ULAM STABILITY AND DATA DEPENDENCE FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH CAPUTO DERIVATIVE

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1 Electronic Journl of Qulittive Theory of Differentil Equtions 2, No. 63, -; ULAM STABILITY AND DATA DEPENDENCE FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH CAPUTO DERIVATIVE JinRong Wng,, Linli Lv Yong Zhou 2. Deprtment of Mthemtics, Guizhou University, Guiyng, Guizhou 5525, P.R. Chin 2. Deprtment of Mthemtics, Xingtn University, Xingtn, Hunn 45, P.R. Chin ABSTRACT. In this pper, Ulm stbility nd dt dependence for frctionl differentil equtions with Cputo frctionl derivtive of order α re studied. We present four types of Ulm stbility results for the frctionl differentil eqution in the cse of < α < nd b = + by virtue of the Henry-Gronwll inequlity. Menwhile, we give n interesting dt dependence results for the frctionl differentil eqution in the cse of < α < 2 nd b < + by virtue of generlized Henry-Gronwll inequlity with mixed integrl term. Finlly, exmples re given to illustrte our theory results. Keywords. Frctionl differentil equtions; Cputo derivtive; Ulm stbility; Dt dependence; Gronwll inequlity.. Introduction Frctionl differentil equtions hve been proved to be strong tools in the modelling of mny physicl phenomen. It drws gret ppliction in nonliner oscilltions of erthqukes, mny physicl phenomen such s seepge flow in porous medi nd in fluid dynmic trffic model. There hs been significnt development in frctionl ordinry differentil equtions nd prtil differentil equtions. For more detils on frctionl clculus theory, one cn see the monogrphs of Kilbs et l. [7], Miller nd Ross [2], Podlubny [23], Trsov [26] nd the ppers of Agrwl et l. [, 2], Ahmd nd Nieto [3], Blchndrn et l. [5], Bi [6], Benchohr et l. [7], Henderson nd Ouhb [3], Li et l. [8, 9], Mophou nd N Guérékt [2], Wng et l. [27, 28, 29, 3, 3], Zhng [34] nd Zhou et l. [35, 36]. On the other hnd, numerous monogrphs hve discussed the dt dependence in the theory of ordinry differentil equtions (see for exmple [4, 9,, 4, 22, 24]). Menwhile, there re some specil dt dependence in the theory of functionl equtions such s Ulm-Hyers, Ulm-Hyers-Rssis nd Ulm- Hyers-Bourgin. The stbility properties of ll kinds of equtions hve ttrcted the ttention of mny mthemticins. Prticulrly, the Ulm-Hyers-Rssis stbility ws tken up by number of mthemticins nd the study of this re hs the grown to be one of the centrl subjects in the mthemticl nlysis re. For more informtion, we cn see the monogrphs Cdriu [8], Hyers [5] nd Jung [6]. The first uthor cknowledges the support by the Tinyun Specil Funds of the Ntionl Nturl Science Foundtion of Chin(262) nd Key Projects of Science nd Technology Reserch in the Ministry of Eduction(269); the third uthor cknowledges the support by Ntionl Nturl Science Foundtion of Chin(9773). Corresponding uthor. Emil ddresses: wjr9668@26.com (J. Wng); lvlinli28@26.com (L. Lv); yzhou@xtu.edu.cn (Y. Zhou). EJQTDE, 2 No. 63, p.

