- Research
- Open Access

# A new approach to $(\alpha ,\psi )$-contractive nonself multivalued mappings

- Muhammad Usman Ali
^{1}, - Tayyab Kamran
^{2}and - Erdal Karapınar
^{3, 4}Email author

**2014**:71

https://doi.org/10.1186/1029-242X-2014-71

© Ali et al.; licensee Springer. 2014

**Received:**18 December 2013**Accepted:**31 January 2014**Published:**13 February 2014

## Abstract

In this paper, we introduce the notions of *α*-admissible and *α*-*ψ*-contractive type condition for nonself multivalued mappings. We establish fixed point theorems using these new notions along with a new condition. Moreover, we have constructed examples to show that our new condition is different from the corresponding existing conditions in the literature.

**MSC:** 47H10, 54H25.

## Keywords

*α*-admissible maps*α*-*ψ*-contractive type condition- nonself
*α*-admissible maps - nonself $(\alpha ,\psi )$-contractive type condition

## 1 Introduction and preliminaries

In the last decades, metric fixed point theory has been appreciated by a number of authors who have extended the celebrated Banach fixed point theorem for various contractive mapping in the context of different abstract spaces; see, for example, [1–32]. Among them, we mention the interesting fixed point theorems of Samet *et al.* [20]. In this paper [20], the authors introduced the notions of *α*-*ψ*-contractive mappings and investigated the existence and uniqueness of a fixed point for such mappings. Further, they showed that several well-known fixed point theorems can be derived from the fixed point theorem of *α*-*ψ*-contractive mappings. Following this paper, Karapınar and Samet [21] generalized the notion *α*-*ψ*-contractive mappings and obtained a fixed point for this generalized version. On the other hand, Asl *et al.* [22] characterized the notions of *α*-*ψ*-contractive mapping and *α*-admissible mappings with the notions of ${\alpha}_{\ast}$-*ψ*-contractive and ${\alpha}_{\ast}$-admissible mappings to investigate the existence of a fixed point for a multivalued function. Afterward, Ali and Kamran [23] generalized the notion of ${\alpha}_{\ast}$-*ψ*-contractive mappings and obtained further fixed point results for multivalued mappings. Some results in this direction in the context of various abstract spaces were also given by the authors [24–28, 33–36]. The purpose of this paper is to prove fixed point theorems for nonself multivalued $(\alpha ,\psi )$-contractive type mappings using a new condition.

Let Ψ be the family of functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, known in the literature as Bianchini-Grandolfi gauge functions (see, *e.g.*, [30–32]), satisfying the following conditions:

(${\psi}_{1}$) *ψ* is nondecreasing;

(${\psi}_{2}$) ${\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty}$ for all $t>0$, where ${\psi}^{n}$ is the *n* th iterate of *ψ*.

Notice that such functions are also known as $(c)$-comparison functions in some sources (see, *e.g.*, [29]).

*e.g.*, [20, 29]). Let $(X,d)$ be a metric space. A mapping $G:X\to X$ is called

*α*-

*ψ*-contractive type if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that

*α*-admissible [20] if

*X*and by $\mathit{CL}(X)$ the space of all nonempty closed subsets of

*X*. For $A\in N(X)$ and $x\in X$, $d(x,A)=inf\{d(x,a):a\in A\}$. For every $A,B\in \mathit{CL}(X)$, let

Such a map *H* is called a generalized Hausdorff metric induced by *d*. We use the following lemma in our results.

**Lemma 1.1** [23]

*Let*$(X,d)$

*be a metric space and*$B\in \mathit{CL}(X)$.

*Then*,

*for each*$x\in X$

*with*$d(x,B)>0$

*and*$q>1$,

*there exists an element*$b\in B$

*such that*

Let $(X,\u2aaf,d)$ be an ordered metric space and $A,B\subseteq X$. We say that $A{\prec}_{r}B$ if for each $a\in A$ and $b\in B$, we have $a\u2aafb$.

## 2 Main results

We begin this section with the following definition which is a modification of the notion of *α*-admissible.

**Definition 2.1**Let $(X,d)$ be a metric space and let

*D*be a nonempty subset of

*X*. A mapping $G:D\to \mathit{CL}(X)$ is called

*α*-admissible if there exists a mapping $\alpha :D\times D\to [0,\mathrm{\infty})$ such that

for each $u\in Gx\cap D$ and $v\in Gy\cap D$.

