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Coupled fixed point theorems with applications to fractional evolution equations
Advances in Difference Equations volume 2017, Article number: 239 (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.
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
Let \(\Psi: E\rightarrow E\) be a mapping. For any \(x\in E\), we define an orbit \(\Phi(x; \Psi)\) by
If for any sequence \(\{x_{n}\}\subset\Phi(x; \Psi)\), \(x_{n}\rightarrow x^{*}\) implies \(\Psi x_{n}\rightarrow\Psi x^{*}\) for each \(x\in E\), Ψ is said to be Φorbitally continuous in E. A normed linear space \((E,\\cdot\_{E})\) is called Φorbitally complete if every Cauchy sequence \(\{x_{n}\}\subset\Phi(x; \Psi)\) converges to a point \(x^{*}\) in E.
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
A mapping \(\Psi: E\rightarrow E\) is called partially nonlinear \(\mathfrak{D}\)Lipschitz if for all comparable elements \(x, y\in E\), there is a \(\mathfrak{D}\)function \(\phi: {\mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}\) such that
Furthermore, if \(\phi(r)< r\) for \(r>0\), Ψ is called a partially nonlinear \(\mathfrak{D}\)contraction in E.
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
Let E be a partially ordered complete normed linear space with the partial order ≤ and the norm \(\\cdot\_{E}\) such that ≤ and \(\ \cdot\_{E}\) are compatible. Suppose that \(A_{1}, A_{2}: E\rightarrow E\) are two monotone nondecreasing mappings satisfying:
 (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.
Then \(x=A_{1}x+A_{2}x\) has a solution in E.
3 Fixed point theorems
Let \((E,\\cdot\_{E})\) be a Φorbitally complete normed linear space. Define a positive cone K in E by
Then E becomes now a partially ordered Φorbitally complete normed linear space with the partial order ≤ induced by K. It is well known that the partially order ≤ and the norm \(\\cdot\_{E}\) are compatible if cone K is normal.
By Lemma 2.8, we first prove the following fixed point theorem.
Theorem 3.1
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:
 (a)′:

\(A_{1}\) is Φorbitally continuous and
$$\A_{1}xA_{1}y\_{E}< \xy\_{E} $$for all comparable elements \(x, y\in D\) with \(x\not\equiv y\),
 (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.
Then the equation \(x=A_{1}x+A_{2}x\) has a solution in D.
Proof
By Remark 2.7 of [4] and the proof of Lemma 2.8, if the continuity of operators \(A_{1}\) and \(A_{2}\) is replaced by the Φorbitally continuity in conditions (a) and (b) of Lemma 2.8, the conclusion of Lemma 2.8 is still true. On the other hand, since D is a nonempty closed subset of E, it follows that D is Φorbitally complete and has partial order ≤. For any comparable elements \(x, y\in D\), \(x\not\equiv y\), by the condition (a)′, it follows that there is \(\tau\in(0,1)\) satisfying
Let \(\phi(r)=\tau r\). Then ϕ is a \(\mathfrak{D}\)function and \(\phi (r)< r\) for any \(r>0\). This implies that the condition (a) of Lemma 2.8 is satisfied. By Lemma 2.8, we obtain the desired conclusion. □
If the inequality given in assumption (c) of Lemma 2.8 is reverse, more precisely, the condition (c) of Lemma 2.8 is replaced by
 (c)′:

there is \(v_{0}\in D\) satisfying \(v_{0}\geq A_{1}v_{0}+A_{2}y\) for all \(y\in D\).
By Theorem 3.7 of [4], the conclusion of Lemma 2.8 is still true. Hence we can obtain the following fixed point theorem. Since the proof is similar to Theorem 3.1, we omit the details here.
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.
