- Research
- Open Access
Best proximity point results for modified Suzuki α-ψ-proximal contractions
- Nawab Hussain^{1},
- Abdul Latif^{1}Email author and
- Peyman Salimi^{2}
https://doi.org/10.1186/1687-1812-2014-10
© Hussain et al.; licensee Springer. 2014
- Received: 10 August 2013
- Accepted: 13 December 2013
- Published: 9 January 2014
Abstract
In this paper, we introduce a modified Suzuki α-ψ-proximal contraction. Then we establish certain best proximity point theorems for such proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The results presented generalize and improve various known results from best proximity and fixed point theory. Moreover, some examples are given to illustrate the usability of the obtained results.
MSC:46N40, 47H10, 54H25, 46T99.
Keywords
- α-proximal admissible map
- Suzuki α-ψ-proximal contraction
- best proximity point
1 Introduction and Preliminaries
In the last decade, the answers of the following question has turned into one of the core subjects of applied mathematics and nonlinear functional analysis. Is there a point ${x}_{0}$ in a metric space $(X,d)$ such that $d({x}_{0},T{x}_{0})=d(A,B)$ where A, B are non-empty subsets of a metric space X and $T:A\to B$ is a non-self-mapping where $d(A,B)=inf\{d(x,y):x\in A,y\in B\}$? Here, the point ${x}_{0}\in X$ is called the best proximity point. The object of best proximity theory is to determine minimal conditions on the non-self-mapping T to guarantee the existence and uniqueness of a best proximal point. The setting of best proximity point theory is richer and more general than the metric fixed point theory in two senses. First, usually the mappings considered in fixed point theory are self-mappings, which is not necessary in the theory of best proximity. Secondly, if one takes $A=B$ in the above setting, the best proximity point becomes a fixed point. It is well known that fixed point theory combines various disciplines of mathematics, such as topology, operator theory, and geometry, to show the existence of solutions of the equation $Tx=x$ under proper conditions. On the other hand, if T is not a self-mapping, the equation $Tx=x$ could have no solutions and, in this case, it is of basic interest to determine an element x that is in some sense closest to Tx. One of the most interesting results in this direction is the following theorem due to Fan [1].
Theorem F Let K be a non-empty compact convex subset of a normed space X and $T:K\to X$ be a continuous non-self-mapping. Then there exists an x such that $\parallel x-Tx\parallel =d(K,Tx)=inf\{\parallel Tx-u\parallel :u\in K\}$.
Many generalizations and extensions of this result have appeared in the literature (see [2–6] and references therein).
In fact best proximity point theory has been studied to find necessary conditions such that the minimization problem ${min}_{x\in A}d(x,Tx)$ has at least one solution. For more details on this approach, we refer the reader to [7–13] and [5, 14–25].
One of the interesting generalizations of the Banach contraction principle which characterizes the metric completeness is due to Suzuki [26, 27] (see also [28, 29]). Recently, Abkar and Gabeleh [8] studied best proximity point results for Suzuki contractions. The aim of this paper is to introduce modified Suzuki α-ψ-proximal contractions and establish certain best proximity point theorems for such proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity and fixed point theory. Moreover, some examples are given to illustrate the usability of the obtained results.
for all ${x}_{1},{x}_{2}\in A$ and ${y}_{1},{y}_{2}\in B$.
In 2012, Samet et al. [24] introduced the concepts of α-ψ-contractive and α-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces.
Samet et al. [24] defined the notion of α-admissible mappings as follows.
Salimi et al. [22] modified and generalized the notion of α-admissible mappings in the following way.
Note that if we take $\eta (x,y)=1$, then this definition reduces to Definition 1.1.
Definition 1.3 [14]
for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$, where $\alpha :A\times A\to [0,\mathrm{\infty})$.
Clearly, if $A=B$, T is α-proximal admissible implies that T is α-admissible.
Recently Hussain et al. [4] generalized the notion of α-proximal admissible as follows.
for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$. Note that if we take $\eta (x,y)=1$ for all $x,y\in A$, then this definition reduces to Definition 1.3.
