Research Article The General Solution of Impulsive Systems with Caputo-Hadamard Fractional Derivative of Order

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1 Hndw Publhng Corporon Mhemcl Problem n Engneerng Volume 06, Arcle ID 8080, 0 pge hp://dx.do.org/0.55/06/8080 Reerch Arcle The Generl Soluon of Impulve Syem wh Cpuo-Hdmrd Frconl Dervve of Order q C (R(q) (, )) Xnmn Zhng, Xnzhen Zhng, Zuohu Lu, 3 Hu Peng, Tong Shu, nd Shyong Yng School of Elecronc Engneerng, Jung Unvery, Jung, Jngx 33005, Chn School of Chemcl nd Envronmenl Engneerng, Jung Unvery, Jung, Jngx 33005, Chn 3 School of Chemry nd Chemcl Engneerng, Chongqng Unvery, Chongqng , Chn Correpondence hould be ddreed o Xnmn Zhng; z6xm@6.com Receved December 05; Acceped 9 Jnury 06 Acdemc Edor: JoéA. T. Mchdo Copyrgh 06 Xnmn Zhng e l. Th n open cce rcle drbued under he Creve Common Arbuon Lcene, whch perm unrerced ue, drbuon, nd reproducon n ny medum, provded he orgnl wor properly ced. Moved by ome prelmnry wor bou generl oluon of mpulve yem wh frconl dervve, he generlzed mpulve dfferenl equon wh Cpuo-Hdmrd frconl dervve of q C (R(q) (, )) re furher uded by nlyzng he lm ce ( mpule pproch zero) n h pper. The formul of generl oluon re found for he mpulve yem.. Inroducon Hdmrd frconl clculu ey pr of he heory of frconl clculu. The uhor n [ 6] mde n mporn developmen of he frconl clculu whn he frme of Hdmrd frconl dervve. For he generl heory of Hdmrd frconl clculu, one cn ee he monogrph of Klb e l. [7]. Recenly, Jrd e l. mde progre on Hdmrd frconl dervve o preen he defnon of Cpuo- Hdmrdfrconldervven[8]nddevelopedhe fundmenl heorem of h frconl dervve n [8, 9]. Furhermore, mpulve dfferenl equon re ulzed vluble ool o decrbe he dynmc of procee n whch udden, dconnuou ump occur, nd mpulve dfferenl equon wh Cpuo frconl dervve were wdely reerched n [0 6]. Nex, he generl oluon of everl nd of mpulve frconl dfferenl equon hve been found n [7 30], repecvely. Moved by he bove-menoned wor, we wll furher ee he generl oluon of generlzed mpulve yem wh Cpuo-Hdmrd frconl dervve: C-H Dq x () f(, x ()), (, T], (,,...,m), l (l,,...,p), Δx x( ) x( )I (x ( )) C,,,...,m, Δx x ( l l ) x ( l )I l (x ( l )) C, x () x C, x () x C. l,,...,p, Here q C nd R(q) (, ), C-H Dq denoe lef-ded Cpuo-Hdmrd frconl dervve of order q nd > 0, ()