Let \(\widehat{E}:=E\times E\). Define a sum and a scalar multiplication in Ê by
for all \(w=(w_{1},w_{2}), v=(v_{1},v_{2})\in\widehat{E}\) and \(\lambda\in {\mathbb{R}}\). And define a positive cone and a norm in Ê by
Then \((\widehat{E},\\cdot\_{\widehat{E}})\) is a partially ordered normed linear space with the order relation ≤ induced by \(K_{\widehat{E}}\). Let D be a nonempty closed subset of E. Then \(D\times D\) is a nonempty closed subset of Ê. It is clear that cone \(K_{\widehat{E}}\) is normal if K is normal.
Let \(Q: \widehat{E}\rightarrow E\). A pair of elements \((x,y)\in\widehat {E}\) is called a coupled fixed point of \(Q: \widehat{E}\rightarrow E\) if and only if it satisfies
By Theorem 3.1, the following coupled fixed point theorem is obtained.
Theorem 3.3
Let E be a partially ordered Φorbitally complete normed linear space with the partial order ≤ and the norm \(\\cdot\_{E}\), 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
 (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.
Then \(Q(x,y)=A_{1}x+A_{2}y\) has a coupled fixed point in Ê.
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.
Let \(\widehat{D}=D\times D\). Define two operators \(\widehat{A}_{1}, \widehat{A}_{2}: \widehat{D}\rightarrow\widehat{D}\) by
If the operator equation \(u=\widehat{A}_{1}u+\widehat{A}_{2}u\) has a solution \(u=(x,y)\in\widehat{D}\), namely,
then we obtain
This implies that the operator \(Q(x,y)\) has a coupled fixed point in Ê. We will apply Theorem 3.1 to prove that the operator equation \(u=\widehat{A}_{1}u+\widehat{A}_{2}u\) has a solution in Ê. The proof will be given in several steps.
Step I. \(\widehat{A}_{1}\) is Φorbitally continuous and
for all comparable elements \(w, v\in\widehat{D}\) with \(w\not\equiv v\).
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.
For all comparable elements \(w=(w_{1}, w_{2}), v=(v_{1}, v_{2})\in\widehat{D}\) with \(w\not\equiv v\), we have
Hence, we obtain
for all comparable elements \(w, v\in\widehat{D}\) with \(w\not\equiv v\).
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
Since \(A_{2}(C)\subset D\) is uniformly bounded, there is a constant \(\overline{M}>0\) satisfying \(\A_{2}z\_{E}\leq\overline{M}\) for any \(z\in C\). For any \(Z\in\widehat{A}_{2}(\widehat{C})\), there are \(x,y\in C\) satisfying \(u=(x,y)\in\widehat{C}\) such that \(Z=\widehat{A}_{2}u\). Thus,
This implies that \(\widehat{A}_{2}(\widehat{C})\) is uniformly bounded in D̂.
Since \(A_{2}(C)\) is equicontinuous in D, for any \(z\in C\) and \(t_{2}>t_{1}\), we have
as \(t_{2}t_{1}\rightarrow0\). Hence for any \(Z=\widehat{A}_{2}u=\widehat {A}_{2}(x,y)\in\widehat{A}_{2}(\widehat{C})\), we have
as \(t_{2}t_{1}\rightarrow0\). This implies that \(\widehat{A}_{2}(\widehat {C})\) is equicontinuous 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 III. There is an element \(u_{0}\in\widehat{D}\) satisfying
for all \(u\in\widehat{D}\).
Let \(u_{0}=(v_{0}, v_{0})\). For any \(u=(x,y)\in\widehat{D}\), by the condition (iii), we have
Hence we obtain the desired conclusion.
Step IV. Every pair of elements in D̂ has an upper and a lower bound.
For every pair of elements \(w=(w_{1}, w_{2}), v=(v_{1}, v_{2})\in\widehat{D}\), by condition (iv), there exist \(z_{1}, z_{2}, z_{3}, z_{4}\in D\) such that
Thus, we have
Consequently, every pair of elements \(w, v\in\widehat{D}\) 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
Let E be a partially ordered Φorbitally complete normed linear space with the norm \(\\cdot\_{E}\) and the order relation ≤, positive cone K be 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
 (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.
Then \(Q(x,y)=A_{1}x+A_{2}y\) has a coupled fixed point in Ê.
