Sin et al. Advances in Difference Equations 8) 8:6 https://doi.org/.86/s366-8-59-9 R E S E A R C H Open Access Existence and uniqueness of mild solutions to initial value problems for fractional evolution equations Chung-Sik Sin *,Hyok-Choln and Kwang-Chol Kim * Correspondence: chongsik@63.com Faculty of athematics, Kim l Sung University, Pyongyang, Democratic People s Republic of Korea Full list of author information is available at the end of the article Abstract The present paper deals with initial value problems for the fractional evolution equations involving the Caputo fractional derivative. By deriving a property of the three-parametric ittag Leffler function and using the Schauder fixed point theorem, new sufficient conditions for existence and uniqueness of mild solutions are established. SC: Primary 6A33; secondary 34K37; 34A8 Keywords: Fractional evolution equation; ild solution; Three-parametric ittag Leffler function; C semigroup; Existence and uniqueness ntroduction n this paper we consider the fractional evolution equation of the form D q xt)=axt)+f t, xt) ), t [, T],.) subject to the initial condition x) = x,.) where D q denotes the Caputo fractional derivative of order q, ), A : DA) B is the infinitesimal generator of a C semigroup Qt)} t of uniformly bounded linear operators on Banach space B, f :[,T] B B and x B. HereT > and the domain DA) is defined as the set of u B for which the following limit exists: Qt)u u Au = lim. t + t For more details as regards semigroup theory of operators, see []. Fractional differential equations have been widely applied in many important areas, including thermodynamics, porous media, plasma dynamics, cosmic rays, continuum mechanics, electrodynamics, quantum mechanics, biological systems and prime number theory [, 3]. n particular, the fractional diffusion equations have been successfully used in modeling anomalous diffusion processes with continuous time random walks [4]. The Authors) 8. This article is distributed under the terms of the Creative Commons Attribution 4. nternational License http://creativecommons.org/licenses/by/4./), 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.
Sinet al. Advances in Difference Equations 8) 8:6 Page of 3 Theoretical aspects for fractional evolution equations have been investigated by many mathematicians. El-Borai [5 7] used a probability density function to obtain the solutions to Cauchy problems for different fractional evolution equations. n, Hernandez et al. [8] proved that the concepts of mild solutions of fractional evolution equations considered in some previous papers were not appropriate. Based on the new definition of a mild solution obtained by employing the Laplace transform, Zhou et al. [9 4] established the existence and uniqueness results for mild solution of different kinds of fractional evolution equations. Wang et al. [5] revisited the nonlocal Cauchy problem for fractional evolution equations and relaxed the compactness and Lipschitz continuity on the nonlocal item given in the previous existence results. Fan et al. [6] usedthefixed point theorem for condensing maps to obtain the existence results for Eqs..).) under the noncompactness condition. n [7] the authors established the local existence and uniquenessof mild solution of Eqs..).) under Lipschitz condition and proved the continuous dependence of mild solution on the initial value and the fractional order. Ge et al. [8] considered the approximate controllability of the fractional evolution equations with nonlocal and impulsive conditions. Chen et al. [9] studied the existence of mild solutions for a nonautonomous fractional evolution equations with delay in Banach space. Yang and Wang [] established the existence and uniqueness of mild solutions of fractional evolution equations involving the Hilfer derivative by using the noncompact measure method. n [ 3], the authors investigated existence, uniqueness and asymptotic behavior of weak solutions of the initial boundary value problems for time fractional diffusion equations by employing the spectral decomposition of the symmetric uniformly elliptic operator. However, to the best of our knowledge, the existence and uniqueness of mild solution of the initial value problem.).) under compact condition have not been deeply investigated yet. n the present paper, we use properties of the three-parametric ittag Leffler function and fixed point theory to prove the existence and uniqueness of mild solution to Eqs..).). nparticular, theuniquenessresultforamildsolutionisobtainedwhen f satisfies a condition weaker than Lipschitz condition. This paper is organized as follows. n Sect., a new property of the three-parametric ittag Leffler function is established. n Sect. 3 weprovetheexistenceofmildsolutions of Eqs..).) by using the Schauder fixed point theorem and the new property of the three-parametric ittag Leffler function. Section 4 deals with the uniqueness of the mild solution. n Sect. 5, two examples are given for demonstration. Section 6 presents some concluding remarks. A property of the three-parametric ittag Leffler function n this section we prove a new property of the three-parametric ittag Leffler function which plays an important role in our investigation. Definition. [4]) Let β. The Riemann Liouville integral operator of order β is defined by being the identity operator and β yt)= Ɣβ) t s) β ys) ds for β >.