2 Mhemcl Problem n Engneerng f:[,t] C C n ppropre connuou funcon, 0 < < < m < m T,nd 0 < < < p < p T.Herex( ) lm ε 0 x( ε)nd x( )lm ε 0 x( ε)repreen he rgh nd lef lm of u(),repecvely,ndx ( l ) nd x ( l ) hve mlr menng. Le u queue,,..., m,,..., p,t o 0 < < < < Π < Π Tuch h e,..., m,,..., p e Π,,..., Π. () For ech [, ] ( 0,,...,Π),uppoe[, ] [, ] [, ] (here,,...,m)nd[, ] [, ] [, ] (here,,...,p), repecvely. In order o ge he oluon of (), we wll fr conder he followng yem: C-H Dq x () f(, x ()), (3) (, T], (,...,m), l (l,...,p), Δx x( ) x( )I (x ( )) C,,,...,m, Δδx l δx( l ) δx( l )J l (x ( l )) C, l,,...,p, (3b) (3c) x () x C, (3d) δx () x C, where dfferenl operor δ (d/d), δ 0 x() x(). Nex, ome defnon nd concluon re nroduced n Secon, nd he formul of generl oluon wll be gven for ome mpulve dfferenl equon wh Cpuo- Hdmrd frconl dervve n Secon 3.. Prelmnre Defnon (ee [7, p. 0]). Le 0 b be fne or nfne nervl of he hlf-x R.Thelef-dedHdmrd frconl negrl of order α C of funcon φ(x) defned by ( H J α φ) (x) Γ (α) x where Γ( ) he Gmm funcon. (ln x )α φ () d, ( <x<b), Defnon (ee [7, p. 0]). The lef-ded Hdmrd frconl dervve of order α C (R(α) 0)on(, b) defned by ( H D α φ) (x) δn ( H J n α φ) (x) (x d dx ) n x Γ (n α) (ln x )n α φ () d, ( <x<b), (4) (5) where n[r(α)] nd dfferenl operor δ x(d/dx) nd δ 0 y(x) y(x). Lemm 3 (ee [7, p. 4 6]). Le α, β C uch h R(α) > R(β) > 0. For0<<b<,fφ L p (, b) ( p < ), hen H D β H Jα φ H Jα β φ nd H J α H Jβ φ H Jαβ φ. The lef-ded Cpuo-Hdmrd frconl dervve w defned n [8] by C-H Dα φ (x) H D α [φ (x) n 0 δ φ ()! (ln ) ] (x) ; here R(α) 0, n[r(α)], 0<<b<, dfferenl operor δ x(d/dx), δ 0 y(x) y(x),nd φ (x) AC n δ [, b] φ:[, b] C :δ(n ) φ (x) AC[, b], δ x d dx. Theorem 4 (ee [8, p. 4]). Le R(α) 0, n[r(α)], nd φ AC n δ [, b], 0<<b<.Then, C-H Dα φ(x) ex everywhere on [, b] nd () f α N 0, x C-H Dα φ (x) Γ (n α) (b) f αn N 0, In prculr, H J n α δn φ (x), (ln x )n α δ n φ () d (6) (7) (8) C-H Dα φ (x) δn φ (x). (9) C-H D0 φ (x) φ(x). (0) Lemm 5 (ee [8, p. 5]). Le R(α) > 0, n[r(α)] nd φ C[, b].ifr(α) 0or α N,hen C-H Dα ( H Jα φ) (x) φ(x). () Lemm 6 (ee [8, p. 6]). Le φ AC n δ [, b] or le Cn δ [, b] nd α C;hen, n H Jα ( C-H Dα φ) (x) φ(x) δ φ () (ln x! ). () 0

3 Mhemcl Problem n Engneerng 3 Lemm 7 (ee [9, p. 4]). Le w C nd R(w) (0, ),nd le ξ be conn. A funcon u() : [, T] C generl oluon of yem C-H Dw u () g(, u ()), (, T], (,,...,m), Δu u( ) u( )Δ (u ( )) C, u () u, u C, f nd only f u() fe he negrl equon,,...,m, (3) u () u Γ (w) (ln )w g (, u ()) d for (, ], u ξ Δ (u ( )) Γ (w) Δ (u ( )) Γ (w) (ln )w g (, u ()) d [ (ln w ) g (, u ()) d (ln )w d g (, u ()) (4) (ln )w g (, u ()) d ] for (, ],,,...,m provded h he negrl n (4) ex. 3. Mn Reul Frly, le u conder ome lm ce n yem (3), (3b), (3c), nd (3d): lm yem I (x( )) 0,,...,m, J l (x( l )) 0 l,,...,p C-H Dq x () f(, x ()), (, T], (3), (3b), (3c), nd (3d) x () x C, δx () x C, lm yem (3), (3b), (3c), nd (3d) J l (x( l )) 0, l,,...,p C-H Dq x () f(, x ()), (, T], (,...,m), Δx x( ) x( )I (x ( )) C,,,...,m, x () x C, δx () x C, lm yem (3), (3b), (3c), nd (3d) I (x( )) 0,,,...,m C-H Dq x () f(, x ()), (, T], l (l,...,p), Δδx l δx( l ) δx( l )J l (x ( l )) C, l,,...,p, x () x C, δx () x C. (5) (6) (7)

4 4 Mhemcl Problem n Engneerng Thu, () () lm he oluon of yem (3), )) 0,,...,m, J l (x( l )) 0 l,,...,p I (x( (3b), (3c), nd (3d) he oluon of yem (5), lm he oluon of yem (3), (3b), (3c), J l (x( l )) 0, l,,...,p nd (3d) he oluon of yem (6), (8) By Theorem 4, we hve for (, ] (here 0,,,...,m). (9) [ C-H D q x ()] (, ] Γ( q) (ln ) q δ [x ( )δx( ) ln Γ(q) () lm he oluon of yem (3), (3b), (3c), I (x( )) 0,,,...,m nd (3d) he oluon of yem (7). Thu, he defnon of oluon of yem (3), (3b), (3c), nd (3d) preened follow. (ln q η ) f (η, x (η)) dη η ] d (, ] Γ ( q) Γ (q) (ln ) q (0) Defnon 8. Afunconz() : [, T] C d o be oluon of (3), (3b), (3c), nd (3d) f z() x nd δz() x,heequoncondon C-H D q z() f(, z()) for ech [, T]/,,..., Π verfed, he mpulve condon Δz I (z( )) (here,,...,m)ndδδz l J l (z( l )) (here l,,...,p) re fed, he rercon of z( ) o he nervl (, ] (here 0,,,...,Π) connuou, nd he condon () () hold. Nex, defne funcon by x () x( )δx( ) ln (ln )q d f (, x ()), δ [ (ln q η ) f (η, x (η)) dη η ] d (, ] f(, x ()) (, ]. Th men h x() fe (3), nd x() fe (3b) (3d). However, x() doe no fy he condon () (), nd no oluon of yem (3), (3b), (3c), nd (3d). Therefore, x() condered n pproxme oluon o ee he exc oluon of (3), (3b), (3c), nd (3d). Nex, le u prove ome ueful concluon. Lemm 9. Le q C, R(q) (, ), ndξ conn. Syem (6) equvlen o he negrl equon x () x x ln (ln )q f (, x ()) d for (, ], x x ln I (x ( )) ξ I (x ( )) Γ(q) [ (ln )q f (, x ()) d (ln ) q f (, x ()) d (ln )q d f (, x ()) () (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d for (, ], m provded h he negrl n () ex.

