Existence uniqueness and stability of mild solutions for semilinear ψ-Caputo fractional evolution equations

In this paper, we study the local and global existence, and uniqueness of mild solution to initial value problems for fractional semilinear evolution equations with compact and noncompact semigroup in Banach spaces. In particular, we derive the form of fundamental solution in terms of semigroup induced by resolvent and ψ-function from Caputo fractional derivatives. These results generalize previous work where the classical Caputo fractional derivative is considered. Moreover, we prove the Mittag-Leffler–Ulam–Hyers stability result. Finally, we give examples of time-fractional heat equation to illustrate the result.


Introduction
Fractional differential equations have been applied in many fields, such as economics, engineering, chemistry, physics, finance, aerodynamics, electrodynamics of complex medium, polymer rheology, control of dynamical systems (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). The research on fractional calculus has become a focus area of study due to the fact that some dynamical models can be described more accurately with fractional derivatives than the ones with integer-order derivatives. In particular, it is shown that fractional calculus provides more realistic models demonstrating hidden aspects in a model of spring pendulum [13], the free motion of a particle in a circular cavity [11] and some epidemic models [15,17].
Several researchers are interested in investigating various aspects of fractional differential equations such as existence and uniqueness of solutions, exact solutions, stability of solutions, and methods for explicit and numerical solutions [17][18][19][20]. The common techniques used to display the existence and uniqueness of solutions are fixed point theorem, upper-lower solutions, iterative method and numerical method. For stability of solutions, there is a concept of data dependence, which becomes one of significant topics in the analysis of fractional differential equations, called the Ulam-Hyers stability (see [21][22][23]).
One of the main research focuses on fractional calculus is the theory of fractional evolution equations since they are abstract formulation for many problems arising in engineering and physics. Evolution equations are commonly used to describe the systems that change or evolve over time. A number of studies have been conducted on the existence and unique of solutions for fractional evolution equations based on semigroup and fixed point theory (see [24][25][26][27][28][29][30][31][32]). On the other hand, there has been some studies about fundamental solution for homogeneous fractional evolution equations [33,34]. Recently, [19] applied the homotopy analysis transform method (HATM) for solving time-fractional Cauchy reaction-diffusion equations. In addition, Wang and Zhou [35] presented four kind of stabilities of the mild solution of the fractional evolution equation in Banach space, namely Mittag-Leffler-Ulam-Hyers stability, generalized Mittag-Leffler-Ulam-Hyers stability, Mittag-Leffler-Ulam-Hyers-Rassias stability and generalized Mittag-Leffler-Ulam-Hyers-Rassias stability.
There is variation in the definition of fractional differential operators found in the literature, including Riemann-Liouville, Caputo, Hilfer, Riesz, Erdelyi-Kober, and Hadamard [2,36] operators. The common definitions that triggered attention from many researchers are Riemann-Liouville and Caputo fractional calculus. In Riemann-Liouville fractional differential modeling, the initial condition involves limit values of fractional derivatives, which is difficult to interpret. The Caputo fractional derivative has the advantage of being suitable for physical models with initial condition because the physical interpretation of the prescribed data is clear and it is in general possible to provide these data by suitable measurements [37].
Almeida [38] generalized the definition of Caputo fractional derivative by considering the Caputo fractional derivative of a function with respect to another function ψ and studied some useful properties of the fractional calculus. The advantage of this new definition of the fractional derivative is that a higher accuracy of the model could be achieved by choosing a suitable function ψ.
Recently, Jarad and Abdeljawad [39] introduced the generalized Laplace transform with respect to another function and the inverse version of the Laplace transform with respect to another function. This can be used to solve some fractional differential equations in the framework of generalized Caputo fractional derivative.
Motivated by the work of [25,39], we consider the following fractional evolution equation in a Banach space E: where 0 < α < 1, T < ∞, A is the infinitesimal generator of a C 0 -semigroup of uniformly bounded linear operators {T(t)} t≥0 on E, u 0 ∈ E and f : [0, ∞) × E → E is given function. The fractional derivative C 0 D α ψ considered in this work is in the sense of Caputo fractional derivative with respect to a function ψ which gives a more general framework to the results in the literature. Moreover, this problem is more general than the work in [39] where we consider the evolution operator A instead of a constant.
In this paper, we aim to establish a mild solution for the problem (1) in terms of semigroup depending on a function ψ from the generalized Caputo derivative. In addition, we prove the existence and uniqueness results of mild solution for the problem (1) in local and global time under the condition that {T(t)} t≥0 is both compact and noncompact operator. The results obtained in this work are in the abstract form which can be applied for further investigation such as the evolution equations with perturbation, delay and nonlocal term. This paper will be organized as follows. In Sect. 2, we will briefly recall some basic definitions and some preliminary concepts about fractional calculus and auxiliary results used in the following sections. We then construct a mild solution by using semigroup for the problem in Sect. 3. We prove the existence and uniqueness of mild solutions of the problem (1) under compact and noncompact analytic semigroup by the Schauder fixed point theorem in Sects. 4 and 5, respectively. In Sect. 6 we present Mittag-Leffler-Ulam-Hyers stability result for the problem (1). Finally, we give some examples to illustrate the application of the results obtained in Sect. 7 and our conclusion in Sect. 8.

