Generalized Gronwall fractional summation inequalities and their applications

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1 Xu ad Zhag Joural of Iequalities ad Applicatios (2015) 2015:242 DOI /s R E S E A R C H Ope Access Geeralized Growall fractioal summatio iequalities ad their applicatios Ru Xu 1* ad Yig Zhag 2 * Correspodece: xuru2005@163com 1 Departmet of Mathematics, Qufu Normal Uiversity, Qufu, Shadog , People s Republic of Chia Full list of author iformatio is available at the ed of the article Abstract Based o a iteratio method, some geeralized discrete fractioal Growall iequalities are developed, which ca be used i the qualitative aalysis of the solutios to fractioal differece equatios ad summatio equatios MSC: 39A12; 26A33 Keywords: Growall s iequality; fractioal differece; fractioal sum; fractioal differece equatio; fractioal sum equatio 1 Itroductio I recet years, the fractioal differetial ad fractioal itegrals have bee adopted i various fields of sciece ad egieerig, which ca be used to describe certai pheomea, reflect some physiochemical properties, ad provide accurate models for the systems uder cosideratio Some applicatios of fractioal calculus iclude fluid flow, rheology, dyamical processes i self-similar ad porous structures, electrical etworks, probability ad statistics, cotrol theory of dyamical systems, chemical physics, optics, ad sigal processig, ecoomics, ad so o Therefore, they are receivig extesive attetio from a variety of domais Refereces [1, 2] itroduced the defiitios of fractioal calculus, theorems ad basic aalytic solutios of the fractioal equatio i detail At the same time, the fractioal differece equatios, fractioal sum equatios, ad fractioal iequalities also play importat roles i may areas I 1989, Miller ad Ross [3] defied a fractioal sum of order α > 0 via the solutio of a liear differece equatio ad proved some basic properties of this operator After that, may authors followed up i various directios [4 17] I 2000, Hirota [18] defied the fractioal order differece operator α where α was a real umber, usig Taylor s series I 2003, Nagai [19] adopted aother defiitio for the fractioal differece by modifyig Hirota s defiitio The defiitio i [19]cotaied the operator ad the term ( 1) j iside the summatio idex Therefore this defiitio was difficult use i studyig the properties of solutios of fractioal differece equatios To avoid this, Deekshitulu ad Moha [20] modified the defiitio ad rearraged the terms i the defiitio of Nagai [19] The they defied the fractioal sum operator α ad the fractioal differece operator α Recetly, they discussed some basic iequalities, 2015 Xu ad Zhag This article is distributed uder the terms of the Creative Commos Attributio 40 Iteratioal Licese ( which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial author(s) ad the source, provide a lik to the Creative Commos licese, ad idicate if chages were made

2 Xu ad ZhagJoural of Iequalities ad Applicatios (2015) 2015:242 Page 2 of 10 compariso theorems, ad qualitative properties of the solutios of fractioal differece equatios i [21 24] Uder the defiitio i [20], they also cosidered a iitial value problem of fractioal order ad obtaied some useful fractioal differece iequalities of Growall-Bellma type [25] I 2009, Atici ad Eloe [26] itroduced the defiitio of the fractioal sum ad the fractioal differece: α, α, α, α I 2012, based o the defiitio, Atici ad Eloe [27] ad Ferreira[28] studied some discrete fractioal Growall s iequalities as regards α, α separately They all adopted the same method: After establishig a compariso theorem, they gave a explicit solutio to the liear discrete fractioal sum equatio of the iitial value problem, which allowed them to state ad prove a aalog of Growall s iequality o discrete fractioal calculus I 2011, Cheg [29] preseted aother form of defiitio He provided some basic properties of fractioal differece ad summatio, ad established the theory symmetrical to the fractioal differetial equatio I this paper, we will itroduce some iequalities ad their applicatios based o the defiitio i [29] todemostratethequalitativepropertiesofsolutiostosomefractioal summatio equatios The proof is based o the iterative method Defiitio 11 [29] Let ν be oegative real umber, defie the ν-order summatio of x()as ν ν x()= x()= ν x(r), r where [ ν] = ν(ν+1) (ν+ 1), is the covolutio operator! r=0 Defiitio 12 [29] Letμ be positive real umber ad m be the miimum positive iteger which is greater tha μ (m 1 μ m) Defie the μ-order differece of x()as μ x()= m (m μ) x(), where m is the mth-order backward differece operator Defiitio 13 [29] Defie the discrete Mittag-Leffler fuctio F α,β (λ, )as F α,β (λ, )= λ k αk + β ( λ ) <1, F α (λ, )=F α,1 (λ, )= λ αk +1 ( λ ) <1 I [11], Cheg gave the discrete fractioal summatio Growall iequality Theorem 11 [29] Suppose that β >0Let u, a, ad g be oegative fuctios, where g is also mootoe ad odecreasig ad satisfies 0 g M (0 M <1),0 N If u a + g β u,

