The purpose of this article is to establish fixed point results for a pair of α∗-dominated fuzzy mappings fulfilling generalized locally new α∗- ψ-Ćirić type rational contractive conditions on a closed ball in complete dislocated metric spaces. Example and application are given to demonstrate the novelty of our results. Our results extend several comparable results in the existing literature.
Let S be a metric space and H : S ⟶ S be a mapping. A point w ∈ S is called a fixed point of S if w = Sw . A lot of fixed point results for contractive mappings defined on the entire space hold. It is possible that H : S ⟶ S is not contractive on whole space while H : Y ⊆ S ⟶ S is a contraction. Shoaib et al. [27], proved the result related with α∗ - ψ-Ćirić type multivalued mappings by intersection of an iterative sequence on closed ball with graph. Recently Rasham et al. [21], proved fixed point results for a pair of multivalued mappings on closed ball for new rational type contraction in dislocated metric spaces. Further fixed point results on closed ball can be observed in [4, 30].
Many authors proved fixed point theorems in complete dislocated metric space. The idea of dislocated topologies has useful applications in the context of logic programming semantics [12]. Dislocated metric space [16] is a generalization of partial metric space [17], which has applications in computer sciences. Nadler [18], started the research of fixed point results for the multivalued contractive mappings in a complete metric space. Asl et al. [5] gave the idea of α∗- ψ contractive multifunctions, α∗-admissible mapping and got some fixed point conclusions for these multifunctions can be seen in [1, 27]. Recently, Senapati and Dey [22], introduced the concept of a pair of multi β∗-admissible mapping and established some common fixed point theorems for multivalued β∗ - ψ-contractive mappings. Recently, Alofi et al. [2] introduced the concept of α-dominated multivalued mappings and established some fixed point results for such mappings on a closed ball in complete dislocated quasi b -metric spaces.
Since the notion of fuzzy sets a lot of work has been done in this area [33] Zadeh. In the field of fixed point theory Weiss [32] and Butnariu [8] presented the notion of fuzzy mappings and proved many concerned results. Heilpern [11] proved a fixed point result for fuzzy mappings which was the further analogue of Nadler’s multivalued result [18] in Hausdorff metric. Motivated by the Heilpern’s results, the fixed point theory for fuzzy contraction using the Hausdorff metric has become more mature in different directions by various authors [23–25].
In this paper, we establish common fixed point of α -dominated mappings for new Ćirić type rational fuzzy contractions on a closed ball in complete dislocated metric spaces. Interesting new results in metric space and partial metric space can be obtained as corollaries of our theorems. An application is derived in the setting of an ordered dislocated metric space for multi ⪯-dominated mappings. Also some new fixed point results with graphic contractions on closed ball for multi graph dominated fuzzy mappings on dislocated metric space are established. Example is given to show the superiority of our result. Our results generalize several comparable results in the existing literature.We give the following concepts which will be used in the paper.
Definition 1.1. Let M be a nonempty set and let dl : M × M → [0, ∞) be a function, called a dislocated metric (or simply dl-metric), if for any c, g, z ∈ M, the following conditions satisfy:
The pair ( M, dl) is called a dislocated metric space.
It is clear that if dl ( c, g) =0, then from (i), c = g. But if c = g, dl ( c, g) may not be 0 . For c ∈ M and ɛ > 0, is a closed ball in ( M, dl) .We use D . L .space instead by dislocated metric space.
Example 1.2. [4] If M = R+ ∪ {0} , then dl ( c, g) = c + g defines a dislocated metric dl on M.
Definition 1.3. [4] Let ( M, dl) be a D . L . space.
(i) A sequence { cn} in ( M, dl) is called Cauchy sequence if given ɛ > 0, there corresponds n0 ∈ N such that for all n, m≥ n0 we have dl ( cm, cn) < ɛ or
(ii) A sequence { cn} dislocated-converges (for short dl -converges) to c if In this case c is called a dl-limit of { cn} .
(iii) ( M, dl) is called complete if every Cauchy sequence in M converges to a point c ∈ M such that dl ( c, c) =0.
