A new type of the Gronwall-Bellman inequality and its application to fractional stochastic differential equations

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1 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE A new ype of he Gronwall-Bellman nequaly and s applcaon o fraconal sochasc dfferenal equaons Receved: 4 Sepember 6 Acceped: 3 December 6 Frs Publshed: January 7 *Correspondng auhor: Qong Wu, Deparmen of Mahemacs, Tufs Unversy, 53 Boson Avenue, Medford, MA 55, USA E-mals: wuqongh@gmalcom, QongWu@ufsedu Revewng edor: Feng Q, Tanjn Polyechnc Unversy, Chna Addonal nformaon s avalable a he end of he arcle Qong Wu * Absrac: Ths paper presens a new ype of Gronwall-Bellman nequaly, whch arses from a class of negral equaons wh a mxure of nonsngular and sngular negrals The new dea s o use a bnomal funcon o combne he nown Gronwall-Bellman nequales for negral equaons havng nonsngular negrals wh hose havng sngular negrals Based on hs new ype of Gronwall-Bellman nequaly, we nvesgae he exsence and unqueness of he soluon o a fraconal sochasc dfferenal equaon SDE wh fraconal order <α< Fnally, he fraconal ype Foer-Planc-Kolmogorov equaon assocaed o he soluon of he fraconal SDE s derved usng Iô s formula Subjecs: Scence; Mahemacs & Sascs; Sascs & Probably; Probably; Probably Theory & Applcaons Keywor: Gronwall-Bellman nequaly; fraconal sochasc dfferenal equaons SDEs; exsence and unqueness; fraconal Foer-Planc equaon ABOUT THE AUTHOR Qong Wu receved BS and MS n Mahemacs from Harbn Insue of Technology, Chna He s a full-me PhD suden n Deparmen of Mahemacs, Tufs Unversy n USA Hs area of neres ncludes he heory of sochasc dfferenal equaons and her applcaons, mahemacal bology, conrol heory and convex opmzaon PUBLIC INTEREST STATEMENT Gronwall-Bellman nequaly plays a sgnfcan role n mahemacal modelng, parcularly n applcaons of negral equaons For a mahemacal model whch arses from a class of negral equaons wh a mxure of nonsngular and sngular negrals, here s lac of a powerful Gronwall-Bellman nequaly o help researchers on hs case To derve such a Gronwall-Bellman nequaly, he new dea s o use a bnomal funcon o combne he nown Gronwall-Bellman nequales for negral equaons havng nonsngular negrals wh hose havng sngular negrals Based on hs new ype of Gronwall-Bellman nequaly, we nvesgae he exsence and unqueness of he soluon o a fraconal sochasc dfferenal equaon SDE wh fraconal order <α< Ths resul generalzes he nown exsence and unqueness heorem relaed o fraconal order <α< 7 The Auhors Ths open access arcle s rbued under a Creave Commons Arbuon CC-BY 4 lcense Page of 3