Sinet al. Advances in Difference Equations 8) 8:6 Page 3 of 3 Definition. [4]) Let β. The Caputo fractional differential operator of order β is defined by D β u = β β D β u, where is the ceiling function and D β is the classical differential operator of order β N. Definition.3 [5]) Let c, d, e >. The three-parametric ittag Leffler function is defined by E e c,d t)= i= e) i t i i!ɣci + d), where e) i = ee +) e + i ). Lemma.4 [5, 6]) Let c, d R, z C. [ )] Ɣz + c) Ɣz + d) = zc d +O argz + c) < π; z ). z Lemma.5 [5, 7]) Let c, d, e, γ >. γ [ Ec,d e λs c ) s d ]} t)=t d+γ Ec,d+γ e λt c ). Lemma.6 Let c, d, e, γ, η, h, r >.Then there exists a real number λ >such that, for t [, h], t η E e c,d+γ λt c ) < re e c,d λt c )..) Proof Firstlywetakeanintegeri N such that ci + d. We choose a real number t >suchthat,fort t, i i= e) i λ i t ci+η i!ɣci + d + γ ) r Ɣd). Set t = mint, r η }.ft t,thenwehave,foranyλ >, t η E e c,d+γ λt c ) = i= e) i λ i t ci+η i i!ɣci + d + γ ) = i= e) i λ i t ci+η i!ɣci + d + γ ) + < r Ɣd) + re) i λitci i!ɣci + d) < ree c,d λt c ). i=i i=i e) i λitci+η i!ɣci + d + γ ) Thus we need to find a real number λ > that satisfies.)fort [t, h]. By Lemma.4, there exists an integer i N such that, for i i, Ɣci + d)h η Ɣci + d + γ ) < r.
Sinet al. Advances in Difference Equations 8) 8:6 Page 4 of 3 There exists a real number λ >suchthat i i= i t ci +h e) i λ i h ci i!ɣci + d + γ ) < e) i λ i!ɣci + d + γ ). Then we have, for any t [t, h], t h Ec,d+γ e λ t c) i = i= e) i λ i t ci+h i!ɣci + d + γ ) + < e) i λ i t ci +h i!ɣci + d + γ ) + < re) i λ i t ci i!ɣci + d) + i=i + e) i λ i t ci+h i!ɣci + d + γ ) i=i i=i + e) i λ i t ci+h i!ɣci + d + γ ) re) i λ i t ci i!ɣci + d) < re) i λ i t ci i!ɣci + d) = ree c,d λ t c), i= which shows that λ satisfies.)fort [, h]. Theorem.7 Let c, e, h, r, γ >.f d < minγ,}, then there exists a real number λ >such that for t [, h], γ [ E e c, d λs c ) s d]} t)= Ɣγ ) t s) γ Ee c, d λsc ) s d ds < rec, d e λt c ). Proof By Lemma.5 and Lemma.6, we can easily prove this result. Remark.8 The above result is a generalization of the result for the two-parametric ittag Leffler function obtained in [8]. For more details as regards the ittag Leffler functions, see [5, 9, 3]. 3 Existence of mild solution n this section we use the property of the three-parametric ittag Leffler function to establish the existence results for mild solutions to the equation.).). Let be the norm of the Banach space B and C[, T],B) be the Banach space of continuous functions from [, T] intob with the supremum norm.letb be the space of all bounded linear operators from B to B with norm F = sup Fu) : u =,u B} for F B.Set = sup t [, ) Qt). Definition 3. [9, 4]) By the mild solution of the fractional evolution equations.).), we mean a function x C[, T], B)satisfying where xt)=s q t)x + S q t)u = t s) q P q t s)f s, xs) ) ds, t [, T], 3.) V q s)q t q s ) uds, P q t)u = qsv q s)q t q s ) uds, u B.
