Monotone iterative technique for impulsive fractional evolution equations with noncompact semigroup
 Lingzhong Zhang^{1}Email author and
 Yue Liang^{2}
https://doi.org/10.1186/s1366201506656
© Zhang and Liang 2015
Received: 27 March 2015
Accepted: 9 October 2015
Published: 19 October 2015
Abstract
This paper deals with the existence of mild solutions for a class of Caputo fractional impulsive evolution equation with nonlocal condition and noncompact semigroup. By using a monotone iterative technique in the presence of coupled lower and upper Lquasisolutions and using Sadovskii’s fixed point theorem, some existence theorems are obtained. The discussion is based on operator semigroup theory.
Keywords
MSC
1 Introduction
On the other hand, as far as we know that there are few papers studied the fractional evolution equations with noncompact semigroup. Recently, Wang et al. [18] discussed the local existence of mild solutions for nonlocal problem of fractional evolution equations under the situation that −A generates a noncompact analytic semigroup. Chen et al. [19] investigated the existence of saturated mild solutions for the initial value problem of fractional evolution equations under the situation that −A generates an equicontinuous \(C_{0}\)semigroup.
In present work, we only assume that −A generates a positive \(C_{0}\)semigroup in Theorem 1 and Theorem 3, which is noncompact and nonanalytic. In Theorem 2, −A generates a positive and equicontinuous \(C_{0}\)semigroup, but the other conditions on f, g, and \(I_{k}\) are much weaker than existing results.
The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional calculus and the measure of noncompactness. The definition of coupled lower and upper Lquasisolutions of the system (2) is also given in this section. In Section 3, we study the existence of coupled mild Lquasisolutions and mild solutions for the system (2). Particularly, a uniqueness result is also obtained in this section. An example is given in Section 4 to illustrate the effectiveness of our results.
2 Preliminaries
Definition 1
A \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)) in X is said to be positive, if the inequality \(S(t)x\geq0\) holds for \(x\geq0\) and \(t\geq0\).
It is clear that for any \(M\geq0\), \((A+MI)\) also generates a \(C_{0}\)semigroup \(S_{1}(t)=e^{Mt}S(t)\) (\(t\geq0\)) in X. \(S_{1}(t)\) (\(t\geq0\)) is a positive \(C_{0}\)semigroup if \(S(t)\) (\(t\geq0\)) is positive. For more details about the positive \(C_{0}\)semigroup, please see [20].
Let us recall the following known definitions in fractional calculus. For more details, see [3, 5, 8–10] and the references therein.
Definition 2
Remark 1
 (1)
The Caputo derivative of a constant is equal to zero.
 (2)
If f is an abstract function with values in X, then the integrals which appear in Definition 2 are taken in Bochner’s sense.
Lemma 1
[9]
A measurable function \(h: [0,a]\rightarrow X\) is Bochner integrable if \(\h\\) is Lebesgue integrable.
Lemma 2
