Abstract
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.
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1 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 \(_{0}^{R}D^{\beta}_{t}\) is the Riemann-Liouville fractional derivative with the order \(\beta \in (0,1)\), \(A:D(A)\subseteq X\rightarrow X\) is the infinitesimal generator of a compact \(C_{0}\)-semigroup \(\{S(t)\}_{t\ge 0}\).
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 [28–31] studied several results on the existence and attractivity for the following fractional differential equations with Riemann-Liouville fractional derivative in \(\mathbb{R}\):
Zhu presented some weakly singular integral inequalities to prove the main results under the following boundedness conditions
where \(\mu \in (0,1],l\), \(k\in C((0,+\infty ),\mathbb{R}_{+})\cap L^{p}_{Loc,1- \beta}([0,+\infty ),\mathbb{R}_{+})\) (\(p>\frac{1}{\beta}\)) and nonnegative nondecreasing function \(\omega \in C([0,+\infty ),\mathbb{R}_{+})\) with \(\lim_{t\rightarrow +\infty }\frac{t}{\omega (t)}=K (0< K\le +\infty )\).
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:
where \(l,k\in C_{1-\beta}((0,+\infty ),\mathbb{R}_{+})\cap L^{p}_{Loc,1- \beta}([0,+\infty ),\mathbb{R}_{+})(p>\frac{1}{\beta })\). 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.
2 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 \(\|\cdot \|_{X}\). For an interval J, let \(C(J,X)\) denote the Banach space of all continuous functions from J into X equipped with the norm \(\|x\|_{C}=\sup_{t\in J}\|x(t)\|_{X}\) and \(L^{p}(J,X)(p>1)\) denote the Banach space of p-th integral functions from J into X equipped with the norm \(\|x\|_{L^{p}}= (\int _{J}\|x(t)\|_{X}^{p}\,dt )^{ \frac{1}{p}}\). Let \(C_{\beta}(J,X)=\{x :y(t)=t^{\beta}x(t),y\in C(J,X)\}\) equipped with the norm \(\|x\|_{C_{\beta}}=\sup \{t^{\beta}\|x(t)\|_{X}:t\in J\}\) and let \(L^{p}_{\beta}(J,X)=\{x :y(t)=t^{\beta}x(t),y\in L^{\beta}(J,X)\}\) equipped with the norm \(\|x\|_{L^{p}_{\beta}}= (\int _{J}t^{\beta}\|x(t)\|_{X}^{p}\,dt)^{\frac{1}{p}}:t\in J\}\). Obviously, the space \(C_{\beta}(J,X)\), \(L^{p}_{\beta}(J,X)\) is Banach spaces. For \(a\ge 0\), let \(C_{0}([a,+\infty ),X)=\{x\in C([a,+\infty ),X): \lim_{t\rightarrow + \infty }x(t)=0\}\). It is clear that \(C_{0}([a,+\infty ),X)\) is a Banach space equipped with the norm \(\Vert x\Vert _{0}=\sup_{a\le t< +\infty }\|x(t)\|\).
Definition 2.1
The Riemann-Liouville fractional integral of order \(\beta \in (0,1)\) for a function \(f:\mathbb{R}_{+}\to X\) is defined by
where Γ is the gamma function.
