Global existence and attractivity for Riemann-Liouville fractional semilinear evolution equations involving weakly singular integral inequalities

In this paper, we obtain several results on the global existence, uniqueness and attractivity for fractional evolution equations involving the Riemann-Liouville type by exploiting some results on weakly singular integral inequalities in Banach spaces. Some boundedness conditions of the nonlinear term are considered to obtain the main results that generalize and improve some well-known works.


Introduction
The aim of this paper is to present several results on the global existence, uniqueness and attractivity of the following fractional differential equation: where R 0 D β t is the Riemann-Liouville fractional derivative with the order β ∈ (0, 1), A : D(A) ⊆ X → X is the infinitesimal generator of a compact C 0 -semigroup {S(t)} t≥0 .
The attractivity of solutions plays a significant role in describing the properties of differential equations.Many researchers have investigated the attractivity of solutions of fractional differential equations.For instance, Furati and Tatar [5] proved that solutions of fractional differential equations with weighted initial data exist globally and decay as a power function.Kassim, Furati, and Tatar [10] studied the asymptotic behavior of solutions for a class of nonlinear fractional differential equations involving two Riemann-Liouville fractional derivatives of different orders.Gallegos and Duarte-Mermoud [6] studied the asymptotic behavior of solutions to Riemann-Liouville fractional systems.Zhou [26] studied the attractivity of solutions for fractional evolution equation with almost sectorial operators.Tuan, Czornik, Nieto and Niezabitowski [22] presented some results for existence of global solutions and attractivity for multidimensional fractional differential equations involving Riemann-Liouville fractional derivative.Sousa, Benchohra, and N'Guérékata [18] considered the attractivity of solutions of the fractional differential equation involving the ψ-Hilfer fractional derivative.For more references, we refer to [1,15,19,20].
Since weakly singular integral inequalities are well-known tools for proving the existence, uniqueness, stability and attractivity of integral evolution equations and fractional differential equations, many scholars have begun to study weakly singular integral inequalities and have obtained several versions of weakly singular integral inequalities.See [3,8,9,12,13,16,21,23,28] for more details.Especially, Zhu [28][29][30][31] studied several results on the existence and attractivity for the following fractional differential equations with Riemann-Liouville fractional derivative in R: Zhu presented some weakly singular integral inequalities to prove the main results under the following boundedness conditions f (t, x) ≤ l(t)|x| μ + k(t), (1.5) where In this paper, by exploiting the Leray-Schauder alternative fixed point theorem and some weakly singular integral inequalities in Banach spaces, we first prove the existence of global mild solutions of problem (1.1).We also prove that there exists a unique mild solution of problem (1.1).Furthermore, we show that the mild solutions of problem (1.1) are globally attractive.Our results generalize and improve the results existing in literature.Finally, we provide several examples to illustrate the applicability of our results.
Below we will describe some of the new features.First, our problem is the natural generalization of many well-known works on the existence and global attractivity for fractional differential equations in finite-dimensional spaces.Second, some boundedness conditions of the nonlinear term are considered to obtain the main results that generalize and improve some well-know works.Instead of conditions (1.3)-(1.6),we deal with more general conditions in the Banach space: ).Third, we obtain several useful nonlinear weakly singular integral inequalities in Banach spaces, which can also be used to control some problems.Fourth, problem (1.1) reduces the problems of first-order and Caputo fractional semilinear evolution equations and can be generalized to more complex forms, for instance, fractional impulsive evolution equations and fractional evolution inclusions.
The outline of this paper is as follows.In Sect.2, we introduce some notations, definitions, and useful lemmas.In Sect.3, we present several nonlinear weakly singular integral inequalities useful to prove the main results.In Sects. 4 and 5, we give some sufficient conditions on the global existence and attractivity of mild solutions of problem (1.1).In Sect.6, some deduced results are given to illustrate our main results.

Preliminaries
In this section, we introduce some notations, definitions and lemmas which will be needed later.
The norm of a Banach space X will be denoted by • X .For an interval J, let C(J, X) denote the Banach space of all continuous functions from J into X equipped with the norm x C = sup t∈J x(t) X and L p (J, X)(p > 1) denote the Banach space of p-th integral functions from J into X equipped with the norm x L p = ( J x(t) p X dt) is a Banach space equipped with the norm x 0 = sup a≤t<+∞ x(t) .Definition 2.1 ([4, 11, 14, 17]) The Riemann-Liouville fractional integral of order β ∈ (0, 1) for a function f : R + → X is defined by where is the gamma function.Definition 2.2 ([4, 11, 14, 17]) The Riemann-Liouville fractional derivative of order β ∈ (0, 1) for a function f : R + → X is defined by ([4, 11, 14, 17]) The Caputo fractional derivative of order β ∈ (0, 1) for a function f : R + → X is defined by , ∀t ∈ [0, ∞). (2.3) where M(t) = t 0 2 p-1 l p (s)ϕ p (s) ds and L(t) = 2 p-1 l p (t)φ p (t). )

