# b-Reich type weak contraction on b-metric spaces endowed with a graph.

1 Introduction and Preliminaries

In 1969, Nadler [8] initiated the study of fixed points for multi-valued contraction maps using the Hausdorff metric and also extended the Banach contraction principle to set-valued mappings. Since then many authors have studied fixed points for set-valued mappings. The theory of set-valued maps has many applications in control theory, convex optimization, differential equations and economics.

Definition 1. [1] Let X be a (nonempty) set and s [greater than or equal to] 1 a given real number. A function d: X x X [right arrow] [[Real part].sup.+] (nonnegative real numbers) is called a b-metric provided that, for all x, y, z [member of] X, the following conditions are satisfied:

1. d(x, y) = 0 if and only if x = y,

2. d(x, y) = d(y, x),

3. d(x, z) [less than or equal to] s[d(x, y) + d(y, z)].

The pair (X, d) is called a b-metric space with parameter s.

We now give some examples of b-metric spaces.

Example 2. [3] The space [l.sub.p](0 < p < 1),

[l.sub.p] = {([x.sub.n]) [member of] [Real part]: [summation] [|[x.sub.n]|.sup.p] < [infinity]},

together with the function d: [l.sub.p] x [l.sub.p] [right arrow] [Real part] defined by

d(x, y) = [([summation] [|[x.sub.n] - [y.sub.n]|.sup.p]).sup.[1/p]] where x = ([x.sub.n]); y = ([y.sub.n]) [member of] [l.sub.p] is a b-metric space with s = [2.sup.[1/p]].

Example 3. [3] The space [L.sub.p](0 < p < 1) of all real functions x(t), t [member of] [0, 1] such that [[integral].sub.0.sup.1] [|x(t)|.sup.p] dt < [infinity], is a b-metric space if we take

d(x, y) = [([[integral].sub.0.sup.1] [|x(t) - y(t)|.sup.p] dt).sup.[1/p]], for each x, y [member of] [L.sub.p].

Remark 4. We note that a metric space is evidently a b-metric space for s = 1. However, in general, a b-metric on X need not be a metric on X as shown in the following example.

Example 5. [1] Let X = {0, 1, 2} and d(2, 0) = d(0, 2) = m [greater than or equal to] 2, d(0, 1) = d(1, 2) = d(1, 0) = d(2, 1) = 1 and d(0, 0) = d(1, 1) = d(2, 2) = 0.

Then d(x, y) [less than or equal to] [m/2] [d(x, z) + d(z, y)] for all x, y, z [member of] X. If m > 2, the ordinary triangle inequality does not hold.

Definition 6. [3] Let (X, d) be a b-metric space. Then a sequence {[x.sub.n]} in X is called a Cauchy sequence if for every [epsilon] > 0, there exists K([epsilon]) [member of] N, such that d([x.sub.n], [x.sub.m]) < [epsilon] for all n, m [greater than or equal to] K([epsilon]).

Definition 7. [3] Let (X, d) be a b-metric space. Then a sequence {[x.sub.n]} in X is said to converge to x [member of] X if for every [epsilon] > 0, there exists K([epsilon]) [member of] N, such that d([x.sub.n], x) < [epsilon] for all n [greater than or equal to] K([epsilon]). In this case, we write [mathematical expression not reproducible].

Definition 8. [3] The b-metric space (X, d) is complete if every Cauchy sequence in X converges in X.

Remark 9. In a b-metric space (X, d) the following assertions hold:

(i) A convergent sequence has a unique limit;

(ii) Every convergent sequence is Cauchy.

Remark 10. The mapping d in a b-metric space (X, d) need not be jointly continuous (see, e.g., [9, 10]).

Definition 11. Let X be any nonempty set. An element x [member of] X is said to be a fixed point of a multi-valued mapping T: X [right arrow] [2.sup.X] if x [member of] Tx, where [2.sup.X] denotes the collection of all nonempty subsets of X.

