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Cirictype δcontractions in metric spaces endowed with a graph
Journal of Inequalities and Applications volume 2014, Article number: 77 (2014)
Abstract
The purpose of this paper is to present some fixedpoint and strict fixedpoint results in a metric space endowed with a graph, using a contractive condition of Ćirić type with respect to the functional δ. The data dependence of the fixedpoint set, the wellposedness of the fixedpoint problem, and the limit shadowing property are also studied.
MSC:47H10, 54H25.
1 Preliminaries
A recent research direction in fixedpoint theory is the study of the fixedpoint problem for singlevalued and multivalued operators in the context of a metric space endowed with a graph. This approach was recently considered by Jachymski in [1], GwóźdźLukawska and Jachymski in [2], and then it was developed in many other papers ([3–5], etc.).
On the other hand, fixed points and strict fixed points (also called endpoints) are important elements in mathematical economics and game theory. It represents optimal preferences in some ArrowDebreu type models or Nash type equilibrium points for some abstract noncooperative games, see, for example, [6] and [7]. From this perspective, it is important to give fixed and strict fixedpoint theorems for multivalued operators.
We shall begin by presenting some notions and notations that will be used throughout the paper.
Let (X,d) be a metric space and Δ be the diagonal of X\times X. Let G be a directed graph such that the set V(G) of its vertices coincides with X and \mathrm{\Delta}\subseteq E(G), E(G) being the set of the edges of the graph. Assuming that G has no parallel edges, we will suppose that G can be identified with the pair (V(G),E(G)).
If x and y are vertices of G, then a path in G from x to y of length k\in \mathbb{N} is a finite sequence {({x}_{n})}_{n\in \{0,1,2,\dots ,k\}} of vertices such that {x}_{0}=x, {x}_{k}=y and ({x}_{i1},{x}_{i})\in E(G), for i\in \{1,2,\dots ,k\}.
Let us denote by \tilde{G} the undirected graph obtained from G by ignoring the direction of edges. Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if \tilde{G} is connected.
Let us consider the following families of subsets of a metric space (X,d):
Let us define the gap functional between the sets A and B in the metric space (X,d) as:
and the (generalized) PompeiuHausdorff functional as
The generalized diameter functional is denoted by \delta :P(X)\times P(X)\to {\mathbb{R}}_{+}\cup \{\mathrm{\infty}\}, and defined by
Denote by diam(A):=\delta (A,A) the diameter of the set A.
Let T:X\to P(X) be a multivalued operator and Graphic(T):=\{(x,y)y\in T(x)\} the graphic of T. x\in X is called fixed point for T if and only if x\in T(x), and it is called strict fixed point if and only if \{x\}=T(x).
The set Fix(T):=\{x\in Xx\in T(x)\} is called the fixedpoint set of T, while SFix(T)=\{x\in X\{x\}=T(x)\} is called the strict fixedpoint set of T. Notice that SFix(T)\subseteq Fix(T).
We will write E(G)\in I(T\times T) if and only if for every x,y\in X with (x,y)\in E(G) we have T(x)\times T(y)\subset E(G).
For the particular case of a singlevalued operator t:X\to X the above notations should be considered accordingly. In particular, the condition E(G)\in I(t\times t) means that the operator t is edge preserving (in the sense of the Jachymski’s definition of a Banach contraction), i.e. for each x,y\in X with (x,y)\in E(G) we have (t(x),t(y))\in E(G) (see [1]). We will also denote by O({x}_{0},n):=\{{x}_{0},t({x}_{0}),{t}^{2}({x}_{0}),\dots ,{t}^{n}({x}_{0})\} the orbit of order n of the operator t corresponding to {x}_{0}\in X.
In this paper we prove some fixedpoint and strict fixedpoint theorems for singlevalued and multivalued operators satisfying a contractive condition of Ćirić type with respect to the functional δ. Our results also generalize and extend some fixedpoint theorems in partially ordered complete metric spaces given in Harjani, Sadarangani [8], Nieto, RodríguezLópez [9] and [10], Nieto et al. [11], O’Regan, Petruşel [12], Petruşel, Rus [13] and Ran, Reurings [14]. For other general results concerning Ćirić type fixedpoint theorems see Rus [15].
In the main section of the paper we give results concerning the existence and uniqueness of the fixed point and of the strict fixed point of a Ćirić type (singlevalued and multivalued) contraction. Then the wellposedness of the fixedpoint problem, the data dependence of the fixedpoint set, and the limit shadowing property are also studied. Our results complement and extend some recent theorems given in [16] for multivalued Reichtype operators.
2 Main results
In this section we present the main results of the paper concerning the fixedpoint problem and, respectively, the strict fixedpoint problem for a singlevalued, respectively, of a multivalued Ćirić type contraction.
Definition 2.1 Let (X,d) be a metric space and T:X\to {P}_{cl}(X) be a multivalued operator. By definition, the fixedpoint problem is wellposed for T with respect to H if:

(i)
SFixT=\{{x}^{\ast}\};

(ii)
If {({x}_{n})}_{n\in \mathbb{N}} is a sequence in X such that H({x}_{n},T({x}_{n}))\to 0, as n\to \mathrm{\infty}, then {x}_{n}\stackrel{d}{\to}{x}^{\ast}, as n\to \mathrm{\infty}.
Definition 2.2 Let (X,d) be a metric space and T:X\to P(X) be a multivalued operator. By definition T has the limit shadowing property if for any sequence {({y}_{n})}_{n\in \mathbb{N}} from X such that D({y}_{n+1},T({y}_{n}))\to 0, as n\to \mathrm{\infty}, there exists {({x}_{n})}_{n\in \mathbb{N}}, a sequence of successive approximation of T, such that d({x}_{n},{y}_{n})\to 0, as n\to \mathrm{\infty}.
In order to prove the limit shadowing property we shall need Cauchy’s lemma.
Lemma 2.1 (Cauchy’s lemma)
Let {({a}_{n})}_{n\in \mathbb{N}} and {({b}_{n})}_{n\in \mathbb{N}} be two sequences of nonnegative real numbers, such that {\sum}_{k=0}^{+\mathrm{\infty}}{a}_{k}<+\mathrm{\infty} and {b}_{n}\to 0, as n\to \mathrm{\infty}. Then
The first main result of this paper is the following result for the case of singlevalued operators. The proof of this result is inspired by the proof of Ćirić’s fixedpoint theorem in [17] and the approach introduced for metric spaces endowed with a graph by Jachymski in [1].
Theorem 2.1 Let (X,d) be a complete metric space and G be a directed graph such that the triple (X,d,G) satisfies the following property:
Let t:X\to X be a singlevalued operator. Suppose the following assertions hold:

(i)
there exists a\in [0,1[ such that
d(t(x),t(y))\le a\cdot max\{d(x,y),d(x,t(x)),d(y,t(y)),d(x,t(y)),d(y,t(x))\},for all (x,y)\in E(G);

(ii)
there exists {x}_{0}\in X such that ({x}_{0},t({x}_{0}))\in E(G);

(iii)
E(G)\in I(t\times t);

(iv)
if (x,y)\in E(G) and (y,z)\in E(G), then (x,z)\in E(G).
In these conditions we have:

(a)
(existence) Fix(t)\ne \mathrm{\varnothing};