Preliminaries
In this section, we introduce preliminary background which is used throughout this paper.
Let E be a Banach space with the norm · and let C(J, E) be the Banach space of continuous functions from J to E with the norm u C = sup t∈J u(t) .
The ψ-Riemann-Liouville fractional integral operator of order α of a function f is defined by It is obvious that when ψ(t) = t, (2) is the classical Riemann-Liouville's fractional integral.
Proposition 2.14 ( [43,44]) The Wright function φ α is an entire function and has the following properties: Next, we introduce the definition for Kuratowski measure of noncompactness, which will be used in the proof of our main results.
The following properties of the Kuratowski measure of noncompactness are well known. ([45, 46]) Let E be Banach spaces and U, V ⊂ E be bounded. Then the noncompactness measure has the following properties:

Lemma 2.23 (Schauder's fixed point theorem) Let E be a Banach space and D ⊂ E, a convex, closed and bounded set. If T : D → D is a continuous operator such that T(D) ⊂ E, T(D) is relatively compact, then T has at least one fixed point in D.
Next, we give some facts about the semigroups of linear operators. These results can be found in [51,52].
For a strongly continuous semigroup (i.e., We denote by D(A) the domain of A, that is, Lemma 2.24 ([51, 52]) Let {T(t)} t≥0 be a C 0 -semigroup, then there exist constants C ≥ 1 and a ≥ 0 such that T(t) ≤ Ce at for all t ≥ 0.

Lemma 2.25 ([51, 52]) A linear operator A is the infinitesimal generator of a C 0 -semigroup if and only if
Throughout this paper, let A be the infinitesimal generator of a C 0 -semigroup of uniformly bounded linear operators {T(t)} t≥0 on E. Then there exists M ≥ 1 such that

Representation of mild solution using semigroup
According to Definition 2.5 and Theorem 2.7, it is suitable to rewrite the Cauchy problem in the equivalent integral equation Proof Let λ > 0. Applying the generalized Laplace transforms to (8), we have It follows that We consider the following one-sided stable probability density in [53]: Using (10), we get Then we get Now, we can invert the Laplace transform to get is the probability density function defined on (0, ∞).
For any u ∈ E, define operators S α ψ (t, s) and T α ψ (t, s) by

Lemma 3.2
The operators S α ψ and T α ψ have the following properties: (iii) If T(t) is compact operator for every t > 0, then S α ψ (t, s) and T α ψ (t, s) are compact for all t, s > 0.
(iv) If S α ψ (t, s) and T α ψ (t, s) are compact strongly continuous semigroup of bounded linear operators for t, s > 0, then S α ψ (t, s) and T α ψ (t, s) are continuous in the uniform operator topology.
Proof The proof follows the argument of [26].
Before starting and proving the main results, we introduce the following hypotheses.