Let (X, d) be a b-metric space with s [greater than or equal to] 1. Let CB(X) be the collection of all nonempty closed bounded subsets of X. For A, B [member of] CB(X), define

H (A, B) = max{[delta](A, B), [delta](B, A)}, where

[delta](A, B) = sup{d(a, B); a [member of] A}, [delta](B, A) = sup{d(b, A); b [member of] B}

with d(a, C) = inf {d(a, x); x [member of] C}, C [member of] CB(X).

Also, define D(A, B) = inf {d(x, B); x [member of] A}.

The mapping H is said to be a Hausdorff metric induced by the metric d.

We cite the following lemmas from Czerwik [4, 5, 6] and Singh et al. [14].

Lemma 12. Let (X, d) be a b-metric space with constant s [greater than or equal to] 1. For any A, B, C [member of] CB(X) and any x, y [member of] X, we have the following:

(i) d(x, B) [less than or equal to] d(x, b) for any b [member of] B,

(ii) [delta](A, B) [less than or equal to] H(A, B),

(iii) d(x, B) [less than or equal to] H(A, B) for any x [member of] A,

(iv) H(A, A)=0,

(v) H(A, B) = H(B, A),

(vi) H(A, C) [less than or equal to] s(H(A, B) + H(B, C)),

(vii) d(x, A) [less than or equal to] s(d(x, y) + d(y, A)).

Lemma 13. Let (X, d) be a b-metric space with constant s [greater than or equal to] 1. Let A and B be in CB(X). Then for each [epsilon] > 0 and for all b [member of] B there exists a [member of] A such that d(a, b) [less than or equal to] H (A, B) + [epsilon].

Lemma 14. Let (X, d) be a b-metric space with constant s [greater than or equal to] 1. For A [member of] CB(X) and x [member of] X, we have d(x, A) = 0 [??] x [member of] [bar.A] = A where [bar.A] denotes the closure of A with respect to the induced metric d. Note that A is closed in (X, d) if and only if [bar.A] = A.

Nadler [8] extended Banach Contraction principle to set-valued mappings as follows.

Theorem 15. Let (X, d) be a complete metric space and T: X [right arrow] CB(X). Assume that there exists k [member of] [0, 1) such that H(Tx, Ty) [less than or equal to] kd(x, y) for all x, y [member of] X. Then there exists z [member of] X such that z [member of] Tz.

Reich [12] generalized the Banach fixed point theorem for single-valued maps and multi-valued maps as the following two theorems.

Theorem 16. [12] Let (X, d) be a complete metric space and let T: X [right arrow] X be a Reich type single-valued (a, b, c)-contraction, that is, there exist non-negative numbers a, b, c with a + b + c < 1 such that

d(T(x), T(y)) [less than or equal to] ad(x, y) + bd(x, T(x)) + cd(y, T(y))

for each x, y [member of] X. Then T has a unique fixed point.

Theorem 17. [12] Let (X, d) be a complete metric space and let a mapping T: X [right arrow] [P.sub.cl](X), where [P.sub.cl](X) is the set of all nonempty closed subsets of X, be a Reich type multi-valued (a, b, c)-contraction, that is, there exist nonnegative numbers a, b, c with a + b + c < 1 such that

H(Tx, Ty) [less than or equal to] ad(x, y) + bD(x, Tx) + cD(y, Ty)

for each x, y [member of] X. Then there exists z [member of] X such that z [member of] Tz.

In 2008, Jachymski [7] introduced the concept of G-contraction and studied this class of generalized Banach contractions on a metric space endowed with a graph. He also showed that many existing results in the literature can be derived by his results.

