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Some generalizations of Darbo’s theorem and applications to fractional integral equations
Fixed Point Theory and Applications volume 2016, Article number: 11 (2016)
Abstract
In this paper, some generalizations of Darbo’s fixed point theorem are presented. An existence result for a class of fractional integral equations is given as an application of the obtained results.
1 Introduction and preliminaries
Let E be a Banach space over \(\mathbb{R}\) (or \(\mathbb{C}\)) with respect to a certain norm \(\\cdot\\). For any subsets X and Y of E, we have the following notations:

X̅ denotes the closure of X;

\(\operatorname{conv}(X)\) denotes the convex hull of X;

\(P(X)\) denotes the set of nonempty subsets of X;

\(X+Y\) and λX (\(\lambda\in\mathbb{R}\)) stand for algebraic operations on sets X and Y.
We denote by \(\mathcal{B}_{E}\) the family of all nonempty bounded subsets of E. Finally, if X is a nonempty subset of E and \(T: X\to X\) is a given operator, we denote by \(\mbox{Fix}(T)\) the set of fixed points of T, that is,
Banaś and Goebel [1] introduced the following axiomatic definition of the concept of a measure of noncompactness.
Definition 1.1
Let \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) be a given mapping. We say that σ is a BGmeasure of noncompactness (in the sense of Banaś and Gobel) on E if the following conditions are satisfied:

(i)
For every \(X\in\mathcal{B}_{E}\), \(\sigma(X)=0\) iff X is precompact.

(ii)
For every pair \((X,Y)\in\mathcal{B}_{E}\times\mathcal {B}_{E}\), we have
$$X\subseteq Y\quad\Longrightarrow\quad\sigma(X)\leq\sigma(Y). $$ 
(iii)
For every \(X\in\mathcal{B}_{E}\), we have
$$\sigma(\overline{X})=\sigma(X)=\sigma\bigl(\operatorname{conv}(X)\bigr). $$ 
(iv)
For every pair \((X,Y)\in\mathcal{B}_{E}\times\mathcal{B}_{R}\) and \(\lambda\in(0,1)\), we have
$$\sigma\bigl(\lambda X+(1\lambda)Y\bigr)\leq\lambda\sigma(X)+(1\lambda) \sigma(Y). $$ 
(v)
If \(\{X_{n}\}\subseteq\mathcal{B}_{E}\) is a decreasing sequence (w.r.t. ⊆) of closed sets such that \(\sigma(X_{n})\to0\) as \(n\to\infty\), then \(X_{\infty}:= \bigcap_{n=1}^{\infty}X_{n}\) is nonempty.
Let \(\mathcal{X}\) be a nonempty, bounded, closed, and convex subset of the Banach space E.
We denote by \(\mathcal{D}_{\mathcal{X}}\) the set of selfmappings \(D: \mathcal{X}\to\mathcal{X}\) satisfying the following conditions:

(i)
D is a continuous mapping.

(ii)
There exist \(\sigma: \mathcal{B}_{E}\to[0,\infty)\), a BGmeasure of noncompactness on E, and a constant \(k\in(0,1)\) such that
$$\sigma(D W)\leq k \sigma(W), \quad W\in P(\mathcal{X}). $$
The following result is known as Darbo’s fixed point theorem (see [1, 2]).
Theorem 1.2
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{D}_{\mathcal{X}}\). Then D has at least one fixed point. Moreover, the set \(\operatorname{Fix}(D)\) is precompact.
Many generalizations and extensions of Darbo’s fixed point theorem can be found in the literature (see, for example, [3–9] and the references therein). Using the BGmeasure of noncompactness, Aghajani et al. [4] obtained the following generalization of Darbo’s theorem. Let \(\mathcal{F}_{\mathcal{X}}\) be the set of selfmappings \(D: \mathcal{X}\to\mathcal{X}\) satisfying the following conditions:

(i)
D is a continuous mapping.

