\(F(\psi,\varphi)\)Contraction in terms of measure of noncompactness with application for nonlinear integral equations
 Farzaneh Nikbakhtsarvestani^{1},
 S Mansour Vaezpour^{2} and
 Mehdi Asadi^{3}Email author
https://doi.org/10.1186/s1366001715452
© The Author(s) 2017
Received: 25 August 2017
Accepted: 13 October 2017
Published: 30 October 2017
Abstract
In this paper, some new generalization of Darbo’s fixed point theorem is proved by using a \(F(\psi,\varphi)\)contraction in terms of a measure of noncompactness. Our result extends to obtaining a common fixed point for a pair of compatible mappings. The paper contains an application for nonlinear integral equations as well.
Keywords
MSC
1 Introduction and preliminaries
A contractive condition in terms of a measure of noncompactness, which was first used by Darbo, is one of the fruitful tools to obtain fixed point and common fixed point theorems. The extensions of these contractions in linear and integral type which are known as generalizations of Darbo’s fixed point theorem, are considered by many authors; see, for example, [1–9] and the references therein.
Recently, Khodabakhshi [6] obtained some new common fixed point results with the technique associated with a measure of noncompactness for two commuting operators.
Inspired by the class of αψ contractive type mappings which was introduced by Samet et al. [10], Ansari [11] presented the weaker class of this contraction named \(F(\psi,\varphi)\)contraction and used it to obtain fixed point and common fixed point results.
This paper mainly aims at employing the \(F(\psi,\varphi)\)contraction and its property in terms of a measure of noncompactness to investigate a fixed point and a common fixed point for a pair of compatible mappings.
Now we present some definitions, notations and results which will be needed later. Throughout this paper we assume that E is an infinite dimensional Banach space. If C is a subset of E then the symbols \(\overline{\operatorname {co}} (C)\) and \(\mathfrak{M}_{E}\) and \(\mathfrak{N}_{E}\) denote the closure of convex hull of C and the family of nonempty bounded subsets of E and the subfamily consisting of all relatively compact subsets of E, respectively.
From now on we will use the following definition for the measure of noncompactness.
Definition 1.1
([13])
 (A1)
\(\emptyset\neq \operatorname {Ker}\mu= \{ X \in \mathfrak {M}_{E} : \mu( X ) = 0 \} \subseteq\mathfrak{N}_{E}\);
 (A2)
\(X \subseteq Y\Rightarrow\mu( X ) \leq\mu( Y )\);
 (A3)
\(\mu(\overline{X}) = \mu(\operatorname {co}X) =\mu( X )\);
 (A4)
\(\mu(\lambda X + ( 1 \lambda) Y ) \leq\lambda\mu( X ) + ( 1 \lambda)\mu( Y )\) for \(\lambda\in[ 0 , 1 ]\);
 (A5)
If \(\{X_{n}\}\) is a sequence of closed sets from \(\mathfrak {M}_{E}\) such that \(X_{n + 1}\subseteq X_{n}\), (\(n \geq1 \)) and if \(\lim_{n\to\infty} \mu( X_{n} ) = 0\) then the intersection set \(X_{\infty}=\bigcap_{n=1}^{\infty}X_{n}\) is nonempty.
The family Kerμ described in (A1) is said to be the kernel of the measure of noncompactness μ. Observe that \(X_{\infty}\in \operatorname {Ker}\mu\), since \(\mu(X_{\infty}) \leq\mu(X _{n})\) for any n.
Definition 1.2
([14])
 (1)
ψ is nondecreasing and continuous;
 (2)
\(\psi(t)=0\) if and only if \(t=0\).
We denote by Ψ the class of altering distance functions.
Definition 1.3
([11])
An ultra altering distance function is a continuous, nondecreasing mapping \(\varphi:[0,\infty) \to[0,\infty)\) such that \(\varphi(t)>0\) for \(t>0\) and \(\varphi(0)\geq0\).
We denote by Φ the class of ultra altering distance functions.
Definition 1.4
([11])
 (1)
\(F(s,t)\leq s\);
 (2)
\(F(s,t)=s\) implies that either \(s=0\) or \(t=0\); for all \(s,t \in[0, \infty)\).
We denote Cclass functions by \(\mathcal{C}\).
Definition 1.5
([15])
A pair of selfmappings F and G on X is weakly compatible if there exists a point \(x \in X\) such that \(F(x)=G(x)\) implies \(FGx=GFx\) i.e., they commute at their coincidence point.
Proposition 1.6
([16, Proposition 1.5])
Let f and g be weakly compatible selfmappings of a set X. If f and g have a unique point of coincidence, \(w=f(x)=g(x)\). Then w is the unique common fixed point of f and g.
Lemma 1.7
([17])
Let X be a nonempty set and \(f: X \to X\) be a function. Then there exists a subset \(E \subseteq X\) such that \(f(E)=f(X)\) and \(f:E \to X\) is one to one.
Now, we mention the following two theorems stated in [13, 18].
Theorem 1.8
(Schauder [18])
Let C be a closed, convex subset of a Banach space E. Then every compact, continuous map \(F : C \to C\) has at least one fixed point.