5 Mhemcl Problem n Engneerng 5 Proof. Necey. LengI (x( )) 0 (,,...,m)n (6), we hve lm yem (6) I (x( )) 0,,,...,m C-H Dq x () f(, x ()), (, T], x () x C, δx () x C. () Th, lm he oluon of yem (6) I (x( )) 0,,,...,m he oluon of yem (5). (3) In fc, we cn verfy h () fe he condon (3). Nex, ng frconl dervve o () for (, ] (here 0,,,...,m), we ge [ C-H D q x ()] ( (, ] Γ( q) (ln ) q δ x x ln I (x ( )) (ln η ) q f (η, x (η)) dη η ξ I (x ( )) Γ(q) [ (ln q η ) f (η, x (η)) dη η (ln q η ) f (η, x (η)) dη η (ln η ) q f (η, x (η)) dη η ] ln (/ ) (ln q η ) f (η, x (η)) dη d η ) (, ] Γ ( q) Γ (q) (ln ) q δ ( (ln q η ) f (η, x (η)) dη η ) d ξ I (x ( )) [ (ln ) q δ ( (ln q η ) f (η, x (η)) dη η ) d (ln ) q (4) δ ( (ln q η ) f (η, x (η)) dη η ) d ] (, ] f(, x ()) (x ( ξi )) [f (, x ()) f(, x ()) ] f(, x ()) (, ]. (, ] So, () fe he condon of frconl dervve n yem (6). Fnlly, ung () for ech (here,,...,m), we hve x( ) x( )lm x () x( )x x ln I (x ( )) f(, x ()) d ξ I (x ( )) Γ(q) [ (ln ) q (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ]ln ( / ) (ln q ) f (, x ()) d x x ln I (x ( )) (ln ) q f(, x ()) d ξ I (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ]ln ( / ) (ln q ) f (, x ()) d I (x ( )). (5) I men h () fe he mpulve condon of (6). Hence, () fe ll condon of yem (6). Suffcency (by mhemcl nducon). By Lemm 6, he oluon of (6) fe x () x x ln (ln )q f (, x ()) d for (, ]. (6)

6 6 Mhemcl Problem n Engneerng Ung (6), we obn x( )x( )I (x ( )) x x ln I (x ( )) δx ( )δx( ) x (ln ) q f (, x ()) d, (ln ) q f (, x ()) d. Thu, he pproxme oluon x() gven by x () x( )δx( ) ln Γ(q) (ln )q d f (, x ()) x x ln I (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) ] ln (/ ) (ln q ) f (, x ()) d for (, ]. (7) (8) Le e () x() x(), for (, ]. By (6), he exc oluonx() of yem (6) fe lm I (x( )) 0 x () x x ln Th how h e () conneced wh I (x( )) nd lm I (x( )) e (). Thu, we ume e () χ(i (x ( ))) lm e () I (x( )) 0 χ(i (x ( ))) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d for (, ], (3) where funcon χ n undeermned funcon wh χ(0). So, x () x () e () x x ln I (x ( )) (ln )q f (, x ()) d [ (ln )q f (, x ()) d, (9) χ(i (x ( )))] for (, ]. Then, lm e () lm x () x () I (x( )) 0 I (x( )) 0 Γ(q) [ (ln q ) f (, x ()) d Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (3) (ln )q d f (, x ()) (30) (ln )q f (, x ()) d ] ln (/ ) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d. (ln q ) f (, x ()) d for (, ].

