Existence of positive mild solutions for a class of fractional evolution equations on the half line
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1 Chen et al. Advances in Difference Equations :279 DOI /s y R E S E A R C H Open Access Existence of positive mild solutions for a class of fractional evolution equations on the half line Yi Chen 1*, Zhanmei Lv 2 and Liang Zhang 3 * Correspondence: mathcyt@163.com 1 School of Mathematics, China University of Mining and Technology, Xuzhou, , China Full list of author information is available at the end of the article Abstract The equivalent integral equation of a new form for a class of fractional evolution equations is obtained by the method of Laplace transform, which is different from those given in the existing literature. By the monotone iterative method without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract fractional evolution equations on the half-line. Keywords: fractional order; abstract evolution equations; mild solutions; monotone iterative method 1 Introduction In this paper, we are concerned with the following fractional evolution equation in the Banach space E: C D + utaut+f t, ut, u μuβ, t, +, 1.1 where C D + is the Caputo fractional derivative, < <1,μ >,β >,Ais the infinitesimal generator of a C semigroup {Tt} t of operators on Banach E,andf :[,+ E E satisfies certain conditions. Fractional calculus, a generalization of the ordinary differentiation and integration, has played a significant role in science, economy, engineering, and other fields see [1 3]. Today there is a large number of papers dealing with the fractional differential equations see [4 16] due to their various applications. One of the branches is the research on the theory about the evolution equations of fractional order, which comes from physics. Recently, fractional evolution equations have attracted increasing attention around the world, see [7 16] and the references therein. Among the existing literature, most of them are focused on the existence of the solutions on the finite interval, see [7 16]. The Authors 217. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.
2 Chenetal. Advances in Difference Equations :279 Page 2 of 14 In [8], El-Sayed investigated the Cauchy problem in a Banach space for a class of fractional evolution equations d u d t Aut+Btut, t [, T], u u, where < 1, T >. The existence and uniqueness of the solution for the above Cauchy problem were studied for a wide class of the family of operators {Bt:t }. As far as we know, for the first time, the equivalent integral equation of the above equation was given in terms of some probability densities by the method of Laplace transform. And since then, most of the research in this direction has been based on this paper. However, many of the previous papers about the existence of solutions of fractional evolution equations are only on the finite interval, and those presenting the existence results on the half-line are still few. Motivated by [8, 9, 17], in this paper, we study the differential equation 1.1undercertain conditions on the unbounded domains. Here, by a method similar to that used in [8, 9], we give a corrected form of the equivalent integral equation of the main problem 1.1, which is different from those obtained in the existing literature. Employing the monotone iterative method, without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract evolution equations of fractional order. And to our best knowledge, there is not any paper to deal with the abstract problems of fractional order on the unbounded domains. Therestofthepaperisorganizedasfollows.InSection2, we introduce the definitions of fractional integral and fractional derivative, some results about fractional differential equations and some useful preliminaries. In Section 3, we obtain the existence result of the solution for problem 1.1 by the monotone iterative method. Then an example is given in Section 4 to demonstrate the application of our result. 2 Preliminaries First of all, we present some fundamental facts on the fractional calculus theory which we will use in the next section. Definition 2.1 [1 3] The Riemann-Liouville fractional integral of order ν >ofafunction h :, R is given by I+ ν htd ν + ht 1 Ɣν t t s ν 1 hs ds, 2.1 provided that the right-hand side is pointwise defined on,. Definition 2.2 [1 3] The Caputo fractional derivative of order ν > of a continuous function h :, R is given by C D ν + ht 1 Ɣn ν t t s n ν 1 h n s ds, 2.2 where n [ν] + 1, provided that the right-hand side is pointwise defined on,.