Definition 2.2
The Riemann-Liouville fractional derivative of order \(\beta \in (0,1)\) for a function \(f:\mathbb{R}_{+}\to X\) is defined by
Definition 2.3
The Caputo fractional derivative of order \(\beta \in (0,1)\) for a function \(f:\mathbb{R}_{+}\to X\) is defined by
Lemma 2.4
([2], Corollary 5.3)
Let u, ϕ, ψ and k be nonnegative continuous functions on \([a, b]\). Let ω be a continuous, nonnegative and nondecreasing function on \([0,+\infty )\), with \(\omega (r)>0\) for \(r>0\), and let \(\Phi (t)=\max_{a\le s\le t}\phi (s)\) and \(\Psi (t)=\max_{a\le s\le t}\psi (s)\). Assume that
Then
where \(W(u)=\int _{u_{0}}^{u}\frac{1}{\omega (\tau )}\,d\tau \), \(u_{0}, u>0\), \(W^{-1}\) is the inverse of W and
Lemma 2.5
([28])
Let \(1\le p<\infty \), φ and ϕ be continuous and nonnegative functions on \([0,\infty )\), function \(l\in L_{Loc}^{p}([0,+\infty ),\mathbb{R}_{+})\), and u be a continuous and nonnegative function with
Then
where \(M(t)=\int _{0}^{t}2^{p-1}l^{p}(s)\varphi ^{p}(s)\mathrm{\,d}s\) and \(L(t)=2^{p-1}l^{p}(t)\phi ^{p}(t)\).
Lemma 2.6
([28])
Let \(0<\beta <1\), \(p>\frac{1}{\beta }\), \(q=\frac{p}{p-1}\), \(\rho \in L^{p}_{1- \beta}([0,1],\mathbb{R})\). Then
Lemma 2.7
([29])
Let \(0<\beta <1\), \(p>\frac{1}{\beta }\), \(q=\frac{p}{p-1}\), \(\rho \in L^{p}_{1- \beta}([0,1],\mathbb{R})\). Then
and if \(0< t_{1}\le t_{2}\le 1\), then
Lemma 2.8
([31])
Let \(0<\beta <1\), \(p>\frac{1}{\beta }\), \(\rho \in L^{p}_{Loc,1-\beta}([1,+ \infty ),\mathbb{R})\) and
Then \(y\in C([1,+\infty ),\mathbb{R})\).
Lemma 2.9
([31])
Let \(0<\beta <1\), \(p>\frac{1}{\beta }\), \(\rho \in L^{p}_{1-\beta }([0,1], \mathbb{R})\) and
Then \(y\in C([1,+\infty ),\mathbb{R})\).
3 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.
Lemma 3.1
Let \(a, b\ge 0\), \(1>\alpha \ge \delta \ge 0\) and \(0<\beta <1\), \(p>\max \{\frac{1}{\beta},\frac{1}{1-\alpha +\delta }\}\) and \(q=\frac{p}{p-1}\). Let \(f:(0,T)\times X\rightarrow X\) be a continuous function, and there exists a function \(l\in C((0,T),\mathbb{R}_{+})\cap L^{p}_{Loc,\alpha -\delta}([0,T), \mathbb{R}_{+})\) and a nondecreasing function \(\omega \in C([0,+\infty ),\mathbb{R}_{+})\) such that
Let \(u\in C_{\alpha}([0,T),\mathbb{R}_{+})\) with
Then
where \(c= \frac{b\Gamma ^{\frac{1}{q}}(q(\beta -1)+1)\Gamma ^{\frac{1}{q}}(q(\delta -\alpha )+1)}{\Gamma ^{\frac{1}{q}}(q(\beta -1)+q(\delta -\alpha )+2)}\), \(W(u)=\int _{u_{0}}^{u}\frac{1}{\omega ^{p}(\tau ^{1/p})}\,d\tau \), \(u_{0}, u>0\) and
Proof
For \(t\in (0,T)\), let \(v(t)=t^{\alpha }u(t)\). We get
Using the Hölder inequality, we obtain
Then
Let \(\mu (t)=\|v(t)\|^{p}\). Then
Using Lemma 2.4, we obtain
and
Thus, we complete the proof. □
The following conclusion is a consequence of Lemma 3.1 when \(\alpha =1-\beta \) and \(\delta =0\).