Definition 1.4. [27] Let K be a nonempty subset of D . L . space M and let c ∈ M . An element g0 ∈ K is called a best approximation in K if
If each c ∈ M has at least one best approximation in K, then K is called a proximinal set.
We denote CP ( M) be the set of all closed proximinal subsets of M . Let Ψ denote the family of all nondecreasing functions ψ : [0, + ∞) → [0, + ∞) such that for all t > 0, where ψn is the nth iterate of ψ . if ψ ∈ Ψ, then ψ ( t) < t for all t > 0 .
Definition 1.5. [22] Let S, T : M → P ( M) be the closed valued mulifunctions and β : M × M → [0, + ∞) be a function. We say that the pair ( S, T) is β★-admissible if for all c, g ∈ M
where β★ ( Tc, Sg) = inf { β ( a, b) : a ∈ Tc, b ∈ Sg} . When S = T, then we obtain the definition of α∗-admissible mapping given in [5].
Definition 1.6. Let ( M, dl) be a D . M .space, S, T : M → P ( M) be multivalued mappings and α : M × M → [0, + ∞). Let A ⊆ M, we say that the S is α∗-dominated on A, whenever α∗ ( c, Sc) ≥1 for all c ∈ A, where α∗ ( c, Sc) = inf { α ( c, b) : b ∈ Sc} . In particular, if A is a closed ball, we say that S is semi α∗-dominated on A. If A = M, then we say that the S is α∗-dominated on M . If S, T : M → M be self mappings, then S is α-dominated on A, whenever α ( c, Sc) ≥1 for all c ∈ A .
Definition 1.7. [27] The function Hdl : P ( M) × P ( M) → R+, defined by
is called dislocated Hausdorff metric on P ( M) .
Lemma 1.8. [27] Let ( M, dl) be a D . L .space and ( P ( M) , Hdl) is a dislocated Hausdorff metric space on P ( M) . If for all H, U ∈ P ( M) and for each h ∈ H, let uh ∈ U satisfying dl ( h, U) = dl ( h, uh) . Then Hdl ( H, U) ≥ dl ( h, uh).
Example 1.9. Let Define the mapping α : M × M → [0, ∞) by
Define the multivalued mappings S, T : M → P ( M) by
and
Suppose c = 3 and g = 2 . As 3 > 2, then α (3, 2) ≥1 . Now, α★ ( S3, T2) = inf { α ( a, b) : this means α★ ( S3, T2) <1, that is, the pair ( S, T) is not α★-admissible. Also, α★ ( S3, S2) ≱1 and α★ ( T3, T2) ≱1 . This implies S and T are not α★-admissible individually. As, α★ ( c, Sc) = inf { α ( c, b) : b ∈ Sc} ≥1, for all c ∈ M . Hence S is α★-dominated mapping. Similarly α★ ( g, Tg) = inf { α ( g, b) : b ∈ Tg} ≥1 . Hence it is clear that S and T are α★-dominated but not α★-admissible.
Definition 1.10. [23] A fuzzy set A is a function with domain M and values in [0, 1], F ( M) is the collection of all fuzzy sets in M. If A is a fuzzy set and c ∈ M, then the function values A ( c) is called the grade of membership of c in A . The β-level set of fuzzy set T, is denoted by [ A] β, and defined as:
Now we select by the family F ( M) of all fuzzy sets, a subfamily with stronger properties, i.e. the subfamily of the approximate quantities, denoted W ( M) and defined by:
Definition 1.11. [11] A fuzzy subset A of M is an approximate quantity iff its β-level set is a compact convex subset of M for each β ∈ [0, 1] and .
At this point, we can introduce a notion of fuzzy mapping, i.e., a mapping with value in the family of approximate quantities.
Definition 1.12. [11] Let M be an arbitrary set and Y any metric linear space. A mapping T : M → W ( Y) is called a fuzzy mapping.
Note that we can see a fuzzy mapping T : M → W ( Y) as a fuzzy subset of M × Y, T : M × Y → [0, 1], in the sense that T ( c, y) = T ( c) ( y) .
Definition 1.13. [23] A point c ∈ M is called a fuzzy fixed point of a fuzzy mapping T : M → W ( M) if there exists 0 < β ≤ 1 such that c ∈ [ Tc] β.