3 Chenetal. Advances in Difference Equations :279 Page 3 of 14 Lemma 2.1 [1, 3] Assume that C D ν + ht L1, +, ν >.Then we have I ν + C D ν + htht+c 1 + C 2 t + + C N t N 1, t >, 2.3 for some C i R, i 1,2,...,N, where N is the smallest integer greater than or equal to ν. If h is an abstract function with values in the Banach space E, then the integrals appearing in Definition 2.1, Definition 2.2 and Lemma 2.1 are taken in Bochner s sense. And a measurablefunction h is Bochner integrable if the norm of h is Lebesgue integrable. Now let us recall some definitions and standard facts about the cone. Let P be a cone in the ordered Banach space E, which defines a partial order on E by x y if and only if y x P. P is normal if there exists a positive constant N such that θ x y implies x N y, whereθ is the zero element of the Banach space E. The infimum of all N with the property above is called the normal constant of P. Formore details on the cone P, we refer readers to [18, 19]. Throughout the paper, we set E to be an ordered Banach space with the norm and the partial order. Let P {x E x θ} be a positive cone, which is normal with normal constant N.LetJ [,+. Set BCJ, E { ut ut is continuous and bounded on J }. Obviously, BCJ, E is a Banach space with the norm u BC sup t J ut.let P BC { u BCJ, E ut θ, t J }. It is easy to see that P BC is also normal with the same normal constant N of the cone P. Besides, BCJ, E is also an ordered Banach space with the partial order inducedbythe positive cone P BC without confusion, we denote by thepartialorderonbothe and BCJ, E. We denote by [v, w] theorderinterval{u P BC v u w, v, w BCJ, E} on BCJ, E, and use [vt, wt] to denote the order interval {z E vt z wt} on E for t J. Next, we give some facts about the semigroups of linear operators. These results can be found in [2, 21]. For a strongly continuous semigroup i.e., C -semigroup {Tt} t, the infinitesimal generator of {Tt} t is defined by Ttx x Ax lim, x E. t + t We denote by DA thedomain of A,thatis, { DA x E Ttx x lim t + t } exists. Lemma 2.2 [2, 21] Let {Tt} t be a C -semigroup, then there exist constants C 1 and R such that Tt Ce t, t.
4 Chenetal. Advances in Difference Equations :279 Page 4 of 14 Lemma 2.3 [2, 21] A linear operator A is the infinitesimal generator of a C -semigroup {Tt} t if and only if i A is closed and DAE. ii The resolvent set ρa of A contains R + and, for every λ >, we have Rλ, A 1 λ, where Rλ, A:λI A 1 + e λt Ttxdt, x E. Definition 2.3 [2, 21] A C -semigroup {Tt} t is said to be uniformly exponentially stable if <,where is the growth bound of {Tt} t, which is defined by inf { R C 1suchthat Tt Ce t, t }. Definition 2.4 [17] A C -semigroup {Tt} t is said to be positive on E if order inequality Ttx θ, x E and t. According to Lemma 2.2 and Definition 2.3, if{tt} t is a uniformly exponentially stable C -semigroup with the growth bound, there exists a constant C 1suchthat Tt Ce t, t, for any, ]. Now, we define a norm in E by x sup e t Ttx. t Evidently, x x C x, that is to say, the norms and are equivalent. We denote by Tt the norm of Ttinducedbythenorm,then Tt e t, t. 2.4 Also, we can define the equivalent norm on BCJ, Eby u BC sup ut, t J u BCJ, E. 3 Main results In this section, we present the existence theorem for the abstract fractional differential equation on the half-line. In order to prove our main result, we need the following facts and lemmas. Consider the one-sided stable probability density [9, 1, 22] ψ θ 1 π n1 n 1 Ɣn +1 θ sinnπ, n! θ,, where < <1.
