J. Math. Anal. Appl. 328 27) 75 8 www.elsevier.com/locate/jmaa A generalized Gronwall inequality and its application to a fractional differential equation Haiping Ye a,, Jianming Gao a, Yongsheng Ding b a Department of Applied Mathematics, Donghua University, Shanghai 262, PR China b College of Information Sciences and Technology, Donghua University, Shanghai 262, PR China Received 7 February 26 Available online 5 July 26 Submitted by William F. Ames Abstract This paper presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation. 26 Elsevier Inc. All rights reserved. Keywords: Integral inequality; Differential equation; Fractional order. Introduction Integral inequalities play an important role in the qualitative analysis of the solutions to differential and integral equations; cf. [. The celebrated Gronwall inequality known now as Gronwall Bellman Raid inequality provided explicit bounds on solutions of a class of linear integral inequalities. On the basis of various motivations, this inequality has been extended and used in various contexts [2 4. Let us recall the standard Gronwall inequality which can be found in [, p. 4. Theorem A. If xt) ht) t ks)xs)ds, t [t,t), * Corresponding author. E-mail addresses: hpye@dhu.edu.cn H. Ye), ysding@dhu.edu.cn Y. Ding). 22-247X/$ see front matter 26 Elsevier Inc. All rights reserved. doi:.6/j.jmaa.26.5.6
76 H. Ye et al. / J. Math. Anal. Appl. 328 27) 75 8 where all the functions involved are continuous on [t,t), T, and kt), then xt) satisfies [ t xt) ht) hs)ks)exp ku)du ds, t [t,t). t If, in addition, ht) is nondecreasing, then t ) xt) ht) exp ks)ds, t [t,t). t s However, sometimes we need a different form, to discuss the weakly singular Volterra integral equations encountered in fractional differential equations. In this paper we present a slight generalization of the Gronwall inequality which can be used in a fractional differential equation. Using the inequality, we study the dependence of the solution on the order and the initial condition for a fractional differential equations with Riemann Liouville fractional derivatives. 2. An integral inequality In this section, we wish to establish an integral inequality which can be used in a fractional differential equation. The proof is based on an iteration argument. Theorem. Suppose β>, at) is a nonnegative function locally integrable on t<tsome T ) and gt) is a nonnegative, nondecreasing continuous function defined on t<t, gt) M constant), and suppose ut) is nonnegative and locally integrable on t<t with ut) at) gt) on this interval. Then ut) at) t s) β us) ds [ gt)γ β)) n t s) nβ as) ds, Γnβ) n= t<t. Proof. Let Bφt) = gt) t s)β φs)ds, t, for locally integrable functions φ. Then ut) at) But) implies n ut) B k at) B n ut). k= Let us prove that B n gt)γ β)) n ut) t s) nβ us) ds ) Γnβ) and B n ut) asn for each t in t<t.
H. Ye et al. / J. Math. Anal. Appl. 328 27) 75 8 77 We know this relation ) is true for n =. Assume that it is true for some n = k. If n = k, then the induction hypothesis implies B k ut) = B B k ut) ) [ s gt) t s) β gs)γ β)) k s τ) kβ uτ) dτ ds. Γkβ) Since gt) is nondecreasing, it follows that B k ut) gt) ) k [ s t s) β τ Γ β)) k Γkβ) s τ)kβ uτ) dτ ds. By interchanging the order of integration, we have B k ut) gt) ) [ t k Γ β)) k Γkβ) t s)β s τ) kβ ds uτ) dτ where the integral τ = gt)γ β)) k Γ k )β) t s)k)β us) ds, t s) β s τ) kβ ds = t τ) kββ z) β z kβ dz = t τ) k)β Bkβ,β) = Γβ)Γkβ) t τ)k)β Γ k )β) is evaluated with the help of the substitution s = τ zt τ) and the definition of the beta function cf. [6, pp. 6 7). The relation ) is proved. Since B n ut ) MΓ β)) n Γnβ) t s) nβ us) ds asn for t [,T),the theorem is proved. For gt) b in the theorem we obtain the following inequality. This inequality can be found in [5, p. 88. Corollary. [5 Suppose b, β>and at) is a nonnegative function locally integrable on t<tsome T ), and suppose ut) is nonnegative and locally integrable on t<t with ut) at) b on this interval; then ut) at) t s) β us) ds [ bγ β)) n t s) nβ as) ds, Γnβ) n= t<t.
