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Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph

Abstract

In this paper, we initiate a study of fixed point results in the setup of partial metric spaces endowed with a graph. The concept of a power graphic contraction pair of two mappings is introduced. Common fixed point results for such maps without appealing to any form of commutativity conditions defined on a partial metric space endowed with a directed graph are obtained. These results unify, generalize and complement various known comparable results from the current literature.

MSC:47H10, 54H25, 54E50.

1 Introduction and preliminaries

Consistent with Jachymski [1], let X be a nonempty set and d be a metric on X. A set {(x,x):xX} is called a diagonal of X×X and is denoted by Δ. Let G be a directed graph such that the set V(G) of its vertices coincides with X and E(G) is the set of the edges of the graph with ΔE(G). Also assume that the graph G has no parallel edges. One can identify a graph G with the pair (V(G),E(G)). Throughout this paper, the letters , R + , ω and will denote the set of real numbers, the set of nonnegative real numbers, the set of nonnegative integers and the set of positive integers, respectively.

Definition 1.1 [1]

A mapping f:XX is called a Banach G-contraction or simply G-contraction if

(a1) for each x,yX with (x,y)E(G), we have (f(x),f(y))E(G),

(a2) there exists α(0,1) such that for all x,yX with (x,y)E(G) implies that d(f(x),f(y))αd(x,y).

Let X f :={xX:(x,f(x))E(G) or (f(x),x)E(G)}.

Recall that if f:XX, then a set {xX:x=f(x)} of all fixed points of f is denoted by F(f). A self-mapping f on X is said to be

  1. (1)

    a Picard operator if F(f)={ x } and f n (x) x as n for all xX;

  2. (2)

    a weakly Picard operator if F(f) and for each xX, we have f n (x) x F(f) as n;

  3. (3)

    orbitally continuous if for all x,aX, we have

    lim k f n k (x)=aimplies lim i f ( f n k ( x ) ) =f(a).

The following definition is due to Chifu and Petrusel [2].

Definition 1.2 An operator f:XX is called a Banach G-graphic contraction if

(b1) for each x,yX with (x,y)E(G), we have (f(x),f(y))E(G),

(b2) there exists α[0,1) such that

d ( f ( x ) , f 2 ( x ) ) αd ( x , f ( x ) ) for all x X f .

If x and y are vertices of G, then a path in G from x to y of length kN is a finite sequence { x n }, n{0,1,2,,k} of vertices such that x 0 =x, x k =y and ( x i 1 , x i )E(G) for i{1,2,,k}.

Notice that a graph G is connected if there is a path between any two vertices and it is weakly connected if G ˜ is connected, where G ˜ denotes the undirected graph obtained from G by ignoring the direction of edges. Denote by G 1 the graph obtained from G by reversing the direction of edges. Thus,

E ( G 1 ) = { ( x , y ) X × X : ( y , x ) E ( G ) } .

Since it is more convenient to treat G ˜ as a directed graph for which the set of its edges is symmetric, under this convention, we have that

E( G ˜ )=E(G)E ( G 1 ) .

If G is such that E(G) is symmetric, then for xV(G), the symbol [ x ] G denotes the equivalence class of the relation R defined on V(G) by the rule:

yRz if there is a path in G from y to z.

A graph G is said to satisfy the property (A) (see also [2]) if for any sequence { x n } in V(G) with x n x as n and ( x n , x n + 1 )E(G) for nN implies that ( x n ,x)E(G).

Jachymski [1] obtained the following fixed point result for a mapping satisfying the Banach G-contraction condition in metric spaces endowed with a graph.

Theorem 1.3 [1]

Let (X,d) be a complete metric space and G be a directed graph and let the triple (X,d,G) have a property (A). Let f:XX be a G-contraction. Then the following statements hold:

  1. 1.

    F f if and only if X f ;

  2. 2.

    if X f and G is weakly connected, then f is a Picard operator;

  3. 3.

    for any x X f we have that f | [ x ] G ˜ is a Picard operator;

  4. 4.

    if fE(G), then f is a weakly Picard operator.

Gwodzdz-Lukawska and Jachymski [3] developed the Hutchinson-Barnsley theory for finite families of mappings on a metric space endowed with a directed graph. Bojor [4] obtained a fixed point of a φ-contraction in metric spaces endowed with a graph (see also [5]). For more results in this direction, we refer to [2, 6, 7].

