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Coupled fixed point theorems with applications to fractional evolution equations
 He Yang^{1}Email authorView ORCID ID profile,
 Ravi P Agarwal^{2} and
 Hemant K Nashine^{2}
https://doi.org/10.1186/s136620171279y
© The Author(s) 2017
Received: 24 January 2017
Accepted: 13 July 2017
Published: 15 August 2017
Abstract
In this paper, we first prove some coupled fixed point theorems in partially ordered Φorbitally complete normed linear spaces. And then apply the obtained fixed point theorems to a class of semilinear evolution systems of fractional order for proving the existence of coupled mild solutions under some weaker monotone conditions. An example is given to illustrate the application of the abstract results.
Keywords
 coupled fixed point theorem
 nonlinear fractional evolution system
 equicontinuous \(C_{0}\)semigroup
 coupled mild solution
 existence
MSC
 34A08
 47H10
1 Introduction
Let E be a nonempty set, \(\Psi: E\rightarrow E\) a mapping. If for \(x\in E\), one has \(\Psi x=x\), then \(x\in E\) is called a fixed point of Ψ in E. Fixed point theory plays an important role in nonlinear functional analysis. Different types of fixed point theorems have been used to prove the existence of solutions for differential and integral equations; see [1–17]. The Banach contraction principle is one of the most powerful fixed point theorems in nonlinear analysis for proving the existence and uniqueness of fixed points in metric spaces. It is interesting to improve and extend the conditions of the Banach contraction principle. Recently, by weakening the requirement on the contraction in partially ordered metric spaces, a series of fixed point theorems are established for monotone mappings by Agarwal et al. [1], Harjani and Sadarangani [7], Nieto and RodriguezLópez [12, 13], O’Regan and Petrusel [14] and Ran and Reurings [15].
We recall some definitions of the monotone mapping. A mapping \(\Psi: E \rightarrow E\) is monotone means it is monotone nondecreasing or monotone nonincreasing. Assume that \((E,\leq)\) is a partially ordered set and \(\Psi: E \rightarrow E\). For \(x, y\in E\), if \(x\leq y\) implies \(\Psi(x)\leq\Psi(y)\), Ψ is called a monotone nondecreasing mapping in E. Similarly, we can define a monotone nonincreasing mapping in E. If for \(x_{1}, x_{2}\in E\), \(x_{1}\leq x_{2}\) implies \(\Psi (x_{1},y)\leq\Psi(x_{2},y)\) for all \(y\in E\), while for \(y_{1}, y_{2}\in E\), \(y_{1}\leq y_{2}\) implies \(\Psi(x,y_{1})\geq\Psi(x,y_{2})\) for all \(x\in E\), then we say that Ψ is a mixed monotone mapping in E.
The above mentioned fixed point theorems, see [1, 7, 12–15], are all for the monotone mapping. In [2], Bhaskar and Lakshmikantham extended the fixed point theorems obtained in [1, 7, 12–15] to the mixed monotone mapping in partially ordered metric spaces. At first, they introduced a definition of the coupled fixed point. And then some coupled fixed point theorems were proved in partially ordered metric spaces. Recently, these coupled fixed point theorems were refined and improved by Lakshmikantham and Ćirić [9], Luong and Thuan [11] and Samet [16].
In the above mentioned results, the assumptions of mixed monotone property and contraction property of the mapping Ψ are essential. The purpose of this paper is to delete or weaken these conditions. In this paper, by utilizing a different technique, we prove some coupled fixed point theorems in a partially ordered Φorbitally complete normed linear space. In our results, we neither assume that the mapping Ψ is mixed monotone, nor assume that it is a contraction. We divide the mapping Ψ into two parts, and assume that every part satisfies some conditions, by using an existing Krasnoselskiitype fixed point theorem, a coupled fixed point theorem for the mapping Ψ is proved. As applications, we apply the obtained coupled fixed point theorem to a certain abstract fractional evolution systems for proving the existence of coupled mild solutions.
The rest of this paper is organized as follows. In Section 2, some definitions are recalled and an existing Krasnoselskiitype fixed point theorem is introduced. In Section 3, coupled fixed point theorems are proved. In Section 4, we apply the obtained coupled fixed point theorem to a certain abstract fractional evolution systems. A specific example is given in Section 5 to illustrate the abstract results.
2 Preliminaries
Let E be a partially ordered normed linear space with partial order ≤ and the norm \(\\cdot\_{E}\). If two elements \(x, y\in E\) satisfy either \(x\leq y\) or \(x\geq y\), we say that they are comparable. If E is complete with respect to the norm \(\\cdot\_{E}\), we called it a partially ordered complete normed linear space.
Definitions 2.12.5 can be found in [4, 5].
Definition 2.1
Definition 2.2
A function \(\phi: {\mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}\) is called a \(\mathfrak {D}\)function if it is a monotone nondecreasing and upper semicontinuous function satisfying \(\phi(0)=0\).
Definition 2.3
Definition 2.4
A mapping \(\Psi: E\rightarrow E\) is said to be partially compact if for all totally ordered sets or chains \(C \subset E\), \(\Psi(C)\) is a relatively compact subset of E.
Definition 2.5
The norm \(\\cdot\_{E}\) and the order relation ≤ on a partially ordered normed linear space \((E,\leq,\\cdot\_{E})\) are said to be compatible if for any monotone sequence \(\{x_{n}\}\) in E, subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) converges to \(x^{*}\) implies that the whole sequence \(\{x_{n}\}\) converges to \(x^{*}\).
Remark 2.6
From Definition 2.1, if the normed linear space \((E, \\cdot\_{E})\) is complete, then it is Φorbitally complete. But the converse expression may not be true.
Remark 2.7
The \(\mathfrak{D}\)functions and the partially nonlinear \(\mathfrak {D}\)Lipschitz conditions are much useful in research of solutions for nonlinear differential equations via fixed point theorems; see [6].
The following Krasnoselskiitype fixed point theorem was proved by Dhage in [4].
Lemma 2.8
 (a):