Main result
Let ( M, dl) be a D . L .space, c0 ∈ M and S, T : M → W ( M) be two fuzzy mappings on M. Moreover let γ, β : M → [0, 1] be two real functions. Let c1 ∈ [ Sc0] γ( c0) be an element such that dl ( c0, [ Sc0] γ( c0)) = dl ( c0, c1) . Let c2 ∈ [ Tc1] β( c1) be such that dl ( c1, [ Tc1] β( c1)) = dl ( c1, c2) . Let c3 ∈ [ Sc2] γ( c2) be such that dl ( c2, [ Sc2] γ( c2)) = dl ( c2, c3) . Continuing this process, we construct a sequence cn of points in M such that c2 n+1 ∈ [ Sc2 n] γ( c2 n) and c2 n+2 ∈ [ Tc2 n+1] β( c2 n+1), where n = 0, 1, 2, . . .. Also dl ( c2 n, [ Sc2] γ( c2)) = dl ( c2 n, c2 n+1) , dl ( c2 n+1, [ Tc2 n+1] β( c2 n+1)) = dl ( c2 n+1, c2 n+2) . We denote this iterative sequence by { TS ( cn)} . We say that { TS ( cn)} is a sequence in M generated by c0 .
Theorem 2.1.Let ( M, dl) be a complete D . L . space. Let α : M × M → [0, ∞) . Let, r > 0, and S, T : M → W ( M) be two fuzzy α∗-dominated mappings on Assume that, for some ψ ∈ Ψ, γ ( c) , β ( g) ∈ (0, 1] and, for a suitable a > 0,
the following hold:
for all with either α ( c, g) ≥1 or α ( g, c) ≥1 . Furthermore, suppese that
Then { TS ( cn)} is a sequence in , α ( cn, cn+1) ≥1 for all n ∈ N ∪ {0} and Also if α ( cn, c∗) ≥1 or α ( c∗, cn) ≥1 for all n ∈ N ∪ {0} the inequality (2.1) holds for c∗ also. Then S and T have common fuzzy fixed point c∗ in .
Proof. Consider a sequence { TS ( c0)} . From (2.2), we get
It follows that,
Let for some j ∈ N. If j = 2 i + 1, where . Since S, T : M → P ( M) be an α∗-dominated mappings on , so α∗ ( c2 i, Sc2 i) ≥1 and α∗ ( c2 i+1, Tc2 i+1) ≥1 . Now by using Lemma 1.8, we obtain,
If max { dl ( c2 i, c2 i+1) , dl ( c2 i+1, c2 i+2)} = dl ( c2 i+1, c2 i+2) , then dl ( c2 i+1, c2 i+2) ≤ ψ ( dl ( c2 i+1, c2 i+2)) . This is the contradiction to the fact that ψ ( t) < t for all t > 0 . So max { dl ( c2 i, c2 i+1) , dl ( c2 i+1, c2 i+2)} = dl ( c2 i, c2 i+1) . Hence, we obtain
As α∗ ( c2 i, Sc2 i) ≥1 and c2 i+1 ∈ Sc2 i, so α ( c2 i, c2 i+1) ≥1 . Similarly we can get α∗ ( c2 i-1, [ Tc2 i-1] β( c2 i-1)) ≥1 and c2 i-1 ∈ [ Tc2 i-1] β( c2 i-1), so α ( c2 i-1, c2 i) ≥1 . Now by using (2.1), and Lemma 1.8, we have
If max { dl ( c2 i, c2 i-1) , dl ( c2 i, c2 i+1)} = dl ( c2 i, c2 i+1) , then
This is the contradiction to the fact that ψ ( t) < t for all t > 0 . If
then
As ψ is nondecreasing function, so
by using the above inequality in (2.3), we obtain
continuing in this way, we obtain
Now, if j = 2 i, where . Then, similarly, we have
Now, by combining (2.4) and (2.5), we obtain
Now,
Thus Hence for all therefore is a sequence in As be two fuzzy semi α∗ −dominated mappings on , so and for all n ∈ N. Now inequality (2.6) can be written as
Fix ε > 0 and let such that Let n, m ∈ N with m > n > n(ε), then, we obtain,
Thus we proved that { TS(c0)} is a Cauchy sequence in . As every closed ball in a complete D.L. space is complete, so there exists such that that is
By assumption for all Since S and T are α∗−dominated, by Definition 1.6 we have that and Now by using Lemma 1.8 and inequality (2.1), we have
Letting , and using the inequalities (2.7) and (2.8), we can easily get that and hence or . Similarly, by using,