- (i)
ψ is non-decreasing;
- (ii)there exist ${k}_{0}\in \mathbb{N}$ and $a\in (0,1)$ and a convergent series of nonnegative terms ${\sum}_{k=1}^{\mathrm{\infty}}{v}_{k}$ such that${\psi}^{k+1}(t)\le a{\psi}^{k}(t)+{v}_{k},$
for $k\ge {k}_{0}$ and any $t\in {\mathbb{R}}^{+}$.
In some sources, the Bianchini-Grandolfi gauge function is known as the $(c)$-comparison function (see e.g. [31]). We denote by Ψ the family of Bianchini-Grandolfi gauge functions. The following lemma illustrates the properties of these functions.
Lemma 1.1 (See [31])
- (i)
${({\psi}^{n}(t))}_{n\in \mathbb{N}}$ converges to 0 as $n\to \mathrm{\infty}$ for all $t\in {\mathbb{R}}^{+}$;
- (ii)
$\psi (t)<t$, for any $t\in (0,\mathrm{\infty})$;
- (iii)
ψ is continuous at 0;
- (iv)
the series ${\sum}_{k=1}^{\mathrm{\infty}}{\psi}^{k}(t)$ converges for any $t\in {\mathbb{R}}^{+}$.
2 Best proximity point results in metric spaces
We start this section with the following definition.
for all $x,y\in A$ where ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$, $\alpha :A\times A\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$.
The following is our first main result of this section.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)
T is α-proximal admissible with respect to $\eta (x,y)=2$;
- (iii)the elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ with$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfy}}\alpha ({x}_{0},{x}_{1})\ge 2;$
- (iv)
T is continuous.
Then T has a unique best proximity point.
which is a contradiction. Hence, $y=z$. This completes the proof of the theorem. □
In the following theorem, we replace the continuity condition on Suzuki α-ψ-proximal contraction T by regularity of the space $(X,d)$.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)
T is α-proximal admissible with respect to $\eta (x,y)=2$;
- (iii)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ with$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfying}}\alpha ({x}_{0},{x}_{1})\ge 2;$
- (iv)
if $\{{x}_{n}\}$ is a sequence in A such that $\alpha ({x}_{n},{x}_{n+1})\ge 2$ and ${x}_{n}\to x\in A$ as $n\to \mathrm{\infty}$, then $\alpha ({x}_{n},x)\ge 2$ for all $n\in \mathbb{N}$.
Then T has a unique best proximity point.
The uniqueness of best proximity point follows as in the proof of Theorem 2.1. □
The following results are nice consequences of Theorem 2.2.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)for a function $\delta :[0,1)\to (0,1/2]$, there exists $r\in [0,1)$ such that$\delta (r){d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\mathit{\text{implies}}\phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$(2.11)
for $x,y\in A$ where ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ and $\psi \in \mathrm{\Psi}$.
Then T has a unique best proximity point.
and so by (2.11) we deduce $d(Tx,Ty)\le \psi (d(x,y))$. Hence all conditions of Theorem 2.2 hold and T has a unique best proximity point. □
If we take $\psi (t)=rt$ in Theorem 2.3, where $0\le r<1$, then we obtain the following result.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)for a function $\delta :[0,1)\to (0,1/2]$, there exists $r\in [0,1)$ such that$\delta (r){d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\mathit{\text{implies}}\phantom{\rule{1em}{0ex}}d(Tx,Ty)\le rd(x,y)$(2.12)
for $x,y\in A$.
Then T has a unique best proximity point.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)define a non-increasing function $\theta :[0,1)\to (1/2,1]$ by$\theta (r)=\{\begin{array}{cc}1\hfill & \mathit{\text{if}}0\le r\le (\sqrt{5}-1)/2,\hfill \\ (1-r){r}^{-2}\hfill & \mathit{\text{if}}(\sqrt{5}-1)/2r{2}^{-1/2},\hfill \\ {(1+r)}^{-1}\hfill & \mathit{\text{if}}{2}^{-1/2}\le r1.\hfill \end{array}$(2.13)
for $x,y\in A$ where ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$.
