Sequentil Frctionl Differentil Equtions with Hdmrd Derivtive M. Klimek Institute of Mthemtics, Czestochow University of Technoy ul. Dbrowskiego 73, 42-200 Czestochow, Polnd e-mil: klimek@im.pcz.pl). Abstrct: A clss of nonliner sequentil frctionl differentil equtions dependent on the bsic frctionl opertor involving Hdmrd derivtive is studied for rbitrry rel noninteger order α R +. The existence nd uniqueness of the solution is proved using the contrction principle nd new, equivlent norm nd metric, introduced in the pper. As n exmple, liner nonhomogeneous FDE is solved explicitly in rbitrry intervl, b nd for nonhomogeneous term given s n rbitrry Fox function. The generl solution consists of the solution of homogeneous counterprt eqution nd prticulr solution corresponding to the nonhomogeneous term nd is given s liner combintion of the respective Fox functions series. Keywords: Frctionl differentil eqution, Hdmrd derivtive, Bnch theorem, existence nd uniqueness of solutions, equivlent metrics, Fox functions series 1. INTRODUCTION During the lst decdes, frctionl differentil equtions FDE) hve become n importnt tool in the mthemticl modelling of mny systems nd processes in mechnics, physics, chemistry, economics, engineering nd bioengineering. The investigtions concerning solving methods, both nlyticl nd numericl, the existence nd uniqueness of solutions, s well s studies of the properties of solutions hve yielded mny importnt results nd mde FDE theory n importnt prt of pplied nd pure mthemtics compre monogrphs nd review ppers 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11) nd the references given therein). In the pper we study clss of nonliner sequentil frctionl differentil equtions SFDE) dependent on the bsic t ) β D α + - opertor with the Hdmrd derivtive of rbitrry rel noninteger order α R + nd rel prmeter β R. Let us note tht the Cuchy problem for FDE contining the Hdmrd derivtives ws discussed in 4) in non-sequentil setting. On the other hnd, results on the existence nd uniqueness of solutions for SFDE with bsic Riemnn-Liouville derivtive cn be found in 4; 12; 13; 14). To prove the existence nd uniqueness of the solutions in n rbitrry finite intervl, b we follow the fixed point method nd pply the Bnch theorem. A crucil point in the proof is the ppliction of newly-introduced clss of one-prmeter equivlent norms nd respective metrics) in the spce of continuous weighted functions. In this spce the SFDE in the form of t ) β m D α +) L t ) ) β D+ α ft) Ψt, ft)), where Ψ C, b R) nd Lλ) : c j λ j, 1) is equivlent to the following frctionl integrl eqution ft) L I+ α t ) ) β ft) + ) + ) t β m Ψt, ft)) + φ 0 t), I α + provided f C n α,, b the function spce described in 5), constnts, b R + rbitrry, function φ 0 belongs to ) t β m the kernel of the ) D α + opertor nd Lλ) : c j λ m j. 2) We shll prove tht the integrl opertor on the right-hnd side of the bove eqution yields contrctive mpping in the spce of continuous weighted functions C n α,, b, when it is endowed with respective metric from the introduced fmily of one-prmeter metrics. As n ppliction, we shll study in detil simple liner nonhomoegeneous FDE in the cse when the nonhomogeneous term is chosen from the fmily of Fox functions. Using the integrtion properties of Fox functions we rrive t n explicit form of the generl solution given s Fox functions series. The convergence of the respective series results from the min theorem.