we can show that Hence S and T have a common fuzzy fixed point c* in Since and (S, T) be the pair of sub α∗-dominated multifunction on , we have so Now,
This implies that,
Corollary 2.2.Let ( M, dl) be a complete D . L . space. Take a function α : M × M → [0, ∞) . Let, r > 0, and S, T : M → W ( M) be two fuzzy semi α∗-dominated mappings on Assume that, for some ψ ∈ Ψ, γ ( c) , β ( g) ∈ (0, 1] and Dl′ ( c, g) = max { dl ( c, g) , dl ( c, [ Sc] γ( c)) , dl ( g, [ Tg] β( g))} , the following hold:
for all with either α ( c, g) ≥1 or α ( g, c) ≥1 . Also
Then { TS ( cn)} is a sequence in and Also, if the inequality (2.9) holds for c∗ and either α ( cn, c∗) ≥1 or α ( c∗, cn) ≥1 for all n ∈ N ∪ {0}. Then S and T have a common fuzzy fixed point c∗ in and dl ( c∗, c∗) =0 .
Corollary 2.3.Let ( M, dl) be a complete D . L . space. Take a function α : M × M → [0, ∞) . Let, r > 0, and S : M → W ( M) be a semi α∗-dominated fuzzy mappings on Assume that, for some ψ ∈ Ψ, γ ( g) ∈ (0, 1] and, for a suitablea > 0,
the following hold:
for all with α ( c, g) ≥1 . Also
Then { S ( cn)} is a sequence in and Also, if the inequality (2.10) holds for c∗ and either α ( cn, c∗) ≥1 or α ( c∗, cn) ≥1 for all n ∈ N ∪ {0}. Then S has a fuzzy fixed point c∗ in and dl ( c∗, c∗) =0 .
Let M be a nonempty set, ⪯ is a partial order on M and A ⊆ M. We say that a ⪯ B whenever for all b ∈ B, we have a ⪯ b . A mapping S : M → W ( M) is said to be prevalent on A if a ⪯ Sa for each a ∈ A ⊆ M . If A = M, then S : M → W ( M) is said to be totally prevalent.
Theorem 2.4.Let ( M, ⪯ , dl) be an ordered complete D . L . space. Let, r > 0, and S, T : M → W ( M) be two fuzzy prevalent mappings on Assume that, for some ψ ∈ Ψ, γ ( c) , β ( g) ∈ (0, 1] and, for a suitablea > 0,
the following hold:for allwith eitherc ⪯ gorg ⪯ c . Also Then { TS ( cn)} is a sequence in and Also if the inequality (2.11) holds for c∗ and either cn ⪯ c∗ or c∗ ⪯ cn for all n ∈ N ∪ {0}. Then S and T have a common fuzzy fixed point c∗ in and dl ( c∗, c∗) =0.
Proof. Let α : M × M → [0, + ∞) be a mapping defined by α ( c, g) =1 for all with c ⪯ g, and α ( c, g) =0 for all other elements c, g ∈ M . As S and T are the fuzzy prevalent mappings on so c ⪯ [ Sc] γ( c) and c ⪯ [ Tc] β( c) for all This implies that c ⪯ b for all b ∈ [ Sc] γ( c) and c ⪯ e for all c ∈ [ Tc] β( c) . So, α ( c, b) =1 for all b ∈ [ Sc] γ( c) and α ( c, e) =1 for all c ∈ [ Tc] β( c) . This implies that inf { α ( c, g) : g ∈ [ Sc] γ( c)} =1, and inf { α ( c, g) : g ∈ [ Tc] β( c)} =1 . Hence α∗ ( c, [ Sc] α( c)) =1, α∗ ( c, [ Tc] β( c)) =1 for all So, S, T : M → W ( M) are the semi α∗-dominated mapping on Moreover, inequality (2.11) can be written as
for all elements c, g in with either α ( c, g) ≥1 or α ( g, c) ≥1 . Also, inequality (2.12) holds. Then, by Theorem 2.1, we have { TS ( cn)} is a sequence in and Now, and either cn ⪯ c∗ or c∗ ⪯ cn implies that either α ( cn, c∗) ≥1 or α ( c∗, cn) ≥1 . So, all the conditions of Theorem 2.1 are satisfied. Hence, by Theorem 2.1, S and T have a common fuzzy fixed point c∗ in and dl ( c∗, c∗) =0.