Nonlinear weakly singular integral inequalities
In this section, we study some nonlinear weakly singular integral inequalities that will be useful to prove the main results.
The following conclusion is a consequence of Lemma 3.1 when α = 1β and δ = 0.

be a continuous function, and there exists a function l
) We can also obtain the following results.
Proof Let v(t) = t α u(t).Using (3.11) and the same procedure as in (3.4), we get and It follows from Lemma 2.5 that which completes the proof.for all t ∈ (0, +∞), where c is defined as in Lemma 3.1.
Proof Let v(t) = t α u(t).Using (3.16) and the same procedure as in (3.4), we get and Then from (3.19), we know Using Lemma 2.4, we get Thus, we complete the proof.

Global existence
In this section, we present the existence and uniqueness results for problem (1.1).
Step 2. We now show that G is continuous.Let x n → x in C 1-β ((0, T], X).Then there exists r > 0 such that x n 1-β ≤ r and x 1-β ≤ r.For every s ∈ (0, T], we have ) in Lemma 2.7, we know that the function is integrable for s ∈ (0, t).Then we deduce that Step 3. We shall prove that the set = {x ∈ C 1-β ((0, T], X) : x = λGx for some 0 < λ < 1} is bounded.Indeed, for x ∈ , one has Using Corollary 3.2, we obtain and Then the set is bounded.Finally, by applying the fixed point theorem in Theorem 6.5.4 in [7], the operator G has a fixed point x ∈ C 1-β ((0, T], X), which is the mild solution of problem (1.1).
Proof Letting μ(t) = ω p (t 1 p ), we know where c is defined as in Theorem 4.3.
For any T > 0, from Theorem 4.3, we know that problem (1.1) has at least one mild solution in C 1-β ((0, T], X).Since T can be chosen arbitrarily large, then problem (1.1) has at least one global mild solution in C 1-β ((0, +∞), X).Thus, we complete the proof of Theorem 4.4.
From Theorem 4.4, we can immediately obtain the following conclusion.

Proof We know
, applying Corollary 4.5, we know that problem (1.1) has at least one mild solution in C 1-β ((0, +∞), X).We suppose that x 1 , x 2 are two global mild solutions of problem (1.1).Then Using Theorem 3.3, we can get x 1 (t) = x 2 (t).Thus, the proof is complete.
The main result in the section reads as follows.
Suppose f : (0, +∞) × X → X is a continuous function and Then problem (1.2) has at least one globally attractive mild solution.
For convenience, we first obtain several lemmas under the assumptions in Theorem 5.2, which will be useful in the proof of the main theorem.
Lemma 5.3 Under the assumptions in Theorem 5.2, problem (1.2) has at least one mild Proof From (5.1), we have Let ω(t) = t μ + 1 and , where u 0 , u > 0, then we get [0, +∞) ⊂ Dom(W -1 ).Using Theorem 4.3, we know that problem (1.2) has at least one mild solution x 1 ∈ C 1-β ((0, T], X) that satisfies the following integral equation ( Now let us define the operator where 2) given in Lemma 5.3, and T is as in Lemma 5.3.For convenience, we denote Let R > 1 be sufficiently larger such that where M 2 is as defined in Lemma 5.3, and M 1 is defined in the following Lemma 5.4.Define a set U as follows It is easy to see that U is a non-empty, closed, convex and bounded subset of C 0 ([T, +∞),

X).
Lemma 5.4 Under the assumptions in Theorem 5.2, F maps U into U.
We now prove that Fx is a continuous function on [T, +∞).Since ).Using Lemma 2.9, we get that , Fx is a continuous function on [T, +∞) when x ∈ U. Now let us prove that (Fx)(t) → 0 as t → +∞.For any x ∈ U, we have (5.11) Using Lemma 2.6, we have then we get that T 0 (ts) β-1 f s, x 1 (s) ds → 0, as t → +∞.
(5.13)Moreover, we know (5.16) Using Lemmas 2.8 and 2.9, we can obtain that FU is equicontinuous on [T, T 1 ].From the inequality (5.10), we know that (Fx)(t) is relatively compact for any t ∈ [T, +∞) and x ∈ U. Using the proof of Lemma 5.4, we can get that lim t→+∞ |(Fx)(t)| = 0 is uniformly for x ∈ U. Therefore, we get that the set FU is relatively compact.
Then problem (1.2) is global attractive.

Deduced results
In this section, we derive some deduced results for the following first-order and Caputo fractional semilinear evolution equations )
F maps U into U.The proof is complete.Under the assumptions in Theorem 5.2, F : U → U is completely continuous.Proof For any T 1 > T > 1 and x ∈ U, let T ≤ t 1 < t 2 ≤ T 1 , then we get