Sinet al. Advances in Difference Equations 8) 8:6 Page 5 of 3 Here V q s) is the ainardi function defined by [, 3]) V q s)= n= s) n n )!Ɣ qn), s C. We recall some properties of the operators S q t)andp q t). Lemma 3. []) For any t >,S q t) and P q t) are bounded linear operators. oreover, for any u B, S q t)u u, P q t)u Ɣq) u. Lemma 3.3 []) S q t)} t and P q t)} t are strongly continuous. That is, for t, t R and u B, S q t )u S q t )u and P q t )u P q t )u as t t. Lemma 3.4 []) f Qt) is a compact operator for any t >,then S q t) and P q t) are also compact operators for any t >. By considering the fixed point problem with J defined by Jxt)=S q t)x + t s) q P q t s)f s, xs) ) ds, we study the existence and uniqueness of solutions to Eqs..).). For the existence theorem, we make the following hypotheses. H3-) For any t >, Qt) is a compact operator. H3-) For a.e. t [, T],thefunctionf t, ):B B is continuous and for any x C[, T], B),thefunctionf, x):[,t] B is strongly measurable. H3-3) There exist T, T], l T, T), a, a, q, q [, q), p, p, ], b, b >, m t) L q [, T ], m t) L q [T, T] such that f t, u) b t a u p + m t) for t, T ] and u B, b t l a u p + m t) for t [T, l) l, T] and u B. Lemma 3.5 Let Y be a bounded subset of C[, T], B) and suppose that H3-), H3-), H3-3) hold. Then Jx : x Y } is equicontinuous. Proof Let H = sup x Y x.wehave,foranyx Y and t < t T, Jxt ) Jxt ) S q t )x S q t )x + t s) q P q t s)f s, xs) ) ds t s) q P q t s)f s, xs) ) ds t Sq t )x S q t )x + t s) q Pq t s)f s, xs) ) ds t
Sinet al. Advances in Difference Equations 8) 8:6 Page 6 of 3 + + t s) q Pq t s)f s, xs) ) P q t s)f s, xs) ) ds t s) q t s) q ) P q t s)f s, xs) ) ds = K + K + K + K 3, where K = Sq t )x S q t )x, K = K = K 3 = t t s) q Pq t s)f s, xs) ) ds, t s) q Pq t s)f s, xs) ) P q t s)f s, xs) ) ds, t s) q t s) q ) P q t s)f s, xs) ) ds. By Lemma 3.3, itisclearthatk ast t. Firstly we consider the case < t < t T. From H3-3), there exists a real number q 3 >suchthata < q 3 < q. ByH3-3), Lemma 3. and the Hölder inequality, we have K ) t s) q b xs) p + m Ɣq) t s a s) ds b H p t ) t s) q q3 q 3 ds s a Ɣq) t t m + Ɣq) L q [t,t ] ) q3 q 3 ds t s) q q q ds), t b H p ) q3 q3 q3 Ɣq) q q 3 q 3 a + ) q q t t ) q q m Ɣq) q q L ) q3 t t ) q q q 3 a 3 q t 3, q [t,t ] t q 3 a q 3 which implies that K ast t. By H3-3), Lemma 3. and the Hölder inequality, we have K 3 Ɣq) b H p Ɣq) + Ɣq) t s) q t s) q ) ) b xs) p + m s) ds b H p q3 Ɣq) q q 3 + q Ɣq) q q s a t s) q t s) q ) ) q3 q t ) 3 ds s a q3 q 3 ds t s) q t s) q ) q ds) q m ) q3 t q q 3 q 3 ) q t q q q q q 3 L q [,t ] q t 3 +t t ) q q 3 ) q q3 q3 3 q 3 a q q q t +t t ) q q ) q q m L q [,t ] ) q3 t q 3 a, ) q3