 (i)For any fixed \(t\geq0\) and any \(x\in X\), one has$$\bigl\Vert U(t)x\bigr\Vert \leq C\x\, \qquad \bigl\Vert V(t)x\bigr\Vert \leq\frac{qC}{\Gamma(q+1)}\x\=\frac{C}{\Gamma(q)}\x\. $$
 (ii)
The operators \(U(t)\) and \(V(t)\) are strongly continuous for all \(t\geq0\).
 (iii)
If \(S(t)\) (\(t\geq0\)) is an equicontinuous semigroup, \(U(t)\) and \(V(t)\) are equicontinuous in X for \(t>0\).
Proof
We denote by \(C(J,X)\) the Banach space of all continuous Xvalue functions on interval J with the norm \(\u\_{C}=\max_{t\in J}\u(t)\\). Let \(\alpha_{X}(\cdot)\) and \(\alpha_{C(J, X)}(\cdot)\) denote the Kuratowski measure of noncompactness of the bounded set in X and \(C(J, X)\), respectively. Let \(B\subset X\) be a bounded set. It is well known that \(0\leq \alpha_{X}(B)<\infty\). \(\alpha(B)\equiv0\) if and only if the set B is precompact. For more details of the definition and properties of measure of noncompactness, see [21]. For any \(B\subset C(J,X)\) and \(t\in J\), set \(B(t)=\{u(t): u\in B\}\subset X\). If B is bounded in \(C(J, X)\), \(B(t)\) is bounded in X, and \(\alpha_{X}(B(t))\leq\alpha_{C(J,X)}(B)\). A mapping \(Q: B\rightarrow B\) is said to be condensing, if \(\alpha_{C(J,X)}(Q(B))<\alpha_{C(J,X)}(B)\). For the measure of noncompactness, the following lemmas will be used in this paper.
Lemma 3
[22]
Lemma 4
[23]
Lemma 5
[24]
Let \(B\subset C(J, X)\) be bounded. Then there exists a countable subset \(B_{0}\) of B such that \(\alpha_{C(J,X)}(B)\leq2\alpha_{C(J,X)}(B_{0})\).
Lemma 6
[25] (Sadovskii’s fixed point theorem)
Let X be a Banach space and Ω be a nonempty bounded convex closed set in X. If \(Q: \Omega\rightarrow\Omega\) is a condensing mapping, then Q has a fixed point in Ω.
Let E be an ordered Banach space with the norm \(\\cdot\\) and the partial order ≤, whose positive cone \(K=\{x\in E: x\geq0\}\) is normal. Let \(\mathit{PC}(J, E)\) = {\(u: J\rightarrow E: u(t)\) is continuous at \(t\neq t_{k}\) and left continuous at \(t=t_{k}\) and \(u(t_{k}^{+})\) exists, \(k=1,2,\ldots, m\)}. Evidently, \(\mathit{PC}(J, E)\) is a Banach space with the norm \(\u\_{\mathit{PC}}=\sup_{t\in J}\u(t)\\). \(\mathit{PC}(J, E)\) is also an ordered Banach space with partial order ≤ reduced by the positive cone \(K_{\mathit{PC}}=\{u\in \mathit{PC}(J, E): u(t)\geq0, {t\in J}\}\). We use \(E_{1}\) to denote the Banach space \(D(A)\) with the graph norm \(\\cdot\_{1}=\ \cdot\+\A\cdot\\). Let \(J'=J\backslash\{t_{1}, t_{2},\ldots, t_{m}\}\). An abstract function \(u\in \mathit{PC}(J, E)\cap C^{1}(J', E)\cap C(J', E_{1})\) is called a solution of the system (2) if \(u(t)\) satisfies all the equalities in (2).
Definition 3
In this paper we adopt the following definition of mild solutions of the system (2), which comes from [17].
Definition 4
In the proof of the main results, we also need the following generalized GronwallBellman inequality, which can be found in [26].
Lemma 7
Remark 2
In Lemma 7, if \(a(t)\equiv0\) for all \(0\leq t< T\), we easily see that \(u(t)=0\).
3 Main results
 (H_{1}):