2 Although, there re some work on the locl stbility nd Mittg-Leffler stbility for frctionl differentil equtions (see [, 8, 9]), to the best of my knowledge, there re very rre works on the Ulm stbility for frctionl differentil equtions. Motivted by [, 25, 32], we will study the Ulm stbility for the following frctionl differentil eqution c D α x(t) = f(t, x(t)), t [, b), b = +, where c D α is the Cputo frctionl derivtive of order α (, ) nd the function f stisfies some conditions will be specified lter. Menwhile, we will study the dt dependence for the following frctionl differentil eqution c D α x(t) = f(t, x(t)), t [, b), b < +, where the Cputo frctionl derivtive of order α (,2). In the present pper, we introduce four types of Ulm stbility definitions for frctionl differentil equtions: Ulm-Hyers stbility, generlized Ulm-Hyers stbility, Ulm-Hyers-Rssis stbility nd generlized Ulm-Hyers-Rssis stbility. We present the four types of Ulm stbility results for frctionl differentil eqution in the cse < α < nd b = + by virtue of Henry-Gronwll inequlity. Menwhile, we give dt dependence results for frctionl differentil eqution in the cse < α < 2 nd b < + by virtue of Henry-Gronwll inequlity with mixed integrl term. Finlly, exmples re given to illustrte our theory results. 2. Preliminries In this section, we introduce nottions, definitions, nd preliminry fcts which re used throughout this pper. We denote (B, ) be Bnch spce. Let R, b R, < b +, Let C([, b), B) be the Bnch spce of ll continuous functions from [, b) into B with the norm y C = sup{ y(t) : t [, b)}. If B := R, we simply denote C([, b), R) by C[, b). We need some bsic definitions nd properties of the frctionl clculus theory which re used further in this pper. For more detils, see [7]. Definition 2.. The frctionl integrl of order γ with the lower limit zero for function f is defined s I γ f(t) = Γ(γ) f(s) ds, t >, γ >, (t s) γ provided the right side is point-wise defined on [, ), where Γ( ) is the gmm function. Definition 2.2. The Riemnn-Liouville derivtive of order γ with the lower limit zero for function f : [, ) R cn be written s d n L D γ f(t) = Γ(n γ) dt n f(s) ds, t >, n < γ < n. (t s) γ+ n Definition 2.3. The Cputo derivtive of order γ for function f : [, ) R cn be written s n X «c D γ f(t) = L D γ t k f(t) k! f(k) (), t >, n < γ < n. k= Let ǫ be positive rel number, f : [, b) B B be continuous opertor nd ϕ : [, b) R + be continuous function. We consider the following differentil eqution (2.) c D α x(t) = f(t, x(t)), α (,) (or (,2)), t [, b), EJQTDE, 2 No. 63, p. 2

3 nd the following inequlities (2.2) c D α y(t) f(t, y(t)) ǫ, t [, b), (2.3) c D α y(t) f(t, y(t)) ϕ(t), t [, b), (2.4) c D α y(t) f(t, y(t)) ǫϕ(t), t [, b). Definition 2.4. The eqution (2.) is Ulm-Hyers stble if there exists rel number c f > such tht for ech ǫ > nd for ech solution y C ([, b), B)(or C 2 ([, b), B)) of the inequlity (2.2) there exists solution x C ([, b), B)(or C 2 ([, b), B)) of the eqution (2.) with y(t) x(t) c f ǫ, t [, b). Definition 2.5. The eqution (2.) is generlized Ulm-Hyers stble if there exists θ f C(R +, R +), θ f () = such tht for ech solution y C ([, b), B)(or C 2 ([, b), B)) of the inequlity (2.2) there exists solution x C ([, b), B)(or C 2 ([, b), B) of the eqution (2.) with y(t) x(t) θ f (ǫ), t [, b). Definition 2.6. The eqution (2.) is Ulm-Hyers-Rssis stble with respect to ϕ if there exists c f,ϕ > such tht for ech ǫ > nd for ech solution y C ([, b), B)(or C 2 ([, b), B)) of the inequlity (2.4) there exists solution x C ([, b), B)(or C 2 ([, b), B)) of the eqution (2.) with y(t) x(t) c f,ϕ ǫϕ(t), t [, b). Definition 2.7. The eqution (2.) is generlized Ulm-Hyers-Rssis stble with respect to ϕ if there exists c f,ϕ > such tht for ech solution y C ([, b), B)(or C 2 ([, b), B)) of the inequlity (2.3) there exists solution x C ([, b), B)(or C 2 ([, b), B)) of the eqution (2.) with y(t) x(t) c f,ϕ ϕ(t), t [, b). Remrk 2.8. It is cler tht: (i) Definition 2.4 = Definition 2.5; (ii) Definition 2.6 = Definition 2.7; (iii) Definition 2.6 = Definition 2.4. Remrk 2.9. A function y C ([, b), B)(or C 2 ([, b), B)) is solution of the inequlity (2.2) if nd only if there exists function g C ([, b), B)(or C 2 ([, b), B)) (which depend on y) such tht (i) g(t) ǫ, t [, b); (ii) c D α y(t) = f(t, y(t)) + g(t), t [, b). One cn hve similr remrks for the inequtions (2.3) nd (2.4). So, the Ulm stbilities of the frctionl differentil equtions re some specil types of dt dependence of the solutions of frctionl differentil equtions. Remrk 2.. Let < α <, if y C ([, b), B) is solution of the inequlity (2.2) then y is solution of the following integrl inequlity y(t) y() (t s) α f(s, y(s))ds (t )α Γ(α + ) ǫ, t [, b). EJQTDE, 2 No. 63, p. 3

