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Fixed Point Results for Multivalued Contractive Mappings Endowed with Graphic Structure.

1. Introduction and Preliminaries

FP theory plays a fundamental role in functional analysis. Banach proved significant result for contraction mappings. After that, a huge number of FP theorems have been established by various authors and they made different generalizations of the Banach's result. Shoaib et al. [1], discussed the result related to [[alpha].sub.*]-[psi]-Ciric type multivalued mappings on an intersection of a closed ball and a sequence with graph. Further FP results on a closed ball can be seen ([2-12]).

Boriceanu [13] proved FP results for multivalued generalized contraction on a set with two b-metrics. After this Aydi et al. [14] established FP theorem for set-valued quasi contraction in b-metric spaces. Nawab et al. [15] established the new idea of dislocated b-metric space as a conception of metric space and proved some common FP results for four mappings fulfilling the generalized weak contractive conditions in partially ordered dislocated b-metric space.

Nadler [16] initiated the study of FP theorems for the multivalued mappings (see also [17]). Several results on multivalued mappings have been observed (see [18-20]). Asl et al. [21] gave the idea of [[alpha].sub.*]-[psi] contractive multifunctions and at [[alpha].sub.*]-admissible mapping and got some FP conclusions for these multifunctions (see also [22, 23]). In 1974, Ciric [24] introduced quasi contraction. Reference [25] established some new common fixed points of generalized rational-contractive mappings in dislocated metric spaces with applications. In this paper, the concept of new Ciric type rational multivalued contraction has been introduced. Now we prove new type of result for a different multivalued rational expression studied by Rasham et al. [6]. Common FP results for such contraction on a closed ball in complete dislocated b-metric space have been established. Example and application have been given. We give the following definitions and results which will be needed in the sequel.

Definition 1 (see [15]). Let M be a nonempty set and let [d.sub.b] : M x M [right arrow] [0, [infinity]) be a function, called a dislocated b-metric (or simply [d.sub.b]-metric), if for any g, q, [??] [member of] M and t [greater than or equal to] 1 the following conditions hold:

(i) if [d.sub.b] (g, q) = 0, then g = q;

(ii) [d.sub.b] (g, q) = [d.sub.b] (q, g);

(iii) [d.sub.b] (g, q) [less than or equal to] t[[d.sub.b] (g, [??]) + [d.sub.b] ([??], q)].

The pair (M, [d.sub.b]) is called a dislocated b-metric space. It should be noted that the class of [d.sub.b] metric spaces is effectively larger than that of [d.sub.l] metric spaces, since [d.sub.b] is a [d.sub.l] metric when t =1.

It is clear that if [d.sub.b](g, q) = 0, then from (i), g = q. But if g = q, [d.sub.b](g, q) may not be 0. For g [member of] M and [epsilon] > 0, [bar.B(g, [epsilon])] = [q [member of] M : [d.sub.b](g, q) [less than or equal to] e} is a closed ball in (M, [d.sub.b]). We use D.B.M space instead of dislocated b-metric space.

Example 2. If M = [R.sup.+] [union] {0}, then [d.sub.b](g, q) = [(g + q).sup.2] defines a D.B.M [d.sub.b] on M.

Definition 3 (see [15]). Let (M, [d.sub.b]) be a D.B.M space.

(i) A sequence {[g.sub.n]} in (M, [d.sub.b]) is called Cauchy sequence if, given [epsilon] > 0, there corresponds [n.sub.0] [member of] N such that for all n, m [greater than or equal to] [n.sub.0] we have [d.sub.b]([g.sub.m], [g.sub.n]) < [epsilon] or [lim.sub.n,m[right arrow]][d.sub.b]([g.sub.n], [g.sub.m]) = 0.

(ii) A sequence {[g.sub.n]} dislocated b-converges (for short [d.sub.b] - converges) to g if [lim.sub.n[right arrow][infinity]] [d.sub.b] ([g.sub.n], g) = 0. In this case g is called a delimit of {[g.sub.n]}.

(iii) (M, [d.sub.b]) is called complete if every Cauchy sequence in M converges to a point g [member of] M such that [d.sub.b](g, g) = 0.