Corollary 3.2
Let \(a, b\ge 0\) and \(0<\beta <1\), \(p>\frac{1}{\beta}\) and \(q=\frac{p}{p-1}\). Let \(f:(0,T)\times X\rightarrow X\) be a continuous function, and there exists a function \(l\in C((0,T),\mathbb{R}_{+}))\cap L^{p}_{Loc,1-\beta}([0,T), \mathbb{R}_{+})\) and a nondecreasing function \(\omega \in C([0,+\infty ),\mathbb{R}_{+})\) such that
Let \(t^{1-\beta}u(t)\) be a continuous, nonnegative function on \([0, T)\) with
Then
where \(c= \frac{b\Gamma ^{\frac{2}{q}}(q(\beta -1)+1)}{\Gamma ^{\frac{1}{q}}(2q(\beta -1)+2)}\), \(W(u)=\int _{u_{0}}^{u}\frac{1}{\omega ^{p}(\tau ^{1/p})}\,d\tau \), \(u_{0}, u>0\) and
We can also obtain the following results.
Lemma 3.3
Let \(a, b\ge 0\), \(1>\alpha \ge \delta \ge 0\) and \(0<\beta <1\), \(p>\max \{\frac{1}{\beta},\frac{1}{1-\alpha +\delta }\}\) and \(q=\frac{p}{p-1}\). Let l be a nondecreasing continuous function on \((0,+\infty )\) with \(l\in L^{p}_{Loc,-\delta}([0,+\infty ),\mathbb{R}_{+}))\). Let \(t^{\alpha }u(t)\) be a continuous, nonnegative function on \([0,+\infty )\) with
Then
where \(M(t)=2^{p-1}a^{p}\int _{0}^{t}s^{-p\delta }l^{p}(s)\mathrm{\,d}s\), \(L(t)=2^{p-1}c^{p}t^{pq -p\delta -1}l^{p}(t)\), and c is defined as in Lemma 3.1.
Proof
Let \(v(t)=t^{\alpha }u(t)\). 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. □
Lemma 3.4
Let \(a, b\ge 0\), \(1>\alpha \ge \delta \ge 0\), \(0<\gamma <1\) and \(0<\beta <1\), \(p>\max \{\frac{1}{\beta},\frac{1}{1-\alpha +\delta }\}\) and \(q=\frac{p}{p-1}\). Let l be a nonnegative nondecreasing continuous function on \((0,+\infty )\) with \(l\in L^{p}_{Loc,(1-\gamma )\alpha -\delta}[0,+\infty )\). Let \(t^{\alpha }u(t)\) be a continuous, nonnegative function on \([0,+\infty )\) with
Then
for all \(t\in (0,+\infty )\), where c is defined as in Lemma 3.1.
Proof
Let \(v(t)=t^{\alpha }u(t)\). 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. □
4 Global existence
In this section, we present the existence and uniqueness results for problem (1.1).
Definition 4.1
A function \(x\in C_{1-\beta}((0,T],X)\) is called a mild solution of problem (1.1) if it satisfies the following fractional integral equation
where
Lemma 4.2
If the \(C_{0}\)-semigroup \(T(t)(t\geq 0)\) is uniformly bounded, then the operator \(S_{\beta}(t)\) has the following properties:
(i)
where \(\sup_{t\in [0,\infty )}\|T(t)\|\leq M<\infty \);
(ii) \(S_{\beta}(t)(t\geq 0)\) is strongly continuous;
(iii) \(S_{\beta}(t)(t> 0)\) is compact if \(S(t)(t> 0)\) is compact.
Theorem 4.3
Let \(p>\frac{1}{\beta}\) and \(q=\frac{p}{p-1}\). Suppose \(f: (0,T]\times X\rightarrow X\) is a continuous function, and there exists a function \(l\in C_{1-\beta}((0,T],\mathbb{R}_{+})\cap L^{p}_{1-\beta}([0,T], \mathbb{R}_{+})\) and a nondecreasing function \(\omega \in C([0,+\infty ),\mathbb{R}_{+})\) such that
Then problem (1.1) has at least one mild solution in \(C_{1-\beta}((0,T],X)\) provided that
where \(c= \frac{M\Gamma ^{\frac{2}{q}}(q(\beta -1)+1)}{\Gamma (q )\Gamma ^{\frac{1}{q}}(2q(\beta -1)+2)}\), \(W(u)=\int _{u_{0}}^{u}\frac{1}{\omega ^{p}(\tau ^{1/p})}\,d\tau \), \(u_{0}, u>0\).
