INTEGRAL SOLUTIONS OF FRACTIONAL EVOLUTION EQUATIONS WITH NONDENSE DOMAIN

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1 Electronic Journal of Differential Equations, Vol. 217 (217, No. 145, pp ISSN: URL: or INTEGRAL SOLUTIONS OF FRACTIONAL EVOLUTION EQUATIONS WITH NONDENSE DOMAIN HAIBO GU, YONG ZHOU, BASHIR AHMAD, AHMED ALSAEDI Communicated by Mokhtar Kirane Abstract. In this article, we study the existence of integral solutions for two classes of fractional order evolution equations with nondensely defined linear operators. First, we consider the nonhomogeneous fractional order evolution equation and obtain its integral solution by Laplace transform and probability density function. Subsequently, based on the form of integral solution for nonhomogeneous fractional order evolution equation, we investigate the existence of integral solution for nonlinear fractional order evolution equation by noncompact measure method. 1. Introduction We consider the nonhomogeneous fractional order evolution equation C D q + u(t Au(t + f(t, t (, b], (1.1 u( u and the nonlinear fractional order evolution equation C D q + u(t Au(t + g(t, u(t, u( u, t (, b], (1.2 where C D q + is the Caputo fractional derivative of order < q < 1, the state u( takes values in a Banach space X with norm, A : D(A X X is a nondensely closed linear operator on X, f and g are given functions satisfying appropriate conditions. For the integer order evolution equation: { u (t Au(t + f(t, u(t, t (, b], u( u, in case A is a Hille-Yosida operator and is densely defined (i.e., D(A X, the problem has been extensively studied (see [15]. When A is a Hille-Yosida operator but its domain is nondensely defined, there have many results (see [2, 8, 18, 19] 21 Mathematics Subject Classification. 26A33, 34K37, 37L5, 47J35. Key words and phrases. Fractional evolution equation; Caputo derivative; integral solution; nondense domain. c 217 Texas State University. Submitted March 11, 217. Published June 19,

2 2 H. B. GU, Y. ZHOU, B. AHMAD, A. ALSAEDI EJDE-217/145 and the references therein. It is noted that Da Prato and Sinestrari are the first to work on equations with nondense domains, see [5]. On the other hand, fractional order differential equations have recently been applied in various areas of engineering, physics and bio-engineering, and other applied sciences. For some fundamental results in the theory of fractional calculus and fractional differential equations, we refer the reader to the monographs by Samko et al. [17], Kilbas et al. [9], Diethelm [4] and Zhou [3], and the papers [1, 1, 29, 24, 27, 25, 26, 28] and the references cited therein. For nonlinear fractional evolution equation (1.2 with initial data or nonlocal condition, when A is densely defined, there have been many results on the existence of mild solutions (see [11, 16, 2, 23]. In [23], by using similar methods due to El-Borai [6, 7], Zhou and Jiao proposed a suitable concept on mild solution by applying probability density function and Laplace transform, which is widely used now. When A is not densely defined, there have been some investigations (see [13, 22]. However, in [13], there was an error in transforming integral solution into an available form. Zhang et al. [22] presented a formula for integral solution by using the similar method described in [23], but the equivalency of integral equations was not proved. Motivated by the above discussion, in this paper, we will firstly give the integral solution for nonhomogeneous fractional evolution equation (1.1 by Laplace transform and probability density function, and subsequently investigate the existence of integral solution for nonlinear fractional order evolution equation (1.2 by Ascoli-Arzela theorem and the measure of noncompactness. In what follows we do not require the C semigroup (will be given later to be compact. The rest of this paper is organized as follows. In Section 2, notation and preinaries are given. The integral solution of nonhomogeneous fractional evolution equation (1.1 is given in Section 3. In Section 4, the existence of integral solution for nonlinear fractional order evolution equation (1.2 is studied. The paper concludes with a problem proposed for further research. 2. Preinaries In this section, we recall some concepts on fractional calculus and present some lemmas and assumptions which are useful in the sequel. Let p >, n p (the least integer greater than or equal to p and u L 1 ([, b], X. The Riemann-Liouville fractional integral is defined by I p + u(t g p(t u(t g p (t su(sds, t >, where denotes convolution and g p (t t p 1 /Γ(p. In case p, we set g (t δ(t, the Dirac measure concentrated at the origin. For u C([, b], X, the Riemann-Liouville fractional derivative is defined by L D p dn + u(t dt n (g n p(t u(t and the Caputo fractional derivative can be defined by C D p + u(t g n p(t dn u(t dt n for all t >. For more details, see [9].

