he problem min { f(x) x R n}. The general descen direcions mehod is x k+ = x k + k d k where x k is he curren

More information

Echocardiography Project and Finite Fourier Series

Echocardiography Project and Finite Fourier Series Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every

More information

Generalized Chebyshev polynomials

Generalized Chebyshev polynomials Generalized Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, 39 86 Roma, Ialy email: c.cesarano@unineunouniversiy.ne ABSTRACT

More information

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3

More information

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3. Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies

More information

LIMIT AND INTEGRAL PROPERTIES OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS. 1. Introduction

LIMIT AND INTEGRAL PROPERTIES OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS. 1. Introduction ARCHIVUM MATHEMATICUM (BRNO) Tomus 43 (2007), 75 86 LIMIT AND INTEGRAL PROPERTIES OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS Mariella Cecchi, Zuzana Došlá and Mauro Marini Absrac. Some

More information

SOME MORE APPLICATIONS OF THE HAHN-BANACH THEOREM

SOME MORE APPLICATIONS OF THE HAHN-BANACH THEOREM SOME MORE APPLICATIONS OF THE HAHN-BANACH THEOREM FRANCISCO JAVIER GARCÍA-PACHECO, DANIELE PUGLISI, AND GUSTI VAN ZYL Absrac We give a new proof of he fac ha equivalen norms on subspaces can be exended

More information

Two Coupled Oscillators / Normal Modes

Two Coupled Oscillators / Normal Modes Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own

More information