7 Mhemcl Problem n Engneerng 7 Leng γ(i (x( ))) χ(i (x( ))),wege Therefore, he pproxme oluon x() provded by x () x x ln I (x ( )) (ln )q f(, x ()) d γ(i (x ( ))) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d Ung (33), we obn x( )x( )I (x ( )) x x ln I (x ( )) I (x ( )) f(, x ()) d γ(i (x ( ))) Γ(q) [ (ln q ) f (, x ()) d for (, ]. (ln ) q (ln q ) f (, x ()) d (ln q ) f (, x ()) d ]ln ( / ) (ln q ) f (, x ()) d, δx ( )δx( ) x f(, x ()) d γ(i (x ( ))) (ln ) q [ (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ]. (33) (34) x () x( )δx( ) ln (ln )q f(, x ()) d x x ln I (x ( )), Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) ] ln (/ ) (ln q ) f (, x ()) d γ(i (x ( ))) Γ(q) [ (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ]ln ( / ) (ln q ) f (, x ()) d γ(i (x ( ))) ln [ (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ] for (, 3 ]. (35) Le e () x() x() for (, 3 ].Moreover,by(33),he exc oluon x() of yem (6) fe lm x () x x ln I (x( )) 0 I (x ( )) Γ(q) (ln )q f (, x ()) d γ(i (x ( ))) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ())

8 8 Mhemcl Problem n Engneerng (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d for (, 3 ], lm x () x x ln I (x( )) 0 I (x ( )) Γ(q) (ln )q f (, x ()) d γ(i (x ( ))) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d lm x () x x ln I (x( )) 0, I (x( )) 0 for (, 3 ], (ln )q lm e () lm x () x () [ I (x( )) 0 I (x( )) 0 γ(i (x ( )))] Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d γ(i (x ( ))) Γ(q) [ (ln ) q (ln )q d f (, x ()) f (, x ()) d (ln )q d f (, x ()) ] ln (/ ) f(, x ()) d for (, 3 ]. (36) (ln q ) f (, x ()) d Then, lm e () lm x () x () [ I (x( )) 0 I (x( )) 0 γ(i (x ( )))] Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d for (, 3 ], lm e () lm x () x () I (x( )) 0, I (x( I (x( )) 0, )) 0 I (x( )) 0 Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d for (, 3 ], for (, 3 ]. (37)

9 Mhemcl Problem n Engneerng 9 By (37), we ge Then, e () [γ(i (x ( ))) γ (I (x ( ))) ] x () x () e () x x ln I (x ( )), Γ(q) [ (ln q ) f (, x ()) d (ln )q f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) γ(i (x ( ))) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln q ) f (, x ()) d γ(i (x ( ))) (38) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d (39) Γ(q) [ (ln ) q (ln )q d f (, x ()) f (, x ()) d γ(i (x ( ))) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q d f (, x ()) ] ln (/ ) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d (ln q ) f (, x ()) d for (, 3 ]. for (, 3 ]. Conder he followng lm ce C-HD q x () f(, x ()), (, 3 ],,, lm Δx x( ) x( )I (x ( )) C,,, x () x C, δx () x C. (40) C-HD q x () f(, x ()), (, 3 ], Δx I (x ( )) I (x ( )) x () x C, δx () x C. (4)

10 0 Mhemcl Problem n Engneerng Ung (33) nd (39) for (4) nd (40), repecvely, we hve χ(i (x ( )) I (x ( ))) χ(i (x ( ))) χ (I (x ( ))) for I (x ( )), I (x ( )) C. Therefore, χ(z) ξz z C;hereξ conn. Thu, x () x x ln I (x ( )) Γ(q) (ln )q f (, x ()) d ξi (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d x () x x ln I (x ( )) I (x ( )) Γ(q) [ for (, ], (ln )q f (, x ()) d ξi (x ( )) (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d ξi (x ( )) Γ(q) [ (ln q ) f (, x ()) d (4) (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d Nex, uppoe x () x x ln I (x ( )) Γ(q) (ln )q f (, x ()) d ξ I (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d Ung (44), we obn for (, 3 ]. (43) for (, ]. x( )x( )I (x ( )) x x ln I (x ( )) f(, x ()) d ξ I (x ( )) Γ(q) [ (ln ) q (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ]ln ( / ) (ln q ) f (, x ()) d, (44)

11 Mhemcl Problem n Engneerng δx ( )δx( ) x (ln q ) f (, x ()) d ξ I (x ( )) [ (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ]. (45) Therefore, he pproxme oluon x() preened by x () x( )δx( ) ln Γ(q) (ln )q d f (, x ()) x x ln (ln q ) f (, x ()) d ξ I (x ( )) ln (/ ) [ (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ] for (, ]. (46) Le e () x() x() for (, ]. In ddon, by (44), he exc oluon x() of yem (6) fe lm x () x x ln I (x( )) 0, (ln )q,,..., I (x ( )) f(, x ()) d for (, ], Γ(q) [ (ln q ) f (, x ()) d lm I (x( )) 0 x () x x ln, I (x ( )) (ln )q d f (, x ()) ]ln (/ ) (ln )q f (, x ()) d (ln q ) f (, x ()) d ξ I (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln q ) f (, x ()) d (ln q ) f (, x ()) d ]ln ( / ) ξ, Γ(q) [ I (x ( )) (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d for (, ],,,...,. (47)