\(f\in C(J\times E\times E, E)\) and there exist \(M>0\) and \(L\geq 0\) such thatfor any \(t\in J\), \(v_{0}(t)\leq x_{1}\leq x_{2}\leq w_{0}(t)\) and \(v_{0}(t)\leq y_{2}\leq y_{1}\leq w_{0}(t)\);$$f(t, x_{2}, y_{2})f(t, x_{1}, y_{1}) \geqM(x_{2}x_{1})+L(y_{2}y_{1}), $$
 (H_{2}):

\(I_{k}\in C(E\times E, E)\) satisfiesfor any \(t\in J\), \(v_{0}(t)\leq x_{1}\leq x_{2}\leq w_{0}(t)\) and \(v_{0}(t)\leq y_{2}\leq y_{1}\leq w_{0}(t)\);$$I_{k}(x_{1}, y_{1})\leq I_{k}(x_{2}, y_{2}), \quad k=1,2,\ldots,m, $$
 (H_{3}):

\(g: [v_{0}, w_{0}]\times[v_{0}, w_{0}]\rightarrow E\) is continuous and satisfiesfor any \(v_{0}\leq x_{1}\leq x_{2}\leq w_{0}\) and \(v_{0}\leq y_{2}\leq y_{1}\leq w_{0}\);$$g(x_{1}, y_{1})\leq g(x_{2}, y_{2}), $$
 (H_{4}):

there exists a constant \(L_{1}>0\) such thatfor \(t\in J\), increasing monotonic sequence \(\{x_{n}\}\subset[v_{0}(t), w_{0}(t)]\) and decreasing sequence \(\{y_{n}\}\subset[v_{0}(t), w_{0}(t)]\);$$\alpha_{E}\bigl(\bigl\{ f(t, x_{n}, y_{n})+f(t, y_{n}, x_{n})\bigr\} \bigr)\leq L_{1} \alpha_{E}\bigl(\{x_{n}\}+\{ y_{n}\}\bigr), $$
 (H_{5}):

\(\{g(x_{n}, y_{n})\}\) is precompact for any monotone sequence \(\{ x_{n}\}, \{y_{n}\}\subset[v_{0}, w_{0}]\).
Theorem 1
Let −A generate a positive \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)). Assume that the system (2) has coupled lower and upper Lquasisolutions \(v_{0}\) and \(w_{0}\) with \(v_{0}\leq w_{0}\). If the assumptions (H_{1})(H_{5}) hold, the system (2) has minimal and maximal coupled mild Lquasisolutions between \(v_{0}\) and \(w_{0}\), which can be obtained by a monotone iterative procedure starting from \(v_{0}\) and \(w_{0}\).
Proof
For any \(t\in J\), \(v_{0}(t)\leq x_{1}(t)\leq x_{2}(t)\leq w_{0}(t)\) and \(v_{0}(t)\leq y_{2}(t)\leq y_{1}(t)\leq w_{0}(t)\), from the assumptions (H_{1})(H_{3}) and the positive property of the operators \(U_{1}(t)\) and \(V_{1}(t)\) for \(t\geq0\), it follows that \(Q(x_{1}, y_{1})(t)\leq Q(x_{2}, y_{2})(t)\) for all \(t\in J\), which means that Q is a mixed monotone operator.
For convenience, let \(B=\{v_{n}: n\in{\mathbb{N}}\}+\{w_{n}: n\in {\mathbb{N}}\}\), \(B_{1}=\{v_{n}: n\in{\mathbb{N}}\}\), \(B_{2}=\{w_{n}: n\in {\mathbb{N}}\}\), \(B_{10}=\{v_{n1}: n\in{\mathbb{N}}\}\), and \(B_{20}=\{ w_{n1}: n\in{\mathbb{N}}\}\). Then \(B=B_{1}+B_{2}\), \(B_{1}=Q(B_{10}, B_{20})\), and \(B_{2}=Q(B_{20}, B_{10})\). From \(B_{10}=B_{1}\cup\{v_{0}\}\) and \(B_{20}=B_{2}\cup\{w_{0}\}\), it follows that \(\alpha _{E}(B_{10}(t))=\alpha_{E}(B_{1}(t))\) and \(\alpha_{E}(B_{20}(t))=\alpha _{E}(B_{2}(t))\) for \(t\in J\). Let \(J_{0}=[0, t_{1}]\), \(J_{k}=(t_{k}, t_{k+1}]\), \(k=1,2,\ldots,m\), and let \(\varphi(t):=\alpha_{E}(B(t))\) for \(t\in J\). Going from \(J_{0}\) to \(J_{m}\) interval by interval, we show that \(\varphi (t)\equiv0\) for \(t\in J\).
 \((\mathrm{H}_{4})'\) :