Example 2.5. Let M = Q+ ∪ {0} and let dl : M × M → M be the complete dislocated metric on M defined by
Define the fuzzy mappings, S, T : M → W ( M) by,
and,
Now, we consider
Considering, then Now we have So we obtain a sequence in M generated by c0 . Let and,
Now take 5, 6 ∈ M, then, we have
So, the contractive condition does not hold on whole space M . Now for all with either α ( c, g) ≥1 or α ( g, c) ≥1, we have
So, the contractive condition holds on Also,
Hence, all the conditions of Theorem 2.1 are satisfied. Now, we have { TS ( cn)} is a sequence in α ( cn, cn+1) ≥1 and Also, α ( cn, 0) ≥1 or α (0, cn) ≥1 for all n ∈ N ∪ {0} . Moreover, 0 is a common fuzzy fixed point of S and T .
Fixed point results for graphic contractions
In this section we presents an application of Theorem 2.1 in graph theory. Jachymski, [15], proved the result concerning for contraction mappings on metric space with a graph. Hussain et al. [14], introduced the fixed points theorem for graphic contraction and gave an application. A graph K is connected if there is a path between any two different vertices (see for detail [7, 31]).
Definition 3.1. Let M be a nonempty set and K = ( V ( K) , F ( K)) be a graph such that V ( K) = M, A ⊆ M. A mapping S : M → P ( M) is said to be multi graph dominated on A if ( c, g) ∈ F ( K) , for all g ∈ Sc and c ∈ A.
Theorem 3.2. Let ( M, dl) be a complete D . L . space endowed with a graph K. Suppose there exist a function α : M × M → [0, ∞) . Let, r > 0, S, T : M → W ( M) and let for a sequence { TS ( cn)} in M generated by c0, with ( c0,c1) ∈ F ( K) . Suppose that the following satisfy:
(i) S and T are graph dominated for all c, g∈
(ii) there exists ψ ∈ Ψ and
where a > 0, such that
for all and ( c,g) ∈ F ( K) or ( g,c) ∈ F ( K);
(iii) for all n ∈ N ∪ {0} .
Then, { TS ( cn)} is a sequence in ( cn, cn+1) ∈ F ( K) as the sequence { TS ( cn)} → c∗ . Also, if ( cn, c∗) ∈ F ( K) or ( c∗, cn) ∈ F ( K) for all n ∈ N ∪ {0} and the inequality (3.1) holds for all Then S and T have common fuzzy fixed point c∗ in .
Proof. Define, α : M × M → [0, ∞) by
As { TS ( cn)} is a sequence in c generated by c0 with ( c0,c1) ∈ F ( K) , we have α ( c0, c1) ≥1 . Let, α ( c, g) ≥1, then ( c, g) ∈ F ( K) . From (i) we have ( c, [ Sc] γ( c)) ∈ F ( K) for all g ∈ [ Sc] γ( c) this implies that α ( c, g) =1 for all g ∈ [ Sc] γ( c) . This further implies that inf { α ( c, g) : g ∈ [ Sc] γ( c)} =1 . Thus S is a α∗-dominated multifunction on Also if ( c, g) ∈ F ( K) , we have α ( c, g) =1 and hence α∗ ( c, [ Sc] γ( c)) =1 . Similarly it can be proved α∗ ( g, [ Tg] β( g)) =1 . Now,condition (ii) can be written as
for all with either α ( c, g) ≥1 or α ( g, c) ≥1 . By including condition (iii), we obtain all the conditions of Theorem 2.1. Now, by Theorem 2.1, we have { TS ( cn)} is a sequence in α ( cn, cn+1) ≥1, that is ( cn, cn+1) ∈ F ( K) and { TS ( cn)}→ c∗ ∈ Also, if ( cn, c∗) ∈ F ( K) or ( c∗, cn) ∈ F ( K) for all n ∈ N ∪ {0} and the inequality (3.1) holds for all Then, we have α ( cn, c∗) ≥1 or α ( c∗, cn) ≥1 for all n ∈ N ∪ {0} and the inequality (2.1) holds for all Again, by Theorem 2.1, S and T have common fuzzy fixed point c∗ in .