**Definition 2.2**Let $(X,d)$ be a metric space and let

*D*be a nonempty subset of

*X*. We say that $G:D\to \mathit{CL}(X)$ is an $(\alpha ,\psi )$-contractive type mapping on

*D*if there exist $\alpha :D\times D\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ satisfying the following conditions:

- (i)
$Gx\cap D\ne \mathrm{\varnothing}$ for all $x\in D$,

- (ii)for each $x,y\in D$, we have$\alpha (x,y)H(Gx\cap D,Gy\cap D)\le \psi (M(x,y)),$(2.1)

where $M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\}$.

Note that if $\psi \in \mathrm{\Psi}$ in the above definition is a strictly increasing function, then $G:D\to \mathit{CL}(X)$ is said to be a strictly $(\alpha ,\psi )$-contractive type mapping on *D*.

**Theorem 2.3**

*Let*$(X,d)$

*be a metric space*,

*let*

*D*

*be a nonempty subset of*

*X*

*which is complete with respect to the metric induced by*

*d*,

*and let*

*G*

*be a strictly*$(\alpha ,\psi )$-

*contractive type mapping on*

*D*.

*Assume that the following conditions hold*:

- (i)
*G**is an**α*-*admissible map*; - (ii)
*there exist*${x}_{0}\in D$*and*${x}_{1}\in G{x}_{0}\cap D$*such that*$\alpha ({x}_{0},{x}_{1})\ge 1$; - (iii)
*G**is continuous*.

*Then* *G* *has a fixed point*.

*Proof*By hypothesis, there exist ${x}_{0}\in D$ and ${x}_{1}\in G{x}_{0}\cap D$ such that $\alpha ({x}_{0},{x}_{1})\ge 1$. If ${x}_{0}={x}_{1}$, then we have nothing to prove. Let ${x}_{0}\ne {x}_{1}$. If ${x}_{1}\in G{x}_{1}\cap D$, then ${x}_{1}$ is a fixed point. Let ${x}_{1}\notin G{x}_{1}\cap D$. From (2.1) we have

*ψ*in (2.5), we have

*G*is an

*α*-admissible mapping, $\alpha ({x}_{1},{x}_{2})\ge 1$. If ${x}_{2}\in G{x}_{2}\cap D$, then ${x}_{2}$ is a fixed point. Let ${x}_{2}\notin G{x}_{2}\cap D$. From (2.1) we have

*ψ*in (2.10), we have

*G*is an

*α*-admissible mapping, $\alpha ({x}_{2},{x}_{3})\ge 1$. If ${x}_{3}\in G{x}_{3}\cap D$, then ${x}_{3}$ is a fixed point. Let ${x}_{3}\notin G{x}_{3}\cap D$. From (2.1) we have

*ψ*in (2.15), we have

*D*such that ${x}_{n+1}\in G{x}_{n}\cap D$, ${x}_{n+1}\ne {x}_{n}$, $\alpha ({x}_{n},{x}_{n+1})\ge 1$, and

*D*. Since

*D*is complete, there exists ${x}^{\ast}\in D$ such that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$. By the continuity of

*G*, we have

□

**Theorem 2.4**

*Let*$(X,d)$

*be a metric space*,

*D*

*be a nonempty subset of*

*X*

*which is complete with respect to the metric induced by*

*d*,

*and let*

*G*

*be a strictly*$(\alpha ,\psi )$-

*contractive type mapping on*

*D*.

*Assume that the following conditions hold*:

- (i)
*G**is an**α*-*admissible map*; - (ii)
*there exist*${x}_{0}\in D$*and*${x}_{1}\in G{x}_{0}\cap D$*such that*$\alpha ({x}_{0},{x}_{1})\ge 1$; - (iii)
*either*- (a)
*for any sequence*$\{{x}_{n}\}$*in**D**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for each*$n\in \mathbb{N}\cup \{0\}$, ${lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},x)\ge 1$,*or* - (b)
*for any sequence*$\{{x}_{n}\}$*in**D**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for each*$n\in \mathbb{N}\cup \{0\}$, $\alpha ({x}_{n},x)\ge 1$*for each*$n\in \mathbb{N}\cup \{0\}$.

- (a)

*Then* *G* *has a fixed point*.