2 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ Inroducon I s well nown ha negral nequales are nsrumenal n sudyng he qualave analyss of soluons o dfferenal and negral equaons Ames & Pachpae, 997 Among hese nequales, he ngushed Gronwall-Bellman ype nequaly from Bellman and Cooe 963, and s assocaed exensons Agarwal & Cho, 6; Agarwal, Deng, & Zhang, 5; Agarwal, Tarboon, & Nouyas, 6; Lpovan, ; Lu, Zhang, Agarwal, & Wang, 6; Mao, 989; Pachpae, 975; Wang, Agarwal, & Chand, 4, are capable of affordng explc boun on soluons of a class of lnear dfferenal equaons wh neger order The followng lemma concerns a sandard Gronwall- Bellman nequaly n Corduneanu 8 for a dfferenal equaon wh order one or equvalenly an negral equaon wh nonsngular negrals Lemma Suppose h,, and x are connuous funcons on < T, < T, wh If x sasfes x h+ sxs, hen Moreover, f h s nondecreasng, hen In order o nvesgae he qualave properes of soluons o dfferenal equaons of fraconal order, here are several generalzaons of Gronwall-Bellman nequales developed by many researchers Aıcı & Eloe, ; Lazarevć & Spasć, 9; Ye, Gao, & Dng, 7; Zheng, 3 Le us recall he followng generalzed Gronwall-Bellman nequaly proposed n Ye e al 7 for a fraconal dfferenal equaon wh order β > or equvalenly an negral equaon wh sngular negrals Lemma Suppose β>, a s a nonnegave funcon whch s locally negrable on < T, < T, and g s a nonnegave, nondecreasng connuous funcon defned on < T wh g M consan If u s nonnegave and locally negrable on < T wh on hs nerval, hen x h+ hss exp u du s x h exp s u a +g u a+ s β us [ ] gγβ n s {nβ } as, Γnβ n= where Γ s he gamma funcon Furhermore, f a s nondecreasng on < T, hen u ae β gγβ β, where E β z s he Mag-Leffler funcon defned by E β z = = z for z > Γβ+ From many real applcaons, such as n physcs, heorecal bology, and mahemacal fnance, here s subsanal neres n a class of fraconal SDEs Jumare, 5a; Mandelbro & Van Ness, 968; Pedjeu & Ladde, The fraconal SDEs ae he form Page of 3

3 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ dx =b, x d + σ, x d α + σ, x db, where he nal value s x =x, <α<, and B s he sandard Brownan moon Accordng o Jumare 5a and Pedjeu and Ladde, he negral equaon correspondng o Equaon s x =x + bs, xs + α s α σ s, xs + Snce <α<, here are nonsngular and sngular negrals n he negral equaon Equaon However, he above-menoned ypes of Gronwall-Bellman nequales, such as Lemmas and, are no applcable o sudyng he qualave properes of he soluon o Equaons or The frs goal of hs paper, presened n Secon, s o derve a new ype of Gronwall-Bellman nequaly whch s applcable o sudy he qualave behavors of he soluon o he fraconal SDE Equaon or he sochasc negral equaon Equaon The second goal, accomplshed n Secon 3, s o apply he resuls from Secon o nvesgae exsence and unqueness of he soluon o he fraconal SDE Equaon of order <α< Fnally, n Secon 4, a fraconal ype Foer- Planc-Kolmogorov equaon assocaed o he soluon of he fraconal SDE Equaon s derved Generalzaon of he Gronwall-Bellman nequaly In hs secon, we develop a new negral nequaly, Equaon 4 below, by verfyng hree clams The frs clam s esablshed by usng he mehod of nducon and ang advanage of he bnomal funcon; he second clam s verfed by ang advanage of properes of he Gamma funcon; he hrd clam s verfed by employng Gamma funcons, Mag-Leffler funcons, and exponenal funcons The esablshed negral nequaly s applcable o he fraconal SDE Equaon or he sochasc negral equaon Equaon Also hs new negral nequaly can be consdered as a generalzaon of he negral nequales n Lemmas and σ s, xs db s Theorem Le <α< and consder he me nerval I =[, T, where T Suppose a s a nonnegave funcon, whch s locally negrable on I and b and g are nonnegave, nondecreasng connuous funcon defned on I, wh boh bounded by a posve consan, M If u s nonnegave, and locally negrable on I and sasfes u a +b hen u a+ Proof us + g n n= = n s α us, b n g [Γα] s {α + n} as Γα + n Le φ be a locally negrable funcon and defne an operaor B on φ as follows 3 4 Bφ: = b From he nequaly, Equaon 3, u a+bu φs + g s α φs, Page 3 of 3