Definition 4. Let [??] be a nonempty subset of D.B.M space M and let g [member of] M. An element [q.sub.0] [member of] [??] is called a best approximation in [??] if

[mathematical expression not reproducible]. (1)

If each g [member of] M has at least one best approximation in [??], then [??] is called a proximinal set.

We denote by P(M) the set of all proximinal subsets of M. Let [PSI], where t [greater than or equal to] 1, denote the family of all nondecreasing functions [psi] : [0, +[infinity]) [right arrow] [0, +[infinity]) such that [mathematical expression not reproducible] and t[psi](u) < u for all u > 0, where [[psi].sup.k] is the [k.sup.th] iterate of [psi]. Also [mathematical expression not reproducible].

Definition 5 (see [26]). Let B, A : M [right arrow] P(M) be the closed valued multifunctions and [beta] : M x M [right arrow] [0, +[infinity]o) be a function. We say that the pair (B, A) is [[beta].sub.*]-admissible if for all g, q [member of] M

[mathematical expression not reproducible], (2)

where [[beta].sub.*] (Ag, Bq) = inf{[beta]([bar.a], b) : [bar.a] [member of] Ag, b [member of] Bq}. When B = A, then we obtain the definition of [[alpha].sub.*]-admissible mapping given in [21].

Definition 6. Let (M, [d.sub.b]) be a D.B.M space, B : M [right arrow] P(M) be multivalued mapping, and [alpha] : M x M [right arrow] [0, +[infinity]). Let [bar.A] [subset] M, and we say that the B is semi-[[alpha].sub.*]-admissible on [bar.A], whenever [alpha](g, q) [greater than or equal to] 1 implies that [[alpha].sub.*] (Bg, Bq) [greater than or equal to] 1 for all g, q [member of] [bar.A], where [[alpha].sub.*] (Bg, Bq) = inf{[alpha]([bar.a], b) : [bar.a] [member of] Bg, b [member of] Bq}. If [bar.A] = M, then we say that the B is [[alpha].sub.*]-admissible on M.

Definition 7. The function [mathematical expression not reproducible], defined by

[mathematical expression not reproducible], (3)

is called dislocated Hausdorff b-metric on P(M).

Lemma 8. Let (M, [d.sub.b]) be a D.B.M space. Let [mathematical expression not reproducible] is a dislocated Hausdorff b-metric space on P(M). Then for all [bar.A], B [member of] P(M) and for each [bar.a] [member of] [bar.A] there exists [b.sub.[bar.a]] [member of] B satisfying [d.sub.b] ([bar.a], B) = [d.sub.b] ([bar.a], [b.sub.[bar.a]]); then [mathematical expression not reproducible].

2. Main Result

Let (M, [d.sub.b]) be a D.B.M space, and let[ .sub.[g.sub.0]] [member of] M and B : M [right arrow] P(M) be the multifunctions on M. Then there exist [g.sub.1] [member of] B[g.sub.0] such that [d.sub.b] ([g.sub.0], B[g.sub.0]) = [d.sub.b]([g.sub.0], [g.sub.1]). Let [g.sub.2] [member of] B[g.sub.1] be such that [d.sub.b]([g.sub.1], B[g.sub.1]) = [d.sub.b]([g.sub.1], [g.sub.2]). Continuing this process, we construct a sequence [g.sub.n] of points in M such that [mathematical expression not reproducible]. We denote this iterative sequence by {MB([g.sub.n])}. We say that {MB([g.sub.n])} is a sequence in M generated by [g.sub.0].

Theorem 9. Let (M, [d.sub.b]) be a complete D.B.M space, [mathematical expression not reproducible], and B : M [right arrow] P(M) be a semi-[[alpha].sub.*] admissible multifunction on [mathematical expression not reproducible] is a sequence in M generated by [g.sub.0], [alpha]([g.sub.0], [g.sub.1]) [greater than or equal to] 1. Assume that, for some [psi] [member of] [PSI] and

[mathematical expression not reproducible] (4)

where [bar.a] > 0, the following hold:

[mathematical expression not reproducible] (5)

[mathematical expression not reproducible]. (6)

Then, {MB([g.sub.n])} is a sequence in [mathematical expression not reproducible], and [mathematical expression not reproducible]. Also if [alpha]([g.sub.n], [g.sup.*]) [greater than or equal to] 1 or [alpha]([g.sup.*], [g.sub.n]) [greater than or equal to] 1, for all n [member of] N [union] {0} and the inequality (5) holds for all g, [mathematical expression not reproducible], then B has a common fixed point [mathematical expression not reproducible].