4 Indeed, by Remrk 2.9 we hve tht c D α y(t) = f(t, y(t)) + g(t), t [, b). Then This implies tht y(t) y() = y(t) = y() + From this it follows tht y(t) y() (t s) α f(s, y(s))ds + (t s) α f(s, y(s))ds + (t s) α f(s, y(s))ds = (t s) α g(s)ds, t [, b). (t s) α g(s)ds, t [, b). ǫ (t )α Γ(α + ) ǫ. We hve similr remrks for the solutions of the inequtions (2.3) nd (2.4). (t s) α g(s)ds (t s) α g(s) ds (t s) α ds In wht follows, we collect the Henry-Gronwll inequlity (see Lemm 7.., [2]), which cn be used in frctionl differentil equtions with initil vlue conditions. Lemm 2.. Let z, ω : [, T) [, + ) be continuous functions where T. If ω is nondecresing nd there re constnts κ nd q > such tht then z(t) ω(t) + z(t) ω(t) + κ " X n= (t s) q z(s)ds, t [, T), # (κγ(q)) n (t s) nq ω(s) ds, t [, T). Γ(nq) If ω(t) = ā, constnt on t < T, then the bove inequlity is reduce to z(t) āe q(κγ(q)t q ), t < T, where E q is the Mittg-Leffler function [7] defined by E β (y) := X k= y k, y C, Re(β) >. Γ(kβ + ) Remrk 2.2. (i) There exists constnt M κ > independent of ā such tht z(t) M κā for ll t < T. (ii) For more generlized Henry-Gronwll inequlities see Ye et l. [33]. To end this section, we collect generlized Henry-Gronwll inequlity with mixed integrl term, which cn be used in boundry vlue problems for frctionl differentil equtions. EJQTDE, 2 No. 63, p. 4

5 Lemm 2.3. Let b < + nd y C([, b], B) stisfy the following inequlity: (2.5) y(t) + b (t s) α y(s) λ ds + c (b s) α y(s) λ ds, where α (,2), λ [, ] for some < p < +,, b, c re constnts. Then there exists p h i b constnt M := (b + c ) p(α )+ p > such tht p(α )+ y(t) ( + )e Mb. Proof. Similr to the proof of Lemm 3.2 in our previous work [32], one cn obtin the result immeditely. 3. Ulm stbility results Let < α <. Without loss of generlity, we consider the eqution (2.) nd the inequlity (2.3) in the cse b = +. We suppose tht: (H ) f C([,+ ) B, B); (H 2) There exists m f > such tht f(t, u ) f(t, u 2) m f u u 2, for ech t [,+ ), nd ll u, u 2 B; (H 3) Let ϕ C([,+ ), R +) be n incresing function. There exists λ ϕ > such tht (t s) α ϕ(s)ds λ ϕϕ(t), for ech t [,+ ). We obtin the following generlized Ulm-Hyers-Rssis stble results. Theorem 3.. In the conditions (H ), (H 2) nd (H 3) the eqution (2.) (b = + ) is generlized Ulm- Hyers-Rssis stble. Proof. Let y C ([,+ ), B) be solution of the inequlity (2.3) (b = + ). Denote by x the unique solution of the Cuchy problem ( c D α x(t) = f(t, x(t)), < α <, t [, + ), (3.) Then we hve x() = y(). x(t) = y() + By differentil inequlity (2.3), we hve y(t) y() (t s) α f(s, x(s))ds, t [,+ ). (t s) α ϕ(s)ds λ ϕϕ(t), t [,+ ). (t s) α f(s, y(s))ds EJQTDE, 2 No. 63, p. 5