On the other hand, Mathews [8] introduced the concept of a partial metric to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle more suitable in this context. For examples, related definitions and work carried out in this direction, we refer to [919] and the references mentioned therein. Abbas et al. [20] proved some common fixed points in partially ordered metric spaces (see also [21]). Gu and He [22] proved some common fixed point results for self-maps with twice power type Φ-contractive condition. Recently, Gu and Zhang [23] obtained some common fixed point theorems for six self-mappings with twice power type contraction condition.

Throughout this paper, we assume that a nonempty set X=V(G) is equipped with a partial metric p, a directed graph G has no parallel edge and G is a weighted graph in the sense that each vertex x is assigned the weight p(x,x) and each edge (x,y) is assigned the weight p(x,y). As p is a partial metric on X, the weight assigned to each vertex x need not be zero and whenever a zero weight is assigned to some edge (x,y), it reduces to a loop (x,x).

Also, the subset W(G) of V(G) is said to be complete if for every x,yW(G), we have (x,y)E(G).

Definition 1.4 Self-mappings f and g on X are said to form a power graphic contraction pair if

  1. (a)

    for every vertex v in G, (v,fv) and (v,gv)E(G),

  2. (b)

    there exists ϕ: R + R + an upper semi-continuous and nondecreasing function with ϕ(t)<t for each t>0 such that

    p δ (fx,gy)ϕ ( p α ( x , y ) p β ( x , f x ) p γ ( y , g y ) )
    (1.1)

for all (x,y)E(G) holds, where α,β,γ0 with δ=α+β+γ(0,).

If we take f=g, then the mapping f is called a power graphic contraction.

The aim of this paper is to investigate the existence of common fixed points of a power graphic contraction pair in the framework of complete partial metric spaces endowed with a graph. Our results extend and strengthen various known results [8, 12, 13, 24].

2 Common fixed point results

We start with the following result.

Theorem 2.1 Let (X,p) be a complete partial metric space endowed with a directed graph G. If f,g:XX form a power graphic contraction pair, then the following hold:

  1. (i)

    F(f) or F(g) if and only if F(f)F(g).

  2. (ii)

    If uF(f)F(g), then the weight assigned to the vertex u is 0.

  3. (iii)

    F(f)F(g) provided that G satisfies the property (A).

  4. (iv)

    F(f)F(g) is complete if and only if F(f)F(g) is a singleton.

Proof To prove (i), let uF(f). By the given assumption, (u,gu)E(G). Assume that we assign a non-zero weight to the edge (u,gu). As (u,u)E(G) and f and g form a power graphic contraction, we have

p δ ( u , g u ) = p δ ( f u , g u ) ϕ ( p α ( u , u ) p β ( u , f u ) p γ ( u , g u ) ) = ϕ ( p α + β ( u , u ) p γ ( u , g u ) ) ϕ ( p α + β ( u , g u ) p γ ( u , g u ) ) = ϕ ( p δ ( u , g u ) ) < p δ ( u , g u ) ,

a contradiction. Hence, the weight assigned to the edge (u,gu) is zero and so u=gu. Therefore, uF(f)F(g). Similarly, if uF(g), then we have uF(f). The converse is straightforward.

Now, let uF(f)F(g). Assume that the weight assigned to the vertex u is not zero, then from (1.1), we have

p δ ( u , u ) = p δ ( f u , g u ) ϕ ( p α ( u , u ) p β ( u , f u ) p γ ( u , g u ) ) = ϕ ( p α + β + γ ( u , u ) ) = ϕ ( p δ ( u , u ) ) < p δ ( u , u ) ,

a contradiction. Hence, (ii) is proved.

To prove (iii), we will first show that there exists a sequence { x n } in X with f x 2 n = x 2 n + 1 and g x 2 n + 1 = x 2 n + 2 for all nN with ( x n , x n + 1 )E(G), and lim n p( x n , x n + 1 )=0.

Let x 0 be an arbitrary point of X. If f x 0 = x 0 , then the proof is finished, so we assume that f x 0 x 0 . As ( x 0 ,f x 0 )E(G), so ( x 0 , x 1 )E(G). Also, ( x 1 ,g x 1 )E(G) gives ( x 1 , x 2 )E(G). Continuing this way, we define a sequence { x n } in X such that ( x n , x n + 1 )E(G) with f x 2 n = x 2 n + 1 and g x 2 n + 1 = x 2 n + 2 for nN.