Proof. Consider a sequence {MB([g.sub.n])} generated by [g.sub.0]. Then, we have [g.sub.n] [member of] B[g.sub.n-1], and [d.sub.b]([g.sub.n-1], B[g.sub.n-1]) = [d.sub.b]([g.sub.n-1], [g.sub.n]), for all n [member of] N. By Lemma 8, we have [mathematical expression not reproducible] for all n [member of] N. If [g.sub.0] = gx, then [g.sub.0] is a fixed point in [mathematical expression not reproducible] of B. Let [g.sub.0] [not equal to] [g.sub.1]. From (6), we have

[mathematical expression not reproducible]. (7)

It follows that

[mathematical expression not reproducible]. (8)

If [g.sub.1] = [g.sub.2], then [g.sub.1] is a fixed point in [mathematical expression not reproducible] of B. Let [g.sub.1] [not equal to] [g.sub.2]. Since [alpha]([g.sub.0], [g.sub.1]) [greater than or equal to] 1 and B is semi-[[alpha].sub.*]-admissible multifunction on [mathematical expression not reproducible], then [mathematical expression not reproducible]. As [mathematical expression not reproducible] and [mathematical expression not reproducible]. Let [mathematical expression not reproducible] for some j [member of] N. As [[alpha].sub.*](B[g.sub.1], B[g.sub.2]) [greater than or equal to] 1, we have [alpha]([g.sub.2], [g.sub.3]) [greater than or equal to] 1, which further implies [[alpha].sub.*](B[g.sub.2], [g.sub.3]) [greater than or equal to] 1. Continuing this process, we have [[alpha].sub.*] (B[g.sub.j-1], B[g.sub.j]) [greater than or equal to] 1. Now, by using Lemma 8

[mathematical expression not reproducible]. (9)

If [mathematical expression not reproducible], then [mathematical expression not reproducible]. This is contradiction to the fact that [psi](u) < u for all u > 0. Hence, we obtain [mathematical expression not reproducible]. Therefore, we have

[mathematical expression not reproducible]. (10)

Now, by using triangular inequality and by (10), we have

[mathematical expression not reproducible]. (11)

Thus [mathematical expression not reproducible]. Hence, by induction, [mathematical expression not reproducible]. As [mathematical expression not reproducible], then we have [alpha]([g.sub.j], [g.sub.j+1]) [greater than or equal to] 1. Also B is semi-[[alpha].sub.*]-admissible multifunction on [mathematical expression not reproducible], and therefore [[alpha].sub.*](B[g.sub.j], B[g.sub.j+1]) [greater than or equal to] 1. This further implies that [alpha]([g.sub.j+1], [g.sub.j+2]) [greater than or equal to] 1. Continuing this process, we have [alpha]([g.sub.n], [g.sub.n]+1) [greater than or equal to] 1 for all n [member of] N. Now, inequality (10) can be written as

[mathematical expression not reproducible]. (12)

Fix [epsilon] > 0 and let [k.sub.1] ([member of]) [member of] N, such that

[mathematical expression not reproducible]. (13)

Let n, m [member of] N with m > n > [k.sub.1] ([member of]). Now, we have

[mathematical expression not reproducible]. (14)

Thus, [MB([g.sub.n])] is a Cauchy sequence in [mathematical expression not reproducible]. As every closed ball in a complete D.B.M space is complete, there exist [mathematical expression not reproducible] such that {MB([g.sub.n])} [right arrow] [g.sup.*], and

[mathematical expression not reproducible]. (15)

By assumption, we have [alpha]([g.sub.n], [g.sup.*]) [greater than or equal to] 1 for all n [member of] N [union] {0}. Thus, [[alpha].sub.*] (B[g.sub.n], B[g.sup.*]) [greater than or equal to] 1. Now, we have

[mathematical expression not reproducible]. (16)

Letting n [right arrow] [infinity] and by using inequality (15), we obtain (1 - t)[d.sub.b]([g.sup.*], B[g.sup.*]) [less than or equal to] 0. So (1 - t) [not equal to] 0, and then [d.sub.b] ([g.sup.*], B[g.sup.*]) = 0. Hence [g.sup.*] [member of] B[g.sup.*]. So B has a fixed point in [mathematical expression not reproducible].

