Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81 DOI 10.1186/s13663-016-0572-x

RESEARCH

Open Access

Some fixed point results via R-functions Antonella Nastasi1 , Pasquale Vetro1 and Stojan Radenovi´c2* *

Correspondence: [email protected] 2 Nonlinear Analysis Research Group and Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minn City, Vietnam Full list of author information is available at the end of the article

Abstract We establish existence and uniqueness of fixed points for a new class of mappings, by using R-functions and lower semi-continuous functions in the setting of metric spaces. As consequences of this results, we obtain several known fixed point results, in metric and partial metric spaces. An example is given to support the new theory. A homotopy result for operators on a set endowed with a metric is given as application. MSC: 47H10; 54H25 Keywords: R-function; R-λ-contraction; fixed point; metric space; partial metric space

1 Introduction Metric fixed point theory is a fundamental topic, which gives basic methods and notions for establish practical problems in mathematics and the other sciences. As an example, we consider the existence of solutions of mathematical problems reducible to equivalent fixed point problems. Thus, we recall that Banach contraction principle [] is at the foundation of this theory. However, the potentiality of fixed point approaches attracted many scientists and hence there is a wide literature available for interested reader; see for instance [–]. We give some details on the notions and ideas used in this study. First, the notion of partial metric space was introduced in  by Matthews [] as a part of the study of denotational semantics of data for networks. Clearly, this setting is a generalization of the classical concept of metric space. Also, some authors discussed the existence of several connections between partial metrics and topological aspects of domain theory; see for instance [–]. Second, the notion of Z -contraction was introduced in  by Khojasteh et al. []. This concept is a new type of nonlinear contraction defined by using a specific function, called simulation function. Consequently, they proved the existence and uniqueness of fixed points for Z -contraction mappings (see [], Theorem .). The notion of R-contraction was introduced in  by Roldán López de Hierro and Shahzad []. Also this notion is a new type of nonlinear contraction defined by using a specific function called R-function. Naturally, they proved the existence and uniqueness of fixed points for R-contraction mappings (see [], Theorem ). We point out that the advantage of these methods is in providing a unifying point of view for several fixed point problems; see recent results in [–]. © 2016 Nastasi et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 2 of 12

Finally, Samet et al. [], and Vetro and Vetro [] discussed fixed point results, by using semi-continuous functions in metric spaces, that generalize and improve many existing fixed point theorems in the literature. As an application of presented results, the authors gave some theorems in the setting of partial metric spaces. In this paper, we use the ideas in [, ] and the notion of R-function to establishing the existence and uniqueness of fixed points that belong to the zero set of a certain function. As consequences of this study, we deduce several related fixed point results, in metric and partial metric spaces. Also, an example is given to support the new theory. As application, a homotopy result for operators on a set endowed with a metric is given.

2 Preliminaries We will start with a brief recollection of basic notions and results in partial metric spaces that can be found in [, , , ]. A partial metric on a non-empty set Z is a function p : Z × Z → [, +∞[ such that, for all u, v, w ∈ Z, we have (p ) (p ) (p ) (p )

u = v ⇔ p(u, u) = p(u, v) = p(v, v); p(u, u) ≤ p(u, v); p(u, v) = p(v, u); p(u, v) ≤ p(u, w) + p(w, v) – p(w, w).

A partial metric space is a pair (Z, p), where Z is a non-empty set and p is a partial metric on Z. Every partial metric p : Z × Z → [, +∞[ generates a T topology τp on Z, which has as a base the family of open p-balls {Up (u, ρ) : u ∈ Z, ρ > }, where Up (u, ρ) = {v ∈ Z : p(u, v) < p(u, u) + ρ} for all u ∈ Z and ρ > . Let (Z, p) be a partial metric space and {uj } ⊂ Z. Then (i) {uj } converges to a point u ∈ Z if and only if p(u, u) = limj→+∞ p(u, uj ); (ii) {uj } is called a Cauchy sequence if there exists limi,j→+∞ p(ui , uj ) (and it is finite); (iii) (Z, p) is said to be complete if every Cauchy sequence {uj } in Z converges, with respect to τp , to a point u ∈ Z such that p(u, u) = limi,j→+∞ p(ui , uj ). It is elementary to verify that the function dp : Z × Z → [, +∞[ defined by dp (u, v) = p(u, v) – p(u, u) – p(v, v)

()

is a metric on Z whenever p is a partial metric on Z. Moreover, limj→+∞ dp (uj , u) =  if and only if p(u, u) = lim p(uj , u) = lim p(ui , uj ). j→+∞

i,j→+∞

()

The following lemma shows that the function λ : Z → [, +∞[ defined by λ(u) = p(u, u) for all u ∈ Z is continuous in (Z, dp ). Lemma . Let (Z, p) be a partial metric space and let λ : Z → [, +∞[ be defined by λ(u) = p(u, u) for all u ∈ Z. Then the function λ is continuous in the metric space (Z, dp ).