Proof
Define the operator \(G: C_{1-\beta}((0,T],X)\rightarrow C_{1-\beta}((0,T],X)\) by
Step 1. We will prove that G is compact. To see this, let \(\Omega \in C_{1-\beta}((0,T],X)\) be bounded and \(\Vert x\Vert _{1-\beta}\le R\) for each \(x\in \Omega \) with some \(R>0\). We will show that \(t^{1-\beta}G(\Omega )\) is uniformly bounded and equicontinuous on \([0, T]\). First, we prove that \(t^{1-\beta}G(\Omega )\) is uniformly bounded. For \(x\in \Omega \), we have
This proves that the set \(t^{1-\beta}G(\Omega )\) is uniformly bounded. Second, we prove that \(t^{1-\beta}G(\Omega )\) is an equicontinuous family. For any \(x\in \Omega \), let \(0\le t_{1}< t_{2}\le T\), we get
Since \(\|f(t, x(t))\|\le l(t)\omega (t^{1-\beta}\|x(t)\|)\le l(t)\omega (R)\) and \(l\in C_{1-\beta}((0,T],\mathbb{R}_{+})\cap L^{p}_{1-\beta}([0,T], \mathbb{R}_{+})\). By Lemma 2.7, we know that the right-hand side of (4.2) tends to zero as \(t_{2}\rightarrow t_{1}\). Therefore, \(t^{1-\beta}G(\Omega )\) is an equicontinuous family. From Lemma 4.2, it follows that \(t^{1-\beta}G(\Omega )\) is relatively compact for each \(t\in [0,T]\).
Step 2. We now show that G is continuous. Let \(x_{n}\rightarrow x\) in \(C_{1-\beta}((0,T],X)\). Then there exists \(r>0\) such that \(\Vert x_{n}\Vert _{1-\beta}\le r\) and \(\Vert x\Vert _{1-\beta}\le r\). For every \(s\in (0, T]\), we have
and
Since \(l\in C_{1-\beta}((0,T],\mathbb{R})\cap L^{p}_{1-\beta}([0,T], \mathbb{R})\), using (2.6) in Lemma 2.7, we know that the function
is integrable for \(s\in (0,t)\). Then we deduce that
Therefore, \(t^{1-\beta}Gx_{n}(t)\rightarrow t^{1-\beta}Gx(t)\) pointwise on [0, T] as \(n\rightarrow +\infty \). With the fact that G is compact, we get that \(G: C_{1-\beta}((0,T],X)\rightarrow C_{1-\beta}((0,T],X)\) is continuous.
Step 3. We shall prove that the set \(\Lambda =\{x \in C_{1-\beta}((0,T],X): x=\lambda Gx\) for some \(0<\lambda <1\}\) is bounded. Indeed, for \(x\in \Lambda \), one has
Using Corollary 3.2, we obtain
and
Then the set Λ is bounded.
Finally, by applying the fixed point theorem in Theorem 6.5.4 in [7], the operator G has a fixed point \(x\in C_{1-\beta}((0,T],X)\), which is the mild solution of problem (1.1). □
Now we investigate the existence of global mild solutions of problem (1.1).