Let M be a nonempty set. Then (M, [less than or equal to], [d.sub.b]) is called a preordered D.B.M space if [d.sub.b] is called D.B.M on M. Let (M, [less than or equal to], [d.sub.b]) be a preordered D.B.M space and H, K [subset or equal to] M. We say that [mathematical expression not reproducible] whenever for each [bar.a] [member of] H there exist b [member of] K such that [bar.a] [less than or equal to] b. Also, we say that [mathematical expression not reproducible] whenever, for each [bar.a] [member of] H and b [member of] K, we have [bar.a] [less than or equal to] b.

Corollary 10. Let (M, [less than or equal to], [d.sup.b]) be a preordered complete D.B.M space,[mathematical expression not reproducible], and B : M [right arrow] P(M) be a multifunction on [mathematical expression not reproducible] is a sequence generated by [g.sub.0], with [g.sub.0] [less than or equal to] [g.sub.1]. Assume that, for some [psi] [member of] [PSI] and

[mathematical expression not reproducible] (17)

where [bar.a] > 0, the following hold:

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible]. (19)

If [mathematical expression not reproducible], such that g [less than or equal to] q implies [mathematical expression not reproducible]. Then {MB([g.sub.n])} is a sequence in [mathematical expression not reproducible], and [mathematical expression not reproducible]. Also if [g.sup.*] [less than or equal to] [g.sub.n] or [g.sub.n] [less than or equal to] [g.sup.*], for all n [member of] N [union] {0}, and inequality (18) holds for all g, [mathematical expression not reproducible]. Then [g.sup.*] is a fixed point of B in [mathematical expression not reproducible].

Corollary 11. Let (M, [less than or equal to], [d.sup.b]) be a preordered complete D.B.M space, [mathematical expression not reproducible], and B : M [right arrow] P(M) be a multifunction on [mathematical expression not reproducible] is a sequence generated by [g.sub.0], with [g.sub.0] [less than or equal to] [g.sub.1]. Assume that, for some k [member of] [0, 1) and

[mathematical expression not reproducible] (20)

where [bar.a] > 0, the following hold:

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

If [mathematical expression not reproducible], such that g [less than or equal to] q implies [mathematical expression not reproducible]. Then {MB([g.sub.n])} is a sequence in [mathematical expression not reproducible], and [mathematical expression not reproducible], for all n [member of] N [union] {0}, and inequality (21) holds for all g, [mathematical expression not reproducible]. Then [g.sup.*] is a fixed point of B in [mathematical expression not reproducible].

Corollary 12. Let (M, [less than or equal to], [d.sub.l]) be a preordered D.M space, [mathematical expression not reproducible], and B : M [right arrow] P(M) be a multifunction on [mathematical expression not reproducible] is a sequence generated by [g.sub.0], with [g.sub.0] [less than or equal to] [g.sub.1]. Assume that, for some [psi] [member of] [PSI] and

[mathematical expression not reproducible] (23)

where [bar.a] > 0, the following hold.

[mathematical expression not reproducible] (24)

[mathematical expression not reproducible]. (25)

If [mathematical expression not reproducible], such that [mathematical expression not reproducible]. Then {MB([g.sub.n])} is a sequence in [mathematical expression not reproducible], and [mathematical expression not reproducible]. Also if [g.sup.*] < [g.sub.n] or [g.sub.n] [less than or equal to] [g.sup.*], for all n [member of] N [union] {0}, and inequality (24) holds for all g, [mathematical expression not reproducible]. Then [g.sup.*] is a fixed point of B in [mathematical expression not reproducible].