Then T has a unique best proximity point.
Proof If we take $\delta (r)=\frac{1}{2}\theta (r)$ in Corollary 2.1, we obtain the required result. □
If we take $\delta (r)=\frac{1}{2(1+r)}$ in Corollary 2.1, we obtain the main result of [8] in the following form.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)define a non-increasing function $\beta :[0,1)\to (1/2,1]$ by$\beta (r)=\frac{1}{2(1+r)}.$(2.14)
for $x,y\in A$.
Then T has a unique best proximity point.
If we take $\delta (r)=\frac{1}{2}$ in Corollary 2.1 we have following result.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le rd(x,y)$
for all $x,y\in A$.
Then T has a unique best proximity point.
3 Best proximity point results in partially ordered metric spaces
Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [2, 9, 32] and references therein). The existence of best proximity and fixed point results in partially ordered metric spaces has been considered recently by many authors [4, 7, 21, 33, 34]. The aim of this section is to deduce some best proximity and fixed point results in the context of partially ordered metric spaces. Moreover, we obtain certain recent fixed point results as corollaries in partially ordered metric spaces.
for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$.
Clearly, if $B=A$, then the proximally order-preserving map $T:A\to A$ reduces to a non-decreasing map.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)
T is proximally order-preserving;
- (iii)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ with$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfying}}{x}_{0}\u2aaf{x}_{1};$
- (iv)
T is continuous;
- (v)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$
for all $x,y\in A$ with $x\u2aafy$ where ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ and $\psi \in \mathrm{\Psi}$.
Then T has a unique best proximity point.
From (v) we get $d(Tx,Ty)\le \psi (d(x,y))$. That is, T is a modified Suzuki α-ψ-proximal contraction. Thus all conditions of Theorem 2.1 hold and T has a unique best proximity point. □
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)
T is proximally ordered-preserving;
- (iii)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ with$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfying}}{x}_{0}\u2aaf{x}_{1};$
- (iv)
T is continuous;
- (v)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le r(d(x,y))$
for all $x,y\in A$ with $x\u2aafy$ where ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ and $0\le r<1$.
Then T has a unique best proximity point.
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)
T is proximally order-preserving;
- (iii)the elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ with$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfy}}{x}_{0}\u2aaf{x}_{1};$
- (iv)
if $\{{x}_{n}\}$ is a non-decreasing sequence in A such that ${x}_{n}\to x\in A$ as $n\to \mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$;
- (v)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$(3.1)
for all $x,y\in A$ with $x\u2aafy$ where ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ and $\psi \in \mathrm{\Psi}$.
Then T has a unique best proximity point.
Proof Defining $\alpha :X\times X\to [0,\mathrm{\infty})$ as in the proof of Theorem 3.1, we find that T is an α-proximal admissible mapping with respect to $\eta (x,y)=2$ and is modified Suzuki α-ψ-proximal contraction. Assume $\alpha ({x}_{n},{x}_{n+1})\ge 2$ for all $n\in \mathbb{N}$ such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$. Then ${x}_{n}\u2aaf{x}_{n+1}$ for all $n\in \mathbb{N}$. Hence, by (iv) we get ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$ and so $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}$. That is, all conditions of Theorem 2.2 hold and T has a unique best proximity point. □
- (i)
$T({A}_{0})\subseteq {B}_{0}$ and $(A,B)$ satisfies the P-property;
- (ii)
T is proximally ordered-preserving;
- (iii)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}$ with$d({x}_{1},T{x}_{0})=d(A,B)\mathit{\text{satisfying}}{x}_{0}\u2aaf{x}_{1};$
- (iv)
if $\{{x}_{n}\}$ is a non-decreasing sequence in A such that ${x}_{n}\to x\in A$ as $n\to \mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$;
- (v)$\frac{1}{2}{d}^{\ast}(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le r(d(x,y))$(3.2)
for all $x,y\in A$ with $x\u2aafy$ where ${d}^{\ast}(x,y)=d(x,y)-d(A,B)$ and $0\le r<1$.
