Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a; b]; (1 6 a < b)
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1 Lypunov-type inequlity for the Hdmrd frctionl boundry vlue problem on generl intervl [; b]; ( 6 < b) Zid Ldjl Deprtement of Mthemtic nd Computer Science, ICOSI Lbortory, Univerity of Khenchel, 40000, Algeri. E-mil zid.ldjl@yhoo.com July 0, 0 Abtrct In thi pper, we tudy n open problem; where we obtined everl reult for Lypunov-type nd Hrtmn-Wintner-type inequlitie for Hdmrd frctionl di erentil eqution on generl intervl [; b]; ( 6 < b) with the boundry vlue condition. Keyword Hdmrd frctionl derivtive; boundry vlue problem; Green function; Lypunov-type inequlity; Hrtmn-Wintner-type inequlity. introduction The rt reult in thi domin i due to Lyponov [], cn be tted follow If nontrivil continuou olution to the following boundry vlue problem < u 00 (t) + q(t)u (t) = 0; < t < b; u = u (b) = 0; exite, where q [; b]! R i continuou function, then Z b jqjd > 4 b Recently, everl rticle from the inequlity of Lypunov hve been publihed bout di erentil eqution of the integer order nd frctionl order, ee [5-0] nd reference therein, for exmple The following reult for the Riemnn-Liouville frctionl boundry vlue problem i found by D. O Regn nd B. Smet [4] < R D u(t) + q(t)u (t) = 0; < t < b; 3 < 6 4; (3) u = u 0 = u 00 = u 00 (b) = 0;
2 h nontrivil continuou olution, then Z b jqjd > ( ) ( 3) 3 (b ) (4) In [] Qinghu, Cho nd Jinxun etblihed Lypunov-type inequlity for di erentil eqution tht depnd on the Hdmrd frctionl derivtive, for the boundry vlue problem < H D u(t) q(t)u (t) = 0; < t < e; < 6 ; (5) u = u (e) = 0; where q [; e]! R i continuou function.they proved tht if nontrivil continuou olution to the bove problem, then Z e jqjd > ( ) exp ; (6) p where = ( ) + nd they hve preented the following open problem for reder How to get the Lypunov inequlity for the following the Hdrmrd frctionl vlue problem (HFBVP) < H D u(t) q(t)u (t) = 0; 6 < t < b; < 6 ; (7) u = u (b) = 0; where H D i the Hdmrd frctionl derivtive, nd q [; b]! R i continuou function. In thi pper we nwered the previou quetion by uing two method, nd lo we get the Hrtmn-Wintner-type inequlitie. Preliminrie De nition [3] Let ; b; R + where < b nd n < 6 n with n N, The Hdmrd frctionl integrl of ordre.for function f L [; b] i de ned with H I t f(t) = i Gmm Euler function t f d ; < t < b De nition [3] Let ; b R + with < b, The Hdmrd frctionl derivtive of ordre R +.for function f L [; b] i de ned Z H Dt t dn f(t) = tn (n ) dt n t n f d ; < t < b (9) where n < 6 n with n N
