Initial Value Problems of Semilinear Supdiffusion Equations

: We consider the existence of a mild solution of supdiffusion equations and obtain some results under some growth and noncompactness conditions of nonlinearity without coefﬁcient restriction; and some new results for Ulam-Hyers-Rassias stability are obtained.


Introduction
The fractional evolution equations with order α ∈ (0, 1) have attracted increasing attention in recent years. In reference [1], Chen and Li concerned the existence of mild solutions for a class of fractional evolution equations; Chen, Zhang and Li dealt with nonlinear time fractional non-autonomous evolution equations with delay; as a result, the existence of a mild solution was obtained in reference [2]; Wang, Zhou and Fečkan discussed Cauchy problems and boundary value problems of nonlinear impulsive problems for fractional differential equations and Ulam stability in reference [3]. Zhou, Wang and Zhang wrote a book [4] about the basic theory of fractional differential equations.
If a coefficient operator is a C 0 -semigroup, ones needs to construct a corresponding operator to deal with the fractional semilinear evolution equations with order α ∈ (0, 1), which are usually described by the C 0 -semigroup and the probability density function. However, at present, this way is very difficult for fractional semilinear evolution equations with order α ∈ (1, 2). Well-posed solutions are obtained for a class of supdiffusion equations of order α ∈ (1, 2) in [5]. To the best of our knowledge, not many results are available for the mild solutions of supdiffusion with order α ∈ (1, 2). In 2001, the solution operator for which there is no semigroup property was firstly introduced by Bajlekova [6] to deal with following fractional evolution equations in Banach space E: (a) T α (t) is strongly continuous for t ≥ 0 and T α (0) = I (the identity operator); is well-posed, and solution u(t) given by and * denotes the convolution of functions. For α ∈ (1, 2), at present, properties of the solution operator have been discussed by several researchers such as Li, Chen and Li [7], who studied the fractional powers of generators of fractional resolvent families. Later, Li, Peng and Jia [8] considered the Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives by developing a resolvent family [6]. In 2015, Li [9] investigated the regularity of mild solutions for a linear, inhomogeneous fractional evolution equation: where D α t is the regularized Caputo fractional derivative for α ∈ (1, 2); u 0 , u 1 ∈ D(A); h is a Bochner integrable function on (0, a]; and when A ∈ G α (M, ω) (the notation G α (M, ω) is introduced in next section), the mild solution u is given by where {S α (t)} t≥0 is the solution operator of the problem (3). Lizama, Pereira and Ponce, in 2016, considered the compactness of fractional resolvent operator functions; see [10]. In particular, in [5], Li, Sun and Feng explored the existence and uniqueness of a classical solution to the linear inhomogeneous problems for order 1 < α < 2 and considered the existence and uniqueness of classical solution for nonlinear problems with a resolvent family that they defined. In [11], Wang and Zhou considered Mittag-Leffler-Ulam stabilities of fractional evolution equations of order 0 < α < 1. Motivated by the above consideration, in this paper, we study the existence of mild solutions of the following problems: where D α t is a regularized Caputo fractional derivative for 1 < α < 2; A : D(A) ⊂ E → E is a densely defined and closed linear operator in Banach space E. The nonlinear map f : × E → E is continuous and = [0, a], a > 0 is a constant, u 0 and u 1 ∈ E. Finally, we consider the stability for problem (4).

Preliminaries
Regarding fractional integrals and derivatives, throughout this paper, they are Riemann-Liouville fractional integrals and regularized Caputo fractional derivatives; we refer to the references Kilbas et al. [12]. Assume u : [0, ∞) → X, where X is a Banach space. The fractional integral of order α > 0 for the function u is defined as The standard Caputo fractional derivative of order 1 < α < 2 for the function u is defined by Further, D α t represents the regularized Caputo fractional derivative of order 1 < α < 2 defined by Let (X, · ) be a Banach space. Use C( , E) for the Banach space of all continuous X-value functions on interval with the norm u C = sup t∈ u(t) , and denote by B(E) the Banach space of all linear and bounded operators in E endowed with the topology defined by the operator norm. Denote If operator S α (t) satisfies (5), we will write (3) is called analytic if S α (t) admits an analytic extension to a sector Σ ϑ 0 for some ϑ 0 ∈ (0, π 2 ]. An analytic solution operator is said to be of analytic type From the proof of Theorem 2.14 and Theorem 2.23 in references [6], for A ∈ A α (ϑ 0 , ω 0 ) and t > 0, the estimate is held.

Remark 1.
If A is a positive definite operator then the condition A ∈ A α (ϑ 0 , ω 0 ) is easily satisfied.
Proof. Define the operator F : From (8), the mild solution of (4) is equivalent to the fixed point of F by (11). Firstly, we prove that F is continuous in C( , E). By the continuity of f with respect to the second variable, for all s ∈ , we know that lim n→+∞ f (s, u n (s)) = f (s, u(s)), (12) where u n ∈ C( , E) and lim n→+∞ u n = u in C( , E). By (7) and assumption (F1), we obtain that Obviously, the above right part of inequality is Lebesgue integrable for s ∈ . From (7), (12) and (13) for t ∈ and the Lebesgue dominated convergence theorem, we know that Next, we show that the set (6) and (7) and assumption (F1), we have that for t ∈ , By the above inequality, we get that u(t) ≤ c * for t ∈ , namely, Ω is a bounded set. Choose R > c * and denote Ω R = {u ∈ C( , E) | u C < R}; then Ω R is a bounded open set and θ ∈ Ω R . Since R > c * , we have that u = νF u, u ∈ ∂Ω R , ν ∈ (0, 1). Let D ⊂ Ω R be a countable set and D ⊂ co({θ} ∪ F (D)). Then For 0 ≤ t < t ≤ a and u ∈ D, we have Given the continuity of solution operator S α (t)(t ≥ 0), the boundedness of (1 * S α )(t)(t ≥ 0), the differentiability of the (g α−1 * S α )(t)(t ≥ 0) and (F1), we easily know for u ∈ D, which means that the F (D) in interval is equicontinuous.
Finally, we show that D is relatively compact. From (7), (11), (14), (F2) and measures of noncompactness properties and Lemma 2, for t ∈ , we get that The Bellman inequality implies that κ(D(t)) ≡ 0 on , which is the D(t) relatively compact on co({θ} ∪ F (D(t))). Namely, D is a relative compact on set Ω R . Therefore, from the Lemma 3, the map F exists as a fixed point in Ω R , which is a mild solution of (4).
Remark 5. In our results, there is no similar restricted condition to the condition about coefficient restriction (5.7) in reference [5].
By above two inequality, we easily see that the conditions (F1) and (F2) of Theorem 1 are satisfied. Then problem (4) has at least one mild solution. Nextly, we show uniqueness of solution. Let u 1 , u 2 ∈ Ω R be two fixed points of the operator F defined by (11). For t ∈ , we get that From the Bellman inequality, we obtain u 1 (t) = u 2 (t), t ∈ , i.e., (4) has a unique mild solution.

Stability
In this section, we prove the stability of solutions for problems (4).
By applying the Bellman inequality for the above inequality, we get that for any t ∈ , where c L f := Me ωa a α−1 Γ(α) ζ ψ exp Me ωa a α−1