The pper is orgnized s follows. In the next section we recll ll the necessry definitions nd properties of frctionl opertors nd Fox functions. We lso construct one-prmeter clss of equivlent norms nd respective metrics in the spce of continuous weighted functions C n α,, b. Then we prove tht certin frctionl integrl opertors re bounded in this spce endowed with n rbitrry norm from the proposed clss. We generlize the bsic integrl opertor to mpping, which ppers to be contrction under the respective ssumptions on prmeter determining the norm nd metric on the function spce. Section 3 contins the min result - theorem on the existence nd uniqueness of the solution to certin nonliner sequentil FDE with the Hdmrd derivtive. In section 4, pplictions to the liner nd nonhomogeneous FDE re given. For the rbitrry order of frctionl derivtive, we derive n explicit generl solution in the form of Fox function series. 2. PRELIMINARIES In the pper we shll study the existence nd explicit form of solutions of certin frctionl differentil eqution in the C γ,, b spce when γ n α. Let us recll the norm γ, nd the generted metric ctive in this spce of weighted continuous functions, when Reγ) 0, 1): f γ, : sup t γ ft) 3) t,b ) df, g) : f g γ,. 4) The C γ, b spce is then given s C γ,, b : {f C, b; f γ, < }. 5) Now, we recll the definitions of left-sided frctionl opertors. In our pper we shll consider frctionl differentil equtions contining Hdmrd derivtives. Both the integrl nd derivtive re defined s follows 4). Definition 2.1 Let Reα) > 0. Then the left-sided Hdmrd integrl of order α is given by the formul I+f)t) α 1 t t ) α 1 fs) ds t > > 0, 6) Γα) s s where Γα) denotes the Euler gmm function. Let Reα) n 1, n). Then the left-sided Hdmrd derivtive is defined s D α +f)t) t d dt ) n I n α + f)t) t > > 0. 7) An importnt nd chrcteristic feture of the bove frctionl opertors is their composition rule which we quote in property below fter the monogrph by Kilbs et l 4). It will be pplied in the trnsformtion of the investigted FDE into its equivlent integrl form s well s in the derivtion of the corresponding initil conditions. Property 2.2 Let Reβ) Reα) > 0. Then the following formul D α +I β +ft) I β α + ft) 8) holds t ny point t, b when f C, b. If f C γ,, b, then the bove composition rule holds t ny point t, b. Let us lso recll some results on Hdmrd integrtion nd differentition which re useful in the derivtion of sttionry functions nd in the construction of solutions to the discussed equtions. Property 2.3 Let Reβ) > 0 nd Reα) > 0. Then the following formuls hold: I α + D α + t ) β 1 Γβ) Γβ + α) t ) β 1 Γβ) Γβ α) t ) β+α 1 9) t ) β α 1. 10) In further considertions we lso pply Fox functions. Such functions re defined vi Mellin-Brnes integrl nd look s follows for integer numbers m, n, p, q fulfilling 0 m q, 0 n p, for complex numbers i, b j C nd for rel prmeters α i, β j R 4; 16): H m,n p,q 1 2πi L z i) 1,p ; α i ) 1,p b j ) 1,q ; β j ) 1,q 11) m Γb j + β j s) n Γ1 i α i s) j1 p in+1 Γ i + α i s) i1 q jm+1 z s ds, Γ1 b j β j s) where i 1, 2,..., p; j 1, 2,..., q nd contour L seprtes the poles of the gmm functions in the numertor of the complex kernel 4; 16). When α i 1, β j 1 for ny i 1,..., p j 1,..., q, the bove Fox function belongs to the subclss of Meijer G-functions: H m,n p,q z i) 1,p ; 1) 1,p G b j ) 1.q ; 1) m,n p,q 1,q z i) 1,p. 12) b j ) 1,q The following prmeters determine the integrtion properties of Fox nd Meijer functions: : µ : n α i i1 q b j j1 p in+1 p i1 α i + i + p q 2 m β j j1 q jm+1 β j 13) 14) In wht follows, we shll study clss of equtions contining the frctionl differentil opertors of rel order. Thus, we ssume order α R + s well s prmeter β R throughout the pper. In the property below, we rewrite the generl theorem on the Riemnn-Liouville integrtion of Fox functions to the cse of rel α nd β compre the definition of the I α 0+ integrl nd Theorem 2.7 from monogrph 16)). Property 2.4 Let us ssume Hp,q m,n C n α 0, b nd > 0 or 0, Reµ) < 1. If α R +, β R nd {α} β > 0, then the following integrtion formul holds: I0+t α β Hp,q m,n t σ i ) 1,p ; α i ) 1,p 15) b j ) 1,q ; β j ) 1,q
