Nonline Anlyi nd Diffeentil Eqution, Vol. 5, 07, no. 6, 7-8 HIKARI Ltd, www.m-hiki.com http://doi.og/0.988/nde.07.796 Hdmd nd Cputo-Hdmd FDE with hee Point Integl Boundy Condition N.I. Mhmudov, M. Awdll nd K. Abub Deptment of Mthemtic Eten Meditenen Univeity Gzimgu, Mein 0, ukey Copyight c 07 N.I. Mhmudov, M. Awdll nd K. Abub. hi ticle i ditibuted unde the Cetive Common Attibution Licene, which pemit uneticted ue, ditibution, nd epoduction in ny medium, povided the oiginl wok i popely cited. Abtct In thi ticle, bed on Bnch nd Schude fixed point theoem, the exitence nd uniquene condition of nonline Hdmd H) / Cputo-Hdmd CH)) fctionl diffeentil eqution of ode < α with thee point integl boundy condition e obtined. Some exmple e intoduced to illutte the pplicbility of the obtined eult. Mthemtic Subject Clifiction: 34A08, 34A60, 34B0, 34B5 Keywod: Hdmd fctionl integl, Cputo deivtive, fixed point Intoduction One of the impotnt chcteitic of fctionl opeto i thei nonlocl ntue. ccounting fo the heedity popetie of mny phenomen nd pocee involved. Duing the lt decde, fctionl clculu h gined emkble impotnce due to the ppliction in lmot ll pplied cience. It hould be pointed tht fctionl opeto e moe convenient fo decibing ome phyicl poblem. On the othe hnd mot of the wok on fctionl diffeentil eqution e bed on Riemnn-Liouville nd Cputo type fctionl opeto. Anothe kind of fctionl opeto tht ppe in the litetue i the
7 N.I. Mhmudov, M. Awdll nd K. Abub fctionl deivtive due to Hdmd. Hdmd deivtive diffe fom the peceding one in the ene tht the kenel of the integl contin logithmic function of bity exponent. Detil nd popetie of the Hdmd fctionl deivtive nd integl cn be found in [], [8], [9], [0], [], [3]. Howeve, diffeentil eqution with Hdmd deivtive i till tudied le thn tht of Riemnn-Liouville nd Cputo fcionl diffeentil eqution, ee [7]-[8], nd efeence theein. In thi ppe, we popoe the exitence nd uniquene of olution fo nonline Hdmd H) / Cputo-Hdmd CH)) fctionl diffeentil eqution of ode < q with thee point integl boundy condition of the fom. Nonline Hdmd fctionl diffeentil eqution nd H D q u t) = g t, u t)), < q, 0 < t, ) ocited with thee point integl boundy condition u ) = 0, u ) = β η u ) d, < η <, β R, ) Nonline Cputo-Hdmd fctionl diffeentil eqution CH D q x t) = g t, x t)), < q, 0 < t, 3) ocited with thee point integl boundy condition x ) = 0, x ) = β η x ) d, < η <, β R. 4) Hee H D q, CH D q denote the Hdmd nd the Cputo-Hdmd fctionl deivtive of ode < q, epectively, g : [, ] R R i continuou function, β i given contnt. Peliminie In thi ection we intoduce ome definition, lemm nd nottion of fctionl clculu Definition he Hdmd fctionl integl of ode q fo continuou function g : [, ) R i defined H I q g t) = t t ) q g ) d, q > 0, Γ q) povided tht the integl exit.