(ii)
There exists \(\sigma: \mathcal{B}_{E}\to[0,\infty)\), a BGmeasure of noncompactness on E, such that for all \(\varepsilon>0\), there exists some \(\delta_{\varepsilon}>0\) for which
$$W\in P(\mathcal{X}), \quad \varepsilon\leq\sigma(W)< \varepsilon+\delta _{\varepsilon}\quad\Longrightarrow\quad\sigma(DW)< \varepsilon. $$
Theorem 1.3
(Aghajani et al. [4])
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{F}_{\mathcal{X}}\). Then D has at least one fixed point.
Observe that \(\mathcal{D}_{\mathcal{X}}\subseteq\mathcal{F}_{\mathcal{X}}\). In fact, let \(D: \mathcal{X}\to\mathcal{X}\) be a given mapping that belongs to \(\mathcal{D}_{\mathcal{X}}\). Let \(\varepsilon>0\). From the definition of \(\mathcal{D}_{\mathcal{X}}\), there is some \(k\in(0,1)\) such that
for any nonempty subset W of \(\mathcal{X}\). Let \(\delta_{\varepsilon}= (\frac{1}{k}1 )\varepsilon\). Then for any nonempty subset W of \(\mathcal{X}\), we have
so \(D\in\mathcal{F}_{\mathcal{X}}\).
In [6], Dhage introduced the following axiomatic definition of the measure of noncompactness.
Definition 1.4
Let \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) be a given mapping. We say that σ is a Dmeasure of noncompactness (in the sense of Dhage) on E if the following conditions are satisfied:

(i)
For every \(X\in\mathcal{B}_{E}\), \(\sigma(X)=0\) iff X is precompact.

(ii)
For every pair \((X,Y)\in\mathcal{B}_{E}\times\mathcal {B}_{E}\), we have
$$X\subseteq Y\quad\Longrightarrow\quad\sigma(X)\leq\sigma(Y). $$ 
(iii)
For every \(X\in\mathcal{B}_{E}\), we have
$$\sigma(\overline{X})=\sigma(X)=\sigma\bigl(\operatorname{conv}(X)\bigr). $$ 
(iv)
If \(\{X_{n}\}\subseteq\mathcal{B}_{E}\) is a decreasing sequence (w.r.t. ⊆) such that \(\sigma(X_{n})\to0\) as \(n\to \infty\), then the \(X_{\infty}:= \bigcap_{n=1}^{\infty}\overline {X_{n}}\) is nonempty.
Observe that if \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) is a BGmeasure of noncompactess on E, then σ is a Dmeasure of noncompactess on E.
In this paper, using the axiomatic definition of the measure of noncompactness given by Dhage, we obtain new generalizations of Theorem 1.2. Finally, an existence result for a certain class of fractional integral equations will be given as an application.
2 Main results
Let \(\mathcal{X}\) be a nonempty, bounded, closed, and convex subset of a Banach space E. We continue to use the same notations presented in the previous section of this paper.
Let \(\mathcal{F}'_{\mathcal{X}}\) be the set of selfmappings \(D: \mathcal {X}\to\mathcal{X}\) satisfying the following conditions:

(i)
D is a continuous mapping.

(ii)
There exists \(\sigma: \mathcal{B}_{E}\to[0,\infty)\), a Dmeasure of noncompactness on E, such that for all \(\varepsilon>0\), there exists some \(\delta_{\varepsilon}>0\) for which
$$W\in P(\mathcal{X}), \quad\varepsilon\leq\sigma(W)< \varepsilon+\delta _{\varepsilon}\quad\Longrightarrow\quad\sigma(DW)< \varepsilon. $$
We have the following result.
Theorem 2.1
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{F}'_{\mathcal{X}}\). Then D has at least one fixed point.
The result of Theorem 2.1 can be obtained using the same arguments of the proof of Theorem 1.3 in [4]. By Theorem 2.1, we want just to mention that Theorem 1.3 is still valid for any Dmeasure of noncompactness.
Let \(\mathcal{G}_{\mathcal{X}}\) be the set of mappings \(D: \mathcal{X}\to \mathcal{X}\) such that

(i)
D is continuous.