\(A_{1}\) is continuous and a partially nonlinear \(\mathfrak{D}\)contraction,
 (b):

\(A_{2}\) is continuous and partially compact,
 (c):

there is an element \(v_{0}\in E\) satisfying \(v_{0}\leq A_{1}v_{0}+A_{2}y\) for all \(y\in E\), and
 (d):

every pair of elements has an upper and a lower bound in E.
3 Fixed point theorems
By Lemma 2.8, we first prove the following fixed point theorem.
Theorem 3.1
 (a)′:

\(A_{1}\) is Φorbitally continuous andfor all comparable elements \(x, y\in D\) with \(x\not\equiv y\),$$\A_{1}xA_{1}y\_{E}< \xy\_{E} $$
 (b)′:

\(A_{2}\) is Φorbitally continuous and partially compact,
 (c):

there exists an element \(v_{0}\in D\) such that \(v_{0}\leq A_{1}v_{0}+A_{2}y\) for all \(y\in D\), and
 (d):

every pair of elements in D has an upper and a lower bound.
Proof
 (c)′:

there is \(v_{0}\in D\) satisfying \(v_{0}\geq A_{1}v_{0}+A_{2}y\) for all \(y\in D\).
Theorem 3.2
Let E be a partially ordered Φorbitally complete normed linear space with the norm \(\\cdot\_{E}\) and the partial order ≤, whose positive cone K is normal, and let D be a nonempty closed subset of E. Assume that \(A_{1}, A_{2}: D\rightarrow D\) are two monotone nondecreasing mappings satisfying the conditions (a)′, (b)′, (c)′ and (d). Then \(x=A_{1}x+A_{2}x\) has a solution in D.
By Theorem 3.1, the following coupled fixed point theorem is obtained.
Theorem 3.3
 (i):

\(A_{1}\) is Φorbitally continuous and a partially nonlinear \(\mathfrak{D}\)contraction,
 (ii):