Corollary 13. Let (M, [less than or equal to], [d.sub.l]) be a preordered complete D.M space, [mathematical expression not reproducible], and B : M [right arrow] P(M) be a multifunction on [mathematical expression not reproducible]; {MB([g.sub.n])} is a sequence generated by [g.sub.0], with [g.sub.0] [less than or equal to] [g.sub.1]. Assume that, for some k [member of] [0, 1) and

[mathematical expression not reproducible] (26)

where [bar.a] > 0, the following hold:

[mathematical expression not reproducible] (27)

[mathematical expression not reproducible]. (28)

If [mathematical expression not reproducible], such that g [less than or equal to] q implies [mathematical expression not reproducible]. Then {MB([g.sub.n])} is a sequence in [mathematical expression not reproducible], and [mathematical expression not reproducible]. Also if [g.sup.*] [less than or equal to] [g.sub.n] or [g.sub.n] [less than or equal to] [g.sup.*], for all n [member of] N [union] {0}, and inequality (27) holds for all g, [mathematical expression not reproducible]. Then [g.sup.*] is a fixed point of B in [mathematical expression not reproducible].

Example 14. Let M = [Q.sup.+] [union] {0} and let [d.sub.b] : M x M [right arrow] M be the D.B.M space on M defined by

[d.sub.b](g, q) = [(g + q).sup.2] [for all]g, q [member of] M (29)

with parameter t > 1. Define the multivalued mappings, B : M x M[right arrow] P(M) by

[mathematical expression not reproducible]. (30)

Considering [g.sub.0] = 1, r = 100, and [bar.a] = 1, b = 2, then [mathematical expression not reproducible]. Now [d.sub.b]([g.sub.0], B[g.sub.0]) = [d.sub.b] (1, B1) = [d.sup.b](1, 1/3) = 16/9. So we obtain a sequence {MB([g.sub.n])} = {1, 1/3, 1/9, 1/27, ....} in M generated by [g.sub.0]. Let t = 1.2, [psi](t) = 4t/5, and then t[psi](t) < t. Define

[mathematical expression not reproducible]. (31)

Now,

[mathematical expression not reproducible]. (32)

So the contractive condition does not hold on M. Now, for all [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (33)

So the contractive condition holds on [mathematical expression not reproducible]. As t = 1.2 > 1, then

[mathematical expression not reproducible]. (34)

Hence, all the conditions of Theorem 9 are satisfied. Now, we have that {MB([g.sub.n])} is a sequence in [mathematical expression not reproducible] and [mathematical expression not reproducible]. Moreover, 0 is a fixed point of B.

3. Fixed Point Results For Graphic Contractions

In this section, we present an application of Theorem 9 in graph theory. Jachymski [27] proved the result concerning contraction mappings on metric space with a graph. Hussain et al. [28] discussed the fixed points theorem for graphic contraction and gave an application. A graph G is affix if there is a way among any two vertices (see for details [29, 30]).

Definition 15. Let M be a nonempty set and [??] = (V([??]), Q(G)) be a graph such that V([??]) = M, and B : M [right arrow] P(M) is said to be multigraph preserving if (g, q) [member of] Q(G), and then (w, p) [member of] Q(G) for all w [member of] Bg and p [member of] Bq.

Theorem 16. Let (M, [d.sub.b]) be a complete D.B.M space endowed with a graph [??]. Suppose there exists a function [alpha]: M x M [right arrow] [0, [infinity]). Let, [mathematical expression not reproducible], and let for a sequence {MB([g.sub.n])} in M generated by [g.sub.0], with ([g.sub.0], [g.sub.1]) [member of] Q(G). Suppose that the following are satisfy:

(i) B is a graph preserving for all g, [mathematical expression not reproducible];

(ii) there exists [psi] [member of] [PSI] and

[mathematical expression not reproducible] (35)

where [bar.a] > 0 such that

[mathematical expression not reproducible], (36)

for all g, [mathematical expression not reproducible] and (g, q) [member of] Q(G);

(iii) [mathematical expression not reproducible] for all n [member of] N [union] {0} and t > 1.

Then, {MB([g.sub.n])} is a sequence in [mathematical expression not reproducible] and {MB([g.sub.n])} [right arrow] [g.sup.*]. Also, if and inequality (36) holds for [g.sup.*] and ([g.sub.n], [g.sup.*]) [member of] Q(G) or ([g.sup.*], [g.sub.n]) [member of] Q(G) for all n [member of] N [union] {0}, then B has a fixed point [g.sup.*] in [mathematical expression not reproducible].