4 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ Ths mples n u B a+b n u = 5 In order o ge he desred nequaly, Equaon 4, from Equaon 5, here are hree clams o be verfed The frs clam provdes a general bound on B n : n n B n u b n g [Γα] s {α + n} us 6 = Γα + n The mehod of nducon wll be used o verfy he nequaly n Equaon 6 Frs le n = Then he nequaly n Equaon 6 s rue Now, suppose ha he nequaly, Equaon 6, hol for n =, and hen compue B n when n = +, B + u =BB u b Le b sg [Γα] s Γα + = + g s α b sg [Γα] s Γα + = s s τ {α + } uτ dτ s s τ {α + } uτ dτ C: = b and = b sg [Γα] s Γα + s s τ {α + } uτ dτ, G: = g s α b sg [Γα] s Γα + Then, compue C and G erm by erm o reach he desred nequaly Equaon 6 Snce b and g are nonnegave and nondecreasng funcons, C = = = Smlarly, compue G = b + g b + g = b + g = = b + [Γα] Γα + [Γα] Γα + s [Γα] Γα + + τ uτ dτ Γ + + b b g = [Γα] Γα + + s s τ {α + } uτ dτ s τ {α + } uτ dτ s τ {α + } uτ dτ τ τ {α } uτ dτ τ {α } uτ dτ 7 Page 4 of 3

5 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ G = b g + = b g + = = b g + = = g + [Γα] + Γ + α + b b g = [Γα] Γα + [Γα] Γα + [Γα] + Γ + α + s α τ {+α } dτ s s τ {α + } uτ dτ s α s τ {α + } uτ dτ τ [Γα] Γα + + τ {+α + } uτ dτ τ {α } uτ dτ 8 Combnng Equaons 7 and 8 yeld B + u =C+G b + τ uτdτ Γ + [ ] + b + b g [Γα] τ α uτ dτ = Γα g + [Γα] + τ +α dτ Γ + α + + = b + g [Γα] τ α uτ dτ = Γα + + Ths mples ha for any n N +, he frs clam, Equaon 6, hol The second clam shows ha B n u vanshes as n ncreases For each n [, T, B n u, as n 9 For he purpose of noaon smplfcaon durng he proof of he second clam, defne n n H n : = b n g [Γα] s {α + n} us = Γα + n Noe ha Γx s posve and decreasng on, ] bu posve and ncreasng on [, Le x = α + n Then, he sequence x s decreasng over [, n] snce x + x = α < when s an neger and [, n] Ths means x mn = nα and x max = n Furhermore, for a fxed α, here exss a large enough n such ha for any n > n, here s n So he sequence sasfes x α for any neger [, n] f n s large enough Thus, for any [, n], Γx mn < Γx, and H n Γnα n n b n g [Γα] = Also for α,, Γα > Therefore, s {α + n} us, n > n Page 5 of 3

6 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ H n [Γα]n Γnα n n = b n g s {α + n} us Le y = α + n Smlar o he sequence x, here s y mn = nα for a large enough n and y max = n Snce [, T, spl he nerval [, T no wo subnervals [, ] and [, T For [, ], s y y mn = nα whle f [, T, s y ymax = n Thus, H n [Γα]n max{ nα, n } Γnα = [Γα]n max{ nα, n } Γnα n n b n g = b+g n us Noce ha b and g are boh bounded by a posve consan M, e b M and g M, and us s locally negrable over < T Ths means ha from Equaon, H n as n because he Gamma funcon, Γnα, s growng faser han a power funcon Therefore, he second clam, Equaon 9, s verfed snce B n u H n for any n N + The hrd clam esablshes ha he rgh-hand sde RHS of Equaon 4 exss on < T In order o show hs saemen, we frs prove ha for < T, he followng nfne sum of sequences denoed by L;τ s convergen where α + n α + s a produc and aes one f α + n < α + Le = n, hen compue us n n L;τ: = b n g [Γα] n= = Γα + n + τα+n n = g [Γα] τ α b n = n= Γα + n + τn = g [Γα] τ α n b n Γα + α + n α + τn, = n= n α + n α + = +!!! α + α + = + +! α + α + = + + α! α + + α α! Subsung = n and Equaon no Equaon gves g [Γα] τ α L;τ Γα + = = b τ = E α! α gγατ α exp α bτ, whch s fne for < T Furhermore, snce b M and g M, defne LM;τ: = = M [Γα] τ α Γα + = M τ = E α! α MΓατ α exp α Mτ, whch means LM;τ s fne and L;τ LM;τ Then, compue he RHS of Equaon 4 Page 6 of 3