(ii)
There exists a function \(\omega: [0,\infty)\to[0,\infty)\) such that
 (\(\omega_{1}\)):

\(\omega(t)=0\) iff \(t=0\);
 (\(\omega_{2}\)):

ω is nondecreasing and right continuous;
 (\(\omega_{3}\)):

for every \(\varepsilon>0\), there exists \(\gamma _{\varepsilon}>0\) such that
$$W\in P(\mathcal{X}),\quad \varepsilon\leq\omega\bigl(\sigma(W)\bigr)< \varepsilon + \gamma_{\varepsilon}\quad\Longrightarrow\quad\omega\bigl(\sigma(DW)\bigr)< \varepsilon, $$where \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) is a Dmeasure of noncompactness.
The following lemma can be proved using a similar argument as in the proof of Theorem 2.6 in [4].
Lemma 2.2
We have
Using Theorem 2.1 and Lemma 2.2, we obtain the following result.
Corollary 2.3
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{G}_{\mathcal{X}}\). Then D has at least one fixed point.
Let Φ be the set of functions \(\varphi: [0,\infty)\to[0,\infty)\) satisfying the conditions:
 (\(\Phi_{1}\)):

\(\varphi\in L^{1}_{\mathrm{loc}}[0,\infty)\);
 (\(\Phi_{2}\)):

for every \(\xi>0\), we have
$$\int_{0}^{\xi}\varphi(s)\,ds >0. $$
Let \(\mathcal{H}_{\mathcal{X}}\) be the set of mappings \(D: \mathcal{X}\to \mathcal{X}\) such that
 (\(\mathcal{H}_{1}\)):

D is continuous;
 (\(\mathcal{H}_{2}\)):

for every \(\varepsilon>0\), there exists some \(\gamma_{\varepsilon}>0\) such that
$$W\in P(\mathcal{X}),\quad \varepsilon\leq \int_{0}^{\sigma(W)}\varphi(s)\,ds< \varepsilon+ \gamma_{\varepsilon}\quad\Longrightarrow\quad \int_{0}^{\sigma(D W)}\varphi(s)\,ds< \varepsilon, $$where \(\varphi\in\Phi\) and \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) is a Dmeasure of noncompactness.
Lemma 2.4
We have
Proof
Take
we obtain the desired result. □
Using Corollary 2.3 and Lemma 2.4, we obtain the following result.
Corollary 2.5
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{H}_{\mathcal{X}}\). Then D has at least one fixed point.
Let \(\mathcal{I}_{\mathcal{X}}\) be the set of mappings \(D: \mathcal{X}\to \mathcal{X}\) such that
 (\(\mathcal{I}_{1}\)):

D is continuous;
 (\(\mathcal{I}_{2}\)):

there exists some \(\varphi\in\Phi\) such that
$$\int_{0}^{\sigma(DW)}\varphi(s)\,ds\leq k \int_{0}^{\sigma(W)}\varphi(s)\,ds, \quad W\in P(\mathcal {X}), $$where \(k\in(0,1)\) is a constant and \(\sigma: \mathcal{B}_{E}\to[0,\infty )\) is a Dmeasure of noncompactness.
Lemma 2.6
We have
Proof
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{I}_{\mathcal{X}}\). Let \(\varepsilon>0\) be fixed. Let \(\gamma_{\varepsilon}= (\frac{1}{k}1 )\varepsilon\). Take \(W\in P(\mathcal{X})\) such that
From (\(\mathcal{I}_{2}\)), we obtain
so \(D\in\mathcal{H}_{\mathcal{X}}\). □
Using Corollary 2.5 and Lemma 2.6, we obtain the following result.
Corollary 2.7
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{I}_{\mathcal{X}}\). Then D has at least one fixed point.
Remark 2.8
Take \(\varphi(t)=1\), \(t\geq0\) in Corollary 2.7, we obtain Theorem 1.2.
Let \(\mathcal{J}_{\mathcal{X}}\) be the set of mappings \(D: \mathcal {X}\to\mathcal{X}\) such that
 (\(\mathcal{J}_{1}\)):