Proof. Define, [alpha] : M x M [right arrow] [0, [infinity]) by

[mathematical expression not reproducible]. (37)

As {MB([g.sub.n])} is a sequence in M generated by [g.sub.0] with ([g.sub.0], [g.sub.1]) [member of] Q(G), we have [alpha]([g.sub.0], [g.sub.1]) [greater than or equal to] 1. Let [alpha](g, q) [greater than or equal to] 1, and then (g, q) [member of] Q(G). From (i), we have (w, p) [member of] Q(G) for all w [member of] Bg and p [member of] Bq. This implies that [alpha](w, p) = 1 for all w [member of] Bg and p v Bq. This implies that inf ([alpha](w, p) : w [member of] Bg, p [member of] Bq} = 1. So, B : M [right arrow] P(M) is a semi-[[alpha].sub.*]-admissible multifunction on [mathematical expression not reproducible]. Moreover, inequality (36) can be written as

[mathematical expression not reproducible], (38)

for all elements g, q in [mathematical expression not reproducible] with either [alpha](g, q) [greater than or equal to] 1 or [alpha](q, g) [greater than or equal to] 1. Also, (iii) holds. Then, by Theorem 9, we have that {MB([g.sub.n])} is a sequence in [mathematical expression not reproducible] and [mathematical expression not reproducible]. Now, [mathematical expression not reproducible] and either ([g.sub.n], [g.sup.*]) [member of] Q(G) or ([g.sup.*], [g.sub.n]) [member of] Q(G) for all n [member of] N [union] {0}, and inequality (36) holds for all g, [mathematical expression not reproducible]. Then we have [alpha]([g.sub.n], [g.sup.*]) [greater than or equal to] 1 or [alpha] ([g.sup.*],[g.sub.n]) [greater than or equal to] 1 for all n [member of] N [union] {0} and inequality (5) holds for all [mathematical expression not reproducible]. So, all the conditions of Theorem 9 are satisfied. Hence, by Theorem 9, B has a common fixed point [g.sup.*] in [mathematical expression not reproducible] and [d.sub.b] ([g.sup.*],[g.sup.*]) = 0.

4. Fixed Point Results for Single-Valued Mapping

In this section we discussed some fixed point results for self-mapping in complete D.B.M space. Let (M, [d.sub.b]) be a D.B.M space, [g.sub.0] [member of] M, and B : M [right arrow] M be a mapping. Let [g.sub.1] = B[g.sub.0], [g.sub.2] = B[g.sub.1]. Continuing this process, we construct a sequence [g.sub.n] of points in M such that [mathematical expression not reproducible]. We denote this iterative sequence by {[g.sub.n]}. We say that {[g.sub.n]} is a sequence in M generated by [g.sub.0].

Theorem 17. Let (M, [d.sub.b]) be a complete D.B.M space, [mathematical expression not reproducible], and B : M [right arrow] M be a semi-[alpha]-admissible function on [mathematical expression not reproducible] is a sequence in M generated by [g.sub.0], [alpha]([g.sub.0], [g.sub.1]) [greater than or equal to] 1. Assume that, for some [psi] [member of] [PSI] and

[mathematical expression not reproducible] (39)

where [bar.a] > 0, the following hold:

[mathematical expression not reproducible] (40)

[mathematical expression not reproducible]. (41)

Then, {[g.sub.n]} is a sequence in [mathematical expression not reproducible], and [mathematical expression not reproducible]. Also if [alpha]([g.sub.n], [g.sup.*]) [greater than or equal to] 1 or a ([g.sup.*], [g.sub.n]) [greater than or equal to] 1, for all n [member of] N [union] {0}, and inequality (40) holds for all g, [mathematical expression not reproducible]. Then B has a common fixed point [g.sup.*] in [mathematical expression not reproducible].

Proof. The proof of the above theorem is similar to Theorem 17.