7 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ n n RHS = a+ b n g [Γα] n= = Γα + n + n n a+ M n M [Γα] Γα + n + = a+ n= = dlm; s as d Snce he Mag-Leffler funcon E α α s an enre funcon n α, see Gorenflo, Loucho, Lucho, and Manard, he exponenal funcon exp s unformly connuous n, and boh α and a are locally negrable over < T, he negral dlm; s as s fne Ths mples ha he RHS of d Equaon 4 s fne So he las clam s also verfed, hereby compleng he proof Corollary Suppose he condons n Theorem are sasfed and furhermore, a s nondecreasng on < T Then d d s{α+n } as d d s{α+n } as u ae α gγα α exp α b Proof From he proof of Theorem, u a+ n n b n g [Γα] Γα + n n= = Snce a s nondecreasng, s {α + n} as n n u a b n g [Γα] s {α + n} n= = Γα + n n n a b n g [Γα] n= = Γα + n + α+n ae α gγα α exp α b Ths complees he proof Remar From Theorem and Corollary, we see ha f α =, Theorem and Corollary are he same as Lemma ; whle f b, Theorem and Corollary become Lemma 3 Exsence and unqueness of he soluon o fraconal SDEs In hs secon, usng he man resuls from Secon, we nvesgae he exsence and unqueness of he soluon o he fraconal SDE Equaon wh fraconal order <α< By applcaon of he classcal Pcard-Lndelöf successve approxmaon scheme and he sandard Gronwall-Bellman nequaly, exsence and unqueness of he soluon o Equaon wh fraconal order <α< s cussed n Pedjeu and Ladde However, he case wh <α remans o be nvesgaed We can apply he generalzed Gronwall-Bellman nequaly developed n Secon o derve exsence and unqueness of he soluon o Equaon when <α< Theorem 3 Le <α<, T >, and B be a m-dmensonal Brownan moon on a complee probably space Ω Ω,, P Assume ha b,, σ, :[, T] R n R n, σ, :[, T] R n R n m are measurable funcons sasfyng he lnear growh condon, Page 7 of 3

8 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ b, x + σ, x + σ, x K + x, 3 for some consan K > and he Lpschz condon, b, x b, y + σ, x σ, y + σ, x σ, y L x y, 4 for some consan L > Le x be a random varable, whch s ndependen of he σ-algebra generaed by {B, } and sasfes E x < Then, he fraconal sochasc dfferenal equaon Equaon has a unque -connuous soluon x, ω wh he propery ha x, ω s adaped o he flraon x generaed by x and {B, }, and T E x d < Proof Exsence From Equaon, he correspondng equvalen sochasc negral equaon of he fraconal sochasc dfferenal equaon Equaon s rewren as x =x + bs, xs + α s α σ s, xs + where < T and <α< For more deals abou hs equvalence beween Equaon and Equaon, we refer o Jumare 5a, 5b, 6 By he mehod of Pcard-Lndelöf successve approxmaons, defne x =x and x =x, ω nducvely as follows σ s, xs db s, x + =x + bs, x s + α s α σ s, x s + Applyng he nequaly x + y + z 3 x + 3 y + 3 z lea o σ s, x s db s 5 E x + x 3E + 3E α + 3E bs, x s bs, x s : = I + I + I 3 Usng he Cauchy Schwarz nequaly on he frs wo erms, I and I, plus Iô s Isomery, see n Osendal 3, n he hrd erm, I 3, produces s σ α s, x s σ s, x s σ s, x s σ s, x s E x + x 3TE + 3α + 3E bs, x s bs, x s : = J + J + J 3 s α E s α σ s, x s σ s, x s σ s, x s σ s, x s Page 8 of 3