D is continuous;
 (\(\mathcal{J}_{2}\)):

there exists a function \(\eta: (0,\infty)\to \mathbb{R}\) such that
 (\(\eta_{1}\)):

for each sequence \(\{\alpha_{n}\}\subset(0,\infty)\), we have
$$\lim_{n\to\infty} \eta(\alpha_{n})=\infty\quad \Longrightarrow\quad \lim_{n\to \infty} \alpha_{n}=0 ; $$  (\(\eta_{2}\)):

there exists \(\tau>0\) such that
$$W\in P(\mathcal{X}),\quad \sigma(W)\sigma(DW)>0 \quad\Longrightarrow\quad\tau + \eta\bigl(\sigma(DW)\bigr)\leq\eta\bigl(\sigma(W)\bigr), $$where \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) is a Dmeasure of noncompactness.
Theorem 2.9
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{J}_{\mathcal{X}}\). Then D has at least one fixed point.
Proof
Consider the sequence \(\{\mathcal{X}_{n}\}\) of subsets of E defined by
By induction, we observe easily that
If for some N, we have \(\sigma(\mathcal{X}_{N})=0\), then by the property (i) of the Dmeasure of noncompactness, \(\mathcal{X}_{N}\) is compact. Since \(D(\mathcal{X}_{N})\subseteq\mathcal{X}_{N}\) (from (2.2)), Schauder’s fixed point theorem applied to the selfmapping \(D: \mathcal{X}_{N}\to\mathcal{X}_{N}\) gives the desired result. So, without loss of the generality, we may assume that
For \(n=0\), since \(\sigma(\mathcal{X}_{0})>0\) and \(\sigma(D \mathcal {X}_{0})=\sigma(\mathcal{X}_{1})>0\), from the property (\(\eta_{2}\)) we have
which yields
Similarly, for \(n=1\), we have
By induction, we obtain
Since
we deduce that
so from the property (\(\eta_{1}\)) we have
From the property (iv) of the Dmeasure of noncompactness, the set \(M:= \bigcap_{n=1}^{\infty}\mathcal{X}_{n}\) is nonempty. Moreover, for every \(p=0,1,2,\ldots\) , we have
which implies from (2.2) that
Then \(D: M\to M\) is well defined. On the other hand, from (2.4) and the property (ii) of the Dmeasure of noncompactness, we have
Passing to the limit as \(p\to\infty\) and using (2.3), we obtain
which implies from the property (i) of the Dmeasure of noncompactness that \(\overline{M}=M\) is compact. Applying Schauder’s fixed point theorem to the mapping \(D: M\to M\), we obtain the desired result. □
Remark 2.10
Observe that \(\mathcal{D}_{\mathcal{X}}\subseteq \mathcal{J}_{\mathcal{X}}\). In fact, if \(D: \mathcal{X}\to \mathcal{X}\) belongs to \(\mathcal{D}_{\mathcal{X}}\), that is,
then
Then \(D\in\mathcal{J}_{\mathcal{X}}\) with \(\eta(t)=\ln t\), \(t>0\). Therefore, Theorem 2.9 is a generalization of Theorem 1.2.
Let \(\mathcal{K}_{\mathcal{X}}\) be the set of mappings \(D: \mathcal {X}\to\mathcal{X}\) such that
 (\(\mathcal{K}_{1}\)):

D is continuous;
 (\(\mathcal{K}_{2}\)):

there exists a function \(\theta: (0,\infty)\to (1,\infty)\) such that
 (\(\theta_{1}\)):

for each sequence \(\{u_{n}\}\subset(0,\infty)\), we have
$$\lim_{n\to\infty} \theta(u_{n})=1 \quad\Longrightarrow\quad \lim _{n\to\infty} u_{n}=0 ; $$  (\(\theta_{2}\)):