78 H. Ye et al. / J. Math. Anal. Appl. 328 27) 75 8 Corollary 2. Under the hypothesis of Theorem,letat) be a nondecreasing function on [,T). Then ut) at)e β gt)γ β)t β ), where E β is the Mittag-Leffler function defined by E β z) = k= n= z k Γkβ). Proof. The hypotheses imply [ gt)γ β)) n ut) at) t s) nβ gt)γ β)t β ) n ds = at) Γnβ) Γnβ ) The proof is complete. 3. Application to dependence of solution on parameters n= = at)e β gt)γ β)t β ). In this section we will show that our main result is useful in investigating the dependence of the solution on the order and the initial condition to a certain fractional differential equation with Riemann Liouville fractional derivatives. It is frequently stated that the physical meaning of initial conditions expressed in terms of Riemann Liouville fractional derivatives is unclear or even nonexistent. But in [9, N. Heymans and I. Podlubny demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann Liouville fractional derivatives and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations. Therefore, our discussion here is meaningful. Let us consider the following initial value problem [6 in terms of the Riemann Liouville fractional derivatives: D α yt) = f t,yt) ), 2) D α yt) t= =, 3) where <α<, t<t, f : [,T) R R and D α denotes Riemann Liouville derivative operator. Riemann Liouville derivative and integral are defined below [6 8. Definition. The fractional derivative of order <α< of a continuous function f : R R is given by D α d fx)= Γ α) dx x x t) α ft)dt provided that the right side is pointwise defined on R. Definition 2. The fractional primitive of order α> of a function f : R R is given by I α fx)= x x t) α ft)dt provided the right side is pointwise defined on R.
H. Ye et al. / J. Math. Anal. Appl. 328 27) 75 8 79 The existence and uniqueness of the initial value problem 2) 3) have been studied in the literature [6. Also the dependence of a solution on initial conditions has been discussed in [6. In this section we present the dependence of the solution on the order and the initial condition. We shall consider the solutions of two initial value problems with neighboring orders and neighboring initial values. It is important to note that here we are considering a question which does not arise in the solution of differential equations of integer order. First, let us reduce the problem 2) 3) to a fractional integral equation. We obtain yt) = tα t τ) α f τ,yτ) ) dτ. 4) It is clear that Eq. 4) is equivalent to the initial value problem 2) 3) cf. [6, pp. 27 28). Theorem 2. Let α> and δ> such that <α δ<α. Let the function f be continuous and fulfill a Lipschitz condition with respect to the second variable; i.e., ft,y) ft,z) L y z for a constant L independent of t,y,z in R. For t h<t,assume that y and z are the solutions of the initial value problems 2) 3) and D α δ zt) = f t,zt) ), D α δ zt) t= =, respectively. Then, for <t h the following holds: t [ ) zt) yt) L n t s) nα δ) At) Γα δ) Γnα δ)) As) ds, n= where At) = Γα δ) tα δ tα t α δ α δ)γ α) t α Γα ) f t α δ [ α δ Γα δ) f and f = max ft,y). t h 5) 6) Proof. The solutions of the initial value problem 2) 3) and 5) 6) are given by and yt) = zt) = tα Γα δ) tα δ Γα δ) t τ) α f τ,yτ) ) dτ t τ) α δ f τ,zτ) ) dτ,
8 H. Ye et al. / J. Math. Anal. Appl. 328 27) 75 8 respectively. It follows that zt) yt) Γα δ) tα δ Γα δ) At) tα t τ) α δ f τ,zτ) ) dτ t τ) α δ f τ,zτ) ) dτ t τ) α δ f τ,yτ) ) dτ t τ) α δ L zτ) yτ) dτ, t τ) α δ f τ,zτ) ) dτ t τ) α δ f τ,yτ) ) dτ t τ) α f τ,yτ) ) dτ where At) = Γα δ) tα δ tα t α δ α δ)γ α) t α Γα ) f t α δ [ α δ Γα δ) f. An application of Theorem yields zt) yt) [ ) L n t s) nα δ) At) Γα δ) Γnα δ)) As) ds and the theorem is proved. n= Remark. It follows from Theorem 2 that for every ɛ between and h small changes in order and initial condition cause only small changes of the solution in the closed interval [ɛ,h which does not contain zero). Remark 2. Existence and uniqueness are granted in the autonomous case, i.e., when f depends only on y. See [7, Lemma 4.2. A general theorem of existence and uniqueness for the nonautonomous case ft,y) can be found in [6, p. 27. Corollary 3. [6 Under the hypothesis of Theorem 2, ifδ =, then zt) yt) t α E α,α Lt α ), for <t h, where E α,α is the Mittag-Leffler function defined by E α,α z) = k= α > ). z k Γαkα)
H. Ye et al. / J. Math. Anal. Appl. 328 27) 75 8 8 Proof. If δ =, then At) = t α ). By Theorem 2, we obtain zt) yt) At) n= for <t h. The proof is complete. Corollary 3 can be found in [6, pp. 34 36. Acknowledgment [ n t s)nα L As) ds = t α Lt α ) n Γnα) Γαn α) The authors are very grateful to referee for his/her valuable suggestions. References n= = t α E α,α Lt α ), [ B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, New York, 998. [2 B.G. Pachpatte, On some generalizations of Bellman s lemma, J. Math. Anal. Appl. 5 975) 4 5. [3 O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl. 252 2) 389 4. [4 R.P. Agarwal, S. Deng, W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput. 65 25) 599 62. [5 D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 84, Springer-Verlag, New York/Berlin, 98. [6 I. Podlubny, Fractional Differential Equations, Academic Press, New York, 999. [7 D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 24 996) 69 625. [8 K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 22) 229 248. [9 N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann Liouville fractional derivatives, Rheol. Acta 37 25) 7. [ C. Corduneanu, Principles of Differential and Integral Equations, Allyn and Bacon, Boston, 97.