3 EJDE-217/145 FRACTIONAL EVOLUTION EQUATIONS 3 Next, we introduce the Hausdorff measure of noncompactness β( defined on each bounded subset Ω of Banach space X by β(ω inf{ɛ >, Ω has a finite ɛ-net in X}. Some basic properties of β( are listed in the following Lemmas. Lemma 2.1 ([3]. The noncompact measure β( satisfies: (i for all bounded subsets B 1, B 2 of X, B 1 B 2 implies β(b 1 β(b 2 ; (ii β({x} B β(b for every x X and every nonempty subset B X; (iii β(b if and only if B is relatively compact in X; (iv β(b 1 + B 2 β(b 1 + β(b 2, where B 1 + B 2 {x + y : x B 1, y B 2 }; (v β(b 1 B 2 max{β(b 1, β(b 2 }; (vi β(λb λ β(b for any λ R. Lemma 2.2 ([14]. Let J [, b] and {u n } n1 be a sequence of Bochner integrable functions from J into X with u n (t m(t for almost all t J and every n 1, where m L(J, R +. Then the function ψ(t β({u n (t} n1 belongs to L(J, R + and satisfies ({ } β u n (sds : n 1 2 ψ(sds. Let X D(A and A be the part of A in D(A defined by D(A {x D(A : Ax D(A}, A x Ax. Proposition 2.3 ([15]. The part A of A generates a strongly continuous semigroup(that is, C semigroup {Q(t} t on X. In the forthcoming analysis, we need the following hypothesis: (H1 The linear operator A : D(A X X satisfies the Hille-Yosida condition, that is, there exist two constant ω R and M such that (ω, + ρ(a and (λi A k L(X M, for all λ > ω, k 1. (λ ω k (H2 Q(t is continuous in the uniform operator topology for t >, and {Q(t} t is uniformly bounded, that is, there exists M > 1 such that sup t [,+ Q(t < M. 3. Integral solution to nonhomogeneous Cauchy problem Here we derive the integral solution for nonhomogeneous fractional order evolution equation (1.1 with the aid of Laplace transform and probability density function. For Cauchy problem (1.1, it is assumed that u X and f : J X is continuous. Definition 3.1. A function u(t is said to be an integral solution of (1.1 if (i u : J X is continuous; (ii I q + u(t D(A for t J and (iii u(t u + AI q + u(t + Iq + f(t, t J. (3.1

4 4 H. B. GU, Y. ZHOU, B. AHMAD, A. ALSAEDI EJDE-217/145 Remark 3.2. If u(t is an integral solution of (1.1, then u(t X for t J. In fact, by I q + u(t D(A, we have I1 +u(t I 1 q + Iq + u(t D(A for t J. Then u(t 1 t+h h + h u(sds X t for t J. Definition 3.3 ([12]. The Wright function M q (θ defined by ( θ n 1 M q (θ (n 1!Γ(1 qn is such that n1 θ δ M q (θdθ Consider the auxiliary problem C D q + u(t A u(t + f(t, Γ(1 + δ, for δ. Γ(1 + qδ u( u. t (, b], By Definition 3.1, the integral solution of (3.2 can be written as (3.2 u(t u + A I q + u(t + Iq + f(t (3.3 for u X and t J. The following Lemma gives an equivalent form of (3.3 by means of Laplace transform. Lemma 3.4. If f take values in X, then the integral equation (3.3 can be expressed as ( u(t I 1 q + K q(t u + K q (t sf(sds, t J, (3.4 where K q (t t q 1 P q (t, P q (t Proof. Let λ >. Applying the Laplace transform χ(λ to (3.3, we obtain e λs u(sds and ω(λ χ(λ λ 1 u + 1 λ q A χ(λ + 1 λ q ω(λ qθm q (θq(t q θdθ. λ q 1 (λ q I A 1 u + (λ q I A 1 ω(λ λ q 1 e λqs Q(su ds + e λs f(sds e λqs Q(sω(λds, (3.5 provided that the integrals in (3.5 exist, where I is the identity operator defined on X. The Laplace transform of ψ q (θ q θ q+1 M q(θ q, is e λθ ψ q (θdθ e λq, (3.6