Definition 18. [7] Let (X, d) be a metric space and G = (V(G), E(G)) a directed graph such that V(G) = X and E(G) contains all loops, i.e., [DELTA] = {(x, x)|x [member of] X} [??] E(G). We say that a mapping T: X [right arrow] X is a G-contraction if T preserves edges of G, i.e., for every x, y [member of] X,

(x, y) [member of] E(G) (T(x), T(y)) [member of] E(G)

and there exists [alpha] [member of] (0, 1) such that, for x, y [member of] X,

(x, y) [member of] E(G) [??] d(T(x), T(y)) [less than or equal to] [alpha]d(x, y).

In 2014, Tiammee and Suantai [15] introduced and studied the concept of g-graph preserving for multi-valued mappings in a complete metric space endowed with a graph.

Definition 19. [15] Let X be a nonempty set and G = (V(G), E(G)) be a graph such that V(G) = X, and let T: X [right arrow] CB(X). T is said to be graph preserving if it satisfies the following:

if (x, y) [member of] E(G), then (u, v) [member of] E(G) for all u [member of] Tx and v [member of] Ty.

Definition 20. [15] Let X be a nonempty set and G = (V (G), E (G)) be a graph such that V(G) = X, and let T: X [right arrow] CB(X), g: X [right arrow] X. T is said to be g-graph preserving if it satisfies the following:

for each x, y [member of] X, if (g(x), g(y)) [member of] E(G), then (u, v) [member of] E(G) for all u [member of] Tx, v [member of] Ty.

Recently, using the concept of g-graph preserving introduced by Tiammee and Suantai [15] and the concept of a Reich type multi-valued contraction defined by Reich[12], Phon-on et al.[11] introduced a new class of Reich type multi-valued contraction on a complete metric space endowed with a graph satisfying the g-graph preserving condition and studied the fixed point theorem for such mappings.

Definition 21. [11] Let (X, d) be a metric space, G = (V(G), E(G)) be a directed graph such that V(G) = X, g: X [right arrow] X and T: X [right arrow] CB(X). T is said to be a Reich type weak G-contraction with respect to g or a (g, [alpha], [beta], [gamma])-G-contraction provided that

(i) T is g-graph preserving;

(ii) there exist nonnegative numbers [alpha], [beta], [gamma] with [alpha] + [beta] + [gamma] < 1 and

H(Tx, Ty) [less than or equal to] ad(g(x), g(y))+ [beta]D(g(x), Tx) + [gamma]D(g(y), Ty) for all x, y [member of] X such that (g(x), g(y)) [member of] E(G).

In this article, we introduce a new class of Reich type multi-valued contraction and establish analogous of results proved in [11] on a complete b-metric space endowed with a graph.

2 Main Results

Let (X, d) be a b-metric space with constant s [greater than or equal to] 1 and [DELTA] be the diagonal of X x X. Let G be a directed graph such that the set V(G) of its vertices coincides with X and [DELTA] [??] E(G), E(G) being the set of the edges of the graph. Assuming that G has no parallel edges we will have 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 [member of] N is a finite sequence ([x.sub.n]), n [member of] {0, 1, 2,..., k} of vertices such that [x.sub.0] = x, [x.sub.k] = y and ([x.sub.i1], [x.sub.i]) [member of] E(G) for i [member of] {1, 2,..., k}. Let us denote by [??] 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 [??] is connected. Let [G.sup.-1] be the graph obtained from G by reversing the direction of edges. Thus, E([G.sup.-1]) = {(x, y) [member of] X x X: (y, x) [member of] E(G)}. Since it is more convenient to treat [??] as a directed graph for which the set of its edges is symmetric, under this convention, we have that E([??]) = E (G) [union] E([G.sup.-1]).

Property A: Let (X, d) be a complete b-metric space with constant s [greater than or equal to] 1 and G be a directed graph. We say that X has property A if for any sequence ([x.sub.n]) in X with [x.sub.n] [right arrow] x, as n [right arrow] [infinity] and ([x.sub.n], [x.sub.n+1]) [member of] E(G), for n [member of] N, there is a subsequence [mathematical expression not reproducible] of {[x.sub.n]} such that [mathematical expression not reproducible] for [n.sub.k] [member of] N.