6 From these reltion it follows y(t) x(t) Z t y(t) y() (t s) α f(s, x(s))ds Z t y(t) y() (t s) α f(s, y(s))ds + (t s) α f(s, y(s))ds (t s) α f(s, x(s))ds Z t y(t) y() (t s) α f(s, y(s))ds + λ ϕϕ(t) + m f (t s) α f(s, y(s)) f(s, x(s)) ds (t s) α y(s) x(s) ds. By Lemm 2. nd Remrk 2.2(i), there exists constnt M f > independent of λ ϕϕ(t) such tht y(t) x(t) M f λ ϕϕ(t) := c f,ϕ ϕ(t), t [,+ ). Thus, the eqution (2.) (b = + ) is generlized Ulm-Hyers-Rssis stble. Corollry 3.2. (i) Under the ssumptions of Theorem 3., we consider the eqution (2.) (b = + ) nd the inequlity (2.4). One cn repet the sme process to verify tht the eqution (2.) (b = + ) is Ulm- Hyers-Rssis stble. (ii) Under the ssumptions (H ) nd (H 2), we consider the eqution (2.) (b = + ) nd the inequlity (2.2). One cn repet the sme process to verify tht the eqution (2.) (b = + ) is Ulm-Hyers stble. 4. Dt Dependence Let < α < 2, we reconsider the eqution (2.) (b < + ) nd the inequlity (2.2). We suppose tht: (H 4) f C([, b] B). (H 5) There exist m f > nd λ [, ] for some < p < such tht p f(t, u ) f(t, u 2) m f u u 2 λ, for ech t [, b], nd ll u, u 2 B. The following result is interesting lthough the proof is not very difficult. Theorem 4.. Assumptions (H 4) nd (H 5) hold. Let y C 2 [, b] be solution of the inequlity (2.2). Denote by x the solution of the following frctionl boundry vlue problem ( c D α x(t) = f(t, x(t)), < α < 2, t [, b], (4.) Then the following reltion holds: x() = y(), x(b) = y(b). (4.2) where y(t) x(t) c f (ǫ + ), t [, b], c f := e Mb mx j (b ) α Γ(α + ), ff > nd M := 2m f» b p(α )+ p(α ) + p. EJQTDE, 2 No. 63, p. 6

7 Proof. By Lemm 3.7 of [], it is cler tht the solution of the frctionl boundry vlue problem (4.) given by x(t) = b t b y() + t b y(b) + t b + (t s) α f(s, x(s))ds. By differentil inequlity (2.2), we hve y(t) b t b y() t b y(b) t b (t s) α f(s, y(s))ds ǫ (b )α ǫ Γ(α + ). From these reltion it follows (t s) α ds (b s) α f(s, x(s))ds (b s) α f(s, y(s))ds y(t) x(t) y(t) b t b y() t b y(b) t Z b (b s) α f(s, x(s))ds b (t s) α f(s, x(s))ds y(t) b t b y() t b y(b) t Z b (b s) α f(s, y(s))ds b (t s) α f(s, y(s))ds Z + t b (b s) α f(s, y(s))ds t Z b (b s) α f(s, x(s))ds b b + (t s) α f(s, y(s))ds (t s) α f(s, x(s))ds (b )α t ǫ + Γ(α + ) b + (b )α Γ(α + ) ǫ + m f + m f (t s) α f(s, y(s)) f(s, x(s)) ds (b s) α f(s, y(s)) f(s, x(s)) ds (b s) α y(s) x(s) λ ds (t s) α y(s) x(s) λ ds. Applying Lemm 2.3 to the bove inequlity nd yields the im inequlity (4.2). 5. Exmple In this section, some exmples re given to illustrte our theory results. Let < α <. We consider in the cse B := R the eqution (5.) c D α x(t) =, t [, b), EJQTDE, 2 No. 63, p. 7