Theorem 4.4
Let \(p>\frac{1}{\beta}\) and \(q=\frac{p}{p-1}\). Suppose \(f: (0,+\infty )\times X\rightarrow X\) is a continuous function, and there exists a nonnegative function \(l\in C((0,+\infty ),\mathbb{R}_{+})\cap L^{p}_{Loc,1-\beta}([0,+ \infty ),\mathbb{R}_{+})\) and a nonnegative nondecreasing function \(\omega \in C([0,+\infty ),\mathbb{R}_{+})\) with \(\lim_{t\rightarrow +\infty }\frac{t}{\omega (t)}=K (0< K\le +\infty )\) such that
Then problem (1.1) has at least one global mild solution in \(C_{1-\beta}((0,+\infty ),X)\).
Proof
Letting \(\mu (t)=\omega ^{p}(t^{\frac{1}{p}})\), we know
Since \(\int _{u_{0}}^{+\infty }\frac{1}{\tau }\,d\tau \) is divergent (\(u_{0}>0\)), from (4.6), we get that \(\int _{u_{0}}^{+\infty }\frac{1}{\mu (\tau )}\,d\tau \) is also divergent. Since \(W(u)=\int _{u_{0}}^{u}\frac{1}{\mu (\tau )}\,d\tau =\int _{u_{0}}^{u} \frac{1}{\omega ^{p}(\tau ^{1/p})}\,d\tau \), then we get \([0,+\infty )\in \mathrm {Dom}(W^{-1})\) and
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 \(C_{1-\beta}((0, T],X)\). Since T can be chosen arbitrarily large, then problem (1.1) has at least one global mild solution in \(C_{1-\beta}((0,+\infty ),X)\). Thus, we complete the proof of Theorem 4.4. □
From Theorem 4.4, we can immediately obtain the following conclusion.
Corollary 4.5
Let \(0<\gamma \le 1\), \(p>\frac{1}{\beta}\) and \(q=\frac{p}{p-1}\). Suppose \(f: (0,+\infty )\times X\rightarrow X\) is a continuous function, and there exist nonnegative functions \(l,k\in C_{1-\beta}((0,+\infty ),\mathbb{R}_{+})\cap L^{p}_{Loc,1- \beta}([0,+\infty ),\mathbb{R}_{+})\) such that
Then problem (1.1) has at least one mild solution in \(C_{1-\beta}((0,+\infty ),X)\).
Proof
Since
then we know
and if \(0<\gamma <1\), then
if \(\gamma =1\), then
Applying Theorem 4.4, we know that problem (1.1) has at least one mild solution in \(C_{1-\beta }(0,+\infty )\). Thus, the proof is complete. □
Theorem 4.6
Let \(p>\frac{1}{\beta}\) and \(q=\frac{p}{p-1}\). If \(f: (0,+\infty )\times X\rightarrow \mathbb{R}\) is a continuous function with and \(f(\cdot ,0)\in L_{1-\beta ,Loc}^{p}[0,+\infty )\), and there exists a function \(l\in C_{1-\beta}((0,+\infty ),\mathbb{R}_{+})\cap L^{p}_{Loc,1- \beta}([0,+\infty ),\mathbb{R}_{+})\) such that
Then problem (1.1) has a unique mild solution on \((0,+\infty )\).
Proof
We know
Since \(f(\cdot ,0)\in L_{Loc,1-\beta}^{p}([0,+\infty ),X)\) and \(l\in C_{1-\beta}((0,+\infty ),\mathbb{R}_{+})\cap L^{p}_{Loc,1- \beta}([0,+\infty ),\mathbb{R}_{+})\), applying Corollary 4.5, we know that problem (1.1) has at least one mild solution in \(C_{1-\beta}((0,+\infty ),X)\). 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. □
5 Global attractivity
Definition 5.1
The mild solution \(x\in C_{1-\beta }((0,+\infty ),X)\) of problem (1.2) is said to be globally attractive if \(\lim_{t\rightarrow +\infty }x(t)=0\).
The main result in the section reads as follows.