We now define a new class of b-Reich type multi-valued contraction on a complete b-metric space as follows.

Definition 22. Let (X, d) be a b-metric space with constant s [greater than or equal to] 1, G = (V(G), E(G)) be a directed graph such that V(G) = X, g: X [right arrow] X and T: X [right arrow] CB(X). T is said to be a b-Reich type weak G-contraction with respect to g or a b-(g, [alpha], [beta], [gamma])-G-contraction provided that

(i) T is g-graph preserving;

(ii) there exist nonnegative numbers [alpha], [beta], [gamma] with s([alpha] + [beta] + [gamma]) < 1 and

H(Tx, Ty) [less than or equal to] [alpha]d(g(x), g(y)) + [beta]D(g(x), Tx) + [gamma]D(g(y), Ty) for all x, y [member of] X such that (g(x), g(y)) [member of] E(G).

Example 23. Let X = N and d(x, y) = [(x - y).sup.2] for all x, y [member of] X. Then (X, d) is a b-metric space with s = 2. Consider the directed graph defined by V(G) = X and E(G) = {(2n - 1, 2n + 1): n [member of] N} [union] {(2n, 2n + 2): n g N ~ {1}} [union] {(2n, 2n + 4): n g N ~ {1}} [union] {(2n, 2n): n [member of] N ~ {1}} [union] {(1, 1), (6, 4)}. Let T: X [right arrow] CB(X) be defined by

[mathematical expression not reproducible]

and g: X [right arrow] X be defined by

[mathematical expression not reproducible]

We will show that T is a b-(g, [alpha], [beta], [gamma])-G-contraction with [alpha] = 0, [beta] = [2/9], [gamma] = [2/9].

Let (g(x), g(y)) [member of] E(G).

Case I: If (g(x), g(y)) = (2k - 1, 2k + 1) for k [member of] N, then (x, y) = (2k + 1, 2k + 3), Tx = {2k + 2, 2k + 4}, and Ty = {2k + 4, 2k + 6}.

We obtain (2k + 2, 2k + 4), (2k + 2, 2k + 6), (2k + 4, 2k + 4), (2k + 4, 2k + 6) [member of] E(G). Also, D(g(x), Tx) = 9, D(g(y), Ty) = 9, and

[mathematical expression not reproducible].

Case II: If (g(x), g(y)) = (2k, 2k + 2) or (2k, 2k + 4) or (2k, 2k) for k [member of] N ~ {1}, then Tx = Ty = {1} and (1, 1) [member of] E(G). It follows that

[mathematical expression not reproducible]

Case III: If (g(x), g(y)) = (1, 1), then x = y = 3, Tx = Ty = {4, 6}, and (4, 4), (4, 6), (6, 4), (6, 6) [member of] E(G). It follows that

[mathematical expression not reproducible].

Case IV: If (g(x), g(y)) = (6, 4), then x = 8, y = 6, and Tx = Ty = {1} and (1, 1) [member of] E(G).

Note that D(g(x), Tx) = 25, D(g(y), Ty) = 9, and so

[mathematical expression not reproducible]

It follows that T is a b-(g, 0, [2/9], [2/9])-G-contraction.

Lemma 24. Let (X, d) be a b-metric space with constant s [greater than or equal to] 1, G a directed graph, [g.sub.1], [g.sub.2]: X [right arrow] X be surjective maps and [T.sub.i]: X [right arrow] CB(X) be [g.sub.i]-graph preserving, i = 1, 2 satisfying the following:

there exist constants [alpha], [beta], [gamma] [greater than or equal to] 0 with s([alpha] + [beta] + [gamma]) < 1 such that, for all x, y [member of] X, if ([g.sub.1](x), [g.sub.2](y)) [member of] E(G), then

H([T.sub.1]x, [T.sub.2]y) [less than or equal to] [alpha]d([g.sub.1](x), [g.sub.2](y)) + [beta]D([g.sub.1](x), [T.sub.1]x) + [gamma]D([g.sub.2](y), [T.sub.2]y) and if ([g.sub.2](x), [g.sub.1](y)) [member of] E(G), then H([T.sub.2]x, [T.sub.1]y) [less than or equal to] [alpha]d([g.sub.2](x), [g.sub.1](y)) + [beta]D([g.sub.2](x), [T.sub.2]x) + [gamma]D([g.sub.1](y), [T.sub.1]y).