t α β H m,n+1 p+1,q+1 t σ β, i ) 1,p ); σ, α i ) 1,p ). b j ) 1,q, β α); β j ) 1,q, σ) In the bove property the results re given for the Riemnn-Liouville frctionl integrl. This formul cn be trnslted for the Hdmrd integrl when we pply the following reltion between both integrls: N ft) : f t ) 16) N I α 0+N 1 Iα +. 17) Using the introduced reltion we rrive t n nous integrtion formul which describes the Hdmrd integrl for the respective Fox function. Property 2.5 Let us ssume Hp,q m,n C n α 0, b nd > 0 or 0, Reµ) < 1. If α R +, β R nd {α} β > 0, then the following integrtion formul holds: I+N α t β Hp,q m,n t σ i ) 1,p ; α i ) 1,p 18) b j ) 1,q ; β j ) 1,q N t α β H m,n+1 p+1,q+1 t σ β, i ) 1,p ); σ, α i ) 1,p ). b j ) 1,q, β α); β j ) 1,q, σ) To solve the discussed eqution in the spce of continuous weighted functions, we shll extend the stndrd norm nd metric 3,4). To this im, we propose to pply two-prmeter composed Mittg-Leffler function compre monogrph 4)). Let us define three-prmeter fmily of functions using such Mittg-Leffler function. For rel numbers α, β, R we define function e α,β, s follows: e α,β, t) : Γ{α} β)e α,{α} β t ) α ), 19) where E denotes the two-prmeter Mittg-Leffler function given in our construction s series z k E α,{α} β z) : Γαk + {α} β), 20) k0 where the z C rbitrry nd {α} denotes the frctionl prt of rel number α. It is esy to check tht functions e α,β, obey the following frctionl integrtion formul: ) t {α} β 1 e α,β, t) 21) I α + ) t {α} β 1 e α,β, t) 1, provided {α} β > 0. Let us observe tht we cn pply the functions defined in 19) to modify norm 3) nd the respective metric in the sense of Bielecki 17). He used exponentil functions to introduce n equivlent metric nd to show the existence of globl solutions of certin ordinry nd prtil differentil equtions in respective function spces. A similr technique ws proposed in 18; 19) for some simple nonliner frctionl differentil equtions. Then Lkshmiknthm et l 1; 20) developed modifiction of the metric in the C0, b spce by mens of one-prmeter Mittg-Leffler function nd pplied it in the solution of the nonliner frctionl differentil eqution of order α 0, 1). Here, we shll construct clss of metrics in the C n α,, b spce which re equivlent to stndrd metric 4) generted by norm 3). These metrics will be pplied in the proof of the existence nd uniqueness of solutions for clss of frctionl differentil equtions in the C n α,, b spce. Definition 2.6 The following formuls define norm nd respective metric in the C n α,, b spce, provided R + nd {α} β > 0 t n α f n α,, : sup ) ft) t,b e α,β, t) 22) d f, g) : f g n α,,. 23) Property 2.7 Metric d is equivlent to stndrd metric 4) generted by norm 3). Proof: The equivlence of the metrics results from the following inequlities fulfilled by norms 3) nd 22): f n α, e α,β, b) f n α,, f n α, 24) for rbitrry function f C n α,, b. ) Let us note tht frctionl integrl opertor I+ α t β is bounded in the C n α,, b; n α,, ) spces when β 0 nd, b R + rbitrry. This property follows from formul 21) nd from the fct tht series e α,β, is bsolutely nd uniformly convergent in ny intervl, b when > 0 nd ssumptions from Definition 2.6 re fulfilled. Property 2.8 If R + nd β 0, then the following inequlities re vlid for ll functions f C n α,, b, > 0: I α + t ) β f n α,, 25) ) b β I α 0+ b f n α,, t ) β ) j f n α,, 26) ) β ) j f n α,,. Now we construct mpping on the spce of weighted continuous functions C n α,, b nd test its properties. Definition 2.9 We define mpping T m s follows T m ft) : L I+ α t ) ) β ft) 27) for ny function f C n α,, b, {α} β 0 nd polynomil L defined in formul 2). The defined mpping is contrction in the C n α,, b; d ) spce of functions when the metric is given by formul 23) nd prmeter is lrge enough. This property is formulted nd proved in the following lemm.