As a significant generalization of Schauder’s fixed point theorem, we have the following fixed point theorem.
Theorem 1.9
(Darbo [13])
2 Main results
This section starts by some of the theorems and corollaries related to fixed point are obtained by using \(F(\psi,\varphi)\)contraction in terms of a measure of noncompactness. Next, for a pair of compatible mappings a common fixed point theorem is considered. In the sequel, theorems are proved in integral type to obtain a fixed point and a common fixed point. Our results generalized Darbo’s fixed point theorem and a fixed point theorem which was recently proved.
Theorem 2.1
Proof
If there exists \(N \in\mathbb{N}\) such that \(\mu(N)=0\) then \(C_{N}\) is compact and Schauder’s fixed point theorem ensures that T has a fixed point in \(C_{N}\) where \(C_{N} \subset C\). Suppose \(\mu(C_{n})>0\) for each \(n \in\mathbb{N}\).
If we let \(F(s,t)=ks\) in Theorem 2.1 we get the following result.
Corollary 2.2
If we let \(\psi(t)=t\) in Theorem 2.1 we get the following.
Corollary 2.3
Theorem 2.4
Proof
If \(\mu(C_{N})=0\) for some \(N \in\mathbb{N}\) then \(C_{N}\) is a compact set and by virtue of Schauder’s theorem the continuous map \(T:C_{N} \to C_{N}\) has a fixed point in \(C_{N} \subset C\).
Now suppose that \(\mu(C_{N})>0\) for every \(n \in\mathbb{N}\). From (4) and condition (A2) of Definition 1.1 we conclude that \(\mu(C_{n+1})\leq\mu(C_{n})\) for every \(n=1,2,\ldots\) . This means that the sequence \(\{\mu(C_{n})\}_{n=0}^{\infty}\) is not increasing, and consequently there exists \(r\geq0\) such that \(\lim_{n \to\infty }\mu(C_{n})=r\).
By taking \(\psi(t)=t\) we have the following.
Corollary 2.5
Theorem 2.6
 (a).
The range of T contains the range of S.
 (b).For any \(M \subset C\):$$ \psi(\mu \bigl( T(M) \bigr)\leq F \bigl(\psi \bigl(\mu \bigl(S(M) \bigr) \bigr),\varphi \bigl(\mu \bigl(S(M) \bigr) \bigr) \bigr), $$(6)
 (i).
The sets \(A=\{x \in C: S(x)=x\}\) and \(B=\{x \in C: T(x)=x \}\) are nonempty and closed.
 (ii).
T and S have a coincidence point.
 (iii).
If T and S are weakly compatible. Then S and T have a unique common fixed point.
Proof
Continuing this process having chosen \(C_{n}\) in E we obtain \(C_{n+1}\) in E such that \(S(C_{n})\subseteq T(C_{n+1})\) and \(C_{n+1}:=\overline{\operatorname {co}} SC_{n}\).
Moreover, since \(C_{n}\) and C are convex, and \(S(C_{n})\subset C _{n}\), \(S:C_{n} \to C_{n}\) for \(n=0,1,2,\ldots\) and so \(S:C_{\infty } \to C_{\infty}\), now Schauder’s fixed point theorem ensures S has a fixed point and the set \(A=\{x \in C: S(x)=x\}\) is nonempty and closed.
Hence T and S have a unique coincidence point thus from Proposition 1.6 and it follows that T and S have a unique common fixed point. □
Example 2.7
 (a).
\(\mathbf{R}_{S}=[0, \frac{2}{3}] \subset\mathbf {R}_{T}=[0,1]\).
By taking \(C_{0}=[0,2]\) and \(C_{1}=[0,\frac{4}{3}]\) we havethe algorithm of \(C_{n}\) follows by$$ S \bigl([0,2] \bigr)= \biggl[0,\frac{2}{3} \biggr]=T \biggl( \biggl[0, \frac{4}{3} \biggr] \biggr), $$such that \(S(C_{n})\subseteq T(C_{n+1})\).$$ C_{3}= \biggl[0,\frac{16}{27} \biggr],\quad\quad C_{4}= \biggl[0,\frac{32}{81} \biggr],\quad\quad C_{5}= \biggl[0, \frac{64}{243} \biggr],\quad\quad \cdots, $$  (b).\(\psi(\mu(T[1,2]))=\frac{1}{2}\), \(\psi(\mu(S[1,2]))= \frac{1}{3}\), \(\varphi(\mu(S[1,2]))=\frac{1}{18}\), and alsotherefore$$ F \biggl(\frac{1}{3},\frac{1}{18} \biggr)=\frac{6}{7}, $$$$ \psi \biggl(\mu \bigl( T(M) \bigr)=\frac{1}{2}\leq F \bigl(\psi \bigl(\mu \bigl(S(M) \bigr) \bigr),\varphi \bigl(\mu \bigl(S(M) \bigr) \bigr) \bigr) \biggr)= \frac{6}{7}, $$(8)
Theorem 2.8
Proof
Remark 2.9
Corollary 2.10
Proof
Corollary 2.11
Proof
If \(\mu(C_{N})=0\) for some \(N \in\mathbb{N}\) then \(C_{N}\) is a compact set and using Schauder’s theorem we conclude that the continuous map \(T:C_{N} \to C_{N}\) has a fixed point in \(C_{N} \subset C\).
Continuing the proof analogously to the proof of the theorem 2.1 we obtain the result. □
Theorem 2.12
 (a).
The range of T contains the range of S.
 (b).For any \(M \subseteq C\):$$ \int_{0}^{\varphi(\mu( T(M))}f(t)\,dt \leq F \biggl( \int_{0}^{\varphi( \mu(S(M)))}f(t)\,dt, \int_{0}^{\psi(\mu(S(M)))}f(t)\,dt \biggr), $$(12)
 (i).
The sets \(A=\{x \in C: S(x)=x\}\) and \(B=\{x \in C: T(x)=x \}\) are nonempty and closed.
 (ii).
T and S have a coincidence point.
 (iii).
If T and S are weakly compatible, then S and T have a common fixed point.
Proof
3 Application
Theorem 3.1
 \((A_{0})\).:

The function \(\alpha:\mathbb{R}^{+} \to\mathbb{R} ^{+}\) is continuous, \(\alpha(t) \to\infty\) as \(t \to\infty\).
 \((A_{1})\).:

The function \(f:\mathbb{R}^{+} \times\mathbb{R} \to\mathbb{R}\) is continuous andmoreover, ψ and φ are an altering distance function and an ultra altering distance function, respectively which ψ satisfies for all \(t,s \in\mathbb{R}^{+}\), \(\psi(t)+\psi(s)\leq \psi(s+t)\) and \(\psi(t)< t\).$$ \bigl\vert f(t,x)f(t,y) \bigr\vert \leq\psi \bigl(\vert xy\vert \bigr), $$
 \((A_{2})\).:

$$ L=\sup \bigl\{ f(t,0): t \in\mathbb{R}^{+} \bigr\} < \infty. $$
 \((A_{3})\).:

The function \(g:\mathbb{R}^{+} \times\mathbb{R} ^{+} \times\mathbb{R} \to\mathbb{R}\) is continuous and there exists a continuous function \(b:\mathbb{R}^{+} \times\mathbb{R}^{+} \to \mathbb{R}^{+}\) which is increasing with respect to the first component, satisfyingfor all \(t,s \in\mathbb{R}^{+}\) and \(x \in\mathbb{R}\) where$$ \bigl\vert g(t,s,x) \bigr\vert \leq b(t,s), $$$$ \lim_{t \to\infty} \int_{0}^{t} b(t,s)\,ds=0. $$
 \((A_{4})\).:

where$$ f \bigl(t,Kx(t) \bigr)=K \bigl(f \bigl(t,x(t) \bigr) \bigr), $$$$ Kx(t)= \int_{0}^{t} g \bigl(t,s,x(s) \bigr)\,ds. $$
For the following remark, by definition, commuting mappings means for a pair of selfmappings \(K,L:X \to X\) that there exists a point \(x \in X\) such that \(KL(x)=LK(x)\).
Remark 3.2
Proof
Khodabakhshi and Vaezpour in [6] have shown that G, K are continuous on Q, bounded, commuting mappings, \(R_{G}\subseteq R_{K}\) and also G, K have a common fixed point. The conditions (a) and (b) of Theorem 2.6 hold.
4 Conclusion
In Theorem 3.2 of [6] the condition \(TS=ST\) for two selfmappings T, S is used. But in this article for achieving a common fixed point from two selfmaps the hypothesis of the common range which is weaker than the hypotheses of a commuting map is utilized.
In this article the measure of noncompactness is used instead of the metric d, which is used in [4] and [11]. Also, contractions associated with a measure of noncompactness in two linear and integral types and the application in solving integral equations are considered, while in the two mentioned article just the linear contractions in terms of the metric d is used for achieving a fixed point.
Declarations
Authors’ contributions
All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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