Suppose that

(A) there exists [x.sub.0] [member of] X such that ([g.sub.1]([x.sub.0]), u) [member of] E(G) for some u [member of] [T.sub.1][x.sub.0], and

(B) if ([g.sub.1](x), [g.sub.2](y)) [member of] E(G), then (z, w) [member of] E(G) for all z [member of] [T.sub.1]x, w [member of] [T.sub.2]y and if ([g.sub.2](x), [g.sub.1](y)) [member of] E(G), then (r, t) [member of] E(G) for all r [member of] [T.sub.2]x, t [member of] [T.sub.1]y, then there exists a sequence [{[x.sub.k]}.sub.k[member of]N[union]{0}] in X such that, for each k [member of] N, [g.sub.1]([x.sub.2k]) [member of] [T.sub.2][x.sub.2k-1], [g.sub.2]([x.sub.2k-1]) [member of] [T.sub.1][x.sub.2k-2], ([g.sub.1]([x.sub.2k-2]), [g.sub.2]([x.sub.2k-1])), ([g.sub.2]([x.sub.2k-1]), [g.sub.1]([x.sub.2k])) [member of] E(G), and a Cauchy sequence {[y.sub.k]} in X where

[mathematical expression not reproducible]

Proof. Since [g.sub.2] is surjective, there exists [x.sub.1] [member of] X such that [g.sub.2]([x.sub.1]) [member of] [T.sub.1][x.sub.0] and ([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1])) [member of] E(G). Using Lemma 13 for [epsilon] = ([alpha] + [beta]), there exists [x.sub.2] [member of] X such that [g.sub.1]([x.sub.2]) [member of] [T.sub.2][x.sub.1] and d([g.sub.2]([x.sub.1]), [g.sub.1]([x.sub.2])) [less than or equal to] H([T.sub.1][x.sub.0], [T.sub.2][x.sub.1]) + ([alpha] + [beta]). By (B), it follows that ([g.sub.2]([x.sub.1]), [g.sub.1]([x.sub.2])) [member of] E(G). Therefore, by assumption, d([g.sub.2]([x.sub.1]), [g.sub.1]([x.sub.2])) [less than or equal to] H([T.sub.1][x.sub.0], [T.sub.2][x.sub.1]) + ([alpha] + [beta])

[less than or equal to] [alpha]d([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1])) + [beta]D([g.sub.1]([x.sub.0]), [T.sub.1][x.sub.0]) + [gamma]D([g.sub.2]([x.sub.1]), [T.sub.2][x.sub.1]) + ([alpha] + [beta]) [less than or equal to] [alpha]d([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1])) + [beta]d([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1]) + [gamma]d([g.sub.2]([x.sub.1]), [g.sub.1]([x.sub.2])) + ([alpha] + [beta]).

Hence, (1 - [gamma])d([g.sub.2]([x.sub.1]), [g.sub.1]([x.sub.2])) [less than or equal to] ([alpha] + [beta])d([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1])) + ([alpha] + [beta]).

It follows that

d([g.sub.2]([x.sub.1]), [g.sub.1]([x.sub.2]) [less than or equal to] [eta]d([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1]) + [eta], (2.1)

where [eta] = [[[alpha]+[beta]]/[1-[gamma]]] < 1.