We may assume that the weight assigned to each edge ( x 2 n , x 2 n + 1 ) is non-zero for all nN. If not, then x 2 k = x 2 k + 1 for some k, so f x 2 k = x 2 k + 1 = x 2 k , and thus x 2 k F(f). Hence, x 2 k F(f)F(g) by (i). Now, since ( x 2 n , x 2 n + 1 )E(G), so from (1.1), we have

p δ ( x 2 n + 1 , x 2 n + 2 ) = p δ ( f x 2 n , g x 2 n + 1 ) ϕ ( p α ( x 2 n , x 2 n + 1 ) p β ( x 2 n , f x 2 n ) p γ ( x 2 n + 1 , g x 2 n + 1 ) ) = ϕ ( p α ( x 2 n , x 2 n + 1 ) p β ( x 2 n , x 2 n + 1 ) p γ ( x 2 n + 1 , x 2 n + 2 ) ) = ϕ ( p α + β ( x 2 n , x 2 n + 1 ) p γ ( x 2 n + 1 , x 2 n + 2 ) ) < p α + β ( x 2 n , x 2 n + 1 ) p γ ( x 2 n + 1 , x 2 n + 2 ) ,

which implies that

p α + β ( x 2 n + 1 , x 2 n + 2 )< p α + β ( x 2 n , x 2 n + 1 ),

a contradiction if α+β=0. So, take α+β>0, and we have

p( x 2 n + 1 , x 2 n + 2 )<p( x 2 n , x 2 n + 1 )

for all nN. Again from (1.1), we have

p δ ( x 2 n + 2 , x 2 n + 3 ) = p δ ( g x 2 n + 1 , f x 2 n + 2 ) = p δ ( f x 2 n + 2 , g x 2 n + 1 ) ϕ ( p α ( x 2 n + 2 , x 2 n + 1 ) p β ( x 2 n + 2 , f x 2 n + 2 ) p γ ( x 2 n + 1 , g x 2 n + 1 ) ) = ϕ ( p α ( x 2 n + 1 , x 2 n + 2 ) p β ( x 2 n + 2 , x 2 n + 3 ) p γ ( x 2 n + 1 , x 2 n + 2 ) ) = ϕ ( p α + γ ( x 2 n + 1 , x 2 n + 2 ) p β ( x 2 n + 2 , x 2 n + 3 ) ) < p α + γ ( x 2 n + 1 , x 2 n + 2 ) p β ( x 2 n + 2 , x 2 n + 3 ) ,

which implies that

p α + γ ( x 2 n + 2 , x 2 n + 3 )< p α + γ ( x 2 n + 1 , x 2 n + 2 ).

We arrive at a contradiction in case α+γ=0. Therefore, we must take α+γ>0; consequently, we have

p( x 2 n + 2 , x 2 n + 3 )<p( x 2 n + 1 , x 2 n + 2 )

for all nN. Hence,

p δ ( x n , x n + 1 )ϕ ( p δ ( x n 1 , x n ) ) < p δ ( x n 1 , x n )
(2.1)

for all nN. Therefore, the decreasing sequence of positive real numbers { p δ ( x n , x n + 1 )} converges to some c0. If we assume that c>0, then from (2.1) we deduce that

0<c lim sup n ϕ ( p δ ( x n 1 , x n ) ) ϕ(c)<c,

a contradiction. So, c=0, that is, lim n p δ ( x n , x n + 1 )=0 and so we have lim n p( x n , x n + 1 )=0. Also,

p δ ( x n , x n + 1 )ϕ ( p δ ( x n 1 , x n ) ) ϕ n ( p δ ( x 0 , x 1 ) ) .
(2.2)

Now, for m,nN with m>n,

p δ ( x n , x m ) p δ ( x n , x n + 1 ) + p δ ( x n + 1 , x n + 2 ) + + p δ ( x m 1 , x m ) p δ ( x n + 1 , x n + 1 ) p δ ( x n + 2 , x n + 2 ) p δ ( x m 1 , x m 1 ) ϕ n ( p δ ( x 0 , x 1 ) ) + ϕ n + 1 ( p δ ( x 0 , x 1 ) ) + + ϕ m 1 ( p δ ( x 0 , x 1 ) )

implies that p δ ( x n , x m ) converges to 0 as n,m. That is, lim n , m p( x n , x m )=0. Since (X,p) is complete, following similar arguments to those given in Theorem 2.1 of [9], there exists a uX such that lim n , m p( x n , x m )= lim n p( x n ,u)=p(u,u)=0. By the given hypothesis, ( x 2 n ,u)E(G) for all nN. We claim that the weight assigned to the edge (u,gu) is zero. If not, then as f and g form a power graphic contraction, so we have

p δ ( x 2 n + 1 , u ) = p δ ( f x 2 n , g u ) ϕ ( p α ( x 2 n , u ) p β ( x 2 n , f x 2 n ) p γ ( u , g u ) ) = ϕ ( p α ( x 2 n , u ) p β ( x 2 n , x 2 n + 1 ) p γ ( u , g u ) ) .
(2.3)