5 Chenetal. Advances in Difference Equations :279 Page 5 of 14 From [9, 1, 22], the Laplace transform of the one-sided stable probability density ψ θ is given by L [ ψ θ ] e λθ ψ θ dθ e λ, < < By Remark 2.8 in [1], for γ 1, one has θ γ ψ θ dθ Ɣ1 + γ Ɣ1 + γ. 3.2 Inthefollowing,weassumethat{Tt} t is a uniformly exponentially stable C - semigroup with the growth bound,and, ]. Lemma 3.1 Define a linear operator V : E Eas Vx μ Ɣ1 ψ θ τ 1 τ 1 T β 1 τ xdθ dτ. Then V is bounded and V μ. Besides, if <μ <1,then I V 1 is a linear bounded operator and I V μ. Proof Since ψ θ Vx μ τ 1 τ 1 β Ɣ1 T 1 τ [ ψ θ μ τ 1 τ 1 e β 1 τ Ɣ1 dθ dτ [ 1 ] μ τ 1 τ 1 ψ θ dθ dτ x Ɣ1 μ x. x dθ dτ ] x Hence, V is bounded and V μ. Lemma 3.2 Set Qht t ψ θ t s 1 T t s hs dθ ds, h BCJ, E. Then Q : BCJ, E BCJ, E and Qht 1 1 Ɣ +1 h BC ; Qh BC 1 1 Ɣ +1 h BC.
6 Chenetal. Advances in Difference Equations :279 Page 6 of 14 Proof Since Qht t ψ θ t s 1 T ψ θ t 1 τ 1 T t s t 1 τ hs dθ ds htτ dθ dτ, then Qht ψ θ t 1 τ 1 t T 1 τ htτ dθ dτ ψ θ t 1 τ 1 e t 1 τ htτ dθ dτ 1 [ 1 h BC e t 1 τ d t 1 τ ] ψ θ dθ Therefore, 1 h BC 1 1 Ɣ +1 h BC. 1 e θ t ψ θ dθ Qh BC 1 1 Ɣ +1 h BC. Lemma 3.3 Let h BCJ, E and u DA. Then the linear fractional evolution equation C D + utaut+ht, t, +, 3.3 u u, has a unique solution u BCJ, E of the following form: t t s 1 t s s ] ut [ ψ θt Ɣ1 u + hs dθ ds ψ θ t τ 1 τ 1 1 τ T u Ɣ1 dθ dτ +Qht. 3.4 Proof In view of Definitions 2.1, 2.2 and Lemma 2.1, equation3.3 canberewrittenby the equivalent integral equation as follows: utu + 1 Ɣ t t s 1[ Aus+hs ] ds. 3.5 Denote by Uλ andhλ thelaplacetransformsofut andht, respectively, using a similar method as that in [7, 8], with the Laplace transform, then we can rewrite the above
7 Chenetal. Advances in Difference Equations :279 Page 7 of 14 equation as Uλ 1 λ u + 1 λ AUλ+ 1 Hλ, λ λ >. 3.6 Hence, λ I A Uλλ 1 u + Hλ. By virtue of 3.1 and Lemma 2.3,weobtain Uλ λ I A 1 λ 1 u + λ I A 1 Hλ λ 1 e λs Tsu ds + e λs TsHλ ds λ 1 + [ λ 1 e λt + e λt [ t e λs1/θ ψ θtsu dθ ds e λs1/θ ψ θtshλ dθ ds t 1 t ] θ ψ θt u dθ dt t s t s 1 ψ θt ] hs dθ ds dt. By the definition of Laplace transforms and the convolution theorem, applying Lemma 3.2 and the inverse Laplace transforms, one can derive that utl 1[ λ 1] [ [ L 1 e λt t 1 t θ ψ θt [ t + L 1 e λt t s 1 t s ψ θt Since t [ Ɣ1 t 1 t θ ψ θt + t t t t s 1 ψ θt s 1 t s Ɣ1 Ɣ1 Ɣ1 t s ] u dθ hs dθ ds s θ ψ θt t s t s 1 s ψ θt ψ θ τ 1 τ 1 T ] ] u dθ dt u dθ ds +Qht t 1 τ Ɣ1 τ 1 τ 1 ψ θ t T 1 τ Ɣ1 τ 1 τ 1 ψ θ e ] hs dθ ds dt u dθ ds +Qht u dθ dτ +Qht. u dθ dτ + Qht t 1 τ u dθ dτ + Qht
8 Chenetal. Advances in Difference Equations :279 Page 8 of 14 Ɣ1 u u τ 1 τ 1 1 Ɣ +1 h BC. ψ θ dθ dτ + Qht Therefore, u BCJ, E. Then we complete the proof. Lemma 3.4 Let h BCJ, E and u DA. Let <μ <1.Then the linear fractional evolution equation C D + utaut+ht, t, +, 3.7 u μuβ, has a unique solution u BCJ, E of the following form: ut L A ht t t s 1 t s s [ : [ ψ θt I V 1 Qhβ ] Ɣ1 ] + hs dθ ds ψ θ τ 1 τ 1 1 τ [I T t V 1 Qhβ ] dθ dτ Ɣ1 +Qht. 3.8 Also, L A isalinearoperatoronthebanachspacebcj, E and L A BC κ, where 1 1 κ : Ɣ +1 1 μ. Proof In view of Lemma 3.3,onecanobtain β β s 1 β s s ] uβ [ ψ θt u + hs dθ ds. Ɣ1 From u μuβ, we have u μ Therefore, μ β Ɣ1 Ɣ1 Vu + μqhβ. I Vu μqhβ. ψ θ s β s 1 T ψ θ τ 1 τ 1 T β s β 1 τ u dθ ds + μqhβ u dθ dτμqhβ