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
Let \((X,\\cdot\)\) be a Φorbitally complete normed linear space. Define its positive cone as \(\overline{K}=\{x\in X: x\geq0\}\). Then X becomes a partially ordered Φorbitally complete normed linear space with the norm \(\\cdot\\) and the partial order ≤ induced by the cone K̅. In this section, we always assume that K̅ is normal. Investigate the existence of coupled mild solutions to the initial value problem of the fractional hybrid evolution system
where \(J=[0,b]\), \(b>0\) is a constant, \({}^{\mathrm{C}}D_{t}^{\sigma}\) denotes the \(\sigma\in(0, 1)\) order Caputo fractional derivative, −A generates a \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)) of uniformly bounded linear operator in X, f and h are given functions.
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
The fractional integral of order \(\sigma>0\) with the lower limits zero for a function \(f\in L^{1}(J, E)\) is defined by
where Γ is the gamma function.
Definition 4.2
The RiemannLiouville derivative of order \(n1<\sigma<n\), \(n\in {\mathbb{N}}\) with the lower limits zero for a function \(f\in L^{1}(J, E)\) can be defined as
The Caputo fractional derivative of order \(0<\sigma<1\) with the lower limits zero for a function \(f\in L^{1}(J, E)\) can be defined as
Define two operator families \(\{\mathcal{U}_{\sigma}(t)\}_{t\geq0}\) and \(\{\mathcal{V}_{\sigma}(t)\}_{t\geq0}\) as
where
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. □
Let \(C(J, X)\) be a set of all continuous Xvalued functions on the interval J and let
Then \(C(J, X)\) is a partially ordered Φorbitally complete normed linear space with the norm \(\x\_{C}:=\sup\{\x(t)\: t\geq0\}\) and the partial order ≤ induced by \(\overline{K}_{C}\). It is clear that \(\overline{K}_{C}\) is normal because K̅ is normal.
Let \(E=C(J, X)\) and \(K=\overline{K}_{C}\). Define two operators \(A_{1}, A_{2}: E\rightarrow E\) by
Definition 4.4
An element \((x,y)\in E\times E\) is called a coupled mild solution of the system (4.1) if and only if it satisfies the following system of operator equations:
We shall use Theorem 3.3 to prove that (4.4) has a coupled fixed point in \(E\times E\). For this purpose, we consider the following hypotheses:
 (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 \(r>0\) satisfying
$$0\leq f(t,u)f(t,v)\leq\rho\phi\bigl( \Vert uv \Vert \bigr),\quad \forall u, v\in X, u\geq v $$for all \(t\in J\).
 (H3):

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

There is an element \(\overline{x}\in E\) satisfying
$$\textstyle\begin{cases} {}^{\mathrm{C}}D_{t}^{\sigma}\overline{x}(t)+A\overline{x}(t)\leq f(t,\overline {x}(t))+h(t,y(t)),\quad t\in J, \\ \overline{x}(0)\leq\tau_{0}\in X \end{cases} $$for all \(y\in E\).
Since, by (H3), \(h(t,x)\) is bounded in x for all \(t\in J\), there is a constant \(\widetilde{M}>0\) such that \(\h(t,x)\\leq\widetilde{M}\) for all \(t\in J\) and \(x\in X\). For
where \(F^{*}=\sup_{t\in J}\f(t,0)\\), we define an open ball \(\mathfrak {B}(x_{0}, r)\) in E by
where \(r=\x_{0}\_{C}+r^{*}\). Let \(D=\overline{\mathfrak{B}}(x_{0}, r)\). Then D is a closed and bounded subset in E. By virtue of the assumptions (H1)(H3), we have the following lemmas.
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
By (H2), (4.2) and (4.5), for any \(x\in E\) with \(\x\_{C}\leq r^{*}\), we have
This implies that \(\A_{1}x\_{C}\leq r^{*}\) for any \(x\in E\) with \(\x\ _{C}\leq r^{*}\). Moreover, we have
This implies that \(A_{1}\) maps D into itself.