there exist \(k\in(0,1)\) and a Dmeasure of noncompactness \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) such that
$$W\in P(\mathcal{X}), \quad\sigma(W)\sigma(DW)>0\quad \Longrightarrow\quad\theta \bigl(\sigma(D W)\bigr)\leq\bigl[\theta\bigl(\sigma(W)\bigr)\bigr]^{k}. $$
We have the following result.
Theorem 2.11
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{K}_{\mathcal{X}}\). Then D has at least one fixed point.
Proof
Consider the sequence \(\{\mathcal{X}_{n}\}\) of subsets of E defined by (2.1). As in the proof of Theorem 2.9, without loss of the generality, we may assume that
For \(n=0\), since \(\sigma(\mathcal{X}_{0})>0\) and \(\sigma(D \mathcal {X}_{0})=\sigma(\mathcal{X}_{1})>0\), we have
that is,
Again, for \(n=1\), since \(\sigma(\mathcal{X}_{1})>0\) and \(\sigma(D \mathcal {X}_{1})=\sigma(\mathcal{X}_{2})>0\), we have
that is,
so
Therefore, by induction, we get
Passing to the limit as \(n\to\infty\), we obtain
so from the property (\(\theta_{1}\)) we have
The rest of the proof is similar to that in the proof of Theorem 2.9. □
Corollary 2.12
Let \(D: \mathcal{X}\to\mathcal{X}\) be a continuous mapping. Suppose that there exist a constant \(k\in(0,1)\) and a Dmeasure of noncompactness \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) such that
for any \(W\in P(\mathcal{X})\) with \(\sigma(W)\sigma(DW)>0 \). Then D has at least one fixed point.
Proof
Taking
in Theorem 2.11, we obtain the desired result. □
Remark 2.13
Observe that \(\mathcal{D}_{\mathcal{X}}\subseteq \mathcal{K}_{\mathcal{X}}\). In fact, if \(D: \mathcal{X}\to \mathcal{X}\) belongs to \(\mathcal{D}_{\mathcal{X}}\), that is,
then
Therefore \(D\in\mathcal{K}_{\mathcal{X}}\) with \(\theta(t)=e^{t}\).
Let \(\mathcal{L}_{\mathcal{X}}\) be the set of mappings \(D: \mathcal {X}\to\mathcal{X}\) such that
 (\(\mathcal{L}_{1}\)):

D is continuous;
 (\(\mathcal{L}_{2}\)):

there exists a function \(\zeta: [0,\infty )\times[0,\infty)\to\mathbb{R}\) such that
 (\(\zeta_{1}\)):

\(\zeta(z_{1},z_{2})< z_{2}z_{1}\), for all \(z_{1},z_{2}>0\);
 (\(\zeta_{2}\)):

if \(\{u_{n}\}\) and \(\{v_{n}\}\) are two sequences in \((0,\infty)\) such that \(\lim_{n\to\infty}u_{n}= \lim_{n\to\infty}v_{n}=\ell>0\), then
$$\limsup_{n\to\infty}\zeta(u_{n},v_{n})< 0, $$  (\(\zeta_{3}\)):

$$\zeta \bigl(\sigma(DW),\sigma(W )\bigr)\geq0,\quad W\in P(\mathcal{X}), $$
where \(\sigma: \mathcal{B}_{E}\to[0,\infty)\) is a Dmeasure of noncompactness.
Theorem 2.14
Let \(D: \mathcal{X}\to\mathcal{X}\) be a mapping that belongs to \(\mathcal{L}_{\mathcal{X}}\). Then D has at least one fixed point.
Proof
Consider the sequence \(\{\mathcal{X}_{n}\}\) of subsets of E defined by (2.1). From the property (\(\zeta_{3}\)), we have
As before, without loss of the generality, we may assume that
From the property (\(\zeta_{1}\)), (2.5) and (2.6), we get
Then there is some \(r\geq0\) such that
If \(r>0\), then from the property (\(\zeta_{2}\)), we have
which contradicts (2.5). As consequence, we have
The rest of the proof is similar to the proof of Theorem 2.9. □
Remark 2.15
Taking
where \(k\in(0,1)\) is a constant, we obtain Theorem 1.2.
Corollary 2.16
Let \(D: \mathcal{X}\to\mathcal{X}\) be a continuous mapping such that
where \(\Phi: [0,\infty)\to[0,\infty)\) is a lower semicontinuous function with \(\Phi^{1}(0)=\{0\}\) and \(\sigma: \mathcal{B}_{E}\to [0,\infty)\) is a Dmeasure of noncompactness. Then D has at least one fixed point.
Proof
Taking
in Theorem 2.14, we obtain the desired result. □
Corollary 2.17
Let \(D: \mathcal{X}\to\mathcal{X}\) be a continuous mapping such that
where \(\psi: [0,\infty)\to[0,\infty)\) is an upper semicontinuous function with \(\psi(t)< t\) for all \(t>0\) and \(\sigma: \mathcal{B}_{E}\to [0,\infty)\) is a Dmeasure of noncompactness. Then D has at least one fixed point.
Proof
Taking
in Theorem 2.14, we obtain the desired result. □
3 An existence result for a fractional integral equation
The measure of noncompactness argument is a useful tool in Nonlinear Analysis. In particular, such argument can be used to obtain existence results for various classes of integral equations. For more details on the applications of the measure of noncompactness concept, we refer the reader to [1, 3, 4, 6, 8–15] and the references therein.
In this section, we discuss the existence of solutions to the fractional integral equation
where \(T>0\), \(\alpha\in(0,1)\), \(u,f: [0,T]\times\mathbb{R}\to\mathbb {R}\) and \(g: [0,T]\to\mathbb{R}\).
We suppose that the following conditions are satisfied.