8 nd the ineqution (5.2) c D α y(t) ǫ, t [, b). Let y C [, b) be solution of the ineqution (5.2). Then there exists g C[, b) such tht: (i) g(t) ǫ, t [, b), (5.3) (ii) c D α y(t) = g(t), t [, b). Integrting (5.3) from to b by virtue of Definition 2.4, we hve We hve, for ll c R, y(t) = y() + y(t) c = y() c + y() c + y() c + ǫ (t s) α g(s)ds, t [, b). (t s) α g(s)ds (t s) α g(s) ds (t s) α ds y() c + (t )α ǫ, t [, b). Γ(α + ) If we tke c := y(), then If b < +, then So, the eqution (5.) is Ulm-Hyers stble. Let b = +. The function y(t) y() (t )α ǫ, t [, b). Γ(α + ) y(t) y() (b )α ǫ, t [, b). Γ(α + ) y(t) = (t )α ǫ Γ(α + ) is solution of the inequlity (5.2) nd y(t) c = (t )α ǫ Γ(α + ) c +, s t +. So, the eqution (5.) is not Ulm-Hyers stble on the intervl [,+ ). Let us consider the ineqution (5.4) c D α y(t) ϕ(t), t [,+ ). Let y be solution of (5.4) nd x(t) = y(), t [,+ ) solution of the eqution (5.). We hve tht If there exists c ϕ > such tht y(t) x(t) = y(t) y() (t s) α ϕ(s)ds, t [,+ ) (t s) α ϕ(s)ds c ϕϕ(t), t [,+ ), then the eqution (5.) is generlized Ulm-Hyers-Rssis stble on [,+ ) with respect to ϕ. EJQTDE, 2 No. 63, p. 8

9 Acknowledgements: This work ws completed when Dr. Wng ws visiting Xingtn University in Hunn, Chin in 2. He would like to thnk Prof. Yong Zhou for the invittion nd providing stimulting working environment. The uthors thnks the referees for their creful reding of the mnuscript nd insightful comments, which help to improve the qulity of the pper. We would lso like to cknowledge the vluble comments nd suggestions from the editors, which vstly contributed to improve the presenttion of the pper. References [] R. P. Agrwl, M. Benchohr, S. Hmni, A survey on existence results for boundry vlue problems of nonliner frctionl differentil equtions nd inclusions, Act. Appl. Mth., 9(2), [2] R. P. Agrwl, Y. Zhou, J. Wng, X. Luo, Frctionl functionl differentil equtions with cusl opertors in Bnch spces, Mth. Comput. Model., 54(2), [3] B. Ahmd, J. J. Nieto, Existence of solutions for nti-periodic boundry vlue problems involving frctionl differentil equtions vi Lery-Schuder degree theory, Topol. Methods Nonliner Anl., 35(2), [4] H. Amnn, Ordinry differentil equtions, Wlter de Gruyter, Berlin, 99. [5] K. Blchndrn, J. Y. Prk, M. D. Julie, On locl ttrctivity of solutions of functionl integrl eqution of frctionl order with deviting rguments, Commun. Nonliner Sci. Numer. Simult., 5(2), [6] Z. Bi, On positive solutions of nonlocl frctionl boundry vlue problem, Nonliner Anl.:TMA, 72(2), [7] M. Benchohr, J. Henderson, S. K. Ntouys, A. Ouhb, Existence results for frctionl order functionl differentil equtions with infinite dely, J. Mth. Anl. Appl., 338(28), [8] L. Cădriu, Stbilitte Ulm-Hyers-Bourgin pentru ecutii functionle, Ed. Univ. Vest Timişor, Timişr, 27. [9] C. Chicone, Ordinry differentil equtions with pplictions, Springer, New York, 26. [] C. Cordunenu, Principles of differentil nd integrl equtions, Chelse Publ. Compny, New York, 97. [] W. Deng, Smoothness nd stbility of the solutions for nonliner frctionl differentil equtions, Nonliner Anl.:TMA, 72(2), [2] D. Henry, Geometric theory of semiliner prbolic equtions, LNM 84, Springer-Verlg, Berlin, Heidelberg, New York, 98. [3] J. Henderson, A. Ouhb, Frctionl functionl differentil inclusions with finite dely, Nonliner Anl.:TMA, 7(29), [4] S.-B. Hsu, Ordinry differentil equtions with pplictions, World Scientific, New Jersey, 26. [5] D. H. Hyers, G. Isc, Th. M. Rssis, Stbility of functionl equtions in severl vribles, Birkhäuser, 998. [6] S.-M. Jung, Hyers-Ulm-Rssis stbility of functionl equtions in mthemticl nlysis, Hdronic Press, Plm Hrbor, 2. [7] A. A. Kilbs, H. M. Srivstv, J. J. Trujillo, Theory nd pplictions of frctionl differentil equtions, in: North-Hollnd Mthemtics Studies, vol. 24, Elsevier Science B.V., Amsterdm, 26. [8] Y. Li, Y. Chen, I. Podlubny, Mittg-Leffler stbility of frctionl order nonliner dynmic systems, Automtic, 45(29), EJQTDE, 2 No. 63, p. 9

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