Since \(S(t)\) (\(t\geq0\)) is a positive \(C_{0}\)semigroup, by (H2) and Lemma 4.3, it follows that \(A_{1}: D\rightarrow D\) is nondecreasing. Take a sequence \(\{x_{n}\}\subset\Phi(x;A_{1})\) for any \(x\in D\) with \(x_{n}\rightarrow x^{*}\) as \(n\rightarrow\infty\). Since f is continuous in x for all \(t\in J\), by assumption (H2) and dominated convergence theorem, we have
This implies that \(A_{1}: D\rightarrow D\) is Φorbitally continuous.
For any comparable elements \(x,y\in D\), without loss of generality, we assume that \(x\geq y\). By (H2), for any \(t\in J\), we have
This implies that
because of \(0<\rho<\frac{\Gamma(\sigma+1)}{Mb^{\sigma}}\). Hence \(A_{1}: D\rightarrow D\) is a partially nonlinear \(\mathfrak{D}\)contraction in E. □
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
By the assumption (H3), (4.3) and (4.5), for any \(x\in E\) with \(\x\ _{C}\leq r^{*}\), we have
This follows that \(\A_{2}x\_{C}\leq r^{*}\) for any \(x\in E\) with \(\x\ _{C}\leq r^{*}\). Further, we have
This implies that \(A_{2}\) maps D into itself.
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.
For any \(t_{1}, t_{2}\in J\) with \(t_{1}< t_{2}\), denote
Then, by Lemma 4.3, we have
This implies that \(G_{1}\rightarrow0\) and \(G_{3}\rightarrow0\) as \(t_{2}t_{1}\rightarrow0\). If \(t_{1}\equiv0\) and \(0< t_{2}\leq b\), it is clear that \(G_{2}\equiv0\). For \(t_{1}>0\) and \(\delta\in(0,t_{1})\) small enough, we have
Hence \(G_{2}\rightarrow0\) as \(t_{2}t_{1}\rightarrow0\) and \(\delta\rightarrow 0\) because of (H1).
Let \(C\subset D\) be an arbitrary chain. For any \(Z\in A_{2}(C)\), there is \(x\in C\) such that \(Z(t)=(A_{2}x)(t)\) for all \(t\in J\). So, for any \(t_{1}, t_{2}\in J\) with \(t_{1}< t_{2}\), by the definition of the operator \(A_{2}\), the inequality
implies that
as \(t_{2}t_{1}\rightarrow0\). This further implies that \(A_{2}(C)\) is equicontinuous on J.
On the other hand, by (H3), we have
It follows that \(\Z\_{C}\leq\frac{M\widetilde{M}b^{\sigma}}{\Gamma (\sigma+1)}\). Hence \(A_{2}(C)\) is uniformly bounded in E. By the ArzelaAscoli theorem, \(A_{2}: D\rightarrow D\) is partially compact. □
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.
To apply Theorem 3.3, it remains to prove that there is an element \(\overline{x}\in D\) satisfying \(\overline{x}\leq A_{1}\overline{x}+A_{2}y\) for all \(y\in D\). By (H4), there is an elements \(\overline{x}\in D\) satisfying
for all \(y\in D\). Let \(F(t)=D^{\sigma}\overline{x}(t)+A\overline{x}(t)\), \(t\in J\). Then we have
for all \(y\in D\) and \(t\in J\). Hence all the conditions of Theorem 3.3 are satisfied. By Theorem 3.3, the system (4.4) has a coupled fixed point in Ê. Therefore, the fractional evolution system (4.1) has a coupled mild solution in Ê. □
By Theorem 4.7, we can obtain the following corollaries easily.
Corollary 4.8
Let the hypotheses (H1), (H3) and (H4) hold. In addition, the following condition is satisfied:
 (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
$$0\leq f(t,u)f(t,v)\leq\beta(uv),\quad \forall u\geq v, t\in J. $$
Then the fractional evolution system (4.1) has a coupled mild solution on J.