9 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ Fnally, usng he Lpschz condon Equaon 4 on all erms, J, J, J 3, evaluang he frs negral n he second erm, J, and combnng he frs and hrd erms, J and J 3, yel E x + x 3L + T + 3L + T E x s x s s α E x s x s Thus, for locally negrable funcon φ, defne an operaor B as follows 6 Bφ: = 3L + T φs + Then, erang Equaon 6 yel s α φs E x + x BE x x B E x x Snce <α< and E x x s nonnegave and locally negrable, from he frs clam, Equaon 6, and he Equaon n he proof of he second clam n Secon, we now ha E x + x B E x x [Γα] max{ α, } Γα Smlarly, apply he Cauchy Schwarz nequaly, he Iô s Isomery, and he lnear growh condon, Equaon 3, nsead of Lpschz condon, Equaon 4, o compue E x x 3 + TK + E x + α [6L + T] E x s x s Ths mples sup T E x + x [Γα] max{t α, T } M [6L + T], Γα where M = 3 + TK + E x T + Tα+ α+ x m x n m x + x = m L P L P =n M where M = 3 + TK + E x T 3 + Tα+ α+ m =n s ndependen of and Thus, for any m > n >, =n T E x + x d [Γα] max{t α, T } [6L + T] Γα, as m, n, s ndependen of and Ths means he successve approxmaons x are mean-square convergen unformly on [, T] I remans now o show ha he sequence of successve approxmaons x s almos surely convergen Frs, apply Chebyshev s nequaly o yeld = { P sup x + x > } T = 4 E sup x + x T 4 E sup x + x T = = 7 Page 9 of 3

10 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ By compuaons smlar o hose leadng o Equaon 7 and Doob s Maxmal Inequaly for marngales, = { P sup x + x > } M T whch s fne Then, applyng he Borel-Canell lemma yel, { } P sup x + x > T for nfnely many = = [Γα] max{t α, T } [6L + T] 4, Γα So here exss a random varable x, whch s he lm of he followng sequence x =x + x n+ x n x as, n= unformly on [, T] Also x s -connuous snce x s -connuous for all Therefore, ang he lm on boh sdes of Equaon 5 as, here s a sochasc process x sasfyng Equaon Unqueness The unqueness s due o he Iô Isomery and he Lpschz condon, Equaon 4 Le x =x, ω and x =x, ω be soluons of Equaon, whch have he nal values, x =y and x =y, respecvely Smlarly, apply he Cauchy Schwarz nequaly, he Iô Isomery, and he Lpschz condon Equaon 4 o compue E x x 4E y y + 4L + T + 4αL T α E x s x s s α E x s x s By applcaon of he generalzed Gronwall-Bellman nequaly n Corollary, we have E x x 4E y y E α 4αL T α Γα α exp α 4L + T Snce x and x boh sasfy he sochasc negral equaon Equaon, he nal values y and y are boh equal o x Ths means E x x = for all > Furhermore, P { x x =, for all T } = Therefore, he unqueness of he soluon o Equaon s proved 4 Fraconal Foer-Planc-Kolmogorov equaon Based on he exsence and unqueness Theorem 3 developed n Secon 3, we derve he fraconal Foer-Planc-Kolmogorov equaon assocaed o he unque soluon of he fraconal SDE, Equaon Before dervng he fraconal Foer-Planc-Kolmogorov equaon, we frs nroduce an Iô formula from Pedjeu and Ladde o he followng Iô process x =x + bs, xs + σ s, xs α + σ s, xs db s, where <α<, B s he m-dmensonal sandard Brownan moon, and funcons b, σ, σ sasfy he condons n Theorem 3 Lemma 4 Le X sasfy he Equaon 8 and furhermore, le V C[R + R n, R m ], and assume ha V, V x, V xx exs and connuous for, x R + R n, where V x, x s an m n Jacoban marx of V, x and V xx, x s an m n Hessan marx Then, 8 Page of 3