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 3 of 12

Proof Let u ∈ Z and {uj } be a sequence which converges to u in the metric space (Z, dp ). By (), we get p(u, u) = lim p(uj , uj ) j→+∞

and hence λ is continuous in (Z, dp ).



The following lemma correlates the Cauchy sequences of the spaces (Z, p) and (Z, dp ). Lemma . ([, ]) Let (Z, p) be a partial metric space. Then () {uj } is a Cauchy sequence in (Z, p) if and only if it is a Cauchy sequence in the metric space (Z, dp ); () a partial metric space (Z, p) is complete if and only if the metric space (Z, dp ) is complete.

3 New fixed point theorems in complete metric spaces In this section, we consider the family R of R-functions introduced by Roldán López de Hierro and Shahzad in []. Precisely, a function η : [, +∞[ ×[, +∞[ → R is called Rfunction if the following conditions hold: (η ) for each sequence {tn } ⊂ ], +∞[ such that η(tn+ , tn ) >  for all n ∈ N, we have limn→+∞ tn = ; (η ) for every two sequences {tn }, {sn } ⊂ ], +∞[ such that limn→+∞ tn = limn→+∞ sn = L ≥ , then L =  whenever L < tn and η(tn , sn ) >  for all n ∈ N. Now, we use R-functions to define a new class of contractions. Let (Z, d) be a metric space. Denote by  the family of lower semi-continuous functions λ : Z → [, +∞[. Let h : Z → Z be a self-mapping and λ ∈ . In the sequel, we will use the following notation D(u, v; λ) := d(u, v) + λ(u) + λ(v) for all u, v ∈ Z. Now, we define the new family of contractions. Definition . Let (Z, d) be a metric space and let h : Z → Z be a mapping. The mapping h is a R-λ-contraction if there exist an R-function η : [, +∞[ ×[, +∞[ → R and a function λ ∈  such that   η D(hu, hv; λ), D(u, v; λ) > 

()

for all u, v ∈ Z with D(u, v; λ) > . In the following theorem, we establish a result of existence and uniqueness of a fixed point for R-λ-contractions that belong to {x ∈ Z : λ(x) = }. Theorem . Let (Z, d) be a complete metric space and let h : Z → Z be a R-λ-contraction. Assume that, at least, one of the following conditions holds: () h is continuous;

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 4 of 12

() for every two sequences {ti }, {si } ⊂ ], +∞[ such that limi→+∞ si =  and η(ti , si ) >  for all i ∈ N, then limi→+∞ ti = ; () η(t, s) ≤ s – t for all t, s ∈], +∞[. Then h has a unique fixed point x ∈ Z such that λ(x) =  and, for any choice of the starting point z ∈ Z, the sequence {zn } defined by zn = hzn– for each n ∈ N converges to the point x. Proof We fix arbitrarily a point z of Z and we consider the Picard sequence {zi } of h starting at z , that is, the sequence defined by zi = hzi– for all i ∈ N. If for some j ∈ N ∪ {} we have zj+ = zj , then zj is evidently a fixed point of h. Also, we claim that λ(zj ) = . First, from zj = zj+ , we deduce that zi = zj for all i ∈ N ∪ {} with i ≥ j. Assume λ(zj ) >  and let ti := D(zj+i , zj+i+ ; λ), that is, a positive real number for all i ∈ N. Since h by hypothesis is an R-λ-contraction, we get   η(ti+ , ti ) = η D(hzj+i , hzj+i+ ; λ), D(zj+i , zj+i+ ; λ) >  for all i ∈ N. By property (η ) of the function η it follows that λ(zj ) = λ(zj+i ) →  as i → +∞ and so λ(zj ) =  and hence the conclusion follows if zj+ = zj for some j ∈ N ∪ {}. Therefore, we can suppose that zi– = zi for all i ∈ N. We shall divide the proof in three parts. First, we show that lim d(zi– , zi ) = 

i→+∞

and

lim λ(zi ) = .