Again, using Lemma 13 for [epsilon] = ([alpha] + [beta])[eta], we can find [x.sub.3] [member of] X such that [g.sub.2]([x.sub.3]) [member of] [T.sub.1][x.sub.2] and

d([g.sub.1]([x.sub.2]), [g.sub.2]([x.sub.3]) [less than or equal to] H([T.sub.2][x.sub.1], [T.sub.1][x.sub.2]) + ([alpha] + [beta])[eta].

Again by (B), we obtain that ([g.sub.1]([x.sub.2]), [g.sub.2]([x.sub.3])) [member of] E(G). Therefore by assumption, we have

[mathematical expression not reproducible].

Hence,

(1 - [gamma])d([g.sub.1]([x.sub.2]), [g.sub.2]([x.sub.3])) [less than or equal to] ([alpha] + [beta])d([g.sub.2]([x.sub.1]), [g.sub.1]([x.sub.2])) + ([alpha] + [beta])[eta].

It follows that

d([g.sub.1]([x.sub.2]), [g.sub.2]([x.sub.3])) [less than or equal to] [eta]d([g.sub.2]([x.sub.1]), [g.sub.1]([x.sub.2])) + [[eta].sup.2].

Thus by inequality (2.1), we have

d([g.sub.1]([x.sub.2]), [g.sub.2]([x.sub.3])) [less than or equal to] [[eta].sup.2]d([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1])) + 2[[eta].sup.2].

Continuing like this, we obtain sequences {[x.sub.k]} and {[y.sub.k]} where

[mathematical expression not reproducible]

and

d([y.sub.k], [y.sub.k+1]) [less than or equal to] [[eta].sup.k]d([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1]) + k[[eta].sup.k].

Next, we show that {[y.sub.k]} is a Cauchy sequence in X.

For n, p [greater than or equal to] 1, consider

[mathematical expression not reproducible]

Hence, as n [right arrow] [infinity], d([y.sub.n], [y.sub.n+p]) [right arrow] 0. Thus the sequence [{[y.sub.k]}.sub.k[member of]N] is a Cauchy sequence in X.

Theorem 25. Let (X, d) be a b-metric space with constant s [greater than or equal to] 1, G a directed graph, [g.sub.1], [g.sub.2]: X [right arrow] X be surjective maps and [T.sub.i]: X [right arrow] CB(X) be [g.sub.i]-graph preserving, i = 1, 2 satisfying the following:

there exist constants [alpha], [beta], [gamma] [greater than or equal to] 0 with s([alpha] + [beta] + [gamma]) < 1 such that, for all x, y [member of] X, if ([g.sub.1](x), [g.sub.2](y)) [member of] E(G), then

H([T.sub.1]x, [T.sub.2]y) [less than or equal to] [alpha]d([g.sub.1](x), [g.sub.2](y)) + [beta]D([g.sub.1](x), [T.sub.1]x) + [gamma]D([g.sub.2](y), [T.sub.2]y) and if ([g.sub.2](x), [g.sub.1](y)) [member of] E(G), then

H([T.sub.2]x, [T.sub.1]y) [less than or equal to] [alpha]d([g.sub.2](x), [g.sub.1](y)) + [beta]D([g.sub.2](x), [T.sub.2]x) + [gamma]D([g.sub.1](y), [T.sub.1]y).

Suppose that

(A) there exists [x.sub.0] [member of] X such that ([g.sub.1]([x.sub.0]), u) [member of] E(G) for some u [member of] [T.sub.1][x.sub.0],

(B) if ([g.sub.1](x), [g.sub.2](y)) [member of] E(G), then (z, w) [member of] E(G) for all z [member of] [T.sub.1]x, w [member of] [T.sub.2]y, and if ([g.sub.2](x), [g.sub.1](y)) [member of] E(G), then (r, t) [member of] E(G) for all r [member of] [T.sub.2]x, t [member of] [T.sub.1]y,

(C) X has property A,

then there exist u, v [member of] X such that [g.sub.1](u) [member of] [T.sub.1]u or [g.sub.2](v) [member of] [T.sub.2]v.