Hdmd nd Cputo-Hdmd FDE 73 Definition he Hdmd fctionl deivtive of ode q fo continuou function g : [, ) R i defined H D q g t) = δ n H I q g ) t) = t d ) n dt Γ n q) t t ) n q g ) d, n < q < n, n = [q] +, δ = t d, [q] denote the intege pt of q. dt Definition 3 Fo t let n-time diffeentible function g : [, ) R, the Cputo-Hdmd fctionl deivtive of ode q i defined CH D q g t) = Lemm 4 Let u, x C n δ Γ n q) t [, ], R). hen H I q H D q u ) t) = u t) t ) n q δ n g ) d. n j= j=0 c j t ) n j, H I q CH D q u ) n t) = u t) c j ) t j. Hee C n δ [, ], R) = {u : [, ] R : δn u C [, ], R)}. Lemm 5 Let h C [, ], R) nd u Cδ [, ], R). he line Hdmd F.D.E given by { H D q u t) = h t), < q, 0 < t, u ) = 0, u ) = β η u ) d, < η <, β R, 5) i equivlent to the integl eqution given by: u t) = Γ q) β whee A = η t η t ) q h ) d + ) q d. Γ q) ) q h ) dd ) t q ) ) q βa ) ) q h ) d, 6) Poof. We conide the Hdmd ce, pplying H I q to both ide of 5) u t) = H I q h t) + c t ) q + c t q. )
74 N.I. Mhmudov, M. Awdll nd K. Abub he fit boundy condition u 0) = 0, implie tht c = 0. hen u t) = H I q h t) + c t ) q. 7) he econd boundy condition u ) = β η H I q h ) + c ) q η η = β H I q h ) d + βc c = ) q β βa u ) d, implie η ) q d, ) H I q h ) d H I q h ). Subtitute 8) in 7) nd expnd the Hdmd fctionl integl we get eqution 6). By diect computtion the convee ttement cn be eily obtined. hi complete the poof. Lemm 6 Let h C [, ], R) nd u Cδ [, ], R). he line Cputo- Hdmd F.D.E given by { CH D q u t) = h t), < q, 0 < t, u ) = 0, u ) = β η u ) d, < η <, β R, 9) i equivlent to the integl eqution given by: u t) = Γ q) β t η t ) q h ) whee A = η η ) +. 8) d + t Γ q) βa ) ) ) q h ) d, 0) ) q h ) dd 3 Exitence eult fo the poblem )-) We define n opeto G H : C [, ], R) C [, ], R) follow G H u) t) = Γ q) β t η t ) q g, u )) d + ) q g, u )) dd Γ q) ) q t ) ) q βa ) ) q g, u )) d.
Hdmd nd Cputo-Hdmd FDE 75 It hould be noticed tht poblem )-) h olution if nd only if the opeto G H h fixed point. Fo the ke of convenience, we et Q = Γ q + ) ) q + ) q β η ) ) Γ q + ) ) q, βa Q = ) q β η ) ) Γ q + ) ) q. βa heoem 7 Let g : [, ] R R be continuou function tifying A) thee exit L g > 0 uch tht g t, u) g t, v) L g u v, t [, ], u, v R. hen the poblem )-) h unique olution on [, ] povided tht QL g <. Poof. Conide the et ω := {u C [, ], R) : u } with MgQ LQ, whee M g = up t g t, 0). It i cle tht g, u )) L g + M g, u ω. Fit we how tht G H ω ω. Fo u ω, t [, ] we hve G H u) t) t t ) q ) g, u )) t q d + Γ q) Γ q) ) q βa η β ) q g, u )) dd + ) ) q g, u )) d L g + M g ) Q, whcih implie tht G H u ω. Next we how tht the opeto G H i contction. Indeed, fo ny u, v ω we hve G H u) t) G H v) t) t t ) q ) g, u )) g, v )) t q d + Γ q) Γ q) ) q βa η β ) q g, u )) g, v )) dd + ) ) q g, u )) g, v )) d L g Q u v.
76 N.I. Mhmudov, M. Awdll nd K. Abub hu G H i contction nd by the Bnch contction mpping theoem the B.V.P h unique olution on [, ]. hi complete the poof. heoem 8 Let g : [, ] R R be continuou function tifying A) nd A) g t, u) y t), t, u) [, ] R, y C [, ], R + ). If L g Q <, then thee i t let one olution fo BVP )-) on [, ]. Poof. We define the opeto G H nd G H on ω, with Q y, follow: G H u ) t) = t t ) q g, u )) d, Γ q) G H u ) ) t q t) = ) ) q Γ q) βa β η Fo u, v ω we hve ) q g, u )) dd G Hu + G Hv Q y, ) ) q g, u )) d tht i G H u + G H v ω. By ou umption, one cn eily how tht G H u G Hv Lg Q u v, which implie tht G H i contction. In ddition, the opeto G H i continuou eult of the continuity of g. Alo, it i unifomly bounded G Hu ) t) y ) q. Γ q + ) Moeove, fo t, t [, ], t < t we hve G Hu ) t ) G Hu ) t ) up { g t, u) : t [, ], u } Γ q) t t ) q t ) ) q d + t t ) q d) t ppoche zeo t t. Note tht G H u) t ) G H u) t ) i independent of u implie tht G H i eltively compct, by Azel-Acoli theoem we conclude tht G H i compct. Hence, the exitence of the olution of the B.V.P hold by Knoelkii fixed point theoem. Next we ue Ley Shude ltentive to pove the exitence of the olution of the BVP.