i→+∞

()

From zi– = zi for all i ∈ N, we deduce that ti– = D(zi– , zi ; λ) >  for all i ∈ N. Thus the sequence {ti } ⊂ ], +∞[. Since h is an R-λ-contraction, from () with u = zi and v = zi+ , we get   η(ti+ , ti ) = η D(zi+ , zi+ ; λ), D(zi , zi+ ; λ)   = η D(hzi , hzi+ ; λ), D(zi , zi+ ; λ) >  for all i ∈ N ∪ {}. The property (η ) of the function η allows one to state that ti →  as i → +∞. Consequently, d(zi– , zi ) →  and λ(zi ) → , that is, () holds. The second part is to show that the sequence {zi } is Cauchy. Let us assume that {zi } is not a Cauchy sequence. Then there exist σ >  and two subsequences {zj(k) } and {zi(k) } of {zi } with k ≤ j(k) < i(k) and d(zj(k) , zi(k)– ) ≤ σ < d(zj(k) , zi(k) ) for all k ∈ N. The above restrictions and limi→+∞ d(zi– , zi ) =  imply lim d(zj(k) , zi(k) ) = lim d(zj(k)– , zi(k)– ) = σ .

k→+∞

k→+∞

Since λ(zi ) →  as i → +∞, we get σ = lim D(zj(k) , zi(k) ; λ) = lim D(zj(k)– , zi(k)– ; λ). k→+∞

k→+∞

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 5 of 12

The previous equality allows us to assume D(zj(k)– , zi(k)– ; λ) >  for each k ∈ N. Now, we consider the sequences {tk }, {sk } given by sk := D(zj(k)– , zi(k)– ; λ) for all k ∈ N.

tk := D(zj(k) , zi(k) ; λ) and

From () with u = zj(k)– and v = zi(k)– , we obtain   η(tk , sk ) = η D(zj(k) , zi(k) ; λ), D(zj(k)– , zi(k)– ; λ)   = η D(hzj(k)– , hzi(k)– ; λ), D(zj(k)– , zi(k)– ; λ) > 

()

for all k ∈ N. Let L = σ ; from L = σ < d(zj(k) , zi(k) ) ≤ D(zj(k) , zi(k) ; λ) = tk and (), by property (η ) of the function η, we obtain σ = L = , which is a contradiction. Hence {zi } is a Cauchy sequence. As (Z, d) is by hypothesis a complete metric space, there exists x ∈ Z such that zi → x as i → +∞. The hypothesis that λ is lower semi-continuous implies that  ≤ λ(x) ≤ lim inf λ(zi ) = , i→+∞

that is, λ(x) = . The third part is to prove that x is a fixed point of h. We consider the following three steps. First step. h is a continuous mapping, that is, condition () holds. From zi+ = hzi → hx, we get x = hx. Second step. Hypothesis () holds. If there exists a subsequence {zi(k) } of {zi } such that hzi(k) = hx for all k ∈ N, then x is a fixed point of h. If this does not happen, then we can assume that zi = x and hzi = hx for all i ∈ N. Now, consider the sequences ti := D(hzi , hx; λ) and

si := D(zi , x; λ)

for all i ∈ N. Such a choice ensures that {ti }, {si } ⊂ ], +∞[. Clearly, by () and λ(x) = , si →  and since h is a R-φ-contraction, we have also   η(ti , si ) = η D(hzi , hx; λ), D(zi , x; λ) >  for all i ∈ N. Then, by condition (), we get ti → . This allows one to state that d(zi+ , hx) = d(hzi , hx) →  and hence x = hx. Third step. Hypothesis () holds, that is, η(t, s) ≤ s – t for all t, s ∈ ], +∞[. Since () ensures that condition () holds, we conclude that x is a fixed point of h. Finally, let us to verify that x is a unique fixed point of h. Proceeding by contradiction, we suppose that there exists z = x such that z = hz. Let ti := D(z, x; λ) >  for all i ∈ N. Therefore   η(ti+ , ti ) = η D(z, x; λ), D(z, x; λ)   = η D(hz, hx; λ), D(z, x; λ) > ,