Then T has a unique best proximity point.
4 Applications
As an application of our results, we deduce new fixed point results for Suzuki-type contractions in the set up of metric and partially ordered metric spaces.
If we take $A=B=X$ in Theorems 2.1 and 2.2, then we deduce the following result.
- (i)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},T{x}_{0})\ge 2$;
- (ii)
either T is continuous or if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 2$ and ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, then $\alpha ({x}_{n},x)\ge 2$ for all $n\in \mathbb{N}$.
Then T has a unique fixed point.
If we take $\psi (t)=kt$ in Theorem 4.1, where $0\le k<1$, then we conclude to the following theorem.
- (i)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},T{x}_{0})\ge 2$;
- (ii)
either T is continuous or if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1})\ge 2$ and ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, then $\alpha ({x}_{n},x)\ge 2$ for all $n\in \mathbb{N}$.
Then T has a unique fixed point.
As a consequence of Theorem 4.2, by taking $\alpha (x,y)=2/\theta (r)$, we derive the following theorem.
for all $x,y\in X$. Then T has a unique fixed point.
Furthermore, if we take $A=B=X$ in Theorems 3.1 and 3.2, then we deduce the following results.
- (i)
T is non-decreasing;
- (ii)
there exists ${x}_{0}$ in X such that ${x}_{0}\u2aafT{x}_{0}$;
- (iii)
T is continuous;
- (iv)$\frac{1}{2}d(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$(4.3)
for all $x,y\in X$ with $x\u2aafy$ where $\psi \in \mathrm{\Psi}$.
Then T has a unique fixed point.
- (i)
T is non-decreasing;
- (ii)
there exists ${x}_{0}$ in X such that ${x}_{0}\u2aafT{x}_{0}$;
- (iii)
if $\{{x}_{n}\}$ is a non-increasing sequence in X such that ${x}_{n}\to x\in X$ as $n\to \mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$;
- (iv)$\frac{1}{2}d(x,Tx)\le d(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d(Tx,Ty)\le \psi (d(x,y))$(4.4)
for all $x,y\in X$ with $x\u2aafy$ where $\psi \in \mathrm{\Psi}$.
Then T has a unique fixed point.
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, for technical and financial support.
Authors’ Affiliations
References
- Ky F: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 1969, 112: 234–240. 10.1007/BF01110225View ArticleMathSciNetGoogle Scholar
- Hussain N, Khan AR, Agarwal RP: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces. J. Nonlinear Convex Anal. 2010, 11(3):475–489.MathSciNetMATHGoogle Scholar
- Hussain N, Khan AR: Applications of the best approximation operator to ∗-nonexpansive maps in Hilbert spaces. Numer. Funct. Anal. Optim. 2003, 24(3–4):327–338. 10.1081/NFA-120022926View ArticleMathSciNetMATHGoogle Scholar
- Hussain N, Kutbi MA, Salimi P: Best proximity point results for modified α - ψ -proximal rational contractions. Abstr. Appl. Anal. 2013., 2013: Article ID 927457Google Scholar
- Sehgal VM, Singh SP: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. Math. Soc. 1988, 102: 534–537.MathSciNetMATHGoogle Scholar
- Takahashi W: Fan’s Existence Theorem for Inequalities Concerning Convex Functions and Its Applications. In Minimax Theory and Applications. Edited by: Ricceri B, Simons S. Kluwer Academic, Dordrecht; 1998:597–602.Google Scholar
- Abkar A, Gabeleh M: Best proximity points for cyclic mappings in ordered metric spaces. J. Optim. Theory Appl. 2011, 150(1):188–193. 10.1007/s10957-011-9818-2View ArticleMathSciNetMATHGoogle Scholar
- Abkar A, Gabeleh M: A best proximity point theorem for Suzuki type contraction non-self mappings. Fixed Point Theory 2013, 14(2):281–288.MathSciNetMATHGoogle Scholar
- Agarwal RP, Hussain N, Taoudi MA: Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations. Abstr. Appl. Anal. 2012., 2012: Article ID 245872Google Scholar
- Amini-Harandi A, Hussain N, Akbar F: Best proximity point results for generalized contractions in metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 164Google Scholar