5 EJDE-217/145 FRACTIONAL EVOLUTION EQUATIONS 5 where q (, 1. Using (3.6, we have and e λqs Q(su ds e λqs Q(sω(λds e λt[ q qt q 1 e (λtq Q(t q u dt e λt[ q qψ q (θe (λtθ Q(t q t q 1 u dθdt ( t qψ q (θe λt q Q e λt t q 1 P q (tu dt qt q 1 e (λtq Q(t q e λs f(sdsdt t q 1 θ q θ q u dθdt ( t q t q 1 ψ q (θq θ q θ q qψ q (θe (λtθ Q(t q e λs t q 1 f(sdθdsdt ( t qψ q (θe λ(t+s q Q t q 1 θ q θ q f(sdθdsdt ( (t s q (t s q 1 ψ q (θq θ q e λt[ ] (t s q 1 P q (t sf(sds dt. Since the Laplace inverse transform of λ q 1 is L 1 (λ q 1 θ q t q Γ(1 q g 1 q(t, therefore, by (3.5, (3.7 and (3.8, for t J, we obtain u(t This completes the proof. ( L 1 (λ q 1 K q (t u + ( I 1 q + K q(t u + ] u dθ dt ] f(sdθds dt K q (t sf(sds K q (t sf(sds. (3.7 (3.8 (3.9 Remark 3.5. Let S q (t I 1 q + K q(t. By the uniqueness of Laplace inverse transform, it is obvious that operators S q (t and P q (t (obtained here are the same as the ones given in [23]. In addition, we also obtain the relationship between S q (t and K q (t; that is, S q (t I 1 q + K q(t for t. So, we can say that {K q (t} t is generated by A. Proposition 3.6 ([3]. With assumption (H2, P q (t is continuous in the uniform operator topology for t >.

6 6 H. B. GU, Y. ZHOU, B. AHMAD, A. ALSAEDI EJDE-217/145 Proposition 3.7 ([23]. With assumption (H2, for any fixed t >, {K q (t} t> and {S q (t} t> are linear operators, and for any x X, K q (tx Mtq 1 x and S q(tx M x. Proposition 3.8 ([23]. With assumption (H2, {K q (t} t> and {S q (t} t> are strongly continuous, that is, for any x X and < t < t b, or K q (t x K q (t x and S q (t x S q (t x, as t t. If we assume that f takes values in X, then (3.4 can be written as u(t S q (tu + u(t S q (tu + K q (t s B λf(sds (3.1 K q (t sb λ f(sds, (3.11 where B λ λ(λi A 1, since B λ x x for x X. When f takes values in X, but not in X, then the it in (3.11 exists (as we will prove. But the it in (3.1 will no longer exist. Lemma 3.9. Any solution of integral equation (3.1 with values in X is represented by (3.11. Proof. Let By applying B λ to (3.1, we have Hence, by Lemma 3.4, we obtain As u(t, u X, we have u λ (t B λ u(t, f λ (t B λ f(t, u λ B λ u. u λ (t u λ + A I q + u λ(t + I q + f λ(t. u λ (t S q (tu λ + K q (t sf λ (sds. u λ (t u(t, u λ u, S q (tu λ S q (tu, as λ +. Thus (3.11 holds. This completes the proof. Let us define Φ q (tx for x X and t. K q (t sb λ x ds K q (sb λ x ds, (3.12 Proposition 3.1. For x X and t, the it in (3.12 exists and defines a bounded linear operator Φ q (t. Proof. Let Φ q(tx K q (t sx ds for x X and t. Then, the definition K q (sx ds, Φ q (t (λi AΦ (t(λi A 1,