\(A_{2}\) is Φorbitally continuous and partially compact,
 (iii):

there is an element \(v_{0}\in D\) satisfying \(v_{0}\leq A_{1}v_{0}+A_{2}y\) for all \(y\in D\), and
 (iv):

every pair of elements in D has an upper and a lower bound.
Proof
Since \((E,\leq,\\cdot\_{E})\) is a partially ordered Φorbitally complete normed linear space, and positive cone K is normal, it follows that \((\widehat{E},\leq,\\cdot\_{\widehat{E}})\) is a partially ordered Φorbitally complete normed linear space and positive cone \(K_{\widehat{E}}\) is normal. Since D is a nonempty closed subset of E, it follows that \(D\times D\) is a nonempty closed subset of Ê.
Since \(A_{1}\) is Φorbitally continuous, by the definition of \(\widehat{A}_{1}\), it is easy to see that \(\widehat{A}_{1}\) is Φorbitally continuous.
Step II. \(\widehat{A}_{2}\) is Φorbitally continuous and partially compact.
Since \(A_{2}\) is Φorbitally continuous, by the definition of \(\widehat{A}_{2}\), it is easy to see that \(\widehat{A}_{2}\) is Φorbitally continuous.
Let \(C\subset D\) be a bounded chain. Since \(A_{2}\) is partially compact in D, it follows that \(A_{2}(C)\) is equicontinuous and uniformly bounded in D. Set \(\widehat{C}=C\times C\). Then Ĉ is a bounded chain in D̂. Next, we claim that \(\widehat {A}_{2}(\widehat{C})\) is equicontinuous and uniformly bounded in D̂
Therefore, by the ArzelaAscoli theorem, \(\widehat{A}_{2}(\widehat {C})\subset\widehat{D}\) is relatively compact. Consequently, \(\widehat {A}_{2}: \widehat{D}\rightarrow\widehat{D}\) is partially compact.
Step IV. Every pair of elements in D̂ has an upper and a lower bound.
Therefore, by Theorem 3.1, the operator equation \(u=\widehat {A}_{1}u+\widehat{A}_{2}u\) has a solution in Ê. □
By Theorems 3.2 and 3.3, the following coupled fixed point theorem is obtained. Because its proof is similar to Theorem 3.3, we omit the details.
Theorem 3.4
 (i):

\(A_{1}\) is Φorbitally continuous and a partially nonlinear \(\mathfrak{D}\)contraction,
 (ii):

\(A_{2}\) is Φorbitally continuous and partially compact,
 (iii)′:

there is an element \(v_{0}\in D\) satisfying \(v_{0}\geq A_{1}v_{0}+A_{2}y\) for all \(y\in D\), and
 (iv):

every pair of elements in D has an upper and a lower bound.
Remark 3.5
The hypothesis (iv) of Theorems 3.3 and 3.4 holds if the partially ordered set E is a lattice. We known that the set \(C(J,X)\) is a lattice, where \(C(J,X)\) is the set of all continuous Xvalued functions on \(J\in {\mathbb{R}}\), X is a partially ordered set. For any \(x,y\in C(J,X)\), \(\max\{x,y\}\) and \(\min\{x,y\}\) are the upper and lower bounds, respectively.
Remark 3.6
The assumptions of mixed monotone property and contractive property of the mapping Q are essential in [2, 9, 11, 16]. But in Theorems 3.3 and 3.4, we neither assume that the mapping Q is mixed monotone, nor assume that the mapping Q is a contraction. We only suppose that the mapping Q is a nondecreasing mapping and a part of Q (namely, the operator \(A_{1}\)) is a partially nonlinear \(\mathcal{D}\)contraction. Plus with other assumptions we obtain the coupled fixed point theorems. Hence Theorems 3.3 and 3.4 extend the main results of [2, 9, 11, 16].
4 Existence results for fractional evolution systems
For the \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)), if \(S(t)x\geq0 \) for all \(x\geq 0\), it is called a positive \(C_{0}\)semigroup. Throughout this section, we always assume that −A generates a positive \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)) of uniformly bounded linear operator in X. Namely, there is a constant \(M>0\) such that \(\S(t)\\leq M\) for all \(t\geq0\).
Definitions 4.1 and 4.2 can be found in [8, 10, 17].
Definition 4.1
Definition 4.2
Lemma 4.3
 (i):