- Amini-Harandi A, Fakhar M, Hajisharifi HR, Hussain N: Some new results on fixed and best proximity points in preordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 263Google Scholar
- Anuradha J, Veeramani P: Proximal pointwise contraction. Topol. Appl. 2009, 156: 2942–2948. 10.1016/j.topol.2009.01.017View ArticleMathSciNetMATHGoogle Scholar
- Basha SS, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. 10.1006/jath.1999.3415View ArticleMathSciNetMATHGoogle Scholar
- Jleli M, Samet B: Best proximity points for α - ψ -proximal contractive type mappings and applications. Bulletin des Sciences Mathématiques 2013. 10.1016/j.bulsci.2013.02.003Google Scholar
- De la Sen M: Fixed point and best proximity theorems under two classes of integral-type contractive conditions in uniform metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 510974Google Scholar
- Pragadeeswarar V, Marudai M: Best proximity points: approximation and optimization in partially ordered metric spaces. Optim. Lett. 2013. 10.1007/s11590-012-0529-xGoogle Scholar
- Proinov PD: A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Anal. 2007, 67: 2361–2369. 10.1016/j.na.2006.09.008View ArticleMathSciNetMATHGoogle Scholar
- Proinov PD: New general convergence theory for iterative processes and its applications to Newton Kantorovich type theorems. J. Complex. 2010, 26: 3–42. 10.1016/j.jco.2009.05.001View ArticleMathSciNetMATHGoogle Scholar
- Raj VS, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 2009, 10: 21–28.View ArticleMathSciNetMATHGoogle Scholar
- Raj VS: A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal. 2011, 74: 4804–4808. 10.1016/j.na.2011.04.052View ArticleMathSciNetMATHGoogle Scholar
- Sadiq Basha S: Best proximity point theorems on partially ordered sets. Optim. Lett. 2012. 10.1007/s11590-012-0489-1Google Scholar
- Salimi P, Latif A, Hussain N: Modified α - ψ -contractive mappings with applications. Fixed Point Theory Appl. 2013., 2013: Article ID 151Google Scholar
- Samet B: Some results on best proximity points. J. Optim. Theory Appl. 2013. 10.1007/s10957-013-0269-9Google Scholar
- Samet B, Vetro C, Vetro P: Fixed point theorem for α - ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014View ArticleMathSciNetMATHGoogle Scholar
- Suzuki T, Kikkawa M, Vetro C: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 2009, 71: 2918–2926. 10.1016/j.na.2009.01.173View ArticleMathSciNetMATHGoogle Scholar
- Suzuki T: A new type of fixed point theorem in metric spaces. Nonlinear Anal. 2009, 71(11):5313–5317. 10.1016/j.na.2009.04.017View ArticleMathSciNetMATHGoogle Scholar
- Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136: 1861–1869.View ArticleMATHGoogle Scholar
- Hussain N, Ðorić D, Kadelburg Z, Radenović S: Suzuki-type fixed point results in metric type spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 126Google Scholar
- Shobe N, Sedghi S, Roshan JR, Hussain N: Suzuki-type fixed point results in metric-like spaces. J. Funct. Spaces Appl. 2013., 2013: Article ID 143686Google Scholar
- Bianchini RM, Grandolfi M: Transformazioni di tipo contracttivo generalizzato in uno spazio metrico. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 1968, 45: 212–216.MathSciNetMATHGoogle Scholar
- Berinde V: Iterative Approximation of Fixed Points. Springer, Berlin; 2007.MATHGoogle Scholar
- Hussain N, Taoudi MA: Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations. Fixed Point Theory Appl. 2013., 2013: Article ID 196Google Scholar
- Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060View ArticleMathSciNetMATHGoogle Scholar
- Hussain N, Al-Mezel S, Salimi P: Fixed points for ψ -graphic contractions with application to integral equations. Abstr. Appl. Anal. 2013., 2013: Article ID 575869Google Scholar
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