3 Lemm 3 [3] Let 0 6 < b nd > 0 where n eqution H D u(t) = 0 h it olution nd moreover j= < 6 n. nd n N The j=n X u(t) = c i t i ; t [; b] (0) j=n X H I H D u(t) = u(t) + c i t i ; where c i R; (i = ; ; n) re contnt. Lemm 4 Let A; B R; we hve j= AB 6 (A + B) 4 3 Min reult Lemm 5 Let u C([; b]; R); the following problem < H D u(t) q(t)u (t) = 0; 6 < t < b; < 6 ; u = u (b) = 0; h equivlent to the frctionl integrl eqution u (t) = 0 (3) G (t; ) qu d (4) where g >< (t; ) = g (t; ) + t ; 6 6 t 6 b; G (t; ) = (5) > ( g (t; ) = ) t ( ) b ( ) b ; 6 t 6 6 b with 6 < b Proof. Uing Lemm 3, we hve u (t) = c t + c t + t where c ; c R uing the boundry condition u = u(b) = 0 we get c = 0 nd c = b Z b b qu d qu d (6) (7) 3
4 Subtituting the vlue of c nd c in (6), we obtin u (t) = = = t b + Z b the proof i complete t t t b Z b b qu d b t qu d t b b # qu d qu d G (t; ) qu d Lemm 6 The Green function G de ned in Lemm 5, h the following propertie i) G(t; ) 6 g (; ) 6 0; for ll (t; ) [; b] [; b] ii) For ny [; b] jg(t; )j 6 jg(; )j = g (; ) 6 b (9) 4 ( ) Proof. We trt by xing n rbitrry [; b] Di erentiting G(t; ) with repect to t, we get For 6 6 t 6 6 b; g = ( ) t t b 6 0; (0) b we obtin while for t 6 b; we hve g (; ) 6 g (t; ) 6 g (; ) g ( ) + t t = = ( ) t ( ) t t t b ( ) b + t t b # b t 4
5 = ( ) t t = ( ) t t by t 6 b we get t b # t t 4 t b b b! 3 5 t > ; (3) nd t b! > (4) uing (3) nd (4) we obtin t 4 t b b b! 3 5 > 0 (5) So thu Uing t 6 b we get We obtin hence We g > 0 (6) g (; ) 6 g (t; ) 6 g (b; ) = 0 (7) g (; ) 6 g (t; ) 6 g (t; ) 6 0; jg(; )j 6 G(t; ) 6 0 (9) 4 b (30) we hve G(; ) = g (; ) = g (; ) 6 g (t; ) 6 g (t; ) 6 0. Uing Lemm 4, we hve jg(; )j = 6 = b b 4 b + b 4 b b # # 5
6 = 6 4 b 4 b Therefore jg(t; )j 6 jg(; )j = g (; ) 6 b (3) 4 ( ) The proof i complete We hve the following Hrtmn-Wintner-type inequlity. Theorem 7 If nontrivil continuou olution to the Hdmrd frctionl boundry vlue problem (7) exite, then b jqj d > b (3) Proof. Let E = C ([; b]; R) be the Bnch pce endowed with the norm we hve which yield ju(t)j 6 kuk 6 kuk kuk = up ju(t)j t[;b] Since u i non trivil, then kuk 6= 0; o 6 b from which the inequlity in (3) follow jg (t; )j jqj ju j d jg (; )j jqj ju j d b jqj d Corollry If nontrivil continuou olution to the Hdmrd frctionl boundry vlue problem exite, then b jqj d > b (33) 6
7 Proof. from theorem 7, we hve nexte we not > thu we get b jqj d > b b jqj d > b (34) We hve the following Lypunov-type inequlity. Theorem 9 If nontrivil continuou olution to the Hdmrd frctionl boundry vlue problem (7) exite, then We prove thi theorem in two method Proof. from the corollry, we hve where jqj d > 4 ( ) b (35) jqj d > b mx h (36) [;b] h = b (37) If = or = b then h = 0 Ele if ]; b[ we di erentite h h 0 = = ( ) b b b ( ) b we hve only one olution of the eqution h 0 = 0 on ]; b[ We obtin 0 = p b (3) mx h = h( 0) = [;b] p b b p b! (39) 7
