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 Open Access
Common fixedpoint results for generalized Berindetype contractions which involve altering distance functions
 Fawzia Shaddad^{1}Email author,
 Mohd Salmi Md Noorani^{1} and
 Saud M Alsulami^{2}
https://doi.org/10.1186/16871812201424
© Shaddad et al.; licensee Springer. 2014
 Received: 22 July 2013
 Accepted: 10 January 2014
 Published: 31 January 2014
Abstract
In this paper, the existence and the uniqueness of a common fixed point for two selfmappings satisfying some generalized Berindetype contractions which involve an altering distance function with $\lambda \in [0,1]$ is established. Our theorems extend, unify, and generalize several existing results in the literature. An application of an integral equation is presented.
MSC:54H25, 54C60, 54E50.
Keywords
 metric space
 common fixed point
 generalized Berindetype contractions
 Urysohn integral equations
1 Introduction and preliminaries
Fixedpoint theory is one of the most fruitful and effective tools in mathematics which has enormous applications within as well as outside mathematics. In 1922, Banach established the famous fixedpoint theorem which is called the Banach contraction principle. This principle is a forceful tool in nonlinear analysis. It has many applications in solving nonlinear equations.
Later, Berinde [1–5] studied many interesting fixedpoint theorems for many kinds of contraction mappings. In [3] and [4], he defined the almost contraction map as follows.
for all $x,y\in X$.
Recently, Babu et al. [6] considered the class of mappings that satisfy ‘condition (B)’.
More recently, Abbas and Ilic in [7] introduced the following definition.
for all $x,y\in X$, where $M(x,y)=max\{d(gx,gy),d(gx,fx),d(fy,gy),\frac{1}{2}(d(fx,gy)+d(gx,fy))\}$.
If $g={I}_{X}$, ${I}_{X}$ is the identity map on X, then they note that f satisfies ‘generalized condition (B)’.
Furthermore, Ćirić et al. [8] introduced the concept of the almost generalized contractive condition as follows.
for all $x,y\in X$.
A new category of contractive fixedpoint problems was addressed by Khan et al. [9]. In their study they introduced the notion of an altering distance function, which is a control function that alters the distance between two points in a metric space.
 (i)
ψ is continuous and nondecreasing;
 (ii)
$\psi (t)=0\iff t=0$.
In the literature, there has been extensive research on common fixed points by using Berindetype contractions, see [10, 11], by using an altering distance function, see [12–14] and by using many other kinds of methods, see [15–17].
The aim of this work is to prove that there is a unique common fixed point for two selfmappings satisfying some generalized Berindetype contractions which involve an altering distance function with $\lambda \in [0,1]$. These results extend and generalize several wellknown compatible recent and classical results in the literature. As an application, the existence of a solution for the Urysohn integral equation is presented.
2 Main results
with $L\ge 0$ and $0\le \lambda \le 1$. Then f and g have a unique common fixed point.
Proof We prove the theorem in several steps.
Step 1. Let ${x}_{0}\in X$ be an arbitrary point.
We will prove that $\{{d}_{n}\}$ is a decreasing sequence which converges to 0.
If ${x}_{n}={x}_{n+1}$ for some $n\ge 0$, then the proof will be finished. Therefore, we suppose that ${x}_{n}\ne {x}_{n+1}$ for all $n\ge 0$. Now, we shall show that $d({x}_{n},{x}_{n+1})\le d({x}_{n1},{x}_{n})$. Arguing by contradiction, we assume $d({x}_{n},{x}_{n+1})>d({x}_{n1},{x}_{n})$. Therefore, we have three cases.
Since ψ is increasing, we have $d({x}_{n},{x}_{n+1})<\lambda d({x}_{n1},{x}_{n})$, which is a contradiction.
Since ψ is increasing, we have $d({x}_{n},{x}_{n+1})<\lambda d({x}_{n},{x}_{n+1})$, which is impossible.
which is impossible since $\lambda /(2\lambda )\le 1$. Hence, from the above we obtain $d({x}_{n},{x}_{n+1})\le d({x}_{n1},{x}_{n})$.
Similarly, we can prove that $d({x}_{n},{x}_{n+1})\le d({x}_{n1},{x}_{n})$ also in the case when n is an odd number.
Next, we want to show that $d=0$. We have two cases.
If $\lambda =0$, then we have $\psi (d)\le 0\Rightarrow d=0$. If $\lambda \ne 0$, then we have $\phi (\lambda d)\le \psi (\lambda d)\psi (d)\le 0$. Thus $\phi (\lambda d)=0$, which implies $d=0$.
Now, we have two subcases.
which leads to a contradiction if $d\ne 0$.
Hence, $\phi (d)=0$ and then $d=0$.
Equations (2.8) and (2.9) show that ${x}^{\ast}$ is a common fixed point of f and g.
Then we obtain ${x}^{\ast}={y}^{\ast}$. □
with $L\ge 0$ and $0\le \lambda \le 1$, then f has a unique fixed point.
Proof Since $M(x,y)\in \{d(x,y),d(x,fx),d(y,gy),\frac{1}{2}(d(fx,y)+d(x,gy))\}$, the result follows from Theorem 2.1. □
By taking $f=g$ and $L=0$ in Corollary 2.2, we obtain the following result.
where $u(x,y)\in \{d(x,y),d(x,fx),d(y,fy),\frac{1}{2}(d(fx,y)+d(x,fy))\}$, $0\le \lambda \le 1$, then f has a unique fixed point.
Remark 2.5 Corollary 2.4 extends the main fixedpoint result of Dutta and Choudhury [[20], Theorem 2.1] and Theorem 2.2 of Doric [18].
If we take $\psi (t)=t$ and $\phi (t)=(1k)t$ with $k<1$ in Theorem 2.1, we have the following corollary.
with $L\ge 0$ and $0\le \alpha <1$, then f and g have a unique common fixed point.
If we take $f=g$ in Corollary 2.6 we obtain the following result.
with $L\ge 0$ and $0\le \alpha <1$, then f has a unique fixed point.
By the aid of Lemma 2.1 of [21], we have the following result as a consequence of Corollary 2.7.
with $L\ge 0$ and $0\le \alpha <1$, then f and g have a unique common fixed point.
Remark 2.9 Corollary 2.8 extends the results of Abbas et al. [[22], Theorem 2.1] and Jleli et al. [[23], Corollary 3.2].
3 Applications: existence of a common solution to Urysohn integral equations
Throughout this section we take $X=C([a,b],\mathbb{R})$ (the set of continuous functions defined in $I=[a,b]$). We define the metric $d:X\times X\to \mathbb{R}$ by $d(x,y)={\parallel xy\parallel}_{\mathrm{\infty}}$ for every $x,y\in X$. Let $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be a function such that

φ is lower semicontinuous;

φ is increasing;

$\phi (t)=0\iff t=0$.
Then there exists a solution to (3.1).
for all $x,y\in X$. Now, all the assumptions of Theorem 2.1 are satisfied with $\psi (t)=t$, for all $t\in {\mathbb{R}}^{+}$, $u(x,y)=d(x,y)$, and $L=0$. Therefore, f and g have a common fixed point, that is, a solution to the integral equation (3.1). □
Declarations
Acknowledgements
The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant UKMDIP201231.
Authors’ Affiliations
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