Sinet al. Advances in Difference Equations 8) 8:6 Page 7 of 3 which implies that K 3 ast t. By H3-) and Lemma 3.,forɛ, t ), K ɛ + t ɛ t s) q P q t s) P q t s) f s, xs) ) ds t s) q Pq t s) P q t s) f s, xs) ) ds sup Pq t s) P q t s) s [,t ɛ] + Ɣq) t ɛ t s) q b xs) ) p + m s) ds s a sup P q t s) P q t s) s [,t ɛ] + Ɣq) t ɛ ) t s) q b xs) p + m s) ds s a t s) q b H p t s) q b H p ) + m s) ds. s a s a ) + m s) ds Similar to K and K 3, by using Lemma 3.4 and the Hölder inequality, we can prove that K ast t, ɛ. Thus if < t < t T,then Jxt ) Jxt ) ast t. n the case t > T and the case t =, by using the same technique as above, we can complete the proof. Lemma 3.6 Let Y be a bounded subset of C[, T], B) and suppose that H3-), H3-), H3-3) hold. Then J is continuous on Y. Proof Let H = sup x Y x and x n } Y be a sequence such that lim n x n = x in C[, T], B). By the continuity of f with respect to the seconde variable, for a.e. t [, T], lim n f t, x n t)) = f t, xt)). Thus lim n t s) q f s, x n s)) = t s) q f s, xs)) for a.e. t [, T]ands [, t]. From H3-), we have t s) q f t, x n s) ) b t s) q s a H p + m t) fors, T ], b t s) q s l a H p + m t) fors [T, l) l, T]. t is easy to prove that the right side of the above inequality is integrable for s [, t]. We have, for t [, T], Jx n t) Jxt) Ɣq) t s) q P q t s)f s, x n s) ) P q t s)f s, xs) ) ds t s) q f s, x n s) ) t s) q f s, xs) ) ds. BytheLebesguedominatedconvergencetheorem,lim n Jx n t) =Jxt) for any t [, T]. Lemma 3.7 Let Y be a bounded subset of C[, T], B) and suppose that H3-), H3-), H3-3) hold. Then, for any t [, T], Jxt):x Y} is relatively compact.
Sinet al. Advances in Difference Equations 8) 8:6 Page 8 of 3 Proof By using the same technique as Theorem 3. of [9], we can prove this result. Nowwewillprovethemainresultofthissection. Theorem 3.8 Suppose that H3-), H3-), H3-3) hold. Then the fractional evolution equations.).) have a mild solution. Proof Firstly, by using the new property of the three-parametric ittag Leffler function obtained in Sect., we will show that there exists a convex bounded closed subset G C[, T], B) suchthatjg G. ByTheorem.7, thereexistλ, λ, λ 3 >suchthat,fort [, T], q [ E 3, a λ s 3) s a ]} t)< q a [ E 3, λ s 3)]} t)< q [ E 3, a λ3 s 3) s a ]} t)< b E 3, a λ t 3), Ɣq) b Ɣq a ) E 3, λ t 3), b E 3, a λ3 t 3). We define a convex bounded closed subset G of C[, T], B) as follows: G = x C [, T], B ) : D E3, a λ t 3 ) fort [, T ], xt) D E3, λ t 3 ) fort [T, l], D 3 E3, a λ 3 t l) 3 ) fort [l, T], where D = Ɣ a ) max, x + D =D E 3, a λ T 3 ) + Ɣq) D 3 = Ɣ a ) D E3, λ l 3) + Ɣq) ) q q q q T m Ɣq) q q L ) q q l T ) q q m q q L ) q q T l) q q m q q By the Hölder inequality, we have, for any x G and t [, T ], t Jxt) Sq t)x + t s) q Pq t s)f s, xs) ) ds x + t s) q b xs) ) p + m Ɣq) s a s) ds x +b D q [ E3, a λ s 3) s a ]} t) + t s) q q q ds) m Ɣq) L q [,T ] D Ɣ a ) + D E 3, a λ t 3) <D E 3, a λ t 3). q [,T ] q [T,l], )},. } L q [l,t]