We deduce, by taking upper limit as n in (2.3), that

p δ ( u , g u ) lim sup n ϕ ( p α ( x 2 n , u ) p β ( x 2 n , x 2 n + 1 ) p γ ( u , g u ) ) ϕ ( p α ( u , u ) p β ( u , u ) p γ ( u , g u ) ) ϕ ( p α + β + γ ( u , g u ) ) < p δ ( u , g u ) ,

a contradiction. Hence, u=gu and uF(f)F(g) by (i).

Finally, to prove (iv), suppose the set F(f)F(g) is complete. We are to show that F(f)F(g) is a singleton. Assume on the contrary that there exist u and v such that u,vF(f)F(g) but uv. As (u,v)E(G) and f and g form a power graphic contraction, so

0 < p δ ( u , v ) = p δ ( f u , f v ) ϕ ( p α ( u , v ) p β ( u , f u ) p γ ( v , g v ) ) = ϕ ( p α ( u , v ) p β ( u , u ) p γ ( v , v ) ) ϕ ( p δ ( u , v ) ) ,

a contradiction. Hence, u=v. Conversely, if F(f)F(g) is a singleton, then it follows that F(f)F(g) is complete. □

Corollary 2.2 Let (X,p) be a complete partial metric space endowed with a directed graph G. If we replace (1.1) by

p δ ( f s x , g t y ) ϕ ( p α ( x , y ) p β ( x , f s x ) p γ ( y , g t y ) ) ,
(2.4)

where α,β,γ0 with δ=α+β+γ(0,) and s,tN, then the conclusions obtained in Theorem 2.1 remain true.

Proof It follows from Theorem 2.1, that F( f s )F( g t ) is a singleton provided that F( f s )F( g t ) is complete. Let F( f s )F( g t )={w}, then we have f(w)=f( f s (w))= f s + 1 (w)= f s (f(w)), and g(w)=g( g t (w))= g t + 1 (w)= g t (g(w)) implies that fw and gw are also in F( f s )F( g t ). Since F( f s )F( g t ) is a singleton, we deduce that w=fw=gw. Hence, F(f)F(g) is a singleton. □

The following remark shows that different choices of α, β and γ give a variety of power graphic contraction pairs of two mappings.

Remarks 2.3 Let (X,p) be a complete partial metric space endowed with a directed graph G.

(R1) We may replace (1.1) with the following:

p 3 (fx,gy)ϕ ( p ( x , y ) p ( x , f x ) p ( y , g y ) )
(2.5)

to obtain conclusions of Theorem 2.1. Indeed, taking α=β=γ=1 in Theorem 2.1, one obtains (2.5).

(R2) If we replace (1.1) by one of the following condition:

(2.6)
(2.7)
(2.8)

then the conclusions obtained in Theorem 2.1 remain true. Note that

  1. (i)

    if we take α=β=1 and γ=0 in (1.1), then we obtain (2.6),

  2. (ii)

    take α=γ=1, β=0 in (1.1) to obtain (2.7),

  3. (iii)

    use β=γ=1, α=0 in (1.1) and obtain (2.8).

(R3) Also, if we replace (1.1) by one of the following conditions:

(2.9)
(2.10)
(2.11)

then the conclusions obtained in Theorem 2.1 remain true. Note that

  1. (iv)

    take α=1 and β=γ=0 in (1.1) to obtain (2.9),

  2. (v)

    to obtain (2.10), take β=1, α=γ=0 in (1.1),

  3. (vi)

    if one takes γ=1, α=β=0 in (1.1), then we obtain (2.11).

Remark 2.4 If we take f=g in a power graphic contraction pair, then we obtain fixed point results for a power graphic contraction.

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Abbas, M., Nazir, T. Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph. Fixed Point Theory Appl 2013, 20 (2013). https://doi.org/10.1186/1687-1812-2013-20

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Keywords

  • partial metric space
  • common fixed point
  • directed graph
  • power graphic contraction pair