3 Xu ad ZhagJoural of Iequalities ad Applicatios (2015) 2015:242 Page 3 of 10 the u a + (g ) k kβ a I [30], Zhag et al have exteded the result ad give the followig coclusio Theorem 12 [30] Let β >0,p r >0,u, a, g be oegative fuctios, where g is also mootoe ad odecreasig ad satisfies 0 g M (0 M <1),0 N If (u ) p a + g β (u ) r, the where { } 1 p u a + D + (Cg ) k kβ D, { r D = g β p k r p p C = r p k r p p, k >1 a + p r } p k p r, I order to prove our results, we eed the followig basic iformatio Lemma 11 [29] Let μ >0,v >0,the μ v x()= (u+v) x() Lemma 12 [29] β [ k = k+β ], k,, β >0 2 Mai results Theorem 21 Suppose that β >0,u, a, g, h are oegative fuctios, g ad h are also mootoe ad odecreasig, u c, g M 1, h M 2 (M 1 M 2 <1,0 N) If u a + g β h u, (21) the u a + g k hk kβ a (22) Proof Let Bu = g β h u WefidthatB is liear ad odecreasig uder the coditios that u, g, h are oegative fuctios ad g, h are also odecreasig Therefore, (21) tursito u a + Bu

4 Xu ad ZhagJoural of Iequalities ad Applicatios (2015) 2015:242 Page 4 of 10 The we have u a + B(a + Bu )=a + Ba + B 2 u m 1 a + B k a + B m u (23) ad Bu = g β 1+β h u cm 1 M 2, B 2 u = g β ( h g β ) h u c(m1 M 2 ) 2 1+2β, B m u = g m hm mβ u c(m 1 M 2 ) m 1+mβ (24) Noticig that M 1 M 2 <1,the (M 1 M 2 ) m 1+mβ = F β,1 (M 1 M 2, ) 1 m=1 Hece we have lim (M 1M 2 ) m m From (24)weget [ 1+mβ ] =0 lim m Bm u 0 Due to the fact that B m u 0, we have lim m Bm u =0 Takig the limit as m o both sides of (23)weget u a + B k a a + g k hk kβ a This completes the proof of Theorem 21 Remark 21 If h 1, (22) becomes Theorem 22 of Chapter 3 i [29]: u a + g k kβ a

5 Xu ad ZhagJoural of Iequalities ad Applicatios (2015) 2015:242 Page 5 of 10 Corollary 21 With the coditios of Theorem 21, let g = a,0<am 2 <1If u a + a β h u, the u a + a k h k kβ a Corollary 22 Suppose that the coditios of Corollary 21 hold, a is mootoe icreasig ad 0<aM 2 <1If u a + a β h u, the u a F β,1 (am 2, ) Proof Uder the coditios of Corollary 21 ad a is mootoe icreasig, ad we get u a + a k h k kβ a a + a a k h k 1 kβ a (am 2 ) k kβ +1 = a F β,1 (am 2, ) (0<aM 2 <1) Corollary 23 Suppose that the coditios of Theorem 21 hold, if a is mootoe icreasig, the from u a + g β h u, we get u a (M 1 M 2 ) k kβ 1 = a F β,1 (M 1 M 2, ) Theorem 22 Suppose that β >0,u, ad a are oegative fuctios, ϕ(t) is mootoe ad odecreasig, ϕ(t + s) ϕ(t)+ϕ(s), ϕ(t) Lt,0<L <1If u a + β ϕ(u ), (25) the u kβ ϕ k (a ) (26)