12 Mhemcl Problem n Engneerng Then, lm I (x( )) 0,,,..., e () lm I (x( )) 0,,,..., x () x () Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) f(, x ()) d ] ln (/ ) (ln q ) f (, x ()) d lm e () I (x( )) 0 ξ, lm I (x( )) 0 x () x () I (x ( )) (ln )q for (, ], Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ]ln (/ ) (ln q ) f (, x ()) d ξ, I (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q d f (, x ()) ] ln (/ ) (ln q ) f (, x ()) d for (, ],,,...,. (48) By (48), we obn e () ξ I (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ]ln (/ ) (ln q ) f (, x ()) d ξ I (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q d f (, x ()) ] ln (/ ) (ln q ) f (, x ()) d for (, ],,,...,. Thu, we hve x () x () e () x x ln I (x ( )) ξ (ln )q f (, x ()) d I (x ( )) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d for (, ]. Then, he oluon of yem (6) fe (). (49) (50)

13 Mhemcl Problem n Engneerng 3 By he proof of Suffcency nd Necey, yem (6) equvlen o (). The proof compleed. Lemm 0. Le q C, R(q) (, ), ndζ conn. Syem (7) equvlen o he negrl equon x () x x ln (ln )q f (, x ()) d, for (, ], x x ln l J (x ( ζ l J (x ( )) Γ(q) [ [ )) ln (ln )q f (, x ()) d (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) ] f (, x ()) d, for ( l, l ], l p (5) provded h he negrl n (5) ex. Remr. For (7), we hve In fc, we cn verfy h (5) fe he condon (53). Moreover, he pproxme oluon x() of yem (7) defned by lm J (x( )) 0,...,J p(x( p )) 0 yem (7) yem (5). (5) x () x( l )δx( l ) ln l (ln )q d f (, x ()) l (54) for ( l, l ]; Then, lm J (x( )) 0,...,J p(x( p )) 0 he oluon of yem (7) he oluon of yem (5). (53) here x( l ) x( l ) nd δx( l ) δx( l )J l(x( l )), l,,...,p. Due o mlry wh Lemm 9, he proof omed. Corollry. Le q C, R(q) (, ), ndξ conn. Afunconx() : [, T] C generl oluon of he yem (6); hen, δx () x (ln )q f (, x ()) d for (, ], x ξ I (x ( )) [ (ln )q f (, x ()) d (ln ) q f (, x ()) d (ln )q d f (, x ()) (55) (ln )q f (, x ()) d ] for (, ], m provded h he negrl n (55) ex.

14 4 Mhemcl Problem n Engneerng Corollry 3. Le q C, R(q) (, ), ndζ conn. Afunconx() : [, T] C generl oluon of he yem (7); hen, δx () x (ln )q f (, x ()) d for (, ], x l ζ l J (x ( )) J (x ( )) [ [ (ln )q f (, x ()) d (ln q ) f (, x ()) d (ln )q d f (, x ()) (56) (ln )q f (, x ()) d ] ] for ( l, l ], l p provded h he negrl n (56) ex. Remr 4. By Corollre nd 3, hown h wo nd of mpule Δx (,,...,m)nd Δδx l (l,,...,p) hve mlr effec on δx() of yem (5). Lemm 5. Le q C, R(q) (, ), ndξ nd ζ re wo conn. A funcon x() : [, T] C generl oluon of heyem(3),(3b),(3c),nd(3d);hen, δx () x (ln )q f (, x ()) d for (, ], x ξ J (x ( )) I (x ( )) [ (ln )q f (, x ()) d ] ζ J (x ( )) [ [ (ln )q f (, x ()) d (ln ) q f (, x ()) d (ln )q d f (, x ()) (ln q ) f (, x ()) d (ln )q d f (, x ()) (57) (ln )q f (, x ()) d ] ] for (, ], Π provded h he negrl n (57) ex. Proof. Accordng o Corollre nd 3, he oluon of yem (3), (3b), (3c), nd (3d) fy δx () x (ln )q f (, x ()) d, for (, ]. (58)