7 EJDE-217/145 FRACTIONAL EVOLUTION EQUATIONS 7 for λ > ω, extends Φ q(t from X to X. This definition is independent of λ because of the resolvent identity. As Φ q (t maps X into X, we have Φ q (tx This completes the proof. B λφ q (tx Φ q(tb λ x. Proposition For x X and t, C D q + Φ q(tx S q (tx and S q (tx AΦ q(tx + x. The proof of the above proposition follows directly from the definitions of S q (t and Φ q(t for t. Lemma (i For x X and t, I q + Φ q(t D(A and (ii For x D(A, Proof. (i For x X and t, let Φ q (tx A ( I q + Φ q(tx + V (t λi q + Φ q(t(λi A 1 x + Clearly V (. By Proposition 3.11, we have C D q + V (t t q x. (3.13 Γ(1 + q Φ q (tax + x S q (tx. (3.14 t q Γ(1 + q (λi A 1 x Φ q(t(λi A 1 x. Then and Thus λφ q(t(λi A 1 x + (λi A 1 x C D q + Φ q(t(λi A 1 x λφ q(t(λi A 1 x + (λi A 1 x S q (t(λi A 1 x λφ q(t(λi A 1 x + (λi A 1 x AΦ q(t(λi A 1 x (λi A 1 x λφ q(t(λi A 1 x AΦ q(t(λi A 1 x (λi AΦ q(t(λi A 1 x Φ q (tx. V (t I q + Φ q(tx + V ( I q + Φ q(tx (λi AV (t (λi AI q + Φ q(tx λi q + Φ q(tx + Φ q (tx A ( I q + Φ q(tx + t q Γ(1 + q x. (ii For x D(A, it follows by Proposition 3.11 that Φ q (tax This completes the proof. K q (sb λ Ax ds A Φ q(tx S q (tx x. A t q Γ(1 + q x Φ q(tx. K q (sb λ x ds

8 8 H. B. GU, Y. ZHOU, B. AHMAD, A. ALSAEDI EJDE-217/145 Theorem u(t is an integral solution of (1.1 if and only if for t J and u X. u(t S q (tu + K q (t sb λ f(sds (3.15 Proof. In view of Lemma 3.9, we only need to show that (3.15 is the integral solution of (1.1. Indeed it is sufficient to prove the theorem for u, because it can easily be proved for the special case f. We complete the proof in two steps. Step I. Assume that f is continuously differentiable, then for t J, we have u λ (t K q (sb λ f(sds K q (sb λ (f( + K q (sb λ f(ds + Φ q(tb λ f( + By Lemma 3.12, for t J, we obtain u(t u λ(t Φ q (tf( + s f (rdr ds K q (sb λ ( s Φ q(t rb λ f (rdr. Φ q (t rf (rdr f (rdr ds A ( I q + Φ q(tf( t q + Γ(1 + q f( + [A ( I q + Φ q(t r (t ] rq + f (rdr Γ(1 + q [ ] A I q + Φ q(tf( + I q + Φ q(t rf (rdr t q + Γ(1 + q f( + 1 (t r q f (rdr Γ(1 + q [ ( A I q + Φ q(tf( + I q t ] + Φ q (t rf (rdr + t q Γ(1 + q f( + 1 Γ(1 + q A ( I q + u(t + I q + f(t. (t r q f (rdr Step II. We approximate f by continuously differentiable functions f n such that Letting sup f(t f n (t, as n. t J u n (t λ K q (sb λ f n (sds,