11 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ dv, X = L V, X d + L V, X d α + L 3 V, X db, where L V, x =V, x+v x, xb, x+ σ, xt V xx, xσ, x and L V, x =V x, xσ, x, L 3 V, x =V x, xσ, x By applyng he exsence and unqueness Theorem 3 and Iô s formula, Lemma 4, he followng fraconal Foer-Planc-Kolmogorov equaon s esablshed Theorem 4 Le B be he m-dmensonal sandard Brownan moon Suppose ha X s he soluon o he fraconal SDE Equaon whose coeffcen funcons b, σ and σ sasfy he condons n Theorem 3 Then he ranson probables P X, x =P X, x, x of X sasfy he followng fraconal ype dfferenal equaon dp X, x =A x PX, x d + B x PX, x d α wh nal condon P X, x =δ x x, he Drac dela funcon wh mass on x, and A, x B x are spaal operaors defned, respecvely, by A x hx = n and B x = n = = [b x, xhx] + j= x [δ hx], j= [ m x x j = = δ δj = ], xhx where b =b,, b n T, δ =δ,, δn T, and δ s an n m marx wh elemens [δ ] j = δ j Proof Le f C c Rn, e f s an nfnely dfferenal funcon on R n wh compac suppor Snce X s he soluon of he sochasc fraconal dfferenal equaon Equaon, hs means X sasfes he sochasc negral equaon Equaon 8 So apply Iô formula Lemma 4 on fx o yeld 9 f X f x = Noce he fac ha + f x Xsbs, Xs + σt s, Xsf xx Xsσ s, Xs f x Xsσ s, Xs α + f x Xsσ s, Xs db s f x Xsσ s, Xs α = α s α f x Xsσ s, Xs, and more deals on hs equaly can be found n Jumare 5a and Pedjeu and Ladde Thus Equaon can be wren as Page of 3

12 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ f X f x = + α f x Xsbs, Xs + σt s, Xsf xx Xsσ s, Xs s α f x Xsσ s, Xs + f x Xsσ s, Xs db s Snce he negral f Xsσ s, Xs db s a marngale wh respec o he flraon x s, ae condonal expecaons on boh sdes of Equaon o oban E[f X X =x ] fx =E f x Xsbs, Xs X =x + E + E α σ T s, Xsf Xsσ s, Xs X =x xx s α f x Xsσ s, Xs X =x By Fubn s Theorem and negraon by pars, he above Equaon can be rewren as f xp X, x dx f x = R n f x xbs, xp X s, x dx R n + Snce f C c Rn s arbrary and C c Rn s dense n L R n, T σ R n s, xf xx xσ s, xpx s, x dx + α s α f x xσ s, xp X s, x dx R n = R n f x + R n f xα A PX s, x dx s α A PX s, x dx P X, x δ x x = A PX s, x + α s α A PX s, x, 3 where δ x x s a generalzed funcon ang value δ x x =P X, x Fnally, ae he dervave wh respec o me on boh sdes of Equaon 3 o yeld he desred resul Equaon 9 5 Concluson In hs paper, a new ype of Gronwall-Bellman nequaly s esablshed for a class of negral equaons wh a mxure of nonsngular and sngular negrals Ths new ype of Gronwall-Bellman nequaly can be consdered as a generalzaon of nown Gronwall-Bellman nequales dealng wh an negral equaon havng nonsngular or sngular negrals, separaely Wh hs new ype of Gronwall- Bellman nequaly, exsence and unqueness of he soluon o a fraconal SDE wh fraconal oder <α< s nvesgaed Furhermore, based on he exsence and unqueness resul, a fraconal ype Foer-Planc-Kolmogorov equaon assocaed o he soluon of a fraconal SDE s derved Page of 3