*Proof*Following the proof of Theorem 2.3, there exists a Cauchy sequence $\{{x}_{n}\}$ in

*D*with ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$ and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$. Suppose that $d({x}^{\ast},G{x}^{\ast})\ne 0$. From (2.1) we have

which is impossible. Thus $d({x}^{\ast},G{x}^{\ast})=0$. □

**Example 2.5**Let $X=(-\mathrm{\infty},-8)\cup [0,\mathrm{\infty})$ be endowed with the usual metric

*d*, and let $D=[0,\mathrm{\infty})$. Define $G:D\to \mathit{CL}(X)$ by

Clearly, $Gx\cap D\ne \mathrm{\varnothing}$ for each $x\in D$. Let $\psi (t)=\frac{t}{2}$ for each $t\ge 0$. To see that *G* is a strictly $(\alpha ,\psi )$-contractive type mapping on *D*, we consider the following cases.

where $M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\}$.

Thus, *G* is a strictly $(\alpha ,\psi )$-contractive type mapping on *D*. For $\alpha (x,y)\ge 1$, we have $x,y\in [0,4]$, then $Gx\cap D,Gy\cap D\subseteq [0,1]$, thus $\alpha (u,v)=1$ for each $u\in Gx\cap D$ and $v\in Gy\cap D$. Further, for any sequence $\{{x}_{n}\}$ in *D* such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$ and $\alpha ({x}_{n},{x}_{n+1})=1$ for each $n\in \mathbb{N}\cup \{0\}$, ${lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},x)=1$. Therefore, all the conditions of Theorem 2.4 hold and *G* has a fixed point.

**Corollary 2.6**

*Let*$(X,\u2aaf,d)$

*be an ordered metric space*,

*let*$(D,\u2aaf)$

*be a nonempty subset of*

*X*

*which is complete with respect to the metric induced by*

*d*.

*Let*$G:D\to \mathit{CL}(X)$

*be a mapping such that*$Gx\cap D\ne \mathrm{\varnothing}$

*for each*$x\in D$

*and for each*$x,y\in D$

*with*$x\u2aafy$,

*we have*

*where*$M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\}$

*and*

*ψ*

*is an increasing function in*Ψ.

*Also*,

*assume that the following conditions hold*:

- (i)
*there exist*${x}_{0}\in D$*and*${x}_{1}\in G{x}_{0}\cap D$*such that*${x}_{0}\u2aaf{x}_{1}$; - (ii)
*if*$x\u2aafy$*then*$Gx\cap D{\prec}_{r}Gy\cap D$; - (iii)
*either*- (a)
*G**is continuous*,*or* - (b)
*for any sequence*$\{{x}_{n}\}$*in**D**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*${x}_{n}\u2aaf{x}_{n+1}$*for each*$n\in \mathbb{N}\cup \{0\}$, ${x}_{n}\u2aafx$*as*$n\to \mathrm{\infty}$,*or* - (c)
*for any sequence*$\{{x}_{n}\}$*in**D**such that*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$*and*${x}_{n}\u2aaf{x}_{n+1}$*for each*$n\in \mathbb{N}\cup \{0\}$, ${x}_{n}\u2aafx$*for each*$n\in \mathbb{N}\cup \{0\}$.

- (a)

*Then* *G* *has a fixed point*.

*Proof*Define $\alpha :D\times D\to [0,\mathrm{\infty})$ by

By using condition (i) and the definition of *α*, we have $\alpha ({x}_{0},{x}_{1})=1$. Also, from condition (ii), we have that $x\u2aafy$ implies $Gx\cap D{\prec}_{r}Gy\cap D$; by using the definitions of *α* and ${\prec}_{r}$, we have that $\alpha (x,y)=1$ implies $\alpha (u,v)=1$ for each $u\in Gx\cap D$ and $v\in Gy\cap D$. Moreover, it is easy to check that *G* is a strictly $(\alpha ,\psi )$-contractive type mapping on *D*. Therefore, all the conditions of Theorem 2.3 (or Theorem 2.4 for (iii)(b), (iii)(c)) hold, hence *G* has a fixed point. □

**Remark 2.7** Condition (a), in the statement of Theorem 2.4, was introduced by Samet *et al.* [20]. In Theorem 2.4 we introduce a new condition (b). The following examples show that (a) and (b) are independent conditions.