Lemm 2.10 If > 1 + c j ) ) b β nd β 0, then mpping T m given in Definition 2.9 is contrction in the C n α,, b spce endowed with the d metric. Proof: Let f, g be pir of rbitrry functions from the C n α,, b spce. We pply Property 2.8 to estimte the d distnce of their imges T m f nd T m g: d T m f, T m g) T m f T m g n α,, c j I α 0+ c j t ) β ) m j ft) gt)) n α,, b ) βm j) m j c j ) b β f g n α,, c j ) b β d f, g). Thus, we obtined inequlity f g n α,, d T m f, T m g) c j ) b β d f, g), 28) which mens T m is contrctive mpping with respect to the d metric s frction c j )/ ) b β 0, 1) by ssumption. This ends the proof. Remrk: Let us note tht the proved result does not predict wht hppens when {α} β > 0 nd β > 0. In this cse norm 3) must be modified using nother type of function. Detiled clcultions will be given in subsequent pper. 3. MAIN RESULTS We shll study the existence nd uniqueness of solutions for sequentil frctionl differentil eqution with the Hdmrd derivtive: t ) β D α +) m L t ) ) β D+ α ft) 29) Ψt, ft)), when {α} β > 0, L is polynomil given in 1), Ψ C, b R) nd t, b. From the composition rules given in Property 2.2, it follows tht the bove eqution is equivlent in the C n α,, b spce to the frctionl integrl eqution written using defined mpping T m Definition 2.9): ft) 30) ) T m ft) + ) t β m Ψt, ft)) + φ 0 t). I α + Let us denote the mpping on the right-hnd side of the bove eqution s follows: T ft) : 31) ) T m ft) + ) t β m Ψt, ft)) + φ 0 t). I α + Function φ 0 in definition 31) is n rbitrry sttionry function of opertor t )β D α +) m, which mens it fulfills the eqution t )β D α +) m φ 0 t) 0. 32) Such n eqution is esy to solve nd the solution is liner combintion of the power functions n φ 0 t) d j,l ) t α β)j+α l, 33) l1 where coefficients d j,l re rbitrry rel numbers. It cn lso be expressed s combintion of composed Meijer G-functions: φ 0 t) 34) n d j,ln G 1,0 α β)j + α l + 1 1,1 t. α β)j + α l l1 Let us observe tht due to ssumption {α} β > 0, ll the sttionry functions given by 33,34) belong to the C n α,, b spce. Thus, mpping T given in 31), mps the weighted continuous functions into functions from the sme spce for rbitrry function φ 0. Proposition 3.1 Let α n 1, n), β 0 nd function Ψ C, b R) fulfill the following Lipschitz condition Ψt, x) Ψt, y) < M x y 35) for t, b nd x, y R. Ech sttionry function φ 0 given by 33,34) genertes unique solution f C n α,, b of frctionl differentil eqution 29). The solution is given s limit of itertions of mpping T 31): ft) lim T k ψt), 36) k where function ψ C n α,, b is rbitrry. Proof: We strt by observing tht Lipschitz condition 35) yields the following inequlity for ech R + : Ψt, ft)) Ψt, gt)) n α,, t sup )n α Ψt, ft)) Ψt, gt)) t,b e α,β, t) t M sup )n α ft) gt) t,b e α,β, t) M f g n α,,. Condition β 0 implies {α} β > 0. Due to this fct nd to the bove clcultions, mpping T defined in 31) is
bounded for ny sttionry function φ 0 nd in ech spce C n α,, b; n α,, ) ccording to Property 2.8. Now we ssume: > 1 + M + c j ) b ) β 37) nd write eqution 29) s n equivlent fixed point condition ft) T ft) 38) in spce C n α,, b; n α,, ). We check tht T is contrction by strightforwrd clcultion: d T f, T g) T f T g n α,, T m f T m g n α,, + + I+ α t ) β ) m Ψt, ft)) Ψt, gt)) n α,, c j b )β f g n α,, + M + b )β ) f g n α,, m M + c j b f g n α,, )β M + c j b d f, g). )β We conclude tht mpping T obeys for ny pir of functions f, g C n α,, b the following condition: d T f, T g) < M + c j b d f, g) 39) )β which mens it is contrction s frction M+ cj 0, 1) by ssumption 37). b )β Hence, unique fixed point in spce C n α,, b exists by the Bnch theorem nd is explicitly given s limit of itertions of mpping T : ft) lim T k ψt), 40) k