9 Chenetal. Advances in Difference Equations :279 Page 9 of 14 So, we obtain u μi V 1 Qhβ. Then 3.8 is followed. By 3.8, one has L A ht ψ θ Ɣ1 τ 1 τ 1 T 1 τ t [ I V 1 μqhβ ] dθ dτ + Qht ψ θ τ 1 τ 1 e t 1 τ Ɣ1 [ I V 1 μqhβ ] dθ dτ + Qht 1 μ 1 μ 1 Ɣ +1 h BC Ɣ +1 1 μ h BC κ h BC. 1 Ɣ +1 h BC Therefore, LA h BC κ h BC. Now, we state the main result on the existence of the positive solutions to problem 1.1 in the following. Theorem 3.1 Let E be a Banach space, and P is its positive normal with N as the normal constant. Let {Tt} t be a uniformly exponentially stable C -semigroup with the growth bound <,andaisthe infinitesimal generator of {Tt} t. Let <μ <1.Provided that f t, u:j E E is continuous and f t, θ θ is bounded on J. If f t, u satisfies the following conditions: a There exists a constant K 1 < such that for f t, y f t, x K 1 y x, θ x y. b There exists a constant K 2 > max{ K 1, } such that for f t, y f t, x K 2 y x, θ x y. c < K 1 + K 2 K 2 < 1 κ. Then problem 1.1 has a unique positive mild solution in BCJ, E.
10 Chenetal.Advances in Difference Equations :279 Page 1 of 14 Proof For simplicity of notation, we denote f tft, θ, then we have f t BCJ, Eand f t θ, t J. In the following, similar to the methods used in [17], we deduce the result of the theorem in four steps. Step 1: Consider the abstract fractional differential equation C D + utaut+k 1ut+f t, t, +, 3.9 u μuβ. By virtue of the theory of semigroups, we can get that {e K1t Tt} t is a uniformly exponentially stable C -semigroup on Banach E generated by A + K 1 I. Besides, the semigroup is positive with the growth bound K 1 + K 1 + <. In view of Lemma 3.4,equation 3.9 has a unique mild solution ϑ BCJ, Eandϑ θ as a result of f t θ, t J. Step 2: For a given function g BCJ, E, consider the abstract fractional differential equation C D + ut+k 2utAut+gt, t, +, 3.1 u μuβ. It is obvious that A K 2 I generates a uniformly exponentially stable C -semigroup {e K2t Tt} t on Banach E. Also, it is positive with the growth bound K 2 + K 2 + <. Based on Lemma 3.4, the unique mild solution of 3.1 isgivenbyu L A K2 Ig, where L A K2 I : BCJ, E BCJ, E is a positive bounded linear operator similar to the operator L A withthepropertythat L A K2 I BC κ K 2, for K 2. Combined with the first step, one can notice that ϑ is the mild solution of problem 3.1forg f + K 1 ϑ + K 2 ϑ,so ϑ L A K2 If + K 1 ϑ + K 2 ϑ Step 3: Take Guf t, u+k 2 u. Evidently, Gθf t, θf t θ and G : BCJ, E BCJ, E is continuous due to conditions a, b and the normality of the cone P BC. By condition b, for θ x y,onecanobtain Gy Gxf t, y+k 2 y f t, x K 2 x f t, y f t, x+k 2 y x θ, that is, G is an increasing operator on the positive cone P BC. Let υ θ. Taking account of a composition operator defined by F L A K2 I G on the order interval [θ, ϑ ], it is easy to see that the fixed point of F is the mild solution of problem 1.1. Now, our task is to demonstrate that the operator F has at least one fixed point.