Theorem 5.2
Let \(0<\beta <\gamma <1\), \(0<\mu \le 1\), \(p>\frac{1}{\beta },l\), \(k\in C_{1-\beta}((0,+ \infty ),\mathbb{R}_{+})\cap L^{p}_{Loc,1-\beta}([0,+\infty ), \mathbb{R}_{+})\) be such that there exists a constant \(K>0\) such that
Suppose \(f: (0,+\infty )\times X\rightarrow X\) 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 solution \(x_{1}\in C_{1-\beta }((0,T],X)\) provided that \(T>1\) and \(M_{2}= \frac{KT^{\beta -\gamma }\Gamma (1-\gamma )}{\Gamma (1+\beta -\gamma )}<1\).
Proof
From (5.1), we have
Let \(\omega (t)=t^{\mu }+1\) and \(W(u)=\int _{u_{0}}^{u}\frac{1}{\omega ^{p}(t^{1/p})}\,dt=\int _{u_{0}}^{u} \frac{1}{(t^{\mu /p}+1)^{p}}\,dt\), where \(u_{0}, u>0\), then we get \([0,+\infty )\subset \mathrm{Dom}(W^{-1})\). Using Theorem 4.3, we know that problem (1.2) has at least one mild solution \(x_{1}\in C_{1-\beta }((0,T],X)\) that satisfies the following integral equation
□
Now let us define the operator \(F: C_{0}([T,+\infty ),X)\rightarrow C_{0}([T,+\infty ),X)\) by the following formula
where \(x_{1}\in C_{1-\beta }((0,T],X)\) is a mild solution of problem (1.2) given in Lemma 5.3, and T is as in Lemma 5.3. For convenience, we denote \(R_{1}=\Vert x_{1}\Vert _{1-\beta }=\sup_{0< t\le T}t^{1-\beta }\|x_{1}(t) \|\).
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 \(C_{0}([T,+\infty ), X)\).
Lemma 5.4
Under the assumptions in Theorem 5.2, F maps U into U.
Proof
For any \(x\in U\), we have
Using Lemma 2.6, we get
Since \(\beta <\lambda \), we get
Using (5.5), (5.7), (5.8), and (5.9), we get
Thus, \(\Vert Fx\Vert _{0}\le R\) for any \(x\in U\).
We now prove that Fx is a continuous function on \([T,+\infty )\). Since
then we have \(f(\cdot ,x_{1}(\cdot ))\in L_{1-\beta }^{p}[0,T], X)\). Using Lemma 2.9, we get that \(\int _{0}^{T}(\cdot -s)^{\beta -1}f(s, x_{1}(s))\,ds\) is continuous on \([T,+\infty )\). Since
where \(x\in U\), then \(f(\cdot , x(\cdot ))\) is continuous on \([T,+\infty )\) and \(f(\cdot , x(\cdot ))\in L_{Loc,1-\beta}^{p}([T,+\infty ),X)\). Using Lemma 2.8, we get that \(\int _{T}^{\cdot}(\cdot -s)^{\beta -1}f(s,x(s))\,ds\) is continuous on \([T,+\infty )\). Therefore, Fx is a continuous function on \([T,+\infty )\) when \(x\in U\).
Now let us prove that \((Fx)(t)\rightarrow 0\) as \(t\rightarrow +\infty \). For any \(x\in U\), we have
Using Lemma 2.6, we have
then we get that
Moreover, we know
Since \(0<\beta <\gamma <1\), we get that
Using (5.13) and (5.15), we get \((Fx)(t)\rightarrow 0\) as \(t\rightarrow +\infty \).
Thus, F maps U into U. The proof is complete. □
Lemma 5.5
Under the assumptions in Theorem 5.2, \(F:U\rightarrow U\) is completely continuous.
Proof
For any \(T_{1}>T>1\) and \(x\in U\), let \(T\le t_{1}< t_{2}\le T_{1}\), then we get
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 \(t\in [T,+\infty )\) and \(x\in U\). Using the proof of Lemma 5.4, we can get that \(\lim_{t\rightarrow +\infty }|(Fx)(t)|=0\) is uniformly for \(x\in U\). Therefore, we get that the set FU is relatively compact.