(i)
The function \(f: [0,T]\times\mathbb{R}\to\mathbb{R}\) is continuous.

(ii)
There exists an upper semicontinuous function \(\psi: [0,\infty)\to[0,\infty)\) such that \(\psi(0)=0\), \(\psi(t)< t\) for all \(t>0\), ψ is nondecreasing, and
$$\biglf(t,x)f(t,y)\bigr\leq\psi\bigl(xy\bigr), \quad(t,x,y)\in[0,T]\times\mathbb {R}\times \mathbb{R}. $$ 
(iii)
The function \(u: [0,\infty)\to[0,\infty)\) is continuous and there exists a nondecreasing function \(\omega: [0,\infty)\to [0,\infty)\) such that
$$\biglu(t,z)\bigr\leq\omega\bigl(z\bigr), \quad(t,z)\in[0,T]\times\mathbb{R}. $$ 
(iv)
The function \(g: [0,T]\to\mathbb{R}\) is \(C^{1}\) and nondecreasing.

(v)
There exists \(r_{0}>0\) such that
$$\bigl(\psi(r_{0})+F\bigr)\omega(r_{0}) \bigl(g(T)g(0) \bigr)^{\alpha}\leq r_{0}\Gamma(\alpha+1) $$and
$$\frac{\omega(r_{0})}{\Gamma(\alpha+1)}\bigl(g(T)g(0)\bigr)^{\alpha}\leq1, $$where \(F=\max\{f(t,0): t\in[0,T]\}\).
Let \(E=C([0,T];\mathbb{R})\) be the set of real continuous functions defined in \([0,T]\). The set E endowed with the norm
is a Banach space. Let W be a nonempty and bounded subset of E. Let us define the mapping \(\gamma: W\times[0,\infty)\to[0,\infty)\) by
Set
Let \(\mathcal{B}_{E}\) be the set of all nonempty bounded subsets of E. Then the mapping
defined by
is a BGmeasure of noncompactness (then it is a Dmeasure of noncompactness) on the space E (see [1]).
We have the following existence result.
Theorem 3.1
Under the assumptions (i)(v), equation (3.1) has at least one solution \(y^{*}\in E\). Moreover, we have \(\y^{*}\\leq r_{0}\).
Proof
Let us consider the operator D defined on E by
At first, we show that the operator D maps E into itself. Set
From the assumption (i), we have just to show that H maps E into itself. In order to prove this fact, let us fix some \(y\in E\). Observe that \(Hy: [0,T]\to\mathbb{R}\) is a welldefined function. In fact, using the assumptions (iii) and (iv), for all \(t\in[0,T]\) we have
that is,
Let us prove the continuity of Hy at 0. To do this, let \(\{t_{n}\}\) be a sequence in \([0,T]\) such that \(t_{n}\to0^{+}\) as \(n\to\infty\). From (3.4), for all n we have
Passing to the limit as \(n\to\infty\) and using the continuity of g at 0, we obtain
Then Hy is continuous at 0.
Now, let \(t\in(0,T]\) be fixed and \(\{t_{n}\}\) be a sequence in \((0,T]\) such that \(t_{n}\to t\) as \(n\to\infty\). Without restriction of the generality, we may assume that \(t_{n}\geq t\) for n large enough. For every n, we have
For n large enough, we can write
Since g is continuous in \([0,T]\), we have
which yields \(\lim_{n\to\infty} (Hy)(t_{n})(Hy)(t)=0\). Then Hy is continuous at t. As consequence, \(Hy\in E\), for all \(y\in E\), and \(D: E\to E\) is well defined.
On the other hand, using the assumptions (ii) and (iii), for an arbitrarily fixed \(y\in E\) and \(t\in[0,T]\), we have
Then
Using the above inequality, the fact that the functions \(\psi,\omega: [0,\infty)\to[0,\infty)\) are nondecreasing, and the assumption (v), we infer that the operator D maps \(\overline{B(0,r_{0})}\) into itself, where