8 We hve b = p b p b, p b = b p b, p b b p b! = 0 p! b, 4 p b p b = b p! b, p b = b 4 p! b, p b b! + p b + b p! b + p b = b b 4 p! b p b b = b ( ) (40) 4 ( ) by (39) nd (40) mx h = h( 0) = [;b] 4 ( ) b ( ) (4) we ubtiting (4) into (36) we obtin jqj d > 4 ( ) b The proof i complete We de ne the contnt = exp [ ( ) + b] r #! 4 ( ) + b ; (4) nd = exp r #! [ ( ) + b] + 4 ( ) + b (43) Lemm 0 The function G de ned in Lemm 5, ti e the following property mx jg(t; )j = t;[;b] b! b ; (44) Proof. we hve where mx jg(t; )j = mx jg (; )j t;[;b] [;b] g (; ) = b b
9 It follow tht we only need to get the mximum vlue of the function f = we oberve tht f = f(b) = 0 If ]; b[; di erentite f f 0 = b ( ) b b # b (45) we hve f 0 = 0, ( ) b = b, [ ( ) + b + ] [( ) + b] ( ) b = 0, [ ( ) + b] + [( ) b + b ] = 0, x [ ( ) + b] x + [( ) b + b ] = 0 where x = we get >< where we hve > x = [( )+ b] p = x = [( x > = 4 ( )+ b]+p = (46) q b + ) + b ) =]; b[ b = b (47) Alo we hve 0 x ( ) + b 0 ( ( ( ) + b = ( ) + b ( ) b = ) > ) + b 4 ( ) b A ) + b A 9
10 nd 0 x ( ) + b 0 ( ) + b = ( ) + b 6 ( ) + b = ( ) + b ( ) b + 4 ( ( ) b A ( ) b j ( )j b ( ) b = b ) b A ) < b we obtient ]; b[ Hence Therefore mx jfj = [;b] b (4) mx jg(t; )j = t;[;b] The proof i complete We hve the following Lypunov-type inequlity. b! b ; (49) Theorem If nontrivil continuou olution to the HFBVP (7) exite, then b! jqj d > b ; (50) where = exp [ ( ) + b] r #! 4 ( ) + b Proof. By Lemm 5, the olution of the HFBVP cn be written u (t) = G (t; ) qu d 0
11 Thu for ll t [; b] we hve which yield ju(t)j 6 6 kuk kuk 6 kuk jg (t; )j jqj ju j d Since u i non trivil, then kuk 6= 0; o 6 jg (t; )j jqj d jg (t; )j jqj d jg (t; )j jqj d New, n ppliction of Lemm 0, we obtin The proof i complete b! jqj d > b Reference [] A. M. Lypunov Problème générl de l tbilité du mouvement. Ann. of Mth. Stud., vol. 7. Princeton Univerity Pre, Princeton, 949 [] Qinghu M, Cho M nd Jinxun Wng A Lypunov-type inequlity for frctionl di erentil eqution with Hdmrd derivtive. Journl of Mthemticl Inequlitie. Vol, Number, 35 4, 07. [3] A. A. Kilb, H. M. Srivtv, J. J. Trujillo Theory nd Appliction of Frctionl Di erentil Eqution. North-Hold Mthemtic Studie 04, Elevier Science B.V, Amterdm, 006. [4] D. O Regn, B. Smet Lypunov-type inequlitie for cl of frctionl di erentil eqution. J. Inequl. Appl. 05, 47, 05. [5] A. Tiryki Recent development of Lypunov-type inequlitie. Adv. Dyn. Syt. Appl. 5, 3-4, 00 [6] X. Yng, Y. Kim, K. Lo Lypunov-type inequlity for cl of liner di erentil ytem. Appl. Mth. Comput. 9, 05-, 0.
12 [7] M. Jleli, B. Smet Lypunov-type inequlitie for frctionl di erentil eqution with mixed boundry condition. Mth. Inequl. Appl., , 05. [] N. Ari, I. Altun, M. Jleli, A. Lhin, B. Smet Lypunov-type inequlitie for frctionl p-lplcin eqution. J. Inequl. Appl. 06, 9, 06. [9] R. A. C. Ferreir On Lypunov-type inequlity nd the zero of certin Mittg-Le er function. J. Mth. Anl. Appl. 4, , 04. [0] R. A. C. Ferreir A Lypunov-type inequlity for frctionl boundry vlue problem. Frct. Clc. Appl. Anl. 6(4), 97-94, 03.
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