9 EJDE-217/145 FRACTIONAL EVOLUTION EQUATIONS 9 we have Then u n (t A ( I q + u n(t + I q + f n(t. u n (t u m (t λ MM MMbq f n f m, K q (sb λ [ fn (s f m (s ] ds (t s q 1 f n (s f m (s ds which implies that {u n } is a Cauchy sequence and its it, denoted by u(t, exists. Taking it on both sides of (3, we obtain u(t A ( I q + u(t + I q + f(t, for t J. Therefore, (3.15 is the integral solution of (1.1. This completes the proof. Remark (i Integrating the last term in (3.15 and using Proposition 3.8, the integral solution (3.15 can be expressed as u(t S q (tu + d dt Φ q (t sf(sds. (ii (λ q I A 1 x λ e λt Φ q (tx ds for x X and λ q > ω. In fact, by taking Laplace transform of (3.13, we obtain L[Φ q (tx] AL[I q + Φ q(tx] + L[ Γ(1 + q x] t q λ q AL[Φ q (tx] + λ q 1 x λ 1 (λ q I A 1 x. (iii We can say that A generates the operator {Φ q (t} t. When q 1, {Φ q (t} t degenerates into {S(t} t, which is integrated semigroup generated by A in [18]. 4. Integral solution to a nonlinear Cauchy problem In this section, we study the existence of integral solution of nonlinear fractional evolution equation (1.2. We need the following assumptions: (H3 for each t J, the function g(t, : X X is continuous and for each x X, the function g(, x : J X is strongly measurable; (H4 there exists a function m L(J, R + such that I q + m(t C(J, R+, t + Iq + m(t, g(t, x m(t for all x X and almost all t J; (H5 there exists a constant l > such that for any bounded D X, β(g(t, D lβ(d, for a.e. t J.

10 1 H. B. GU, Y. ZHOU, B. AHMAD, A. ALSAEDI EJDE-217/145 By Theorem 3.13, it is easy to see that the integral solution of (1.2 is equal to the solution of or u(t S q (tu + d dt u(t S q (tu + For u C(J, X, define an operator where (T u(t (T 1 u(t + (T 2 u(t, (T 1 u(t S q (tu and (T 2 u(t for all t J. Let B r (J {u C(J, X : u r}. Φ q (t sg(s, u(sds (4.1 K q (t sb λ g(s, u(sds. (4.2 K q (t sb λ g(s, u(sds, Lemma 4.1. Suppose that conditions (H1 (H4 hold. Then {T u : u B r (J} is equicontinuous. Proof. By Proposition 3.8, S q (tu is uniformly continuous on J. Consequently, {T 1 u : u B r (J} is equicontinuous. For u B r (J, taking t 1, < t 2 b, we obtain (T 2 u(t 2 (T 2 u( For < t 1 < t 2 b, we have 2 2 MM K q (t sb λ g(s, u(sds (t 2 s q 1 m(sds, as t 2. (T 2 u(t 2 (T 2 u(t 1 2 (t 2 s q 1 P q (t 2 sb λ g(s, u(sds + + t MM + MM + t 1 1 (t 2 s q 1 P q (t 2 sb λ g(s, u(sds (t 1 s q 1 P q (t 2 sb λ g(s, u(sds (t 1 s q 1 P q (t 2 sb λ g(s, u(sds (t 1 s q 1 P q (t 1 sb λ g(s, u(sds (t 2 s q 1 m(sds 1 I 1 + I 2 + I 3, [(t 1 s q 1 (t 2 s q 1 ]m(sds (t 1 s q 1 [P q (t 2 s P q (t 1 s]b λ g(s, u(sds

