Existence of mild solutions for fractional nonlocal evolution equations with delay in partially ordered Banach spaces
- Yue Liang^{1}Email author,
- He Yang^{2} and
- Kun Gou^{3}
https://doi.org/10.1186/s13662-016-1058-1
© The Author(s) 2017
Received: 22 August 2016
Accepted: 7 December 2016
Published: 10 January 2017
The Erratum to this article has been published in Advances in Difference Equations 2017 2017:40
Abstract
This paper deals with the existence of mild solutions for the abstract fractional nonlocal evolution equations with noncompact semigroup in partially ordered Banach spaces. Under some mixed conditions, a group of sufficient conditions for the existence of abstract fractional nonlocal evolution equations are obtained by using a Krasnoselskii type fixed point theorem. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to illustrate the applicability of abstract result.
Keywords
MSC
1 Introduction
2 Preliminaries
Let \((X,\leq,\|\cdot\|)\) be a partially ordered normed linear space. Two elements x, y in X are said to be comparable if either \(x\leq y\) or \(y\leq x\). A function \(\varPsi : X\rightarrow X\) is said to be upper semi-continuous (u.s.c.) if the set \(\{x\in X: \varPsi (x)\cap B\neq \varnothing\}\) is closed for any closed subset B in X. Definitions 1-8 have been introduced in [2, 3, 7] and are frequently used in the subsequent part of the article.
Definition 1
- (i)
if a nondecreasing sequence \(\{x_{n}\}_{n=0}^{\infty}\subset X\) converges to \(x^{*}\), then \(x_{n}\leq x^{*}\) for all \(n\in{\mathbb{N}}\), or
- (ii)
if a nonincreasing sequence \(\{x_{n}\}_{n=0}^{\infty}\subset X\) converges to \(x^{*}\), then \(x_{n}\geq x^{*}\) for all \(n\in{\mathbb{N}}\).
Definition 2
The order relation ≤ and the norm \(\|\cdot\|\) on partially ordered normed linear space \((X,\leq,\|\cdot\|)\) are compatible if for any monotone nondecreasing or monotone nonincreasing sequence \(\{x_{n}\} \subset X\), the convergence of any subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\} \) to \(x^{*}\) implies that the whole sequence \(\{x_{n}\}\) converges to \(x^{*}\).
Definition 3
A mapping \(Q: X\rightarrow X\) is nondecreasing if \(x\leq y\) implies \(Qx\leq Qy\) for all \(x,y\in X\). Similarly, Q is nonincreasing if \(x\leq y\) implies \(Qx\geq Qy\) for all \(x,y\in X\).
Definition 4
A mapping \(Q: X\rightarrow X\) is partially continuous at a point \(a\in X\) if for every \(\epsilon>0\) there exists a \(\delta>0\) such that \(\| Qx-Qa\|<\epsilon\) whenever x is comparable to a and \(\|x-a\| <\delta\). If Q is partially continuous at every point of X, then it is partially continuous on X.
Definition 5
A mapping \(Q: X\rightarrow X\) is partially bounded if \(Q(C)\) is bounded for every chain C in X. Q is bounded if \(Q(X)\) is a bounded subset of X.
Remark 1
If Q is bounded in X, then it is partially bounded in X. However, the reverse description does not hold.
Definition 6
A mapping \(Q: X\rightarrow X\) is partially compact if \(Q(C)\) is relatively compact on X for all totally ordered sets or chains C of X.
Definition 7
A mapping \(\psi: {\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}\) is called a \(\mathfrak{D}\)-function if it is an upper semi-continuous and monotonic nondecreasing function satisfying \(\psi(0)=0\).
Remark 2
Examples of \(\mathfrak{D}\)-functions are \(\psi(r)=kr\) for \(k>0\), \(\psi(r)=\frac{r}{1+r}\) and \(\psi(r)=\tan^{-1}r\) etc. If \(\phi,\psi\) are two \(\mathfrak{D}\)-functions, then (i) \(\phi+\psi\), (ii) \(\lambda \phi,\lambda>0\), and (iii) \(\phi\circ\psi\) are all \(\mathfrak{D}\)-functions.
Definition 8
Our result is based on the following Krasnoselskii type fixed point theorem, which can be found in [3].
Lemma 1
- (a)
A is partially bounded and it is a partially nonlinear \(\mathcal{D}\)-contraction,
- (b)
B is partially continuous and partially compact, and
- (c)
there exists an element \(x_{0}\in X\) such that \(x_{0}\leq Ax_{0}+Bx_{0}\).
At the end of this section, we recall the definitions of fractional calculus. See [5, 12] for more details.
Definition 9
Remark 3
By Definition 9, the Caputo derivative of a constant is equal to zero.
3 Existence theorem
Let E be a partially ordered Banach space with the partial order ≤ and the norm \(\|\cdot\|\), whose positive cone K is defined by \(K=\{ x\in E: x\geq0\}\). It is well known that if cone K is normal, then the order relation ≤ and the norm \(\|\cdot\|\) in E are compatible. Throughout this paper, we assume that \(-A: D(A)\subset E\rightarrow E\) generates a positive \(C_{0}\)-semigroup \(T(t)\) (\(t\geq0\)) of uniformly bounded linear operator in E. Namely, there exists a constant \(M>0\) such that \(\|T(t)\|\leq M\) for all \(t\geq0\). A \(C_{0}\)-semigroup \(T(t)\) (\(t\geq0\)) is said to be positive if the inequality \(T(t)x\geq0 \) holds for all \(x\geq0\). Let \(K\geq0\) be a constant. Then \(-(A+KI)\) generates a positive \(C_{0}\)-semigroup \(S(t)=e^{-Kt}T(t)\) (\(t\geq 0\)) in E. For more properties of operator semigroup and positive \(C_{0}\)-semigroup, please see [4, 8].