We now show that F is continuous, that is \(x_{n}\rightarrow x\) implies \(Fx_{n}\rightarrow Fx\). Since \(x_{n}(t)\rightarrow x(t)\), then \(f(t,x_{n}(t))\rightarrow f(t,x(t))\) for \(t\in [T,+\infty )\). Therefore, we have
Since \(l, k\in C([T,+\infty ),\mathbb{R}_{+})\) and
then we have \((t-\cdot )^{\beta -1}f(\cdot ,x_{n}(\cdot ))\in L^{1}([T,t],X)\). From (5.17) and (5.18), using the Lebesgue dominated convergence theorem, we have
as \(n\rightarrow +\infty \). Therefore, \((Fx_{n})(t)\rightarrow (Fx)(t)\) pointwise on \([T,+\infty )\) as \(n\rightarrow +\infty \). With the fact that F is compact, then \(\Vert Fx_{n}-Fx\Vert _{0}\rightarrow 0\) as \(n\rightarrow +\infty \), which implies F is continuous.
Therefore, \(F:U\rightarrow U\) is completely continuous. □
Lemma 5.6
Under the assumptions in Theorem 5.2, the following integral equation
has at least one mild solution in \(C_{0}([T,+\infty ),X)\), where \(x_{1}\in C_{1-\beta }((0,T],X)\) is the mild solution of problem (5.3), and T is as in Lemma 5.3.
Proof
Using Lemma 5.4, Lemma 5.5, and Theorem 4.3, we have that the integral equation (3.21) has at least one mild solution \(x_{2}\in C_{0}([T,+\infty ),X)\). □
Now we give the proof of Theorem 5.2.
Proof of Theorem 5.2
We denote
where \(x_{1}\in C_{1-\beta }((0,T],X)\) is a mild solution of problem (5.3), and \(x_{2}\in C_{0}([T,+\infty ),X)\) is a mild solution of the integral equation (5.19). From (5.3) and (5.19), we know that x is continuous on \((0,+\infty )\), and we have that x is the mild solution of the following integral equation
From Theorem 4.3, we know that x is also a global mild solution of problem (1.2) and
Thus, the mild solution x of problem (1.2) is globally attractive. □
The following conclusion is a consequence of Theorem 5.2.
Theorem 5.7
Under the assumptions in Theorem 5.2, problem (1.2) has at least one mild solution \(x\in C_{1-\beta }((0,+\infty ),X)\) and
where \(\beta <\gamma _{1}<\gamma <1\).
Proof
From Theorem 5.2, we get that \(x\in C_{1-\beta }((0,+\infty ),X)\) is a globally attractive mild solution of problem (1.2). Since \(0<\beta <\gamma _{1}<\gamma <1\), for \(t>T\), then we have
Using (5.12) and (5.14), we get
Thus, \(x(t)=t^{\beta -1}x_{0}+o(t^{\beta -\gamma _{1}})\) as \(t\rightarrow +\infty \). □
Remark 5.8
In fact, from (5.12) and (5.14), we get that the mild solution x of problem (1.2) satisfies
where \(K_{1}\) and \(K_{2}\) are nonnegative constants.
Theorem 5.9
Let \(0<\mu \le 1\), \(0<\beta <1\), \(\gamma >\beta \), \(p>1\), \(\beta >\frac{1}{p}>2 \beta -1\), \(l\in C_{(1-\mu )(1-\beta )}((0,+\infty ), \mathbb{R}_{+})\cap L^{p}_{Loc,(1- \mu )(1-\beta )}([0,+\infty ),\mathbb{R}_{+})\). Suppose that there exists a constant \(K>0\) such that
and \(f: (0,+\infty )\times X\rightarrow X\) is a continuous function with
Then problem (1.2) is global attractive.