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 6 of 12

for all i ∈ N. Then by the property (η ) of the function η, we obtain ti → , which contradicts the fact that d(z, x) = . Therefore z = x and so h has a unique fixed point.  Now, we present some particular results of fixed point in metric spaces, by choosing an appropriate R-function. The first corollary is a generalization of Geraghty’s fixed point theorem [] and it is obtained by taking in Theorem . as R-function η(t, s) = ψ(s)s – t for all t, s ∈ [, +∞[, where ψ is endowed with a suitable property. Corollary . Let (Z, d) be a complete metric space and h : Z → Z be a mapping. Suppose that there exists a function λ ∈  such that   D(hu, hv; λ) ≤ ψ D(u, v; λ) D(u, v; λ) for all u, v ∈ Z with D(u, v; λ) > , where ψ : [, +∞[ → [, [ is a function such that limi→+∞ ψ(ti ) =  implies limi→+∞ ti = , for all {ti } ⊂ [, +∞[. Then h has a unique fixed point x ∈ Z such that λ(x) =  and, for any choice of the initial point z ∈ Z, the sequence {zi } defined by zi = hzi– for each i ∈ N converges to the point x. Remark . From Corollary ., we obtain Geraghty fixed point theorem [], if the function λ ∈  is defined by λ(u) =  for all u ∈ Z. Clearly, the Geraghty result is a generalization of Banach’s contraction principle. In the following corollary we give a result inspired by well-known results in [, , ]. It is obtained by taking in Theorem . as R-function η(t, s) = ψ(s)s – t for all t, s ∈ [, +∞[, where ψ is endowed with a suitable property. Corollary . Let (Z, d) be a complete metric space and h : Z → Z be a mapping. Suppose that there exists a function λ ∈  such that   D(hu, hv; λ) ≤ ψ D(u, v; λ) D(u, v; λ) for all u, v ∈ Z with D(u, v; λ) > , where ψ : [, +∞[ → [, [ is a function such that lim supt→r+ ψ(t) < , for all r > . Then h has a unique fixed point x ∈ Z such that λ(x) =  and, for any choice of the initial point z ∈ Z, the sequence {zi } defined by zi = hzi– for each i ∈ N converges to the point x. If in Theorem . we consider as R-function η(t, s) = s – ψ(t) for all t, s ∈ [, +∞[, where ψ is a right continuous function, then we deduce the following corollary. Corollary . Let (Z, d) be a complete metric space and h : Z → Z be a mapping. Suppose that there exists a function λ ∈  such that   ψ D(hu, hv; λ) ≤ D(u, v; λ) for all u, v ∈ Z with D(u, v; λ) > , where ψ : [, +∞[ → [, [ is a right continuous function such that ψ(t) > t, for all t > . Then h has a unique fixed point x ∈ Z such that λ(x) =  and, for any choice of the initial point z ∈ Z, the sequence {zi } defined by zi = hzi– for each i ∈ N converges to the point x. From the previous corollary, we deduce the following result of integral type.

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 7 of 12

Corollary . Let (Z, d) be a complete metric space and h : Z → Z be a mapping. Suppose that there exists a function λ ∈  such that 

D(hu,hv;λ)

ξ (τ ) dτ ≤ D(u, v; λ) for all u, v ∈ Z with D(u, v; λ) > ,

()



t t where ξ : [, +∞[ → [, +∞[ is a function such that  ξ (τ ) dτ exists and  ξ (τ ) dτ > t, for every t > . Then h has a unique fixed point x ∈ Z such that λ(x) =  and, for any choice of the initial point z ∈ Z, the sequence {zi } defined by zi = hzi– for each i ∈ N converges to the point x. ] ∪ {} endowed with the usual metric d(u, v) = |u – v| for all Example . Let Z = [,   u, v ∈ Z. Obviously, (Z, d) is a complete metric space. Consider the function h : Z → Z defined by  hu =

if u ∈ [,  ],  if u = .

u   

Clearly, h satisfies condition () with respect to the function ξ : [, +∞[ → [, +∞[ given by ξ (t) =  +

 (t + )

for all t ∈ [, +∞[

and the lower semi-continuous function λ : Z → [, +∞[ defined by λ(u) = u for all u ∈ Z. ], then Indeed, if u ≤ v and u, v ∈ [,   

D(hu,hv;λ)

ξ (τ ) dτ = 

v+ v ≤ v = D(u, v; λ). v+

] and v = , or u = v = , then If u ∈ [,   

D(hu,h;λ)

ξ (τ ) dτ = 

+  ≤  = D(u, ; λ). +

Since all the conditions of Corollary . are satisfied, the mapping T has a unique fixed point x =  in Z. Clearly, λ(x) = . From d(h, h) = / and d(, ) = , we deduce that 

d(h,h)

ξ (τ ) dτ = 

 ≥  = d(, ). 