- (H1)
\(\|e^{-K\theta(t_{2}^{q}-t_{1}^{q})}T(t_{2}^{q}\theta)-T(t_{1}^{q}\theta)\| \rightarrow0\) as \(t_{2}-t_{1}\rightarrow0\) for every \(\theta\in(0,+\infty)\).
- (H2)The function \(f(t,x): J\times C([-r, 0], E)\rightarrow E\) is continuous in x for all \(t\in J\) and satisfies the following conditions:
- (i)There exist a constant \(0<\sigma\leq\frac{\varGamma (q+1)}{Mb^{q}}\) and a \(\mathcal{D}\)-function ψ satisfying \(\psi(r)< r\) for \(r>0\) such thatfor all \(t\in J\) and \(x,y\in C([-r,b], E)\) with \(x\geq y\), where \(K>0\) is a constant.$$0\leq\bigl[f(t,x_{t})+Kx(t)\bigr]-\bigl[f(t,y_{t})+Ky(t) \bigr]\leq\sigma\psi(x-y) $$
- (ii)
\(f(t,x_{t})+Kx(t)\) is bounded for all \(t\in J\).
- (i)
- (H3)
The function \(h(t,x): J\times C([-r, 0], E)\rightarrow E\) is continuous, bounded and nondecreasing in x for all \(t\in J\).
- (H4)
The function \(g: C([-r, b], E)\rightarrow E\) is continuous, bounded and nondecreasing.
- (H5)
There exists a function \(v\in C([-r, b], E)\) such that \(D_{t}^{q}v(t)+Av(t)\leq f(t, v_{t})+h(t, v_{t})\) for \(t\in J\) and \(v_{0}(t)\leq \varphi(t)+g(v)\) for \(t\in[-r,0]\).
Lemma 2
Let the assumption (H2) hold. Then the operator A is nondecreasing, partially bounded, and it is a partially nonlinear \(\mathcal{\mathcal {D}}\)-contraction in X.
Proof
Lemma 3
Let assumptions (H1), (H3), and (H4) hold. Then the operator B is nondecreasing, partially continuous and partially compact in X.
Proof
Since \(T(t)\) (\(t\geq0\)) is a positive \(C_{0}\)-semigroup, it follows that \(\{U(t)\}_{t\geq0}\) and \(\{V(t)\}_{t\geq0}\) are positive operators. Since the functions h and g are nondecreasing, it is easy to see that B is nondecreasing in X.
Theorem 1
Let \((E,\leq,\|\cdot\|)\) be a partially ordered Banach space, whose positive cone P is normal, and let −A generate a positive \(C_{0}\)-semigroup \(T(t)\) (\(t\geq0\)) in E. Assume that the conditions (H1)-(H5) hold. Then the fractional nonlocal evolution equation (1.1) has a solution \(x^{*}\in C([-r,b], E)\) and the sequence \(\{x_{n}\}\) defined by \(x_{0}=v\), \(x_{n+1}=Ax_{n}+Bx_{n}\), \(n=0,1,2,\ldots\) , converges monotonically to \(x^{*}\).
Proof
Remark 4
If \(g\equiv0\), \(A\equiv0\) and without delay in equation (1.1), Theorem 1 still extends the main result in [2], because the condition (H2) in this paper is much weaker than the condition (A_{1}) in [2] and the condition (H1) is not needed if \(A\equiv0\) in equation (1.1). On the other hand, in the case of \(A\neq0\), we do not use the assumptions of the compactness and equi-continuity of the \(C_{0}\)-semigroup in this paper, just assume that the \(C_{0}\)-semigroup \(T(t)\) (\(t\geq0\)) is positive and satisfies the condition (H1). Hence, our result also extends some existing results of evolution equation.
Remark 5
In this paper, the order relation in the partially ordered Banach space is induced by positive cone. By the closed property of positive cone, we obtain the regularity (see Definition 1) of the partially ordered Banach space. And by the normal property of positive cone, we see that the order relation and the norm in the partially ordered Banach space are compatible (see Definition 2). Hence, it is different from [1–3, 7].
4 Application
At the end of this paper, we give an example of the fractional parabolic equation to illustrate the applicability of the abstract result.
Let \(\lambda_{1}\) be an eigenvalue of the Laplace operator −Δ under the Dirichlet boundary condition \(x|_{\partial \varOmega }=0\) and \(e_{1}\in E\) the corresponding eigenvector. Assume that \(\lambda _{1}e_{1}(y)\leq f(t,y,e_{1}(y))+h(t,y,e_{1}(y))\), \(t\in[0,b]\), \(y\in \varOmega \) and \(e_{1}(y)\leq\varphi(t)+g(e_{1})(y)\), \(t\in[-r,0]\), \(y\in \varOmega \). Then the condition (H5) holds with \(v=e_{1}(y)\) for all \(y\in \varOmega \).
Notes
Declarations
Acknowledgements
The first author is thankful to the Sheng Tong-sheng Technological Innovation Fund of Gansu Agricultural University (GSAU-STS-1423) and the second author is thankful to the NSFC (11261053).
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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