15 Mhemcl Problem n Engneerng 5 By he defnon of Cpuo-Hdmrd frconl dervve, yem (3), (3b), (3c), nd (3d) fe yem (3), (3b), (3c), nd (3d) C-H Dq (δx ()) f(, x ()), J (, T], (,...,m), l (l,...,p), Δx x( ) x( )I (x ( )) C,,,...,m, Δδx l δx( l ) δx( l )J l (x ( l )) C, l,,...,p, (59) x () x C, δx () x C. Moreover, reonble h mpule Δx (,,...,m) re condered pecl mpule Δδx l (l,,...,p)nyem(59)byremr4.therefore,ung Lemm 7 for yem (59) ( (, ],here,,...,π), we hve δx () x f(, x ()) d ξ I (x ( )) J (x ( )) (ln )q [ (ln q ) f (, x ()) d (ln )q d f (, x ()) f(, x ()) d ] ζ J (x ( )) [ [ (ln )q d f (, x ()) (ln )q (ln q ) f (, x ()) d (ln )q (60) repecvely, we ge ξ ξ(for ll,,..., )ndζ ζ (for ll,,..., )bycorollrend3.thu, δx () x f(, x ()) d ξ J (x ( )) (ln )q I (x ( )) [ (ln )q d f (, x ()) (ln )q f (, x ()) d ] ζ J (x ( )) [ [ (ln )q d f (, x ()) (ln q ) f (, x ()) d (ln q ) f (, x ()) d (6) f(, x ()) d ], ] where ξ (,,..., )ndζ (,,..., )re undeermned conn. Leng J (x( )) 0 (for ll,,..., )ndi (x( )) 0 (for ll,,..., ), (ln )q f (, x ()) d ], ] Th proof compleed. for (, ], Π.

16 6 Mhemcl Problem n Engneerng Theorem 6. Le q C, R(q) (, ), ndξ nd ζ re wo conn. Syem (3), (3b), (3c), nd (3d) equvlen o he negrl equon x () x x ln (ln )q f (, x ()) d for (, ], x x ln I (x ( )) J (x ( )) ln (ln )q f (, x ()) d ξ I (x ( )) Γ(q) [ (ln )q f (, x ()) d ] ln (/ ) ζ J (x ( )) Γ(q) [ [ (ln ) q f (, x ()) d (ln )q d f (, x ()) (ln ) q f (, x ()) d (ln q ) f (, x ()) d (ln )q d f (, x ()) (6) (ln )q f (, x ()) d ] ln (/ ) (ln q ) ] f (, x ()) d for (, ], Π provded h he negrl n (6) ex. Proof. Necey. We cn verfy h (6) fe condon () () by Lemm 9, 0, nd 6. Nex, ng he frconl dervve o (6) for (, ] (here 0,,,...,Π), we ge [ C-H D q x ()] (, ] Γ( q) (ln q η ) δ (x x ln η I (x ( )) J (x ( )) ln η η (ln η )q f (, x ()) d ξ I (x ( )) Γ(q) [ (ln q ) f (, x ()) d η (ln η )q η d f (, x ()) (ln η )q f (, x ()) d ] ln (η/ ) (ln q ) f (, x ()) d ζ J (x ( )) Γ(q) [ [ Γ( q)γ(q) (ln q ) f (, x ()) d η (ln η )q d f (, x ()) (ln η ) q δ η η (ln η )q f (, x ()) d ] ln (η/ ) ] (ln q ) f (, x ()) d ) dη η (, ] (ln η )q f (, x ()) d ξ I (x ( η )) [ (ln η )q η d f (, x ()) (ln η )q f (, x ()) d ]ζ J (x ( )) (63) η [ (ln η )q η d f (, x ()) ζ (ln η )q f (, x ()) d ] dη η (, ] J (x ( )) [f (, x ()) f(, x ()) ] f(, x ()) (, ]. (, ] f (, x ()) ξ I (x ( )) [f (, x ()) f(, x ()) ] So, (6) fe (3).

17 Mhemcl Problem n Engneerng 7 Fnlly, rghforwrd o verfy h (6) fe (3b) nd (3c). So, (6) fe ll condon of yem (3), (3b), (3c), nd (3d). Suffcency. AccordngoLemm9nd0,heoluonof yem (3), (3b), (3c), nd (3d) fy ξ η I (x ( )) [ (ln )q η (ln q ) η f (, x ()) d f (, x ()) d x () x x ln (ln )q f (, x ()) d, for (, ]. (64) η ζ (ln η d )q f (, x ()) η ] J (x ( )) Nex, by Lemm 5, he oluon of yem (3), (3b), (3c), nd (3d) fy δx () x J (x ( )) (ln )q f (, x ()) d ξ I (x ( )) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) [ [ η η (ln q ) d f (, x ()) η (ln η )q η x ln f (, x ()) d (ln η d )q f (, x ()) η ] dηc ] J (x ( )) ln (ln )q (ln )q f (, x ()) d ]ζ J (x ( )) [ [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ], ] Ung (65), we hve for (, ], Π. x () C x η J (x ( )) η η (ln η d )q f (, x ()) η (65) f(, x ()) d ξi (x ( )) ln (ln q ) f (, x ()) d ζj (x ( )) ln ξi (x ( )) Γ(q) (ln q ) f (, x ()) d [ (ln )q d f (, x ()) (ln η )q f (, x ()) d ] ζj (x ( )) Γ(q) [ (ln )q d f (, x ()) (ln )q f (, x ()) d ]. (66)