11 Chenetal.Advances in Difference Equations :279 Page 11 of 14 Consider the following two sequences: ϑ n Fϑ n 1, n 1,2,3,..., 3.12 and υ n Fυ n 1, n 1,2,3, By condition a, we have f t, ϑ t f t, θ K 1 ϑ t, then f t, ϑ t K 1 ϑ t+f t. So, we can get Gϑ f t, ϑ t + K 2 ϑ t K 1 ϑ t+k 2 ϑ t+f t. By the fact that G is an increasing operatoron the cone P, therefore, we can obtain θ f tgθ Gϑ K 1 ϑ + K 2 ϑ + f Combining 3.11, 3.14 and the positivity of the linear bounded operator L A K2 I,one can get θ L A K2 I GθFθ L A K2 I Gϑ Fϑ L A K2 IK 1 ϑ + K 2 ϑ + f ϑ, that is, θ υ ϑ 1 ϑ As F is an increasing operator on the order interval [θ, ϑ ], by the definition of F and 3.15, we can get two sequences {ϑ n } and {υ n } n,1,2,3,...satisfying θ υ υ 1 υ 2 υ n ϑ n ϑ 2 ϑ 1 ϑ. According to the above facts, by condition a, one can get that θ ϑ n υ n Fϑ n 1 Fυ n 1 L A K2 I Gϑ n 1 L A K2 I Gυ n 1 [ ] L A K2 I f, ϑn 1 +K 2 ϑ n 1 f, υ n 1 K 2 υ n 1 K 1 + K 2 L A K2 Iϑ n 1 υ n 1.
12 Chenetal.Advances in Difference Equations :279 Page 12 of 14 Thus, by induction, we have θ ϑ n υ n K 1 + K 2 n L A K2 Iϑ υ K 1 + K 2 n L n A K 2 I ϑ. On account of the fact that the cone P BC is normal with the normal constant N,byvirtue of condition c, we get ϑ n υ n BC NK 1 + K 2 n L n A K 2 I ϑ BC NK 1 + K 2 n L n A K2 I BC ϑ BC NK 1 + K 2 n L A K2 I n BC ϑ BC NK 1 + K 2 n κ n ϑ BC K 2 κk1 + K 2 n N ϑ BC, K 2 n Thus, from 3.16, analogous to the nested interval method, there exists a unique u n1 [υ n, ϑ n ]suchthatu lim n ϑ n lim n υ n. By 3.12and3.13, taking limit of n,weobtainthat u F u, namely, u is a fixed point of the operator F.Thus,u is a mild positive solution of problem 1.1. Step 4: In this part, we certify the uniqueness of the mild solution for problem 1.1. By using reduction to absurdity, suppose that u 1 and u 2 are two different positive mild solutions for the fractional evolution equation 1.1, thus, u 1 u 2 BC >. Replace ϑ by u 1 and u 2 in 3.12, respectively. Following the same steps as above, for each u i i 1,2,wecangetthatu i Fu i, u i υ n BC n andϑ n u i for each n N i 1, 2. Therefore, < u 1 u 2 BC u 1 υ BC n + u 2 υ BC n, n, which is a contradiction. Hence, problem 1.1 has a unique positive solution. The proof is completed. 4 Examples To illustrate our main result, we present an example. Consider the following partial fractional differential equation. Example 4.1 t zt, x 2 x zt, x+ft, zt, x, zt, zt, π, z, xμzβ, x, t [, +, t [, +, x [, π], 4.1
13 Chenetal.Advances in Difference Equations :279 Page 13 of 14 where t is the Caputo fractional partial derivative of order, 1. Set E L 2 [, π], R andaz x 2 z, according to [23], then A : DA E is a linear operator with domain DA{u E u E, u uπ}. Besides, the operator A generates a uniformly exponentially stable C -semigroup {Tt} t with the growth bound 1. Let utzt,, f t, ut Ft, zt,, then problem 4.1canbewrittenas C D + utaut+f t, ut, t, +, 4.2 u μuβ. Take 1/2, μ 3/4, β 1,thenwecanget 1 1 κ Ɣ +1 1 μ 8. π Consider the following function: f t, x K x, a 2 t where a C[, + is bounded and K 1 2, K κ 1. It is easy for us to certify that < K 1 + K 2 2 K 2 3κ 1 < 1 κ. Since K 2 κ κ 1 κ 1 2 K 1, then, for θ x y, K 2 f t, y f t, x [ ] K 2 + y x 1 + a 2 tκ 1 K 2 y x κ 1 K 1 y x. Noting that f t, θθ. Thereby,f satisfies the conditions of Theorem 3.1, wecanconclude that problem 4.1 has a unique positive mild solution.