Corollary 18. Let (M, [less than or equal to], [d.sup.b]) be a preordered complete D.B.M space, [mathematical expression not reproducible], and B : M [right arrow] M be a self-mapping on [mathematical expression not reproducible] is a sequence generated by [g.sub.0], with [g.sub.0] [less than or equal to] [g.sub.1]. Assume that, for some k [member of] [0, 1) and

[mathematical expression not reproducible] (42)

where [bar.a] > 0, the following hold:

[mathematical expression not reproducible] (43)

[mathematical expression not reproducible]. (44)

If [mathematical expression not reproducible], such that g [less than or equal to] q implies [mathematical expression not reproducible]. Then {[g.sub.n]} is a sequence in [mathematical expression not reproducible], and [mathematical expression not reproducible]. Also if [g.sup.*] [less than or equal to] [g.sub.n] or [g.sub.n] [less than or equal to] [g.sup.*], for all n [member of] N [union] {0}, and inequality (43) holds for all g, [mathematical expression not reproducible]. Then [g.sup.*] is a fixed point of B in [mathematical expression not reproducible].

Corollary 19. Let (M, [less than or equal to], [d.sub.l]) be a preordered complete D.M space, [mathematical expression not reproducible], and B : M [right arrow] M be a self-mapping on [mathematical expression not reproducible] is a sequence generated by [g.sub.0], with [g.sub.0] [less than or equal to] [g.sub.1]. Assume that, for some [psi] [member of] [PSI] and

[mathematical expression not reproducible] (45)

where [bar.a] > 0, the following hold:

[mathematical expression not reproducible] (46)

[mathematical expression not reproducible]. (47)

If g, [mathematical expression not reproducible], such that g [less than or equal to] q implies [mathematical expression not reproducible]. Then {[g.sub.n]} is a sequence in [mathematical expression not reproducible], and [mathematical expression not reproducible]. Also if [g.sup.*] [less than or equal to] [g.sub.n] or [g.sub.n] [less than or equal to] [g.sup.*], for all n [member of] N [union] {0}, and inequality (46) holds for all g, [mathematical expression not reproducible]. Then [g.sup.*] is a fixed point of B in [mathematical expression not reproducible].

Recall that if (M, [less than or equal to]) is a preordered set and A : M [right arrow] M is such that for g, q [member of] M, with g [less than or equal to] q implying Ag [less than or equal to] Aq, then the mapping A is said to be nondecreasing.

Corollary 20. Let (M, [less than or equal to], [d.sub.l]) be a preordered complete D.M space, r > 0, [g.sub.0] be an arbitrary point in [mathematical expression not reproducible] be a self-mapping on [mathematical expression not reproducible], and {[g.sub.n]} be a Picard sequence in M with initial guess [g.sub.0], with [g.sub.0] [less than or equal to] [g.sub.1]. For some k [member of] [0, 1) and

[mathematical expression not reproducible], (48)

where [bar.a] > 0, the following hold:

[mathematical expression not reproducible] (49)

[mathematical expression not reproducible]. (50)

Then, {[g.sub.n]} is a sequence in [mathematical expression not reproducible], such that [g.sub.n] [less than or equal to] [g.sub.n+1] and [mathematical expression not reproducible]. Also if [g.sup.*] [less than or equal to] [g.sub.n] or [g.sub.n] [less than or equal to] [g.sup.*], for all n [member of] N [union] {0}, and inequality (49) holdsfor all g, [mathematical expression not reproducible]. Then [g.sup.*] is a fixed point of B in [mathematical expression not reproducible].

Data Availability

The data used to support the findings of this study are included within the article.

https://doi.org/10.1155/2018/5816364

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

Each author equally contributed to this paper and read and

approved the final manuscript.

Acknowledgments

The authors acknowledge with thanks the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, for financial support.

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Tahair Rasham (iD), (1) Abdullah Shoaib (iD), (2) Badriah A. S. Alamri, (3) and Muhammad Arshad (iD) (1)

(1) Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan

(2) Department of Mathematics and Statistics, Riphah International University, Islamabad 44000, Pakistan

(3) Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Tahair Rasham; tahir_resham@yahoo.com

Received 9 August 2018; Accepted 27 November 2018; Published 19 December 2018

Academic Editor: Ali Jaballah
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Title Annotation:Research Article
Author:Rasham, Tahair; Shoaib, Abdullah; Alamri, Badriah A.S.; Arshad, Muhammad
Publication:Journal of Mathematics
Date:Jan 1, 2018
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