Now, we claim that the operator \(D: \overline{B(0,r_{0})}\to\overline {B(0,r_{0})}\) is continuous. From (3.2), we can write D in the form
where
and Hy is defined by (3.3). In order to prove our claim, it is sufficient to show that the operators G and H are continuous on \(\overline{B(0,r_{0})}\). First of all, we show that G is a continuous operator on \(\overline {B(0,r_{0})}\). To do this, we take a sequence \(\{y_{n}\}\subset\overline {B(0,r_{0})}\) and \(y\in\overline{B(0,r_{0})}\) such that \(\y_{n}y\\to0\) as \(n\to\infty\), and we have to prove that \(\Gy_{n}Gy\\to0\) as \(n\to \infty\). In fact, for all \(t\in[0,T]\), using the condition (ii), we have
Thus we have
Passing to the limit as \(n\to\infty\) in the above inequality, we obtain
This proves that G is a continuous operator on \(\overline{B(0,r_{0})}\). Next, we show that H is a continuous operator on \(\overline {B(0,r_{0})}\). To do this, we fix a real number \(\varepsilon>0\) and we take arbitrary functions \(x, y \in\overline{B(0,r_{0})}\) such that \(\ xy\<\varepsilon\). For all \(t\in[0,T]\), we have
where
Therefore,
Since u is uniformly continuous on the compact \([0,T]\times [r_{0},r_{0}]\), we have \(u(r_{0},\varepsilon)\to0\) as \(\varepsilon\to0^{+}\) and, therefore, the last inequality gives us
Then H is continuous on \(\overline{B(0,r_{0})}\) and D maps continuously the set \(\overline{B(0,r_{0})}\) into itself.
Further, let W be a nonempty subset of \(\overline{B(0,r_{0})}\). Let \(\rho> 0\) be fixed, \(y\in W\), and \(t_{1},t_{2}\in[0,T]\) be such that \(t_{1}t_{2}\leq\rho\). Without restriction of the generality, we may assume that \(t_{1}\geq t_{2}\). We have
where
Therefore,
Passing to the limit superior as \(\rho\to0^{+}\) and using the fact that ψ is upper semicontinuous, we obtain
Then, from the assumption (v), we obtain
As a consequence, for any nonempty subsets W of \(\overline {B(0,r_{0})}\), we have
where \(\zeta: [0,\infty)\times[0,\infty)\to\mathbb{R}\) is defined by
Under the assumptions on the function ψ, the operator \(D: \overline {B(0,r_{0})}\to\overline{B(0,r_{0})}\) belongs to the family of operators \(\mathcal{L}_{\mathcal{X}}\), where \(\mathcal{X}=\overline{B(0,r_{0})}\). Then by Theorem 2.14, we deduce that D has at least one fixed point \(y^{*}\in\overline{B(0,r_{0})}\), which is a solution to equation (3.1). □
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The first and fourth authors would like to extend their sincere appreciations to the Deanship of Scientific Research at King Saud University for its funding of this Prolific Research Group Project No. PRG143610.
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Jleli, M., Karapinar, E., O’Regan, D. et al. Some generalizations of Darbo’s theorem and applications to fractional integral equations. Fixed Point Theory Appl 2016, 11 (2016). https://doi.org/10.1186/s1366301604974
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DOI: https://doi.org/10.1186/s1366301604974