Proof. By (A) let [x.sub.0] [member of] X be such that ([g.sub.1]([x.sub.0]), [g.sub.2]([x.sub.1])) [member of] E(G) for some [g.sub.2]([x.sub.1]) [member of] [T.sub.1][x.sub.0]. Now, using Lemma 24, there exists a sequence {[x.sub.k]} in X such that, for each k [member of] N,

[g.sub.1]([x.sub.2k]) [member of] [T.sub.2][x.sub.2k-1], [g.sub.2]([x.sub.2k-1]) [member of] [T.sub.1][x.sub.2k-2],

([g.sub.1]([x.sub.2k-2]), [g.sub.2]([x.sub.2k-1])), ([g.sub.2]([x.sub.2k-1]), [g.sub.1]([x.sub.2k])) [member of] E(G),

and a Cauchy sequence {[y.sub.k]} where

[mathematical expression not reproducible]

Now, since X is complete, there exists w in X such that [y.sub.k] [right arrow] w as k [right arrow] [infinity].

Let u, v [member of] X be such that [g.sub.1](u) = w = [g.sub.2](v). By Property A, there is a subsequence [mathematical expression not reproducible] such that [mathematical expression not reproducible] for any n [member of] N.

We will show that [g.sub.1](u) [member of] [T.sub.1]u or [g.sub.2](v) [member of] [T.sub.2]v.

Let A = {[k.sub.n]: [k.sub.n] is even} and B = {[k.sub.n]: [k.sub.n] is odd}. Since A [union] B is infinite, atleast A or B must be infinite. If A is infinite, for each [mathematical expression not reproducible], where [k.sub.n] [member of] A, we have

[mathematical expression not reproducible]

Since [mathematical expression not reproducible] and [mathematical expression not reproducible] are subsequences of {[y.sub.k]}, they converge to [g.sub.2](v) = w as n [right arrow] [infinity], and hence D([g.sub.2](v), [T.sub.2]v) = 0. Since [T.sub.2]v is closed, we conclude that [g.sub.2](v) [member of] [T.sub.2]v. Similarly, if B is infinite, we can show that [g.sub.1](u) [member of] [T.sub.1]u. Note that if A and B are both infinite then [g.sub.2](v) [member of] [T.sub.2]v and [g.sub.1](u) [member of] [T.sub.1]u.

Now we give example in support of above theorem.

Example 26. Let (X, d) be a b-metric space where X = [0, 1] and d = [(x - y).sup.2] for all x, y [member of] X. Here s = 2.

Consider the directed graph G = (V (G), E (G)) defined by V (G) = X and E(G) = {(0, 0), (1, 1), (0, [1/2]), ([1/2], 0), ([1/2], [1/2])}[union]{(0, [1/4]), ([1/4], 0), ([1/4], [1/4]), ([1/2], [1/4]), ([1/4], [1/2])}.

Let T, S: X [right arrow] CB(X) and [g.sub.1], [g.sub.2]: X [right arrow] X be defined by

[mathematical expression not reproducible]

[g.sub.1](x) = [x.sub.2] and [g.sub.2](x) = x for all x [member of] X.

Clearly, [T.sub.1], [T.sub.2] are [g.sub.1], [g.sub.2]-graph preserving, respectively. Also, it can be easily verified that the conditions (A), (B), (C) of Theorem 25 are satisfied. Next, we will show that for all x, y [member of] X, if ([g.sub.1](x), [g.sub.2](y)) [member of] E(G), then we have

H([T.sub.1]x, [T.sub.2]y) [less than or equal to] [alpha]d([g.sub.1](x), [g.sub.2](y)) + [beta]D([g.sub.1](x), [T.sub.1]x) + [gamma]D([g.sub.2](y), [T.sub.2]y) and if ([g.sub.2](x), [g.sub.1](y)) [member of] E(G), then we have

H([T.sub.2]x, [T.sub.1]y) [less than or equal to] [alpha]d([g.sub.2](x), [g.sub.1](y)) + [beta]D([g.sub.2](x), [T.sub.2]x) + [gamma]D([g.sub.1](y), [T.sub.1]y), where [alpha] = 0, [beta] = [1/16], [gamma] = [1/4].