Hdmd nd Cputo-Hdmd FDE 77 heoem 9 Let g : [, ] R R be continuou function tifying A) nd A3) thee exit function w C [, ], R + ) nd nondeceing function µ : R + R + uch tht g t, u) w t) µ u ), t, u) [, ] R, A4) thee exit contnt M > 0 uch tht M > w µ M) Q. hen thee i t let one olution fo BVP )-) on [, ]. Poof. Fit tep we how tht the opeto mp bounded et into bounded et of. t t ) q w ) µ u ) d + G H u) t) Γ q) η β ) q w ) µ u ) dd + w µ ) Q ) t q ) q βa Γ q) ) ) q w ) µ u ) d Poof. Next we how tht the opeto mp bounded et into equicontinuou et of. fo t, t [, ], t < t we hve G Hu ) t ) G Hu ) t ) w µ ) Γ q) t t ) q t ) ) q t d + ) t q ) t q + Γ q) ) q βa η β ) q w ) µ u ) dd + t t ) q d) ) ) q w ) µ u ) d ppoche zeo t t. Note tht the ight hnd ide of the bove inequlity i independent of u,by Azel-Acoli theoem we conclude tht i completely continuou. he lt tep to complete the umption of Ley-Schude nonline ltentive theoem i to how the boundedne of the et of ll olution to
78 N.I. Mhmudov, M. Awdll nd K. Abub eqution u = δg H u, 0 δ. Aume tht u i olution, then by me mnne befoe we how the opeto i bounded: u t) = δ G H u) t) w µ u ) Q, u w µ u ) Q. But by A4) thee exit contnt M > 0 uch tht M u. Contuct the et Ω = {u C [, ], R) : u < M}, it i obviou tht the opeto G H : Ω C [, ], R) i continuou nd completely continuou. By the contucted Ω, thee i no u Ω uch tht u = δg H u fo ome 0 < δ <. Conequently, by the nonline ltentive of Ley-Schude type, we deduce tht G H h fixed point u Ω which i olution of the BVP. hi complete the poof. 4 Exitence eult fo the poblem 3)-4) In thi ection, ome exitence eult e intoduced fo the poblem 3)- 4). he poof e omitted ince they e imil to the one employed in the peviou ection. We define fixed point opeto G CH : C [, ], R) C [, ], R) ocited with the poblem 3)-4) follow G CH u) t) = t t ) q g, u )) t d + Γ q) Γ q) βa ) η β ) q g, u )) dd ) ) q g, u )) d. Fo computtionl convenience, we let R = ) q ) q+ β η ) ) + Γ q + ) Γ q + ) βa, ) q+ R β η ) ) = Γ q + ) βa. Uing the method of poof fo the theoem obtined by the peviou ection nd the opeto we cn intoduce the following theoem. heoem 0 Let g : [, ] R R be continuou function tifying A) thee exit L g > 0 uch tht g t, u) g t, v) L g u v, t [, ], u, v R. hen the poblem 3)-4) h unique olution on [, ] povided tht RL g <.
Hdmd nd Cputo-Hdmd FDE 79 heoem Let g : [, ] R R be continuou function tifying A) nd A) g t, u) y t), t, u) [, ] R, y C [, ], R + ). If L g R <, then thee i t let one olution fo BVP )-) on [, ]. heoem Let g : [, ] R R be continuou function tifying A) nd A3) thee exit function w C [, ], R + ) nd nondeceing function µ : R + R + uch tht g t, u) w t) µ u ), t, u) [, ] R, A4) thee exit contnt M > 0 uch tht M > w µ M) R. hen thee i t let one olution fo BVP )-) on [, ]. 5 Exmple Exmple : Conide the following non line fctionl Hdmd diffeentil eqution { H D 3/ u t) = g t, u t)), t e, u ) = 0, u e) = β η u ) d, η =, β = 3. ) Fo the pplicbility of heoem 7 conide the function g t, u) = t u e t t + 4) 3 u +, t e. It i cle tht the function g i jointly continuou nd Lipchitzin wtih L g = 5e. One cn eily compute Q =.079. hu L gq < nd ll condition of the heoem 7 tified which how the exitence of uniquene olution of BVP. given by ). Exmple : o illutte theoem 9 conide It i jointly continuou nd g t, u) = e t tn u π t + 44, t e. g t, u) e t t + 44 = w t), Q =.07 4. king M > 0.0,we obeve tht ll the condition of heoem 9 e tified, which implie the exitence of the olution of the BVP )
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8 N.I. Mhmudov, M. Awdll nd K. Abub [0] X. Zhng, On impulive ptil diffeentil eqution with Cputo- Hdmd fctionl deivtive, Adv. in Diffeence Eqution, 8 06), -. http://doi.og/0.86/366-06-008-y Received: Octobe, 07; Publihed: Octobe 3, 07