11 EJDE-217/145 FRACTIONAL EVOLUTION EQUATIONS 11 where I 1 MM 2 (t 2 s q 1 m(sds 1 1 (t 1 s q 1 m(sds, I 2 2MM [ (t 1 s q 1 (t 2 s q 1] m(sds, 1 I 3 (t 1 s q 1 [P q (t 2 s P q (t 1 s]b λ g(s, u(sds. By condition (H4, one can deduce that t2 t 1 I 1. Noting that [ (t1 s q 1 (t 2 s q 1] m(s (t 1 s q 1 m(s, and 1 (t 1 s q 1 m(sds exists, it follows by Lebesgue dominated convergence theorem that t1 [(t 1 s q 1 (t 2 s q 1 ]m(sds, as t 2 t 1, which implies that t2 t 1 I 2. For ε > small enough, by (H4, we have where I 3 M + M M 1 ε 1 t 1 ε 1 (t 1 s q 1 P q (t 2 s P q (t 1 s g(s, u(s ds (t 1 s q 1 P q (t 2 s P q (t 1 s g(s, u(s ds (t 1 s q 1 m(sds sup 1 s [,t 1 ε] + 2MM (t 1 s q 1 m(sds t 1 ε I 31 + I 32 + I 33, I 32 2MM I 31 r MM 1 (t 1 s q 1 m(sds I 33 2MM 1 ε P q (t 2 s P q (t 1 s sup P q (t 2 s P q (t 1 s, s [,t 1 ε] 1 ε (t 1 ε s q 1 m(sds, [(t 1 ε s q 1 (t 1 s q 1 ]m(sds. By Proposition 3.6, it follows that I 31 as t 2 t 1. Applying the arguments similar to the ones employed in proving that I 1, I 2 tend to zero, we obtain I 32 and I 33 as ε. Thus, I 3 tends to zero independently of u B r (J as t 2 t 1, ε. Therefore, (T 2 u(t 2 (T 2 u(t 1 independently of u B r (J as t 2 t 1, which implies that {T 2 u : u B r (J} is equicontinuous. Therefore, {T u : u B(J} is equicontinuous. The proof is complete. Lemma 4.2. Assume that (H1 (H4 hold. Then T maps B r (J into B r (J, and is continuous in B r (J.

12 12 H. B. GU, Y. ZHOU, B. AHMAD, A. ALSAEDI EJDE-217/145 Proof. Claim I. T maps B r (J into B r (J. Obviously, by (H4, there exists a constant r > such that ( { M t } M u + sup (t s q 1 m(sds r. t J For any u B r (J, by Proposition 3.7, we have (T u(t S q (tu + M u + MM ( M u + sup t J Hence T u r for any u B r (J. { M K q (t sb λ g(s, u(sds (t s q 1 g(s, u(s ds } (t s q 1 m(sds r. Claim II. T is continuous in B r (J. For any u m, u B r (J, m 1, 2,..., with m u m u, by (H3, we have g(t, u m (t g(t, u(t as m, for t J. On the one hand, using (H4, for each t J, we obtain (t s q 1 g(s, u m (s g(s, u(s 2(t s q 1 m(s, a.e. in [, t. As the function s 2(t s q 1 m(s is integrable for s [, t and t J, by Lebesgue dominated convergence theorem, we obtain For t J, we obtain (t s q 1 g(s, u m (s g(s, u(s ds as m. (T u m (t (T u(t K q (t sb λ (g(s, u m (s g(s, u(sds MM (t s q 1 g(s, u m (s g(s, u(s ds as m. Therefore, T u m T u pointwise on J as m. Hence it follows by Lemma 4.1 that T u m T u uniformly on J as m and so T is continuous. The proof is complete. Theorem 4.3. Assume that (H1 (H5 hold. Then the Cauchy problem (1.2 has at least one integral solution in B r (J. Proof. Let y (t S q (tu for all t J and y m+1 T y m, m, 1, 2,. Consider the set H {y m : m, 1, 2, }, and show that it is relatively compact. By Lemmas 4.1 and 4.2, H is uniformly bounded and euqicontinuous on J. Next, for any t J, we just need to show that H (t {y m (t, m, 1, 2, } is relatively compact in X. By the assumption (H5 together with Lemmas 2.1 and 2.2, for any t J, we have β ( H (t ( ( ( β {y m (t} m β {y (t} {y m (t} m1 β {y m (t} m1