6 Xu ad ZhagJoural of Iequalities ad Applicatios (2015) 2015:242 Page 6 of 10 Proof Let Bu = β ϕ(u ) Uder the coditios that ϕ(a + b) ϕ(a)+ϕ(b) adϕ(t) is mootoe ad odecreasig, we have ϕ ( β ϕ(u ) ) ( ν = ϕ ϕ ( u(r) )) r r=0 ϕ 2( β ϕ(u ) ) β ϕ 3 (u ), ν ϕ 2( u(r) ) = β ϕ 2 (u ), r r=0 the for all N + 0, u a + Bu = a + β ϕ(u ) a + β ϕ(a + Bu )=a + β ϕ ( a + β ϕ(u ) ) a + β ϕ(a )+ β ϕ ( β ϕ(u ) ) a + β ϕ(a )+ 2β ϕ 2 (u ) a + β ϕ(a )+ 2β ϕ 2( a + β ϕ(u ) ) a + β ϕ(a )+ 2β ϕ 2 (a )+ 3β ϕ 3 (u ) a + β ϕ(a )+ 2β ϕ 2 (a )+ 3β ϕ 3 (a )+ + (m 1)β ϕ m 1 (a )+ mβ ϕ m (u ), that is, m 1 u kβ ϕ k (a )+ mβ ϕ m (u ) (27) Noticig that ϕ(u ) Lu, L <1,adu is bouded for 0 N, supposethat 0<u C,theweget β ϕ(u ) L β u L β C = CL 2β ϕ 2 (u ) CL 2 1+2β, 3β ϕ 3 (u ) CL 3 1+3β, [ 1+β ], (28) mβ ϕ m (u ) CL m 1+mβ

7 Xu ad ZhagJoural of Iequalities ad Applicatios (2015) 2015:242 Page 7 of 10 From the fact that the series L m 1+mβ m=0 coverges to F β,1 (L, ), L <1,wehave lim m Lm [ 1+mβ ] =0 (29) Due to the fact that mβ ϕ m (u ) 0, from (28)ad(29)wehave lim m mβ ϕ m (u )=0, takig the limit as m o both sides of (27), we have u kβ ϕ k (a ) This completes the proof of Theorem 22 By choosig some particular fuctios ϕ(t), we ca get the correspodig results For example, suppose that a, u, β are the same as i Theorem 22Letϕ(t)= 1 2 si t (0 t π 2 ), 0 u π 2,theϕ(t)satisfiesϕ(t + s) ϕ(t)+ϕ(s), ϕ(t) 1 2 tif the u a β si u, u ( ) 1 k kβ si k a 2 Furthermore, if a 1, the from si t t,wehave u ( ) 1 k kβ 1 2 ( 1 2 ) k [ 1+kβ ] 3 Applicatios Cosider the fractioal differece equatio α y()=f (, y() ), α 1 y() = 1 = η, (31) where 0 < α <1,0 N < From Propositio 31 i Chapter 3 of [29], we kow that the problem (31)isequivalet to the summatio equatio α y()=η + α f (, y() ) (32)

8 Xu ad ZhagJoural of Iequalities ad Applicatios (2015) 2015:242 Page 8 of 10 Suppose that z is the solutio of the iitial value problem α z()=f (, z() ), α 1 z() = 1 = η, (33) 0<α <1,0 N <,the(33) is equivalet to the summatio equatio α z()= η + α f (, z() ) (34) Theorem 31 Suppose that f (, t) satisfies the coditio f (, z) f (, y) h z y, (35) where h is a mootoe odecreasig positive fuctio, h <1The the solutios of (31) rely o the iitial value cotiuously Proof From (32), (34), ad (35)wehave z() y() α η η + α ( ) ( ) f, z() f, y() α η η + α h z() y() From Corollary 21,wehave z() y() α η η + h k α kα η η = η η h k kα + α = η η F α,α (h, ), h <1 The the solutios of (31) rely o the iitial value cotiuously Remark 31 The coditio (35) geeralizes the Lipschitz coditio, the Theorem 31is a promotio of Theorem 32 i Chapter 3 of [29] Theorem 32 Suppose that f (, t) satisfies the coditio f (, z) f (, y) ϕ ( z y ), (36) where ϕ(t) is the same as i Theorem 22 The the solutios of (31) rely o the iitial value cotiuously