Existence and uniqueness of mild solution under compact analytic semigroup
In this section, we begin by proving a theorem concerning the existence and uniqueness of mild solution for the problem (1) under the condition of compact analytic semigroup. The discussions are based on fractional calculus and Schauder fixed point theorem. Our main results are as follows.
Proof For any r > 0, let Step 1: We will prove that K : Ω r → Ω r , that is, there exists r > 0 such that K(Ω r ) ⊂ Ω r , We assume that for each r > 0, there exists u r ∈ Ω r and t ∈ [0, T], such that (Ku)(t) > r. According to Lemma 3.2(i) and (H 3 ), we have Dividing to both side by r and taking the limit supremum as r → ∞, we obtain which is contradiction. Therefore K : Ω r → Ω r .
Step 3: We will prove that K(Ω r ) is equicontinuous. For any u ∈ Ω r and 0 ≤ t 1 < t 2 ≤ T, we have
By Lemma 3.2, it is clear that I 1 → 0 as t 1 → t 2 and we obtain α and hence I 2 → 0 and I 3 → 0 as t 2 → t 1 . For t 1 = 0 and 0 < t 2 ≤ T, it easy to see that I 4 = 0. Then, for any ε ∈ (0, t 1 ), we have It follows that I 4 → 0 as t 2 → t 1 and ε → 0 by Lemma 3.2(iv) and (iii). Therefore, which means that K(Ω r ) is equicontinuous.
Obviously, K(0) is relatively compact in E. Let 0 ≤ t ≤ T be fixed. Then, for every ε > 0 and δ > 0, let u ∈ Ω r and define an operator K ε,δ on Ω r by Then, by the compactness of T(ε α δ) for ε α δ > 0, we see that the set K ε,δ (t) = {(K ε,δ u)(t) : u ∈ Ω r } is relatively compact in E for all ε > 0 and δ > 0. Furthermore, for any u ∈ Ω r , we have Therefore, there are relatively compact sets arbitrarily close to the set K(t) for t > 0. Hence, K(t) is relatively compact in E.

Therefore, by the Arzelá-Ascoli theorem K(Ω r ) is relatively compact in C([0, T], E).
Thus, the continuity of K and relatively compact of K(Ω r ) imply that K is a completely continuous. By the Schauder fixed point theorem, we see that K has a fixed point u * in Ω r , which is a mild solution of (1). The proof is complete.
Remark 4.2 From Theorem 4.1, we notice that if ψ is bijection function then the problem (1) has at least mild solution provided that

Theorem 4.3 Assume (H 4 ) holds. Then the problem (1) has a unique mild solution.
Proof Let u 1 and u 2 be the solutions of the problem (1) in Ω r . Then, for each i ∈ {1, 2}, the solution u i satisfies Then, for any t ∈ [0, T], we have where k * = sup 0≤t≤T |k(t)|. By using the Gronwall inequality (Lemma 2.11), we obtain which implies that u 1 ≡ u 2 . Therefore, the problem (1) has a unique mild solution u * ∈ Ω r .

Theorem 4.4 Suppose that conditions (H 1 )-(H 3 ) hold. Then, for any u 0 ∈ E, the problem (1) has a mild solution u on a maximal interval of existence
Proof We notice that a mild solution u of the problem (1) defined on [0, T] can be extended to a larger interval [0, Therefore, repeating the procedure and using the methods of steps in Theorem 4.1, we can prove that there exists a maximal interval [0, T max ) such that the mild solution u of the problem (1). We want to prove that if T max < ∞ then lim t→Tmax u(t) = ∞. First, we will prove that lim sup t→Tmax u(t) = ∞. Assume by contradiction that