Sinet al. Advances in Difference Equations 8) 8:6 Page 9 of 3 We have, for any x G and t [T, l], T Jxt) Sq t)x + t s) q Pq t s)f s, xs) ) ds + t s) q P q t s)f s, xs) ) ds T D E3, a 3 λ T ) + t s) q b xs) ) p + m Ɣq) T l s) a s) ds D E3, a 3 λ T ) + t s) q q q ds) m Ɣq) T L q [T,l] + b Ɣq a ) [ D q a E Ɣq) 3, λ s 3)]} t) D E3, λ t 3). We have, for any x G and t [l, T], l Jxt) Sq t)x + t s) q Pq t s)f s, xs) ) ds + t s) q P q t s)f s, xs) ) ds l D E3, λ l 3) + t s) q b xs) ) p + m Ɣq) l s l) a s) ds D E3, λ l 3) + t s) q m s) ds + l b Ɣq) l Ɣq) s t l s)q xs + l) p ds a D E3, λ l 3) + t s) q q q ds) m Ɣq) L q [l,t] +b D 3 q [ E 3, a λ3 s 3)]} t l) D 3 E 3, a λ3 t l) 3). l Thus JG G. By Lemma 3.5, Lemma 3.7 and the Ascoli theorem, Jx : x G} is relatively compact. Lemma 3.6 and relatively compactness of Jx : x G} imply that J is a compact operator. Thus, from the Schauder fixed point theorem, J has at least one fixed point in G. Remark 3.9 The condition H3-3) of Theorem 3.8 can be replaced by the following condition. There exist n N, =T < T < < T n = T, l i T i, T i )fori =,...,n, a j, q j q [, q), p j, ], b j >,m j t) L j [T j, T j ]forj =,...,n such that f t, u) b t a u p + m t) fort, T ]andu B, b i t l i a i u p i + m i t) fort [T i, l i ) l i, T i ]andu B, where i =,...,n.
Sinet al.advances in Difference Equations 8) 8:6 Page of 3 4 Uniqueness of mild solution This section discusses the uniqueness of mild solutions of.).). For the uniqueness theorem, we make the following hypotheses. H3-) f :[,T] B B is continuous. H3-3) There exist constants a, a [, q), b, b >, T, T], l T, T),suchthat f t, u) f t, v) b t a u v for t, T ], u, v B, b t l a u v for t [T, l) l, T], u, v B. Theorem 4. Suppose that H3-), H3-) and H3-3) hold. Then the fractional evolution equations.).) have a unique mild solution. Proof From the conditions H3-) and H3-3), we can easily prove that H3-3) holds. Thus, by Theorem 3.8,the fractionalevolutionequations.).)have at least one mild solution. By using the method of proof by contradiction, we will establish a uniqueness result for mild solutions of Eqs..).). Assume that.).) have two solutions. Then the operator J has also two fixed points x, y such that x y >. By Theorem.7, there exists a real number λ >suchthat,fort [, T], q [ E 3, a λ s 3) s a ]} t)< b E 3, a λ t 3). We define W and L as follows: W = inf w : xt) yt) we 3, a λ t 3), t [, T ] }, L = inf t [, T ]: xt) yt) = W E 3, a λ t 3)}. f W,thenwehave W E 3, a λ L 3 ) = xl ) yl ) L Ɣq) b Ɣq) L s) q P q L s) f s, xs) ) f s, ys) ) ds L L < W E 3, a λ L 3 ), L s) q b xs) ys) ds s a L s) q s a W E 3, a λ s 3) ds which implies that W =. Therefore xt)=yt), t [, T ]. By Theorem.7,thereexistsa real number λ >suchthat,fort [, T], q a [ E 3, λ s 3)]} t)< We define W and L as follows: Ɣq) b Ɣq a ) E 3, λ t 3). W = inf w : xt) yt) we 3, λ t 3), t [T, l] }, L = inf t [T, l]: xt) yt) = W E 3, λ t 3)}.