(b)
(uniqueness) If, in addition, the following implication holds
{x}^{\ast},{y}^{\ast}\in Fix(t)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}({x}^{\ast},{y}^{\ast})\in E(G),
then Fix(t)=\{{x}^{\ast}\}.
Proof (a) Let {x}_{0}\in X such that ({x}_{0},t({x}_{0}))\in E(G). By (iii) we obtain ({t}^{i}({x}_{0}),{t}^{i+1}({x}_{0}))\in E(G), for all i\in \mathbb{N}. Then, by (iv), we get
Since \mathrm{\Delta}\subset E(G) we get
For n\in {\mathbb{N}}^{\ast}, let i,j\in \mathbb{N} with 1\le i<j\le n. Then
Notice now that, from the above relation, it follows that there exists k\in {\mathbb{N}}^{\ast} with k\le n such that
In order to show that the sequence {({t}^{n}({x}_{0}))}_{n\in \mathbb{N}} is Cauchy, let us consider n,m\in \mathbb{N} with n<m. Then we have
By (2.2) there exists p\in \mathbb{N} with p\le mn+1 such that
Then
Thus, we get
Hence
Notice now that
Indeed, if we take m\in \mathbb{N} arbitrary, then by (2.2), there exists r\in \mathbb{N} such that 1\le r\le m and d({x}_{0},{t}^{r}({x}_{0}))=diam(O({x}_{0};m)). Then, using (2.1), we get
Hence we have
proving (2.3). Now, we can conclude that
Thus, (2.4) shows that the sequence {({t}^{n}({x}_{0}))}_{n\in \mathbb{N}} is Cauchy and, as a consequence of the completeness of the space, it converges to an element {x}^{\ast}\in X.
We will show now that {x}^{\ast}\in Fix(t).
Notice first that, by the property (P), there exists a subsequence {({x}_{{k}_{n}})}_{n\in \mathbb{N}} of {({x}_{n})}_{n\in \mathbb{N}} such that ({x}_{{k}_{n}},{x}^{\ast})\in E(G) for each n\in \mathbb{N}. Now, we can write
Hence
Letting n\to +\mathrm{\infty} we get {x}^{\ast}\in Fix(t).
(b) Suppose that there exist {x}^{\ast},{y}^{\ast}\in Fix(t) with {x}^{\ast}\ne {y}^{\ast}.
Using (i) and the additional hypothesis of (b), we obtain
which is a contradiction. □
Remark 2.1 Notice that, from the proof of the above theorem, it follows that the sequence {({t}^{n}({x}_{0}))}_{n\in \mathbb{N}} converges to {x}^{\ast} in (X,d).
Remark 2.2 If in the above theorem, instead of property (P) we suppose that t has closed graphic, then we can reach the same conclusion. Moreover if we suppose that
then we get again Fix(t)=\{{x}^{\ast}\}.
Based on the above theorem, we can prove now our second main result.
Theorem 2.2 Let (X,d) be a complete metric space and G be a directed graph such that the triple (X,d,G) satisfies the following property:
Let T:X\to {P}_{b}(X) be a multivalued operator. Suppose the following assertions hold:

(i)
there exists a\in [0,1[ such that
\delta (T(x),T(y))\le a\cdot max\{d(x,y),\delta (x,T(x)),\delta (y,T(y)),D(x,T(y)),D(y,T(x))\},for all (x,y)\in E(G);

(ii)
there exists {x}_{0}\in X such that, for all y\in T({x}_{0}), we have ({x}_{0},y)\in E(G);

(iii)
E(G)\in I(T\times T);

(iv)
if (x,y)\in E(G) and (y,z)\in E(G), then (x,z)\in E(G).
In these conditions we have:

(a)
(existence) Fix(T)=SFix(T)\ne \mathrm{\varnothing};

(b)
(uniqueness) If, in addition, the following implication holds:
{x}^{\ast},{y}^{\ast}\in Fix(T)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}({x}^{\ast},{y}^{\ast})\in E(G),
then Fix(T)=SFix(T)=\{{x}^{\ast}\};

(c)
(wellposedness of the fixedpoint problem) If T has closed graphic and for any sequence {({x}_{n})}_{n\in \mathbb{N}}, {x}_{n}\in X with H({x}_{n},T({x}_{n}))\to 0, as n\to \mathrm{\infty}, we have ({x}_{n},{x}^{\ast})\in E(G), then the fixedpoint problem is wellposed for T with respect to H;