Thus h is not a R-contraction with respect to the R-function η : [, +∞[ ×[, +∞[ → R defined by 

t

ξ (τ ) dτ ,

η(t, s) = s – 

for all t, s ∈ [, +∞[.

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 8 of 12

It follows that Theorem  of [] cannot be used to deduce that h has a fixed point with respect to this R-function. The previous consideration also shows that the role of the function λ is decisive in enlarging the class of self-mappings satisfying condition () and hence condition ().

4 Fixed points in partial metric spaces In this section, from our Theorem ., we deduce easily various fixed point theorems on partial metric spaces including the Matthews fixed point theorem. Theorem . Let (Z, p) be a complete partial metric space and let h : Z → Z be a mapping. Suppose that there exists a R-function η such that   η p(hu, hv), p(u, v) > 

for all u, v ∈ Z, u = v.

()

Assume that, at least, one of the following conditions holds: (j) h is continuous with respect to metric dp ; (jj) for every two sequences {ti }, {si } ⊂ ], +∞[ such that limi→+∞ si =  and η(ti , si ) >  for all i ∈ N, then limi→+∞ ti = ; (jjj) η(t, s) ≤ s – t for all t, s ∈ ], +∞[. Then h has a unique fixed point x ∈ Z such that p(x, x) = . Proof From (), we say that p(u, v) =

dp (u, v) + p(u, u) + p(v, v) 

for all u, v ∈ Z.

()

The hypothesis that (Z, p) is complete, by Lemma ., ensures that the metric space (Z, – dp ) is complete. Also, by Lemma ., the function λ : Z → [, +∞[ defined by λ(u) = – p(u, u) is continuous and so lower semi-continuous in (Z, – dp ). Now, from () and (), we see that for the mapping h the condition   η – dp (hu, hv) + λ(hu) + λ(hv), – dp (u, v) + λ(u) + λ(v) >  holds for all u, v ∈ Z, u = v. Consequently, for the mapping h all the conditions of Theorem . hold with respect to the metric space (Z, – dp ). To conclude that the mapping h has a unique fixed point x ∈ Z such that p(x, x) =  λ(x) = .  From Theorem . if consider as R-function η(t, s) = ks – t for all t, s ∈ [, +∞[ with k ∈ [, [, we obtain the Matthews fixed point theorem. Corollary . Let (Z, p) be a complete partial metric space and let h : Z → Z be a mapping. Suppose that there exists k ∈ [, [ such that p(hu, hv) ≤ k p(u, v) for all u, v ∈ Z, u = v. Then h has a unique fixed point x ∈ Z such that p(x, x) = .

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 9 of 12

From Theorem . if consider as R-function η(t, s) = ψ(s) s – t for all t, s ∈ [, +∞[, then we obtain a result of Geraghty type in partial metric spaces. Corollary . Let (Z, p) be a complete partial metric space and let h : Z → Z be a mapping. Suppose that there exists a function ψ : [, +∞[ → [, [ such that limj→+∞ ψ(tj ) =  implies limj→+∞ tj = , for all {tj } ⊂ [, +∞[. If   p(hu, hv) ≤ ψ p(u, v) p(u, v)

for all u, v ∈ Z,

then h has a unique fixed point such that p(x, x) =  and, for any choice of the initial point z ∈ Z, the sequence {zj } defined by zj = hzj– for each j ∈ N converges to the point x. Other known results of fixed point in the setting of partial metric spaces we can get considering suitable simulation functions.