18 8 Mhemcl Problem n Engneerng Leng J (x( )) 0 (for ll,,...,p)ndi (x( )) 0 (for ll,,...,m)n(66),repecvely,bylemm9 nd 0, we obn Thu, Cx x ln ξi (x ( )) I (x ( )) J (x ( )) (ln q ) f (, x ()) d ln (ln q ) f (, x ()) d ζj (x ( )) ln (ln q ) f (, x ()) d (ln q ) f (, x ()) d. x () x x ln I (x ( )) J (x ( )) ln ξ I (x ( )) (ln )q f (, x ()) d (67) Γ(q) [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) (ln q ) f (, x ()) d ζ J (x ( )) Γ(q) [ [ (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln )q f (, x ()) d ] ln (/ ) ] (ln q ) f (, x ()) d for (, ], Π. (68) So, he oluon of yem (3), (3b), (3c), nd (3d) fy (6). Th proof compleed. Corollry 7. Le q C, R(q) (, ), ndξ nd ζ re wo conn. Syem () equvlen o he negrl equon x () x x ln (ln )q f (, x ()) d for (, ], x x ln I (x ( )) I (x ( ξ I (y ( )) Γ(q) [ (ln )q f (, x ()) d ] ln (/ ) ζ )) ln (ln ) q f (, x ()) d I (x ( )) Γ(q) [ [ (ln )q f (, x ()) d ] ln (/ ) ] (ln )q f (, x ()) d (ln )q d f (, x ()) (ln ) q f (, x ()) d (ln q ) f (, x ()) d (ln )q d f (, x ()) (ln q ) f (, x ()) d for (, ], Π (69) provded h he negrl n (69) ex.

19 Mhemcl Problem n Engneerng 9 Remr 8. Subung x x nd J (x( )) I (x( )) no (6), (69) cn be obned. Nex, le u nlyze he lmed ce of yem (): lm yem () q δ (x ()) f(, x ()), J (, T], (,...,m), l (l,...,p), Δx x( ) x( )I (x ( )) C,,,...,m, Δx x ( l l ) x ( l )I l (x ( l )) C, l,,...,p, x () x C, x () x C. (70) On he oher hnd, by (69), we hve lm q x () x x ln x x ln I (x ( )) I (x ( )) ln ln d f (, x ()), for (, ], ln d f (, x ()) for (, ], Π. (7) I cn be verfed h (7) he oluon of (70), whch ndrecly uppor our concluon. Compeng Inere The uhor declre h hey hve no compeng nere. Acnowledgmen The wor decrbed n h pper fnnclly uppored by he Nonl Nurl Scence Foundon of Chn (Grn no ), he Nurl Scence Foundon of Jngx Provnce (Grn no. 05BAB0703), he Reerch Foundon of Educon Bureu of Jngx Provnce, Chn (Grn no. GJJ4738), nd Jung Unvery Reerch Foundon (Grn no ). Reference [] A. A. Klb, Hdmrd-ype frconl clculu, Journl of he Koren Mhemcl Socey,vol.38,no.6,pp.9 04,00. []P.L.Buzer,A.A.Klb,ndJ.J.Trullo, Compoon of Hdmrd-ype frconl negron operor nd he emgroup propery, Journl of Mhemcl Anly nd Applcon,vol.69,no.,pp ,00. [3]P.L.Buzer,A.A.Klb,ndJ.J.Trullo, Mellnrnform nly nd negron by pr for Hdmrd-ype frconl negrl, Journl of Mhemcl Anly nd Applcon, vol.70,no.,pp. 5,00. [4] M. Klme, Sequenl frconl dfferenl equon wh Hdmrd dervve, Communcon n Nonlner Scence nd Numercl Smulon, vol. 6, no., pp , 0. [5] B.AhmdndS.K.Nouy, AfullyHdmrdypenegrl boundry vlue problem of coupled yem of frconl dfferenl equon, Frconl Clculu nd Appled Anly, vol.7,no.,pp ,04. [6] P.Thrmnu,S.K.Nouy,ndJ.Trboon, Exencend unquene reul for Hdmrd-ype frconl dfferenl equon wh nonlocl frconl negrl boundry condon, Abrc nd Appled Anly, vol. 04, Arcle ID 90054, 9 pge, 04. [7] A. A. Klb, H. H. Srvv, nd J. J. Trullo, Theory nd Applcon of Frconl Dfferenl Equon, Elever, Amerdm, The Neherlnd, 006. [8]F.Jrd,T.Abdelwd,ndD.Blenu, Cpuo-ypemodfcon of he Hdmrd frconl dervve, Advnce n Dfference Equon,vol.0,rcle4,8pge,0. [9]Y.Y.Gmbo,F.Jrd,D.Blenu,ndT.Abdelwd, On Cpuo modfcon of he Hdmrd frconl dervve, Advnce n Dfference Equon,vol.04,rcle0,04. [0] B. Ahmd nd S. Svundrm, Exence reul for nonlner mpulve hybrd boundry vlue problem nvolvng frconl dfferenl equon, Nonlner Anly. Hybrd Syem,vol. 3, no. 3, pp. 5 58, 009.