Similar to
Step 3 of Theorem 4.1, we can prove that u(t )u(t) → 0 as t , t → T max Therefore, by the Cauchy criteria we see that lim t→Tmax u(t) = u 1 exists. By the first part of the proof, there exists a δ > 0 such that the solution can be extended to [0, T max + δ) and we know that to the fractional evolution equation there exists a mild solution on [T max , T max + δ). This means that the mild solution of the problem (1) can be extended to [0, T max + δ), which contradicts with the maximal interval [0, T max ). Hence, lim sup t→Tmax u(t) = ∞. Now, we will prove that if T max < ∞, then lim t→Tmax u(t) = ∞. If this is not true, then there exist a constant K > 0 and a sequence t n → T max such that u(t n ) ≤ K for all n. Since t → u(t) is continuous and lim sup t→Tmax u(t) = ∞, we can find a sequence a n such that a n → 0 as n → ∞, u(t) ≤ M(K + 1) for t n ≤ t ≤ t n + a n and u(t n + a n ) = M(K + 1) for all n sufficiently large. But we have M(K + 1) = u(t n + a n ) ≤ S α ψ (a n , 0)u(t n ) + t n +a n t n ψ(t n + a n )ψ(s) α-1 T α ψ (t n + a n , s)f s, u(s) ψ (s) ds t n +a n t n ψ(t n + a n )ψ(s) α-1 ψ (s) ds which implies that M(K + 1) ≤ MK as a n → 0, a contradiction. Therefore, we find that if T max < ∞, then lim t→Tmax u(t) = ∞.
Next, we discuss the existence of a global mild solution for the problem (1). To this end, we need replace the assumption (H 3 ) by (H 5 ). Proof It is clearly that (H 5 ) implies (H 3 ). Therefore, by Theorem 4.4 we know that the problem (1) has a mild solution u on a maximal interval of existence [0, T max ). By the proof process of Theorem 4.4, we can see that the problem (1) has a global mild solution if u(t) is bounded for every t in the interval of existence of u. If suffices to show that u(t) is bounded for every t ∈ [0, T max ) with T max < ∞.
Then for any 0 ≤ t ≤ T max we have and By Corollary 2.12, we obtain which means that u(t) is bounded for every t ∈ [0, T max ).

Existence and uniqueness of mild solution under noncompact analytic semigroup
In this section, we will prove the existence of mild solution for the problem (1) under the condition of a noncompact analytic semigroup.
Proof For any r > 0, let Then, Ω r is bounded closed convex subset of C([0, T], E). Define an operator K : Using the same argument in Theorem 4.1, we obtain K : Ω r → Ω r is continuous and K(Ω r ) is equicontinuous. Then it is sufficient to prove that K : Ω r → Ω r is condensing.
where Co is the closure of convex hull. Then, by Lemma 2.21 we obtain Co K(Ω r ) ⊂ Ω r is bounded and equicontinuous. Now, we will prove that K : D → D is a condensing operator. For any D ⊂ Co K(Ω r ), by Lemma 2.17, we see that there exists a countable set D 0 = {u n } ⊂ D such that By the equicontinuity of D, we know that D 0 ⊂ D is also equicontinuous. Therefore, by Lemma 2.20, we have Since K(D 0 ) ⊂ D is bounded and equicontinuous, we obtain by Lemma (2.18). It follows that Thus, K : D → D is a condensing operator. Therefore, by Lemma 2.22, K has at least one fixed point u * in Ω r , which is a mild solution of (1). The proof is complete.
Remark 5.2 From Theorem 5.1, we notice that if ψ is bijection function then the problem (1) has at least one mild solution provided that Proof The proof uses the same argument as in Theorem 4.5.

Proof
Let v ∈ C 1 ([0, T], ∞) be a solution of inequality (16). Then we get for all t ∈ [0, ∞). Let us denote by u ∈ C([0, T], ∞) the unique mild solution of the Cauchy problem We have By Corollary 2.12, we obtain The proof is complete.

Examples
In this section, we give examples of fractional differential equation of compact and noncompact semigroup cases. The main results can be applied for a larger class of Caputo fractional derivative with respect to ψ. In particular, our results can be reduced to the examples in [25,32] when ψ(t) = t.
Then for t ∈ [0, 1] we have , and C 2 is the set of all is the set of all continuous defined on (0, 1) which have continuous partial derivatives of order less than or equal to 2, and H 1 0 (0, 1) is the completion of C 1 (0, 1) with respect to the norm u H 1 (0,1) .

Conclusion
We construct a mild solution for fractional evolution equation based on Laplace transform with respect to ψ-function. We obtain the local and global existence and uniqueness of mild solution for the problem with ψ-Caputo fractional derivative, which can be reduced to the classical Caputo fractional derivative in previous work. Furthermore, the form of a fundamental solution obtained in this work is a foundation result for further investigation such as the problem with perturbation, delay and a nonlocal term.