For any \(x\in X\) and fixed \(t\geq0\), one has$$\bigl\Vert \mathcal{U}_{\sigma}(t)x \bigr\Vert \leq M \Vert x \Vert , \qquad \bigl\Vert \mathcal{V}_{\sigma}(t)x \bigr\Vert \leq \frac{M}{\Gamma(\sigma)} \Vert x \Vert . $$
 (ii):

If \(S(t)\) (\(t\geq0\)) is an equicontinuous semigroup, \(\mathcal {V}_{\sigma}(t)\) is equicontinuous in X for \(t>0\).
 (iii):

If \(S(t)\) (\(t\geq0\)) is a positive \(C_{0}\)semigroup, \(\mathcal {U}_{\sigma}(t)\) and \(\mathcal{V}_{\sigma}(t)\) are positive operators for all \(t\geq0\).
Proof
(i) and (ii) can be found in reference [10, 17]. (iii) is easily seen from the definitions of \(\mathcal{U}_{\sigma}(t)\) and \(\mathcal{V}_{\sigma}(t)\). So, we omit the details here. □
Definition 4.4
 (H1):

The positive \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)) is equicontinuous.
 (H2):

The function \(f: J\times X\rightarrow X\) is continuous in x for all \(t\in J\) and there exist a constant \(\rho\in {\mathbb{R}}\) with \(0<\rho <\frac{\Gamma(\sigma+1)}{Mb^{\sigma}}\) and a \(\mathfrak{D}\)function \(\phi: {\mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}\) with \(\phi(r)< r\) for any \(r>0\) satisfyingfor all \(t\in J\).$$0\leq f(t,u)f(t,v)\leq\rho\phi\bigl( \Vert uv \Vert \bigr),\quad \forall u, v\in X, u\geq v $$
 (H3):

The function \(h(t,x): J\times X\rightarrow X\) is continuous, nondecreasing and bounded in x.
 (H4):

There is an element \(\overline{x}\in E\) satisfyingfor all \(y\in E\).$$\textstyle\begin{cases} {}^{\mathrm{C}}D_{t}^{\sigma}\overline{x}(t)+A\overline{x}(t)\leq f(t,\overline {x}(t))+h(t,y(t)),\quad t\in J, \\ \overline{x}(0)\leq\tau_{0}\in X \end{cases} $$
Lemma 4.5
Assume that the hypothesis (H2) holds. Then the operator \(A_{1}: D\rightarrow D\) is Φorbitally continuous, nondecreasing and a partially nonlinear \(\mathfrak{D}\)contraction in E.
Proof
Lemma 4.6
Let the hypotheses (H1) and (H3) hold. Then the operator \(A_{2}: D\rightarrow D\) is Φorbitally continuous, nondecreasing and partially compact in E.
Proof
Since \(S(t)\) (\(t\geq0\)) is a positive \(C_{0}\)semigroup, by (H3) and Lemma 4.3, a similar proof as in Lemma 4.5 shows that \(A_{2}: D\rightarrow D\) is Φorbitally continuous and nondecreasing.
Theorem 4.7
Let the hypotheses (H1)(H4) hold. Then the fractional evolution system (4.1) has a coupled mild solution on J.
Proof
Define two operators \(A_{1}\) and \(A_{2}\) as in (4.2) and (4.3). By Lemmas 4.5 and 4.6, we deduce that \(A_{1}: D\rightarrow D\) is Φorbitally continuous, nondecreasing and a partially nonlinear \(\mathfrak {D}\)contraction as well as \(A_{2}: D\rightarrow D\) is Φorbitally continuous, nondecreasing and partially compact.
By Theorem 4.7, we can obtain the following corollaries easily.
Corollary 4.8
 (H2)′:

The function \(f: J\times X\rightarrow X\) is continuous in x for all \(t\in J\) and there is a constant \(\beta\in(0,\frac{\Gamma (\sigma+1)}{Mb^{\sigma}})\) such that$$0\leq f(t,u)f(t,v)\leq\beta(uv),\quad \forall u\geq v, t\in J. $$
Corollary 4.9
 (H2)″:

The function \(f: J\times X\rightarrow X\) is continuous in x for all \(t\in J\) and there is a constant \(\gamma\in(0,\frac{\Gamma (\sigma+1)}{Mb^{\sigma}})\) such that$$0\leq f(t,u)f(t,v)\leq\frac{\gamma\uv\}{1+\uv\},\quad \forall u\geq v, t\in J. $$
5 Applications
Theorem 5.1
 (P1):

The function \(f: I\times I\times {\mathbb{R}}\rightarrow {\mathbb{R}}\) is continuous and there exist a constant \(\rho\in(0,\frac{\sqrt{\pi}}{2})\) and a \(\mathcal{D}\)function \(\phi: {\mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}\) with \(\phi(r)< r\) for \(r>0\) such thatfor all \((z,t)\in I\times I\) and \(u,v\in C(I\times I, {\mathbb{R}})\) with \(u\geq v\).$$0\leq f \bigl(z,t,u(z,t) \bigr)f \bigl(z,t,v(z,t) \bigr)\leq\rho\phi \bigl( \bigl\vert u(z,t)v(z,t) \bigr\vert \bigr) $$
 (P2):

There exists a function \(\hat{x}\in C(I\times I,{\mathbb{R}})\) such thatfor any \(y\in C(I\times I,{\mathbb{R}})\).$$\textstyle\begin{cases} {}^{\mathrm{C}}D_{t}^{\frac{1}{2}}\hat{x}(z,t)+\frac{\partial\hat {x}(z,t)}{\partial z}\leq f(z,t,\hat{x}(z,t))+h(z,t,y(z,t)),\quad (z,t)\in I\times I, \\ \hat{x}(z,0)\leq\tau_{0}(z),\quad z\in(0,1), \end{cases} $$
Proof
By the assumptions (P1) and (P2), the conditions (H2) and (H4) hold. Hence by Theorem 4.7, the abstract fractional evolution system (4.1) has a coupled mild solution, which is also the coupled mild solution of the fractional hybrid dynamic system (5.1). □
Similarly, using Corollaries 4.8 and 4.9, we can obtain the following theorems.
Theorem 5.2
 (P3):

The function \(f: I\times I\times {\mathbb{R}}\rightarrow {\mathbb{R}}\) is continuous and there exist a constant \(\beta\in(0,\frac{\sqrt{\pi}}{2})\) such thatfor all \((z,t)\in I\times I\) and \(u,v\in C(I\times I, {\mathbb{R}})\) with \(u\geq v\).$$0\leq f \bigl(z,t,u(z,t) \bigr)f \bigl(z,t,v(z,t) \bigr)\leq\beta \bigl(u(z,t)v(z,t) \bigr) $$
Theorem 5.3
 (P4):

The function \(f: I\times I\times {\mathbb{R}}\rightarrow {\mathbb{R}}\) is continuous and there exist a constant \(\gamma\in(0,\frac{\sqrt{\pi}}{2})\) such thatfor all \((z,t)\in I\times I\) and \(u,v\in C(I\times I, {\mathbb{R}})\) with \(u\geq v\).$$0\leq f \bigl(z,t,u(z,t) \bigr)f \bigl(z,t,v(z,t) \bigr)\leq\frac{\gamma\uv\_{C}}{1+\uv\_{C}} $$
Declarations
Acknowledgements
The first author is thankful to the NSF (No. 11661071) and the third author is thankful to the United StatesIndia Education Foundation, New Delhi, India and IIE/CIES, Washington, DC, USA for FulbrightNehru PDF Award (No. 2052/FNPDR/2015).
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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