**Example 2.8**Let $X=\{\frac{1}{n}:n\in \mathbb{N}\}\cup \{0\}$. Consider ${x}_{n}=\frac{1}{n+1}$ for each $n\in \mathbb{N}\cup \{0\}$, then ${x}_{n}\to 0={x}^{\ast}$ as $n\to \mathrm{\infty}$. Define $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Now, we have $\alpha ({x}_{n},{x}_{n+1})=\alpha (\frac{1}{n+1},\frac{1}{n+2})=n+2>1$ for each $n\in \mathbb{N}\cup \{0\}$ and $\alpha ({x}_{n},{x}^{\ast})=\alpha (\frac{1}{n+1},0)=n+1\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$. Thus condition (a) holds but ${lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},{x}^{\ast})={lim}_{n\to \mathrm{\infty}}(n+1)=\mathrm{\infty}$. Thus condition (b) does not hold.

**Example 2.9**Let $X=\{\frac{1}{n}:n\in \mathbb{N}\}\cup \{0\}$. Consider ${x}_{n}=\frac{1}{n+1}$ for each $n\in \mathbb{N}\cup \{0\}$, then ${x}_{n}\to 0={x}^{\ast}$ as $n\to \mathrm{\infty}$. Define $\alpha :X\times X\to [0,\mathrm{\infty})$ by

Now, we have $\alpha ({x}_{n},{x}_{n+1})=\alpha (\frac{1}{n+1},\frac{1}{n+2})=n+2>1$ for each $n\in \mathbb{N}\cup \{0\}$ and $\alpha ({x}_{n},{x}^{\ast})=\alpha (\frac{1}{n+1},0)=\frac{2n+2}{2n+3}$. Then ${lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},{x}^{\ast})={lim}_{n\to \mathrm{\infty}}\frac{2n+2}{2n+3}=1$. Thus condition (b) holds but for $n=0$, we have $\alpha ({x}_{n},{x}^{\ast})=\frac{2}{3}$; for $n=1$, we have $\alpha ({x}_{n},{x}^{\ast})=\frac{4}{5}$; for $n=2$, we have $\alpha ({x}_{n},{x}^{\ast})=\frac{6}{7}$, which implies that $\alpha ({x}_{n},x)\ngeqq 1$ for each $n\in \mathbb{N}\cup \{0\}$. Thus condition (a) does not hold.

## Declarations

### Acknowledgements

The authors are grateful to the reviewers for their careful reviews and useful comments.