13 Wu, Cogen Mahemacs 7, 4: 7978 hp://dxdoorg/8/ Acnowledgemens The auhor wshes o han Dr Marjore Hahn for her advce, fruful cusson, encouragemen, and paence wh my research, and Dr Xaozhe Hu for hs helpful cusson Fundng The auhor receved no drec fundng for hs research Auhor deals Qong Wu E-mals: wuqongh@gmalcom, QongWu@ufsedu Deparmen of Mahemacs, Tufs Unversy, 53 Boson Avenue, Medford, MA 55, USA Caon nformaon Ce hs arcle as: A new ype of he Gronwall- Bellman nequaly and s applcaon o fraconal sochasc dfferenal equaons, Qong Wu, Cogen Mahemacs 7, 4: 7978 References Agarwal, P, & Cho, J 6 Ceran fraconal negral nequales assocaed wh Pahway fraconal negral operaors Bullen of he Korean Mahemacal Socey, 53, 8 93 Agarwal, P, Tarboon, J, & Nouyas, S K 6 Some generalzed Remann-Louvlle -fraconal negral nequales Journal of Inequales and Applcaons, 6, Agarwal, R P, Deng, S, & Zhang, W 5 Generalzaon of a rearded Gronwall-le nequaly and s applcaons Appled Mahemacs and Compuaon, 65, Ames, W F, & Pachpae, B 997 Inequales for dfferenal and negral equaons Vol 97 Academc Press Aıcı, F M, & Eloe, P W Gronwall s nequaly on cree fraconal calculus Compuers & Mahemacs wh Applcaons, 64, Bellman, R E, & Cooe, K L 963 Dfferenal-dfference equaons Sana Monca, CA: Rand Corporaon Corduneanu, C 8 Prncples of dfferenal and negral equaons Vol 95 Amercan Mahemacal Socey Gorenflo, R, Loucho, J, Lucho, Y, & Manard, D T F Compuaon of he Mag-Leffler funcon Eα, β z and s dervave Fraconal Calculus and Appled Analyss Jumare, G 5a On he soluon of he sochasc dfferenal equaon of exponenal growh drven by fraconal brownan moon Appled Mahemacs Leers, 8, Jumare, G 5b On he represenaon of fraconal brownan moon as an negral wh respec o d α Appled Mahemacs Leers, 8, Jumare, G 6 New sochasc fraconal models for malhusan growh, he Possonan brh process and opmal managemen of populaons Mahemacal and Compuer Modellng, 44, 3 54 Lazarevć, M P, & Spasć, A M 9 Fne-me sably analyss of fraconal order me-delay sysems: Gronwall s approach Mahemacal and Compuer Modellng, 49, Lpovan, O A rearded Gronwall-le nequaly and s applcaons Journal of Mahemacal Analyss and Applcaons, 5, Lu, X, Zhang, L, Agarwal, P, & Wang, G 6 On some new negral nequales of Gronwall-Bellman-Bhar ype wh delay for connuous funcons and her applcaons Indagaones Mahemacae, 7, Mandelbro, B B, & Van Ness, J W 968 Fraconal brownan moons, fraconal noses and applcaons SIAM revew,, Mao, X 989 Lebesgue-Seljes negral nequales and sochasc sables The Quarerly Journal of Mahemacs, 4, 3 3 Osendal, B 3 Sochasc dfferenal equaons: An nroducon wh applcaons Sprnger Scence & Busness Meda Pachpae, B 975 On some generalzaons of Bellman s lemma Journal of Mahemacal Analyss and Applcaons, 5, 4 5 Pedjeu, J C, & Ladde, G S Sochasc fraconal dfferenal equaons: Modelng, mehod and analyss Chaos, Solons & Fracals, 45, Wang, G, Agarwal, P, & Chand, M 4 Ceran Grüss ype nequales nvolvng he generalzed fraconal negral operaor Journal of Inequales and Applcaons, 4, 8 Ye, H, Gao, J, & Dng, Y 7 A generalzed Gronwall nequaly and s applcaon o a fraconal dfferenal equaon Journal of Mahemacal Analyss and Applcaons, 38, 75 8 Zheng, B 3 Some new Gronwall-Bellman-ype nequales based on he modfed Remann-Louvlle fraconal dervave Journal of Appled Mahemacs, 3 7 The Auhors Ths open access arcle s rbued under a Creave Commons Arbuon CC-BY 4 lcense You are free o: Share copy and rerbue he maeral n any medum or forma Adap remx, ransform, and buld upon he maeral for any purpose, even commercally The lcensor canno revoe hese freedoms as long as you follow he lcense erms Under he followng erms: Arbuon You mus gve approprae cred, provde a ln o he lcense, and ndcae f changes were made You may do so n any reasonable manner, bu no n any way ha suggess he lcensor endorses you or your use No addonal resrcons You may no apply legal erms or echnologcal measures ha legally resrc ohers from dong anyhng he lcense perms Page 3 of 3