5 An application to homotopy In this section, we give an application of our results to homotopy theory. Denote by  the family of nondecreasing upper semi-continuous functions ρ : [, +∞[ → [, +∞[ such that ρ(s) < s for all s >  and with the following property:   lim si,j – ρ(si,j ) = 

i,j→+∞

implies

lim si,j = 

i,j→+∞

()

for every sequence {si,j } ⊂ [, +∞[. Theorem . Let (Z, d) be a complete metric space, let U an open subset of Z and V a closed subset of Z with U ⊂ V . Assume that the operator Q : V × [, ] → Z satisfies the following conditions: (i) u = Q(u, s) for each u ∈ V \ U and all s ∈ [, ]; (ii) there exists ρ ∈  such that, for each s ∈ [, ] and all u, v ∈ V , we have     d Q(u, s), Q(v, s) ≤ ρ d(u, v) ;

()

(iii) there exists a continuous function f : [, ] → R such that   d Q(u, t), Q(u, s) ≤ f (t) – f (s) for all t, s ∈ [, ] and every u ∈ V . Then Q(·, ) has a fixed point if Q(·, ) has a fixed point. Proof Assume that Q(·, ) has a fixed point and consider the following set:

 A := s ∈ [, ] : u = Q(u, s) for some u ∈ U . By (i) we get  ∈ A and this implies that A is a non-empty set. We claim that A is both open and closed in [, ]. Since [, ] is connected, we deduce that A = [, ].

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 10 of 12

First, we prove that A is a closed subset of [, ]. Let {sj } be a sequence in A and assume that sj → s ∈ [, ] as j → +∞. Now, we establish that s ∈ A. The definition of the set A ensures that there exists uj ∈ U such that uj = Q(uj , sj ) for all j ∈ N. Then we get   d(ui , uj ) = d Q(ui , si ), Q(uj , sj )     ≤ d Q(ui , si ), Q(ui , sj ) + d Q(ui , sj ), Q(uj , sj )   ≤ f (si ) – f (sj ) + ρ d(ui , uj ) . Thus   d(ui , uj ) – ρ d(ui , uj ) ≤ f (si ) – f (sj ) for all i, j ∈ N. Letting i, j → +∞ in the previous inequality, we get    lim d(ui , uj ) – λ d(ui , uj ) = ,

i,j→+∞

and by (), we get d(ui , uj ) →  as i, j → +∞. This relation ensures that {uj } is a Cauchy sequence. Since Z is a complete metric space, there exists some x ∈ V such that uj → u. From     d uj , Q(x, s ) = d Q(uj , sj ), Q(x, s )     ≤ d Q(uj , sj ), Q(uj , s ) + d Q(uj , s ), Q(x, s )   ≤ f (sj ) – f (s ) + ρ d(uj , x) ≤ f (sj ) – f (s ) + d(uj , x), letting j → +∞, we obtain     d x, Q(x, s ) = lim d uj , Q(x, s ) = . j→+∞

This implies that x = Q(x, s ) and by (i), we deduce that x ∈ U. Consequently, s ∈ A and hence A is a closed subset of [, ]. Now, we prove that A is an open subset of [, ]. If s ∈ A, then there exists u ∈ U such that u = Q(u , s ). Since U is open in Z, there exists σ >  such that B(u , σ ) = {u ∈ Z : d(u , v) ≤ σ } ⊂ U. Because f is continuous at s , in correspondence to δ = σ – ρ(σ ) >  there exists ε = ε(δ) >  such that |f (s) – f (s )| < δ for all s ∈ ]s – ε, s + ε[. Let s ∈ ]s – ε, s + ε[ and u ∈ B(u , σ ), we have     d Q(u, s), u = d Q(u, s), Q(u , s )     ≤ d Q(u, s), Q(u, s ) + d Q(u, s ), Q(u , s )   ≤ f (s) – f (s ) + λ d(u, u )   ≤ σ – ρ(σ ) + ρ d(u, u ) ≤ σ .

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 11 of 12

Therefore, Q(·, s) is a self-mapping on B(u , σ ) for every fixed s ∈ ]s – ε, s + ε[. From (), we deduce that Q(·, s) is an R-ξ -contraction on B(u , σ ) with respect to the R-function η : [, +∞[ ×[, +∞[ → R defined by η(t, s) = ρ(s) – t for all s, t ≥  and the function ξ ∈  defined by ξ (u) =  for all u ∈ B(u , σ ). This ensures that Q(·, s) has a fixed point in B(u , σ ) and hence in U, since all hypotheses of Theorem . are satisfied. So ]s – ε, s + ε[⊂ A and thus we see that A is an open subset of [, ]. 