(d)
(limit shadowing property of T) If a<\frac{1}{3} and {({y}_{n})}_{n\in \mathbb{N}} is a sequence in X such that the following implication holds:
D({y}_{n+1},T({y}_{n}))\to 0,\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.25em}{0ex}}n\to \mathrm{\infty}\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}({y}_{n},{x}^{\ast})\in E(G),\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\in \mathbb{N},
then T has the limit shadowing property.
Proof (a) Let 0<q<1 be an arbitrary real number. Notice first that, for any x\in X, there exists u\in T(x) such that {a}^{q}\delta (x,T(x))\le d(x,u). In this way, we get an operator t:X\to X which assigns to each x\in X the element t(x)\in T(x) with
Then, for (x,y)\in E(G), we have
Thus, the operator t satisfies all the hypotheses of Theorem 2.1 and, as a consequence, it has a fixed point {x}^{\ast}\in X. Then {x}^{\ast}\in Fix(T). If we suppose now that there exists x\in Fix(T)\setminus SFix(T), then, since (x,x)\in \mathrm{\Delta}, from the condition (i) (with y=x), we get \delta (T(x))\le a\delta (T(x)), which implies (since a<1) that \delta (T(x))=0. Thus T(x)=\{x\}. This is a contradiction with x\in Fix(T)\setminus SFix(T), proving that Fix(T)=SFix(T)\ne \mathrm{\varnothing}.
(b) Suppose that there exist {x}^{\ast},{y}^{\ast}\in Fix(T)=SFix(T).
Using (i) we obtain
which implies that d({x}^{\ast},{y}^{\ast})=0. Hence Fix(T)=SFix(T)=\{{x}^{\ast}\}.
(c) Let {({x}_{n})}_{n\in \mathbb{N}}, {x}_{n}\in X, be a sequence with the property that H({x}_{n},T({x}_{n}))\to 0, as n\to \mathrm{\infty}. It is obvious that H({x}_{n},T({x}_{n}))=\delta ({x}_{n},T({x}_{n})).
Let {x}^{\ast}\in Fix(T)=SFix(T). Then
Hence
and, as a consequence, the fixedpoint problem is wellposed for T with respect to H.
(d) Let {({y}_{n})}_{n\in \mathbb{N}} be a sequence in X such that D({y}_{n+1},T({y}_{n}))\to 0, as n\to \mathrm{\infty} and let {({x}_{n})}_{n\in \mathbb{N}} be a sequence of successive approximation of T starting from {x}_{0}\in X, constructed as in the proof of Theorem 2.1. We shall prove that d({x}_{n},{y}_{n})\to 0, as n\to \mathrm{\infty}.
Let {x}^{\ast}\in Fix(T)=SFix(T). Then
In what follows we shall prove that d({x}^{\ast},{y}_{n})\to 0, as n\to \mathrm{\infty}. Then we have
Now, we can write
Thus
If we replace this in the above relation, we get
Hence
Continuing this process we shall obtain
If we consider {a}_{n}={(\frac{2a}{1a})}^{n} and {b}_{n}=D({y}_{n+1},T({y}_{n})) and using the fact that a<\frac{1}{3}, we obtain, via Cauchy’s lemma, d({x}^{\ast},{y}_{n})\to 0, as n\to \mathrm{\infty}. Thus d({x}^{\ast},{y}_{n})\to 0, as n\to \mathrm{\infty}, and hence, the operator T has the limit shadowing property. □
Remark 2.3 If in Theorem 2.1, instead of property (P), we suppose that every selection t of T has closed graphic, then we obtain Fix(T)=SFix(T)\ne \mathrm{\varnothing}. Moreover if we suppose that the following implication holds:
then Fix(T)=SFix(T)=\{{x}^{\ast}\}.
Proof Since every selection t of T has closed graphic, the conclusion follows by Theorem 2.1, via Remark 2.2. □
Definition 2.3 Let (X,d) be a metric space and T:X\to {P}_{cl}(X) be a multivalued operator. By definition the fixedpoint problem is wellposed in the generalized sense for T with respect to H if

(i)
SFixT\ne \mathrm{\varnothing};