If ([g.sub.1](x), [g.sub.2](y)), ([g.sub.2](x), [g.sub.1](y)) [member of] E (G)\{(1, 1)}, then H(Tx, Sy) = 0 = H(Sx, Ty). So the above inequalities are satisfied.

If ([g.sub.1](x), [g.sub.2](y)) = (1, 1), then [g.sub.2](x) = 1 = [g.sub.2](y), which implies that x = 1 = y. Thus we have [T.sub.1]x = [1/4] and [T.sub.2]y = [1/2] and hence H([T.sub.1]x, [T.sub.2]y) = H({[1/4]}, {[1/2]}) = [1/16]

[less than or equal to] 0(0) + [1/16]([9/16]) +[1/4]([1/4]) = [alpha]d([g.sub.1](x), [g.sub.2](y)) + [beta]D([g.sub.1](x), Tx) + [gamma]D([g.sub.2](y), Sy).

If ([g.sub.2](x), [g.sub.1](y)) = (1, 1), then [g.sub.2](x) = 1 = [g.sub.1](y), which gives x = 1 = y.

Thus we have [T.sub.1]x = [1/4] and [T.sub.2]y = [1/2] and hence

[mathematical expression not reproducible]

Therefore, by Theorem 25, there are u, v [member of] X such that [g.sub.1](u) [member of] Tu or [g.sub.2](v) [member of] Sv.

Here,

choose u = [1/4]. Since [g.sub.2](u) = u, we have u [member of] Su = {u}.

From Theorem 25 we deduces the following two corollaries.

Corollary 27. Let (X, d) be a b-metric space with constant s [greater than or equal to] 1, the directed graph G, g: X [right arrow] X be surjective map and [T.sub.1], [T.sub.2]: X [right arrow] CB(X) be g-graph preserving satisfying:

H([T.sub.1]x, [T.sub.2]y) [less than or equal to] [alpha]d(g(x), g(y)) + [beta]D(g(x), [T.sub.1]x) + [gamma]D(g(y), [T.sub.2]y) for all x,y [member of] X with (g(x), g(y)) [member of] E (G) and constants [alpha], [beta], [gamma] [greater than or equal to] 0 with s([alpha] + [beta] + [gamma]) < 1.

Suppose that

(A) there exists [x.sub.0] [member of] X such that (g([x.sub.0]), u) [member of] E(G) for some u [member of] [T.sub.1][x.sub.0],

(B) X has property A,

then there exist u, v [member of] X such that g(u) [member of] [T.sub.1]u or g(v) [member of] [T.sub.2]v.

Proof. Take [g.sub.1] = [g.sub.2] = g in Theorem 25.

Corollary 28. Let (X, d) be a complete b-metric space, G = (V(G), E(G)) be a directed graph such that V(G) = X, and let g: X [right arrow] X be a surjective map. If T: X [right arrow] CB(X) is a multi-valued mapping satisfying the following properties:

(A) T is a b-(g, [alpha], [beta], [gamma])-G-contraction;

(B) the set [X.sub.T] = {x [member of] X|(g(x), y) [member of] E(G) for some y [member of] Tx} [not equal to] [empty set];

(C) X has Property A,

then there exists u [member of] X such that g(u) [member of] Tu.

Proof. Take [T.sub.1] = [T.sub.2] = T and [g.sub.1] = [g.sub.2] = g in Theorem 25. *

References

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Nidhi Malhotra

Department of Mathematics,

University of Delhi, Delhi-110007, India.

(*) Corresponding author

Bindu Bansal

Department of Mathematics,

Hindu College, University of Delhi, Delhi-110007, India.

AMS Subject Classification: Primary 47H10; Secondary 54H25.