where ψ C n α,, b is n rbitrry strting function nd the convergence with respect to the d metric is equivlent to the convergence with respect to stndrd d metric 4). This ends the proof. By crefully nlysing the bove proposition, we notice one to one correspondence between the choice of the generting sttionry function nd the unique solution of problem 29). Thus, the initil conditions t t re lso determined by function φ 0. This reltion leds to the following formultion nd solution of the respective Cuchy problem in the C n α,, b spce. Theorem 3.2 Let α n 1, n), β 0 nd let function Ψ obey the ssumptions of Proposition 3.1. Then frctionl differentil eqution 29) hs unique solution f in the C n α,, b spce fulfilling the set of initil conditions: D α l + t )β D α +) j ft) t d j,l, 41) where l 1,..., n nd j 0,..., m 1. This solution is limit of the itertions of mpping T 31) generted by sttionry function in the form of n l1 φ 0 t) 42) d j,l Γα β)j + α l + 1) t )α β)j+α l. 4. APPLICATIONS - EXAMPLE We shll now discuss the ppliction of the bove results to nonhomogeneous liner frctionl differentl eqution with the Hdmrd derivtive: t )β D α + λ ft) N H m,n p,q t σ ), 43) where t, b, function H is n rbitrry Fox function from the C n α 0, b clss, fulfilling the ssumptions of Property 2.5 nd σ > 0 compre formul 11)): Hp,q m,n t σ ) Hp,q m,n t σ i ) 1,p ; α i ) 1,p. 44) b j ) 1,q ; β j ) 1,q Let us denote the vectors defining the bove Fox function s follows: i ) 1,p α α i ) 1,p 45) b b j ) 1,q β β j ) 1,q. 46) The solution of eqution 43) is constructed using the integrtion formul from Property 2.5. The convergence of the respective series is implied nd ensured by Proposition 3.1. In this pper we shll restrict the exmple to generl solution to problem 43), dependent on n constnt coefficients. As we re discussing liner eqution, we know tht using the itertions of mppings T, generted by the corresponding sttionry functions, we cn split the solution into f 0, which solves the homogenous counterprt of eqution 43): t )β D+ α λ f 0 t) 0 47) nd f s which is generted by sttionry function φ 0 0. The full solution of eqution 43) is then the following sum: ft) f 0 t) + f s t). 48) Prt f s of the full solution is generted by the φ 0 0 sttionry function when we lso tke ψ 0 in 31,40). In this cse, mpping T is given for ny function g C n α,, b by the formul: T g : λi α + t ) β g + I α + t ) β N H m,n p,q t σ ). 49) This implies tht f s is given s series: f s t) λ k I+ α t ) β ) k+1 N Hp,q m,n t σ ) 50) k0 which is bsolutely convergent with respect to the n α, norm. Function f s cn be explicitly clculted
using integrtion formul 18) nd it is the following series of Fox functions: f s t) λ k t )k+1)α β) 51) k0 N H m,n+k+1 p+k+1,q+k+1 N t σ E k+1 ; E k+1, F k+1 ; F k+1 where the component vectors look s follows for k 1: E k βe k + β α)j k ; R k +p F k b; β α)e k + j k ) R k +q E k σe k ; α R k +p 52) 53) 54) F k β; σe k R k +q. 55) Vectors, α, b, β re given in 45,46) nd we hve denoted: e k 1,..., 1 R k 56) j k 0, 1,..., k 1 R k. 57) The f 0 prt of the solution is liner combintion of bsic solutions: n f 0 t) l f l t). 58) l1 Ech of the f l bsic components of the solution is generted by the t )α l component of the sttionry function nd is given s the following series of Meijer G- functions: f l t) λ k t )α β)k N G 1,k k+1,k+1 t A k,l 59) B k,l k0 with defining vectors A k,l, B k,l A k,l βe k + β α)j k ; α l + 1 R k+1 60) B k,l α l; β α)e k + j k ) R k+1. 61) In the bove formuls e k, j k re given in 56,57). 