There exists a constant \(\overline{L}_{1}>0\) such thatfor any \(t\in J\) and monotone sequences \(\{x_{n}\}, \{y_{n}\}\subset [v_{0}(t), w_{0}(t)]\).$$\alpha_{E}\bigl(\bigl\{ f(t, x_{n}, y_{n})\bigr\} \bigr)\leq\overline{L}_{1}\bigl(\alpha_{E}\bigl( \{x_{n}\} \bigr)+\alpha_{E}\bigl(\{y_{n}\}\bigr) \bigr), $$
Corollary 1
Let −A generate a positive \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)). Assume that the system (2) has coupled lower and upper Lquasisolutions \(v_{0}\) and \(w_{0}\) with \(v_{0}\leq w_{0}\). If the conditions (H_{1})(H_{3}), \((\mathrm{H}_{4})'\), and (H_{5}) hold, the system (2) has minimal and maximal coupled mild Lquasisolutions between \(v_{0}\) and \(w_{0}\), which can be obtained by a monotone iterative procedure starting from \(v_{0}\) and \(w_{0}\).
 (H_{6}):

there exists a constant \(L_{2}>0\) such thatfor any \(t\in J\) and countable subsets \(\{x_{n}\}, \{y_{n}\}\subset [v_{0}(t),w_{0}(t)]\);$$\alpha_{E}\bigl(\bigl\{ f(t, x_{n}, y_{n})\bigr\} \bigr)\leq L_{2}\bigl(\alpha_{E}\bigl(\{x_{n}\} \bigr)+\alpha_{E}\bigl(\{ y_{n}\}\bigr)\bigr), $$
 (H_{7}):

there exist \(M_{k}>0\), \(k=1,2,\ldots,m\) with \(\sum_{k=1}^{m}M_{k}<\frac{1}{4C}\) such thatfor any countable subsets \(\{x_{n}\}, \{y_{n}\}\subset[v_{0},w_{0}]\);$$\alpha_{E}\bigl(\bigl\{ I_{k}\bigl(x_{n}(t_{k}), y_{n}(t_{k})\bigr)\bigr\} \bigr)\leq M_{k}\bigl[ \alpha_{E}\bigl(\bigl\{ x_{n}(t_{k})\bigr\} \bigr)+ \alpha_{E}\bigl(\bigl\{ y_{n}(t_{k})\bigr\} \bigr) \bigr],\quad k=1,2,\ldots,m, $$
 (H_{8}):

\(\{g(x_{n}, y_{n})\}\) is precompact for any countable subsets \(\{ x_{n}\}, \{y_{n}\}\subset[v_{0},w_{0}]\).
Then we obtain the following existence result.
Theorem 2
Let −A generate a positive and equicontinuous \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)) in E. Assume that the system (2) has coupled lower and upper Lquasisolutions \(v_{0}\) and \(w_{0}\) with \(v_{0}\leq w_{0}\). If the conditions (H_{1})(H_{3}) and (H_{6})(H_{8}) hold, the system (2) has minimal and maximal coupled mild Lquasisolutions \(\underline{u}\) and u̅ between \(v_{0}\) and \(w_{0}\), and has at least one mild solution between \(\underline{u}\) and u̅.
Proof
 (i)
If \(4C\sum_{k=1}^{m}M_{k}+\frac{4C(M+2L_{2})}{\Gamma (q+1)}a^{q}<1\), then the \(T: [v_{0}, w_{0}]\rightarrow[v_{0}, w_{0}]\) is a condensing mapping. By Lemma 6, T has at least one fixed point u in \([v_{0}, w_{0}]\).
 (ii)If \(4C\sum_{k=1}^{m}M_{k}+\frac{4C(M+2L_{2})}{\Gamma (q+1)}a^{q}\geq1\), then divided \(J=[0, a]\) into n equal parts. Let \(\triangle_{n}: 0=t_{0}'< t_{1}'<\cdots<t_{n}'=a\) and \(t_{i}'\) (\(i=1,2,\ldots ,n1\)) be not the impulsive points such that$$ 4C\sum_{k=1}^{m}M_{k}+ \frac{4C(M+2L_{2})}{\Gamma(q+1)}\ \triangle_{n}\^{q}< 1. $$(12)
Remark 3
Analytic semigroup and differentiable semigroup are equicontinuous semigroup [27]. In applications of partial differential equations, such as parabolic and strongly damped wave equations, the corresponding solution semigroup is an analytic semigroup. So Theorem 2 has extensive applicability.
 (H_{9}):

there exist \(M_{3}>0\) and \(L_{3}>0\) such thatfor any \(t\in J\), \(v_{0}(t)\leq x_{1}\leq x_{2}\leq w_{0}(t)\) and \(v_{0}(t)\leq y_{2}\leq y_{1}\leq w_{0}(t)\);$$f(t, x_{2}, y_{2})f(t, x_{1}, y_{1}) \leq M_{3}(x_{2}x_{1})L_{3}(y_{2}y_{1}), $$
 (H_{10}):