Corollary 4.9
Let the hypotheses (H1), (H3) and (H4) hold. In addition, the following condition is satisfied:
 (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
$$0\leq f(t,u)f(t,v)\leq\frac{\gamma\uv\}{1+\uv\},\quad \forall u\geq v, t\in J. $$
Then the fractional evolution system (4.1) has a coupled mild solution on J.
5 Applications
In this section, we apply the obtained abstract results to the following fractional hybrid dynamic system:
where \(I=[0,1]\), \(h: I\times I\times {\mathbb{R}}\rightarrow {\mathbb{R}}\) is defined by
It is clear that \(h: I\times I\times {\mathbb{R}}\rightarrow {\mathbb{R}}\) is continuous, nondecreasing and
This implies that the condition (H3) holds.
Theorem 5.1
Suppose that the following conditions are satisfied:
 (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 that
$$0\leq f \bigl(z,t,u(z,t) \bigr)f \bigl(z,t,v(z,t) \bigr)\leq\rho\phi \bigl( \bigl\vert u(z,t)v(z,t) \bigr\vert \bigr) $$for all \((z,t)\in I\times I\) and \(u,v\in C(I\times I, {\mathbb{R}})\) with \(u\geq v\).
 (P2):

There exists a function \(\hat{x}\in C(I\times I,{\mathbb{R}})\) such that
$$\textstyle\begin{cases} {}^{\mathrm{C}}D_{t}^{\frac{1}{2}}\hat{x}(z,t)+\frac{\partial\hat {x}(z,t)}{\partial z}\leq f(z,t,\hat{x}(z,t))+h(z,t,y(z,t)),\quad (z,t)\in I\times I, \\ \hat{x}(z,0)\leq\tau_{0}(z),\quad z\in(0,1), \end{cases} $$for any \(y\in C(I\times I,{\mathbb{R}})\).
Then the fractional hybrid dynamic system (5.1) has a coupled mild solution.
Proof
Let \(X=C(I, {\mathbb{R}})\). Then X is a Φorbitally complete normed linear space with the norm \(\x(t)\_{C}=\max_{t\in I}x(t)\). Define a positive cone in X by \(\overline{K}=\{x\in X: x\geq0\}\). Then K̅ is a closed convex cone in X, which is normal. Define an operator \(A: D(A)\subset X\rightarrow X\) by
It is well known that A generates a \(C_{0}\)semigroup \(S(t)\) (\(t\geq0\)) given by
Then \(S(t)\) (\(t\geq0\)) is an equicontinuous \(C_{0}\)semigroup, but it is not compact, and \(\sup_{t\in I}\S(t)\\leq1\). This implies that the condition (H1) holds.
Let
Then the fractional hybrid dynamic system (5.1) can be rewritten into the abstract fractional evolution system (4.1).
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
Assume that the condition (P2) and the following condition are satisfied:
 (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 that
$$0\leq f \bigl(z,t,u(z,t) \bigr)f \bigl(z,t,v(z,t) \bigr)\leq\beta \bigl(u(z,t)v(z,t) \bigr) $$for all \((z,t)\in I\times I\) and \(u,v\in C(I\times I, {\mathbb{R}})\) with \(u\geq v\).
Then the fractional hybrid dynamic system (5.1) has a coupled mild solution.
Theorem 5.3
Let the condition (P2) and the following condition hold:
 (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 that
$$0\leq f \bigl(z,t,u(z,t) \bigr)f \bigl(z,t,v(z,t) \bigr)\leq\frac{\gamma\uv\_{C}}{1+\uv\_{C}} $$for all \((z,t)\in I\times I\) and \(u,v\in C(I\times I, {\mathbb{R}})\) with \(u\geq v\).
Then the fractional hybrid dynamic system (5.1) has a coupled mild solution.
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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).
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Yang, H., Agarwal, R.P. & Nashine, H.K. Coupled fixed point theorems with applications to fractional evolution equations. Adv Differ Equ 2017, 239 (2017). https://doi.org/10.1186/s136620171279y
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DOI: https://doi.org/10.1186/s136620171279y