(ii)
If {({x}_{n})}_{n\in \mathbb{N}} is a sequence in X such that H({x}_{n},T({x}_{n}))\to 0, as n\to \mathrm{\infty}, then there exists a subsequence {({x}_{{k}_{n}})}_{n\in \mathbb{N}} of {({x}_{n})}_{n\in \mathbb{N}} such that {x}_{{k}_{n}}\stackrel{d}{\to}{x}^{\ast}, as n\to \mathrm{\infty}.
Remark 2.4 If in Theorem 2.1 instead of (c) we suppose the following assumption, (c′):
(c′) If every selection t of T has closed graphic and, in addition, we suppose that for any sequence {({x}_{n})}_{n\in \mathbb{N}}\subset X with H({x}_{n},T({x}_{n}))\to 0, as n\to \mathrm{\infty}, there exists a subsequence {({x}_{{k}_{n}})}_{n\in \mathbb{N}} such that ({x}_{{k}_{n}},{x}^{\ast})\in E(G) and H({x}_{{k}_{n}},T({x}_{{k}_{n}}))\to 0, then the fixedpoint problem is wellposed in the generalized sense for T with respect to H.
In what follows we shall present some examples of operators satisfying the hypotheses of our main results.
Example 2.1 Let X=\{0,1\}\cup \{\frac{1}{{2}^{k}}:k\in {\mathbb{N}}^{\ast}\} and f:X\to X given by
Let V(G):=X and E(G):=\{(0,1),(1,0)\}\cup \{(0,\frac{1}{{2}^{k}}),(\frac{1}{{2}^{k}},0),(1,\frac{1}{{2}^{k}}),(\frac{1}{{2}^{k}},1):k\in {\mathbb{N}}^{\ast}\}\cup \mathrm{\Delta}.
Then all the hypotheses of Theorem 2.1 are satisfied, Fix(t)=\{0\} and, if we take {x}_{0}=\frac{1}{2}, then {({t}^{n}({x}_{0}))}_{n\in {\mathbb{N}}^{\ast}} converges to {x}^{\ast}=0.
Example 2.2 Let X=\{(0,0),(0,1),(1,0)\} and T:X\to {P}_{b}(X) given by
Let E(G):=\{((0,1);(1,0)),((1,0);(0,0))\}\cup \mathrm{\Delta}.
Notice that Fix(T)=SFix(T)=\{(0,0)\} and all the hypotheses in Theorem 2.2 are satisfied (the condition (i) is verified for a\ge 0,71).
Remark 2.5 It is also important to notice that, if we suppose that there exists x\in Fix(T)\setminus SFix(T), then, since (x,x)\in \mathrm{\Delta}, from the condition (i) in the above theorem (with y=x), we get \delta (T(x))\le a\delta (T(x)), which implies \delta (T(x))=0 (since a<1). This is a contradiction with x\in Fix(T)\setminus SFix(T), showing that we cannot get fixed points which are not strict fixed points in the presence of the condition (i) of the above theorem. It is an open problem to prove a similar theorem to the above one for a more general class of multivalued operators T.
The next result presents the data dependence of the fixedpoint set of a multivalued operator which satisfies a contractive condition of Ćirić type.
Theorem 2.3 Let (X,d) be a complete metric space and G be a directed graph such that the triple (X,d,G) satisfies property (P). Let {T}_{1},{T}_{2}:X\to {P}_{b}(X) be two multivalued operators. Suppose the following assertions hold:

(i)
for i\in \{1,2\}, there exist {a}_{i}\in [0,1) such that
\delta ({T}_{i}(x),{T}_{i}(y))\le a\cdot max\{d(x,y),\delta (x,{T}_{i}(x)),\delta (y,{T}_{i}(y)),D(x,{T}_{i}(y)),D(y,{T}_{i}(x))\},for all (x,y)\in E(G);

(ii)
for each x\in X and each y\in T(x) we have (x,y)\in E(G), for i\in \{1,2\};

(iii)
E(G)\in I(T\times T);

(iv)
if (x,y)\in E(G) and (y,z)\in E(G), then (x,z)\in E(G);

(v)
there exists \eta >0 such that H({T}_{1}(x),{T}_{2}(x))\le \eta for all x\in X.
Under these conditions we have:

(a)
Fix({T}_{i})=SFix({T}_{i})\ne \mathrm{\varnothing}, i\in \{1,2\};

(b)
If, in addition, for i\in \{1,2\}, the following implication holds:
{x}_{i}^{\ast},{y}_{i}^{\ast}\in Fix({T}_{i})\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}({x}_{i}^{\ast},{y}_{i}^{\ast})\in E(G),then Fix({T}_{i})=SFix({T}_{i})=\{{x}_{i}^{\ast}\}, for each i\in \{1,2\};

(c)
H(SFix({T}_{1}),SFix({T}_{2}))\le \frac{\eta}{1max\{{a}_{1},{a}_{2}\}}.
Proof Conclusions (a) and (b) are immediate if we apply Theorem 2.2.
For (c) notice that we can prove that for every {x}_{1}^{\ast}\in SFix({T}_{1}), there exists {x}_{2}^{\ast}\in SFix({T}_{2}), such that
A second relation of this type will be obtained by interchanging the role of {T}_{1} and {T}_{2}. Hence, the conclusion follows by the properties of the functional H. □
References
Jachymski J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 2008, 136: 1359–1373.
GwóźdźLukawska G, Jachymski J: IFS on a metric space with a graph structure and extensions of the KeliskyRivlin theorem. J. Math. Anal. Appl. 2009, 356: 453–463. 10.1016/j.jmaa.2009.03.023
Nicolae A, O’Regan D, Petruşel A: Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph. Georgian Math. J. 2011, 18: 307–327.
Beg I, Butt AR, Radojevic S: The contraction principles for setvalued mappings on a metric space with a graph. Comput. Math. Appl. 2010, 60: 1214–1219. 10.1016/j.camwa.2010.06.003
Dinevari T, Frigon M: Fixed point results for multivalued contractions on a metric space with a graph. J. Math. Anal. Appl. 2013, 405: 507–517. 10.1016/j.jmaa.2013.04.014
Border K: Fixed Point Theorems with Applications to Economic and Game Theory. Cambridge University Press, London; 1985.
Yuan GXZ: KKM Theory and Applications in Nonlinear Analysis. Dekker, New York; 1999.
Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240
Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s1108300590185
Nieto JJ, RodríguezLópez R: Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23: 2205–2212. 10.1007/s1011400507690
Nieto JJ, Pouso RL, RodríguezLópez R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc. 2007, 135: 2505–2517. 10.1090/S0002993907087291
O’Regan D, Petruşel A: Fixed point theorems in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026
Petruşel A, Rus IA: Fixed point theorems in ordered L spaces. Proc. Am. Math. Soc. 2006, 134: 411–418.
Ran ACM, Reurings MC: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204
Rus IA: Generalized Contractions and Applications. Transilvania Press, ClujNapoca; 2001.
Chifu C, Petruşel G, Bota M: Fixed points and strict fixed points for multivalued contractions of Reich type on metric spaces endowed with a graph. Fixed Point Theory Appl. 2013., 2013: Article ID 203 10.1186/168718122013203
Ćirić LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45: 267–273.
Petruşel A, Rus IA: Wellposedness of the fixed point problem for multivalued operators. In Applied Analysis and Differential Equations. Edited by: Carja O, Vrabie II. World Scientific, Hackensack; 2007:295–306.
Petruşel A, Rus IA, Yao JC: Wellposedness in the generalized sense of the fixed point problems for multivalued operators. Taiwan. J. Math. 2007, 11: 903–914.
Acknowledgements
The work of the second author is supported by financial support of a grant of the Romanian National Authority for Scientific Research, CNCSUEFISCDI, project number PNIIIDPCE201130094.
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Chifu, C., Petruşel, A. Cirictype δcontractions in metric spaces endowed with a graph. J Inequal Appl 2014, 77 (2014). https://doi.org/10.1186/1029242X201477
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DOI: https://doi.org/10.1186/1029242X201477
Keywords
 fixed point
 strict fixed point
 metric space
 connected graph
 wellposed problem
 limit shadowing property