\(g: [v_{0}, w_{0}]\times[v_{0}, w_{0}]\rightarrow E\) is continuous and satisfiesfor any \(v_{0}\leq x_{1}\leq x_{2}\leq w_{0}\) and \(v_{0}\leq y_{2}\leq y_{1}\leq w_{0}\); particularly, for any \(x_{1}, x_{2}\in[v_{0}, w_{0}]\) with \(x_{1}\leq x_{2}\), one has$$g(x_{1}, y_{1})\leq g(x_{2}, y_{2}), $$$$g(x_{2}, x_{1})g(x_{1}, x_{2})=0, $$
Theorem 3
Let −A generate a positive \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)) in E. Assume that the system (2) has coupled lower and upper Lquasisolutions \(v_{0}\) and \(w_{0}\) with \(v_{0}\leq w_{0}\). If the conditions (H_{1}), (H_{2}), (H_{5}), (H_{9}), and (H_{10}) hold, the system (2) has a unique mild solution between \(v_{0}\) and \(w_{0}\), which can be obtained by a monotone iterative procedure starting from \(v_{0}\) or \(w_{0}\).
Proof
Continuing such a process interval by interval up to \(J_{m}\), we see that \(\overline{u}(t)\equiv\underline{u}(t)\) over the whole J. Hence, \(\tilde{u}:=\overline{u}=\underline{u}\) is the unique mild solution of the system (2) on \([v_{0}, w_{0}]\), which can be obtained by the monotone iterative procedure starting from \(v_{0}\) or \(w_{0}\). □
4 An example
 (P_{1}):

\(0\leq I_{k}(0, w(t_{k}, y))\) and \(I_{k}(w(t_{k}, y), 0)\leq\Delta w_{t=t_{k}}\), \(k=1,2,\ldots, m\), \(y\in[0, \pi]\);
 (P_{2}):

\(0\leq u_{0}(y)+g(0, w(t, y))\) and \(u_{0}(y)+g(w(t, y), 0)\leq w(0, y)\), \((t, y)\in[0, 1]\times(0,\pi)\);
 (P_{3}):

\(Lw(t, y)\leq f(t, y, 0, w(t, y))\) and \(f(t, y, w(t, y), 0)\leq D_{t}^{q}w(t, y)+(ALI)w(t, y)\), \((t, y)\in[0, 1]\times[0,\pi]\), \(t\neq t_{k}\).
Therefore, if the functions f, g, and \(I_{k}\) (\(k=1,2,\ldots, m\)) satisfy the conditions (H_{1})(H_{5}) on the interval \([0, w]\), the system (13) has minimal and maximal coupled mild Lquasisolutions between 0 and w.
If the functions f, g, and \(I_{k}\) (\(k=1,2,\ldots, m\)) satisfy the conditions (H_{1})(H_{3}) and (H_{6})(H_{8}) on the interval \([0, w]\), the system (13) has at least mild Lquasisolutions on \([0, w]\).
If the functions f, g, and \(I_{k}\) (\(k=1,2,\ldots, m\)) satisfy the conditions (H_{1}), (H_{2}), (H_{5}), (H_{9}), and (H_{10}) on the interval \([0, w]\), the system (13) has a unique mild solution on \([0, w]\).
Declarations
Acknowledgements
Thanks to the reviewers for their helpful comments and suggestions. The second author is supported by the Sheng Tongsheng Technological Innovation Fund of Gansu Agricultural University (Grant No. GSAUSTS1423).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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