6 Conclusions Fixed point theory in various metric settings is largely studied as a useful tool for solving problems arising in mathematics and the other sciences. Here, we proved existence and uniqueness of fixed point by using the notion of an R-function in metric and partial metric spaces. This kind of result is helpful to cover existing theorems in the literature from a unifying point of view. An homotopy result for certain operators supports the new theory. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Author details 1 Department of Mathematics and Computer Sciences, University of Palermo, Via Archirafi 34, Palermo, 90123, Italy. 2 Nonlinear Analysis Research Group and Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minn City, Vietnam. Received: 9 February 2016 Accepted: 20 July 2016 References 1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133-181 (1922) ´ c, L, Samet, B, Vetro, C, Abbas, M: Fixed point results for weak contractive mappings in ordered K-metric spaces. 2. Ciri´ Fixed Point Theory 13, 59-72 (2012) 3. Cosentino, M, Salimi, P, Vetro, P: Fixed point results on metric-type spaces. Acta Math. Sci. 34, 1237-1253 (2014) 4. Reich, S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 5, 26-42 (1972) 5. Reich, S, Zaslavski, AJ: A fixed point theorem for Matkowski contractions. Fixed Point Theory 8, 303-307 (2007) 6. Reich, S, Zaslavski, AJ: A note on Rakotch contractions. Fixed Point Theory 9, 267-273 (2008) 7. Vetro, C, Vetro, F: Common fixed points of mappings satisfying implicit relations in partial metric spaces. J. Nonlinear Sci. Appl. 6, 152-161 (2013) 8. Matthews, SG: Partial metric topology. In: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci., vol. 728, pp. 183-197 (1994) 9. Bukatin, MA, Scott, JS: Towards computing distances between programs via Scott domains. In: Adian, S, Nerode, A (eds.) Logical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 1234, pp. 33-43. Springer, Berlin (1997) 10. Bukatin, MA, Shorina, SY: Partial metrics and co-continuous valuations. In: Nivat, M (ed.) Foundations of Software Science and Computation Structures. Lecture Notes in Computer Science, vol. 1378, pp. 33-43. Springer, Berlin (1998) 11. O’Neill, SJ: Partial metrics, valuations and domain theory. In: Proc. 11th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci., vol. 806, pp. 304-315 (1996) 12. Romaguera, S, Schellekens, M: Partial metric monoids and semivaluation spaces. Topol. Appl. 153, 948-962 (2005) 13. Khojasteh, F, Shukla, S, Radenovi´c, S: A new approach to the study of fixed point theorems via simulation functions. Filomat 29, 1189-1194 (2015) 14. Roldán López de Hierro, AF, Shahzad, N: New fixed point theorem under R-contractions. Fixed Point Theory Appl. 2015, 98 (2015) 15. Argoubi, H, Samet, B, Vetro, C: Nonlinear contractions involving simulation functions in a metric space with a partial order. J. Nonlinear Sci. Appl. 8, 1082-1094 (2015) 16. Nastasi, A, Vetro, P: Fixed point results on metric and partial metric spaces via simulation functions. J. Nonlinear Sci. Appl. 8, 1059-1069 (2015) 17. Roldán, A, Karapinar, E, Roldán, C, Martínez-Moreno, J: Coincidence point theorems on metric spaces via simulation functions. J. Comput. Appl. Math. 275, 345-355 (2015) 18. Samet, B, Vetro, C, Vetro, F: From metric spaces to partial metric spaces. Fixed Point Theory Appl. 2013, Article ID 5 (2013) 19. Vetro, C, Vetro, F: Metric or partial metric spaces endowed with a finite number of graphs: a tool to obtain fixed point results. Topol. Appl. 164, 125-137 (2014)

Nastasi et al. Fixed Point Theory and Applications (2016) 2016:81

Page 12 of 12

20. Oltra, S, Valero, O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste 36, 17-26 (2004) 21. Paesano, D, Vetro, P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl. 159, 911-920 (2012) 22. Geraghty, M: On contractive mappings. Proc. Am. Math. Soc. 40, 604-608 (1973) 23. Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458-464 (1969) 24. Mizoguchi, N, Takahashi, W: Fixed point theorems for multi-valued mappings on complete metric spaces. J. Math. Anal. Appl. 141, 177-188 (1989)