14 Chenetal.Advances in Difference Equations :279 Page 14 of 14 Acknowledgements This research is supported by the Fundamental Research Funds for the Central Universities No. 214QNA52, Natural Science Foundation of Jiangsu Province of China No. BK214176, Shandong Provincial Natural Science Foundation No. ZR214AP11 and the NNSF No Competing interests The authors declare that they have no competing interests. Authors contributions All of the authors contributed equally in writing this paper. All authors read and approved the final manuscript. Author details 1 School of Mathematics, China University of Mining and Technology, Xuzhou, , China. 2 School of Business, Central South University, Changsha, 4183, China. 3 School of Mathematical Sciences, University of Jinan, Jinan, 2522, China. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 1 July 217 Accepted: 26 August 217 References 1. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York Lakshmikantham, V, Leela, S, Vasundhara Devi, J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge Henderson, J, Luca, R: Systems of Riemann-Liouville fractional equations with multi-point boundary conditions. Appl. Math. Comput. 39, Cabada, A, Kisela, T: Existence of positive periodic solutions of some nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 5, Ahmad, B, Ntouyas, BK, Tariboon, J: Nonlocal fractional-order boundary value problems with generalized Riemann-Liouville integral boundary conditions. J. Comput. Anal. Appl. 23, Mei, DZ, Peng, JG, Gao, JH: General fractional differential equations of order 1, 2 and type β [, 1] in Banach spaces. Semigroup Forum 94, El-Borai, MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, Zhou, Y, Jiao, F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, Wang, J, Zhou, Y: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, Chen, P, Zhang, PX, Li, Y: Study on fractional non-autonomous evolution equations with delay. Comput. Math. Appl. 73, Chen, P, Li, Y, Li, Q: Existence of mild solutions for fractional evolution equations with nonlocal initial conditions. Ann. Pol. Math. 11, Chen, P, Li, Y, Zhang, X: On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Commun. Pure Appl. Anal. 14, Wang, R, Ma, Q: Some new results for multi-valued fractional evolution equations. Appl. Math. Comput. 257, Zhao, J, Wang, R: Mixed monotone iterative technique for fractional impulsive evolution equations. Miskolc Math. Notes 17, Jabeena, T, Lupulescu, V: Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions. J. Nonlinear Sci. Appl. 1, Chen, P, Li, Y, Zhang, X: Existence and uniqueness of positive mild solutions for nonlocal evolution equations. Positivity 19, Deimling, K: Nonlinear Functional Analysis. Springer, New York Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cone. Academic Press, Orlando Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York Engel, K, Nagel, R: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York Mainardi, F, Paradisi, P, Gorenflo, R: Probability distributions generated by fractional diffusion equations. In: Kertesz, J, Kondor, I eds. Econophysics: An Emerging Science. Kluwer Academic, Dordrecht Hernandez, E, Sakthivel, R, Tanaka, AS: Existence results for impulsive evolution differential equations with state-dependent delay. Electron. J. Differ. Equ. 28, ArticleID28 28
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