9 Xu ad ZhagJoural of Iequalities ad Applicatios (2015) 2015:242 Page 9 of 10 Proof From (32), (34), ad (36)wehave z() y() α η η + α ( ) ( ) f, z() f, y() α η η + α ϕ ( ) z() y() By Theorem 22 we get ) z() y() kα ϕ ( η k α η η η L k kα + α = η η F α,α (L, ) The the solutios of (31) rely o the iitial value cotiuously Competig iterests The authors declare that they have o competig iterests Authors cotributios RX studied the geeralized discrete fractioal Growall iequalities ad completed the correspodig proof YZ obtaied the results of Sectio 3: Applicatios All authors read ad approved the fial mauscript Author details 1 Departmet of Mathematics, Qufu Normal Uiversity, Qufu, Shadog , People s Republic of Chia 2 Basic Departmet of Shadog Water Polytechic, Rizhao, Shadog , People s Republic of Chia Ackowledgemets The authors are very grateful to the aoymous referees for their valuable suggestios ad commets, which helped to improve the quality of the paper The author RX is supported by the Natioal Sciece Foudatio of Chia ( ad ) Received: 6 May 2015 Accepted: 16 July 2015 Refereces 1 Miller, KS, Ross, B: A Itroductio to the Fractioal Calculus ad Fractioal Differetial Equatios Wiley, New York (1993) 2 Samko, SG, Kilbas, AA, Maitchev, OI: Fractioal Itegrals ad Derivatives Theory ad Applicatios Nauka i Techika, Misk (1987) 3 Miller, KS, Ross, B: Fractioal differece calculus I: Uivalet Fuctios, Fractioal Calculus, ad Their Applicatios, pp Ellis Horwood, Chichester (1989) 4 Atici, FM, Segul, SS: Modelig with discrete fractioal equatios J Math Aal Appl 369, 1-9 (2010) 5 Abdeljawad, T, Baleau, D: Caputo q-fractioal iitial value problems ad a q-aalogue Mittag-Leffler fuctio Commu Noliear Sci Numer Simul 16(12), (2011)doi:101016/jcss Abdeljawad, T, Baleau, D: Fractioal differeces ad itegratio by parts J Comput Aal Appl 13(3), (2011) 7 Atici, FM, Eloe, PW: A trasform method i discrete fractioal calculus It J Differece Equ 2(2), (2007) 8 Atici, FM, Eloe, PW: Iitial value problems i discrete fractioal calculus Proc Am Math Soc 137(3), (2009) 9 Atici, FM, Segul, S: Modelig with fractioal differece equatios J Math Aal Appl 369(1),1-9 (2010) 10 Atici, FM, Eloe, PW: Two-poit boudary value problems for fiite fractioal differece equatios J Differ Equ Appl 17, (2011) 11 Bastos, NRO, Ferreira, RAC, Torres, DFM: Necessary optimality coditios for fractioal differece problems of the calculus of variatios Discrete Coti Dy Syst 29(2), (2011) 12 Cheg, J, Chu, Y: O the fractioal differece equatios of order (2, q) Abstr Appl Aal 2011, Article ID (2011) doi:101155/2011/ Cheg, J: The solutio of fractioal differece equatios of order (k, q) Acta Math Appl Si 34(2), (2011) 14 Cheg, J, Chu, Y: Fractioal differece equatios with real variable Abstr Appl Aal 2012, Article ID (2012) 15 Goodrich, CS: Solutios to a discrete right-focal fractioal boudary value problem It J Differece Equ 5, (2010) 16 Goodrich, CS: Cotiuity of solutios to discrete fractioal iitial value problems Comput Math Appl 59(11), (2010)

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