13 EJDE-217/145 FRACTIONAL EVOLUTION EQUATIONS 13 and β Thus ( ( {y m (t} m1 β {(T y m (t} m ({ β S q (tu + ({ β 2MM 2MMl } K q (t sb λ g(s, y m (sds m K q (t sb λ g(s, y m (sds } m ( (t s 1 q β g(s, {y m (s} m ds (t s 1 q β ( {y m (s} m ds. β(h (t 2MMl (t s 1 q β(h (sds. Therefore, by generalized Grownwall s inequality [21], we infer that β(h (t. In consequence, H (t is relatively compact. Hence, it follows from Ascoli- Arzela theorem that H is relatively compact. Therefore, there exists a convergent subsequence of {y m } m. For the sake of clarity, let m y m y B r (J. Thus, by continuity of the operator T, we have y m y m m T y m 1 T ( m y m 1 T y, which implies the Cauchy problem (1.2 has least an integral solution. 5. An example As an application of our results we consider the fractional time partial differential equation q 2 z(t, x z(, x + G(t, z(t, x, tq x2 x [, π], t (, b], < q < 1, z((t, z(t, π, t (, b], z(, x z, x [, π], where G : [, b] R R is a given function. Let u(t(x z(t, x, g(t, u(x G(t, u(x, t [, b], x [, π], t [, b], x [, π]. (5.1 We choose X C([, π], R endowed with the uniform topology and consider the operator A : D(A X X defined by: D(A {u C 2 ([, π], R : u( u(π }, Au u. It is well known that the operator A satisfies the Hille-Yosida condition with (, + ρ(a, (λi A 1 1 λ for λ >, and D(A {u X : u( u(π } X. We can show that problem (1.2 is an abstract formulation of problem (5.1. Under suitable conditions, Theorem 4.3 implies that problem (5.1 has a unique solution z on [, b] [, π].

14 14 H. B. GU, Y. ZHOU, B. AHMAD, A. ALSAEDI EJDE-217/145 Concluding remarks. In this article, we have obtained the integral solution for nonhomogeneous Cauchy problem (1.1 and established the relationship between {S q (t} t and {K q (t} t. Also sufficient conditions ensuring the existence of integral solutions to nonlinear Cauchy problem (1.2, involving a linear closed operator A of Hille-Yosida type with not densely defined domain, are presented. For further research, we propose the following open problem: How to establish the existence of an integral solution to fractional evolution equation (1.2 when linear closed operator A is not a Hille-Yosida type and its domain is not densely defined? Acknowledgements. The authors would like to express their thanks to the editor and anonymous referees for their suggestions and comments that improved the quality of the paper. The second author is supported by National Natural Science Foundation of China ( References [1] A. Alsaedi, B. Ahmad, M. Kirane; Maximum principle for certain generalized time and space fractional diffusion equations, Quarterly of Applied Mathematics, 73.1 (215, [2] N. Abada, M. Benchohra, H. Hammouche; Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations, 246 (29, [3] J. Banas, K. Goebel; Measure of noncompactness in Banach Spaces, in: Lecture Notes in Pure and App. Math., vol. 6, Marcel Dekker, New York, (198. [4] K. Diethelm; The Analysis of Fractional Differential Equations, Lect. Note. Math., Springer, (21. [5] G. Da Prato, E. Sinestrari; Differential operators with nondense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987, [6] M. M. El-Borai; Semigroups and some nonlinear fractional differential equations, Applied Math. Comput., 149 (24, [7] M. M. El-Borai; Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solutions & Fractals, 149 (24, [8] X. L. Fu; On solutions of neutral nonlocal evolution equations with nondense domain, J. Math. Anal. Appl., 299 (24, [9] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, (26. [1] M. Kirane, B. Ahmad, A. Alsaedi, et al.; Non-existence of global solutions to a system of fractional diffusion equations, Acta Applicandae Mathematicae, (214, [11] M. Li, C. Chen, F. B. Li; On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (21, [12] F. Mainardi, P. Paradisi, R. Gorenflo; Probability Distributions Generated by Fractional Diffusion Equations, in: J. Kertesz, I. Kondor (Eds., Econophysics: An Emerging Science, Kluwer, Dordrecht, (2. [13] G. M. Mophou, G. M. N Guérékata; On integral solutions of some nonlocal fractional differential equations with nondense domain, Nonlinear Anal., 71 (29, [14] H. Mönch; Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (198, [15] A. Pazy; Semigroups of Linear Operator and Applications to Partial Differential Equations, Springer-Verlag, New York, (1983. [16] J. G. Peng, K. X. Li; A novel characteristic of solution operator for the fractional abstract Cauchy problem, J. Math. Anal. Appl., 385 (212, [17] S. G. Samko, A. A. Kilbas, O. I. Marichev; Fractional integrals and derivatives: Theory and Applications, Gordon and Breach, Yverdon, (1993. [18] H. R. Thieme; Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (199,

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