6 Deduced results
In this section, we derive some deduced results for the following first-order and Caputo fractional semilinear evolution equations
Definition 6.1
A function \(x\in C([0,T],X)\) is called a mild solution of problem (6.1) if it satisfies the following fractional integral equation
Definition 6.2
A function \(x\in C([0,T],X)\) is called a mild solution of problem (6.2) if it satisfies the following fractional integral equation
where
Theorem 6.3
Suppose \(f: (0,T]\times X\rightarrow X\) is a continuous function, and there exists a function \(l\in C([0,T],\mathbb{R}_{+})\) and a nondecreasing function \(\omega \in C([0,+\infty ),\mathbb{R}_{+})\) such that
Then problem (6.1) has at least one mild solution in \(C([0,T],X)\) provided that
where \(W(u)=\int _{u_{0}}^{u}\frac{1}{\omega (\tau )}\,d\tau \), \(u_{0}, u>0\).
Proof
Define the operator \(G_{1}: C([0,T],X)\rightarrow C([0,T],X)\) by
Similar to the proof of Theorem 4.3, we only prove that the set \(\Lambda _{1} =\{x \in C([0,T],X): x=\lambda G_{1}x\) for some \(0<\lambda <1\}\) is bounded. Indeed, for \(x\in \Lambda _{1} \) one has
Using Lemma 2.4, we obtain
which shows that the set \(\Lambda _{1} \) is bounded. The proof is complete. □
Theorem 6.4
Let \(p>\frac{1}{\beta}\) and \(q=\frac{p}{p-1}\). Suppose \(f: (0,T]\times X\rightarrow X\) is a continuous function, and there exists a function \(l\in C((0,T],\mathbb{R}_{+})\cap L^{p}([0,T],\mathbb{R}_{+})\) and a nondecreasing function \(\omega \in C([0,+\infty ),\mathbb{R}_{+})\) such that
Then problem (6.2) has at least one mild solution in \(C([0,T],X)\) provided that
where \(c= \frac{M\Gamma ^{\frac{1}{q}}(q(\beta -1)+1)\Gamma ^{\frac{1}{q}}(q+1)}{\Gamma (q )\Gamma ^{\frac{1}{q}}(q(\beta -1)+q+2)}\), \(W(u)=\int _{u_{0}}^{u}\frac{1}{\omega ^{p}(\tau ^{1/p})}\,d\tau \), \(u_{0}, u>0\).
Proof
Define the operator \(G_{2}: C([0,T],X)\rightarrow C([0,T],X)\) by
Similar to the proof of Theorem 4.3, we only prove that the set \(\Lambda _{2} =\{x \in C([0,T],X): x=\lambda G_{2}x\) for some \(0<\lambda <1\}\) is bounded. Indeed, for \(x\in \Lambda _{2} \) one has
By Lemma 3.1 for \(\alpha =\delta =0\), we obtain
which shows that the set \(\Lambda _{2}\) is bounded. The proof is complete. □
Remark 6.5
Theorems 6.3 and 6.4 generalize and improve the results on the existence of mild solutions in [24, 25, 27].
Data Availability
No datasets were generated or analysed during the current study.
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The First author is supported by the Basic Ability Improvement Project for Middle-Aged and Young Teachers of Universities in Guangxi (No. 2021KY0172), the Special Fund for Science and Technological Bases and Talents of Guangxi (No. GUIKE AD21220103) and the Start-up Project of Scientific Research on Introducing Talents at School Level in Guangxi University for Nationalities (No. 2019KJQD04). The second author is supported by the Basic Ability Improvement Project for Middle-Aged and Young Teachers of Universities in Guangxi (No. 2022KY1210).
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Jiang, C., Xu, K. Global existence and attractivity for Riemann-Liouville fractional semilinear evolution equations involving weakly singular integral inequalities. J Inequal Appl 2024, 64 (2024). https://doi.org/10.1186/s13660-024-03137-x
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DOI: https://doi.org/10.1186/s13660-024-03137-x