5. FINAL REMARKS In the pper we derived n explicit solution for clss of SFDE with the Hdmrd derivtive. It is given s limit of contrctions in the spce of continuous weighted functions with respective metric. As ll the newly constructed metrics re equivlent to the stndrd metric, we observe tht the proved convergence lso provides the convergence with respect to the initil metric. The cse of simple liner nonhomogeneous frctionl eqution is discussed in detil nd the solutions re Fox functions series. This form of the solution ppers to be similr to tht obtined erlier for FDE 11). Let us note tht the proposed method of proving the existence nd uniqueness of solution to SFDE cn be esily extended to the cse when frctionl opertor of nother type is involved. Our result, vlid in the spce of continuous weighted functions, nd tht obtined in 1; 18; 19; 20) for continuous functions, imply tht the extension of the Bielecki method 17) to FDE theory requires creful construction of the equivlent norms. The respective function spce, the form of the bsic frctionl opertor nd the properties of Mittg -Leffler functions determine new norms which llow us to prove the globl existence of solution in n rbitrry finite intervl. REFERENCES 1 V. Lkshmiknthm, S. Leel, J. Vsundhr Devi, Theory of Frctionl Dynmic Systems, Cmbridge Scientific Publishers, Cmbridge, 2009. 2 I. Podlubny, Frctionl Differentil Equtions, Acdemic Press, Sn Diego, 1999. 3 K.S. Miller, B. Ross, An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions, Wiley nd Sons, New York, 1993. 4 A.A. Kilbs, H.M. Srivstw, J.J. Trujillo, Theory nd Applictions of Frctionl Differentil Equtions, Elsevier, Amsterdm, 2006. 5 V. Kirykov, Generlized Frctionl Clculus nd Applictions, Longmnn-Wiley, New York, 1994. 6 A.A. Kilbs, J.J. Trujillo, Differentil eqution of frctionl order: methods, results nd problems. I, Appl. Anl. 78 2001) 153-192. 7 A.A. Kilbs, J.J. Trujillo, Differentil eqution of frctionl order: methods, results nd problems. II, Appl. Anl. 81 2002) 435-493. 8 R. Mgin, Frctionl Clculus in Bioengineering, Begell House Publisher, Redding, 2006. 9 B.J. West, M. Bon, P. Grigolini, Physics of Frctionl Opertors, Springer-Verlg, Berlin, 2003. 10 M.W. Michlski, Derivtives of noninteger order nd their pplictions, Disserttiones Mthemtice CCCXXVIII, Institute of Mthemtics, Polish Acd. Sci., Wrsw 1993. 11 M. Klimek, On Solutions of Liner Frctionl Differentil Equtions of Vritionl Type, The Publishing Office of the Czestochow University of Technoy, Czestochow, 2009. 12 A.A. Kilbs, M. Rivero, L. Rodriguez-Germá, J.J. Trujillo, α-anlytic solutions of some liner frctionl differentil equtions with vrible coefficients, Appl. Mth. Comp. 187 2007) 239-249. 13 M. Rivero, L. Rodriguez-Germá, J.J. Trujillo, Liner frctionl differentil equtions with vrible coefficients, Appl. Mth. Lett. 21 2008) 892-897. 14 Zhongli Wei, Qingdong Li, Junling Che, Initil vlue problems for frctionl differentil equtions involving Riemnn-Liouville sequentil frctionl derivtive, J. Mth. Anl. & Appl. 367 2010) 260-272. 15 S.G. Smko, A.A, Kilbs, O.I. Mrichev, Frctionl Integrls nd Derivtives, Theory nd Applictions, Gordon nd Brech, Yverdon, 1993. 16 A.A. Kilbs, M. Sigo, H-Trnsforms, Theory nd Applictions, Chpmn & Hll/CRC, Boc Rton, 2004. 17 A. Bielecki, Une remrque sur l methode de Bnch- Ccciopoli-Tikhonov dns l theorie des equtions differentielles ordinires, Bull. Acd. Polon. Sci. Cl. III - Vol. IV 1956) 261-264. 18 D. Blenu, O.G. Mustf, On the globl existence of solutions to clss of frctionl differentil equtions, Comp. Mth. Appl. 59 2010) 1835-1841. 19 Z.F.A. El-Rheem, Modifiction of the ppliction of contrction mpping method on clss of frctionl differentil eqution, Appl. Mth. & Comput. 137 2003) 371-374. 20 V. Lkshmiknthm, J. Vsundhr Devi, Theory of frctionl differentil equtions in Bnch spce, Europen J. Pure nd Appl. Mth. 1 2008) 38-45.