## Authors’ Affiliations

## References

- Azam A, Mehmood N, Ahmad J, Radenovic S:
**Multivalued fixed point theorems in cone**b**-metric spaces.***J. Inequal. Appl.*2013.,**2013:**Article ID 582Google Scholar - Shukla S, Radojevic S, Veljkovic Z, Radenovic S:
**Some coincidence and common fixed point theorems for ordered Presic-Reich type contractions.***J. Inequal. Appl.*2013.,**2013:**Article ID 520Google Scholar - Shukla, S, Sen, R, Radenovic, S: Set-valued Presic type contraction in metric spaces. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. (2012, in press)Google Scholar
- Long W, Shukla S, Radenovic S, Radojevic S:
**Some coupled coincidence and common fixed point results for hybrid pair of mappings in 0-complete partial metric spaces.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 145Google Scholar - Kadelburg Z, Radenovic S:
**Some results on set-valued contractions in abstract metric spaces.***Comput. Math. Appl.*2012,**62:**342–350.MathSciNetView ArticleMATHGoogle Scholar - Khojasteh F, Karapınar E, Radenovic S: θ
**-Metric spaces: a generalization.***Math. Probl. Eng.*2013.,**2013:**Article ID 504609Google Scholar - Shatanawi W, Rajic VC, Radenovic S, Rawashdeh AA:
**Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces.***Fixed Point Theory Appl.*2012.,**2012:**Article ID 106Google Scholar - Ali, MU: Mizoguchi-Takahashi’s type common fixed point theorem. J. Egypt. Math. Soc. (in press)Google Scholar
- Ali MU, Kamran T:
**Hybrid generalized contractions.***Math. Sci.*2013.,**7:**Article ID 29Google Scholar - Chandok S, Postolache M:
**Fixed point theorem for weakly Chatterjea-type cyclic contractions.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 28Google Scholar - Shatanawi W, Postolache M:
**Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 60Google Scholar - Shatanawi W, Postolache M:
**Some fixed point results for a**G**-weak contraction in**G**-metric spaces.***Abstr. Appl. Anal.*2012.,**2012:**Article ID 815870Google Scholar - Choudhury BS, Metiya N, Postolache M:
**A generalized weak contraction principle with applications to coupled coincidence point problems.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 152Google Scholar - Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N:
**Theorems for Boyd-Wong-type contractions in ordered metric spaces.***Abstr. Appl. Anal.*2012.,**2012:**Article ID 359054Google Scholar - Shatanawi W, Postolache M:
**Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 271Google Scholar - Shatanawi W, Pitea A:
**Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 275Google Scholar - Miandaragh MA, Postolache M, Rezapour S:
**Some approximate fixed point results for generalized**α**-contractive mappings.***Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys.*2013,**75**(2):3–10.MathSciNetMATHGoogle Scholar - Haghi RH, Postolache M, Rezapour S:
**On**T**-stability of the Picard iteration for generalized**ϕ**-contraction mappings.***Abstr. Appl. Anal.*2012.,**2012:**Article ID 658971Google Scholar - Shatanawi W, Postolache M:
**Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 54Google Scholar - Samet B, Vetro C, Vetro P:
**Fixed point theorems for**α**-**ψ**-contractive type mappings.***Nonlinear Anal.*2012,**75:**2154–2165. 10.1016/j.na.2011.10.014MathSciNetView ArticleMATHGoogle Scholar - Karapınar E, Samet B:
**Generalized**α**-**ψ**-contractive type mappings and related fixed point theorems with applications.***Abstr. Appl. Anal.*2012.,**2012:**Article ID 793486Google Scholar - Asl JH, Rezapour S, Shahzad N:
**On fixed points of**α**-**ψ**-contractive multifunctions.***Fixed Point Theory Appl.*2012.,**2012:**Article ID 212Google Scholar - Ali MU, Kamran T: On $({\alpha}^{\ast},\psi )$-contractive multi-valued mappings.
*Fixed Point Theory Appl.*2013.,**2013:**Article ID 137Google Scholar - Mohammadi B, Rezapour S, Shahzad N:
**Some results on fixed points of**α**-**ψ**-Ciric generalized multifunctions.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 24Google Scholar - Amiri P, Rezapour S, Shahzad N:
**Fixed points of generalized**α**-**ψ**-contractions.***Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat.*2013. 10.1007/s13398-013-0123-9Google Scholar - Minak G, Altun I:
**Some new generalizations of Mizoguchi-Takahashi type fixed point theorem.***J. Inequal. Appl.*2013.,**2013:**Article ID 493Google Scholar - Ali MU, Kamran T, Karapınar E: $(\alpha ,\psi ,\xi )$-Contractive multi-valued mappings.
*Fixed Point Theory Appl.*2014.,**2014:**Article ID 7Google Scholar - Ali MU, Kamran T, Sintunavarat W, Katchang P:
**Mizoguchi-Takahashi’s fixed point theorem with**α**,**η**functions.***Abstr. Appl. Anal.*2013.,**2013:**Article ID 418798Google Scholar - Rus IA:
*Generalized Contractions and Applications*. Cluj University Press, Cluj-Napoca; 2001.MATHGoogle Scholar - Bianchini RM, Grandolfi M:
**Transformazioni di tipo contracttivo generalizzato in uno spazio metrico.***Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat.*1968,**45:**212–216.MathSciNetMATHGoogle Scholar - Proinov PD:
**A generalization of the Banach contraction principle with high order of convergence of successive approximations.***Nonlinear Anal. TMA*2007,**67:**2361–2369. 10.1016/j.na.2006.09.008MathSciNetView ArticleMATHGoogle 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.2009.05.001MathSciNetView ArticleMATHGoogle Scholar - Karapınar E:
**Discussion on**α**-**ψ**-contractions in generalized metric spaces.***Abstr. Appl. Anal.*2014.,**2014:**Article ID 962784Google Scholar - Berzig, M, Karapınar, E: On modified α-ψ-contractive mappings with application. Thai J. Math. 12 (2014)Google Scholar
- Jleli M, Karapınar E, Samet B:
**Best proximity points for generalized**α**-**ψ**-proximal contractive type mappings.***J. Appl. Math.*2013.,**2013:**Article ID 534127Google Scholar - Jleli M, Karapınar E, Samet B:
**Fixed point results for**α**-**${\psi}_{\lambda}$**-contractions on gauge spaces and applications.***Abstr. Appl. Anal.*2013.,**2013:**Article ID 730825Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.