Sinet al.advances in Difference Equations 8) 8:6 Page of 3 f W,thenwehave W E 3, λ L 3 ) = xl ) yl ) L Ɣq) L s) q P q L s) f s, xs) ) f s, ys) ) ds L L s) q b xs) ys) ds T l s) a L b L s) q a W E Ɣq) 3, λ s 3) ds < W E3, 3 λ L ), which implies that W =. Therefore xt)=yt), t [, l]. By Theorem.7, thereexistsa real number λ 3 >suchthat,fort [, T], q [ E 3, a λ3 s 3) s a ]} t)< b E 3, a λ3 t 3). We define W 3 and L 3 as follows: W 3 = inf w : xt) yt) we 3, a λ3 t l) 3), t [l, T] }, L 3 = inf t [l, T]: xt) yt) = W 3 E 3, a λ3 t l) 3)}. From the assumption x y >,itisclearthatw 3.Thenwehave W 3 E 3, a λ3 L 3 l) 3) = xl 3 ) yl 3 ) L3 l Ɣq) b Ɣq) L 3 s) q P q L 3 s) f s, xs) ) f s, ys) ) ds L3 l L3 l L 3 s) q b xs) ys) ds s l) a < W 3 E 3, a λ3 L 3 l) 3). L 3 l s) q s a W 3E 3, a λ3 s 3) ds This contradiction shows that Eqs..).) have a unique mild solution. Remark 4. The condition H3-) of Theorem 4. can be replaced by the following condition. There exists a real number q [, q)suchthat f, ) L q [, T] and H3-) holds. Here is the zero vector of the Banach space B. Remark 4.3 The condition H3-3) of Theorem 4. can be replaced by the following condition. There exist n N,=T < T < < T n = T, l i T i, T i )fori =,...,n, a j [, q), b j >forj =,...,n such that f t, u) f t, v) b t a u v for t, T ]andu, v B, b i t l i a u v for t [T i i, l i ) l i, T i ]andu, v B, where i =,...,n.
Sinet al.advances in Difference Equations 8) 8:6 Page of 3 Remark 4.4 f the exponential function is used instead of the ittag Leffler function in proving the uniqueness result, we can only establish the uniqueness result when f satisfies the Lipschitz condition. Theorem 3.8 and Theorem 4. canbe proved onlyif the ittag Leffler function is employed. 5 Applications n this section we discuss existence and uniqueness of mild solutions of time fractional diffusion equations as an application of main results. The existence of solutions of the following equations cannot be proved by previous results. Example Consider the fractional diffusion equation of the form D.7 t xt, u)= xt, u)+ 3x.8 t,u), <t T, u 3, t.5 xt, ) =xt, 3) =, t [, T], x, u)=, u [, 3]. 5.) Define the operator A by A = with the domain DA) =x ) B : x, x are absolutely u continuous, x B,andx) = x3) = } where B = L [, 3]. Then the operator A generates a strongly continuous semigroup. For more details as regards this conclusion, please refer to []. Since f t, y)= 3y.8 and q =.7 >.5, by Theorem 3.8, Eq.5.) hasamildsolution t.5 in C[, T], L [, 3]). Example Consider the fractional diffusion equation of the form D.8 t xt, u)= xt, u)+ 8xt,u), <t T, u, t. xt, u) u =, t [, T], x, u)=x u), u, 5.) where is a bounded domain with smooth boundary in R 3 and x H ) H ). We denote A = + + and B = L ). Then the operator A is a strongly elliptic u u u 3 operator defined in H ) H ) and the operator A generates an analytic semigroup on L )see[]). By Theorem 4.,Eq.5.) has a unique mild solution in C[, T], L )). 6 Conclusion n this paper the existence and uniqueness of mild solutions of the initial value problems of fractional evolution equations are proved under some appropriate conditions by using a fantastic property of the ittag Leffler function. n particular, the uniqueness result of mild solution is established when f satisfies the condition close to Nagumo-type condition. n the future, we will investigate the initial value problems for different fractional differential equations by employing the proof technique used in the present paper. Acknowledgements The authors would like to thank referees for their valuable advices for the improvement of this article. Competing interests The authors declare that they have no competing interests. Authors contributions SCS, HC and GCK participated in obtaining the main results of this manuscript and drafted the manuscript. All authors read and approved the final manuscript.
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