20 0 Mhemcl Problem n Engneerng [] B. Ahmd nd S. Svundrm, Exence of oluon for mpulve negrl boundry vlue problem of frconl order, Nonlner Anly: Hybrd Syem, vol. 4, no., pp. 34 4, 00. [] B. Ahmd nd G. Wng, A udy of n mpulve four-pon nonlocl boundry vlue problem of nonlner frconl dfferenl equon, Compuer & Mhemc wh Applcon, vol. 6, no. 3, pp , 0. [3] Y. Tn nd Z. B, Exence reul for he hree-pon mpulve boundry vlue problem nvolvng frconl dfferenl equon, Compuer & Mhemc wh Applcon,vol.59, no. 8, pp , 00. [4] J. Co nd H. Chen, Some reul on mpulve boundry vlue problem for frconl dfferenl ncluon, Elecronc Journl of Qulve Theory of Dfferenl Equon,vol.00,no., pp. 4, 00. [5] G. Wng, B. Ahmd, nd L. Zhng, Impulve n-perodc boundry vlue problem for nonlner dfferenl equon of frconl order, Nonlner Anly: Theory, Mehod & Applcon,vol.74,no.3,pp ,0. [6] G. Wng, L. Zhng, nd G. Song, Syem of fr order mpulve funconl dfferenl equon wh devng rgumen nd nonlner boundry condon, Nonlner Anly: Theory, Mehod & Applcon, vol. 74, no. 3, pp , 0. [7] G. Wng, B. Ahmd, nd L. Zhng, Some exence reul for mpulve nonlner frconl dfferenl equon wh mxed boundry condon, Compuer & Mhemc wh Applcon, vol. 6, no. 3, pp , 0. [8] X. Wng, Impulve boundry vlue problem for nonlner dfferenl equon of frconl order, Compuer & Mhemc wh Applcon, vol. 6, no. 5, pp , 0. [9] M. Fecn, Y. Zhou, nd J. R. Wng, On he concep nd exence of oluon for mpulve frconl dfferenl equon, Communcon n Nonlner Scence nd Numercl Smulon,vol.7,no.7,pp ,0. [0] I. Smov nd G. Smov, Sbly nly of mpulve funconl yem of frconl order, Communcon n Nonlner Scence nd Numercl Smulon, vol.9,no.3,pp , 04. [] S. Abb nd M. Benchohr, Impulve hyperbolc funconl dfferenl equon of frconl order wh e-dependen dely, Frconl Clculu nd Appled Anly,vol.3,pp.5 4, 00. [] S. Abb nd M. Benchohr, Upper nd lower oluon mehod for mpulve prl hyperbolc dfferenl equon wh frconl order, Nonlner Anly: Hybrd Syem, vol. 4, no. 3, pp , 00. [3] S. Abb, R. P. Agrwl, nd M. Benchohr, Drboux problem for mpulve prl hyperbolc dfferenl equon of frconl order wh vrble me nd nfne dely, Nonlner Anly: Hybrd Syem,vol.4,no.4,pp.88 89,00. [4] S. Abb, M. Benchohr, nd L. Gòrnewcz, Exence heory for mpulve prl hyperbolc funconl dfferenl equon nvolvng he Cpuo frconl dervve, Scene Mhemce Jponce,vol.7,no.,pp.49 60,00. [5] M. Benchohr nd D. Seb, Impulve prl hyperbolc frconl order dfferenl equon n bnch pce, Journl of Frconl Clculu nd Applcon, vol.,no.4,pp., 0. [6] T. L. Guo nd K. Zhng, Impulve frconl prl dfferenl equon, Appled Mhemc nd Compuon,vol.57, pp ,05. [7] X. Zhng, X. Zhng, nd M. Zhng, On he concep of generl oluon for mpulve dfferenl equon of frconl order q (0,), Appled Mhemc nd Compuon,vol.47,pp. 7 89, 04. [8] X. Zhng, On he concep of generl oluon for mpulve dfferenl equon of frconl-order q (,), Appled Mhemc nd Compuon,vol.68,pp.03 0,05. [9] X. Zhng, The generl oluon of dfferenl equon wh Cpuo-Hdmrd frconl dervve nd mpulve effec, Advnce n Dfference Equon,vol.05,rcle5,05. [30] X. Zhng, P. Agrwl, Z. Lu, nd H. Peng, The generl oluon for mpulve dfferenl equon wh Remnn-Louvlle frconl-order q ε (,), Open Mhemc, vol.3,pp , 05.

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