Fixed point results for multi-valued mapping
In this section, we show that Theorem 2.1 can be utilized to derive a common fixed point for a multivalued mapping in a dislocated metric space.
Theorem 4.1.Let ( M, dl) be a complete D . L . space. Suppose there exist a function α : M × M → [0, ∞) . Let, r > 0, and G, H : M → P ( M) be a α∗-dominated mappings on Assume that, for some ψ ∈ Ψ andwherea > 0 the following hold:
for allwith eitherα ( c, g) ≥1 orα ( g, c) ≥1 . Also Then { HG ( cn)} is a sequence in , α ( cn, cn+1) ≥1 for all n ∈ N ∪ {0} and Also if α ( cn, c∗) ≥1 or α ( c∗, cn) ≥1 for all n ∈ N ∪ {0} and the inequality (2.1) holds for c∗ also. Then G and H have common fixed point c∗ in .
Proof. Let δ : M → (0, 1] be an arbitrary mapping. Consider S, T : M → W ( M) be the two fuzzy mappings defined as
and
We get
and
So, the conditions (4.1) and (4.2) of Theorem 4.1 becomes the conditions (2.1) and (2.2) of Theorem 2.1. This implies that there exists c∗ ∈ [ Sc] γ( c) ∩ [ Tc] β( c) = Gc ∩ Hc .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each author equally contributed to this paper, read and approved the final manuscript.
Footnotes
Acknowledgment
This article was supported by Universita della Calabria, Dipartimento di Matematica e Informatica, Arcavacata di Rende (CS), Italy. Article processing charges will be given by Universita della Calabria (No. DEMACS EX 60% 2019). Therefore, the authors acknowledge with thanks to Universita della Calabria, for the financial support.
References
1.
AliM.U.,KamranT. and KarapnarE.,Further discussion on modified multivalued α* - ψ-contractive type mapping,Filomat29(8) (2015),1893–1900.
2.
AloA.S.M.,Al-MazrooeiA.E.,LeyewB.T. and AbbasM.,Common fixed points of α-dominated multivalued mappings on closed balls in a dislocated quasi b-metric space,J Nonlinear Sci Appl10(7) (2017),3456–3476.
3.
ArshadM.,KadelburgZ.,RadenovićS.,ShoaibA. and ShuklaS.,Fixed Points of α-Dominated Mappings on Dislocated Quasi Metric Spaces,Filomat31(11) (2017),3041–3056.
4.
ArshadM.,ShoaibA. and VetroP.,Common Fixed Points of a Pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Dislocated Metric Spaces,Journal of Function Spaces2013 (2013), Article ID 63818.
5.
AslJ.H.,RezapourS. and ShahzadN.,On fixed points of α - ψ contractive multifunctions,Fixed Point Theory Appl2012 (2012),212.
6.
BotaM.F.,ChifuC. and KarapinarE.,Fixed point theorem for generalized ( α* - ψ) Ćirić-type contractive multivalued operator in b- metric spaces,J Nonlinear Sci Appl9 (2016),1165–1177.
7.
BojorF.,Fixed point theorems for Reich type contraction on metric spaces with a graph,Nonlinear Anal75 (2012),3895–3901.
8.
ButnariuD.,Fixed point for fuzzy mapping,Fuzzy Sets Systems7 (1982),191–207.
9.
CiricL.B.,A Generalization of Banach’s Contraction Principle,Proc Amer Math Soc45 (1974),267–273.
10.
CiricL.B.,Generalized contractions and fixed point theorems,Publications of the Institute of Mathematics12 (1971),19–26.
11.
HeilpernS.,Fuzzy mappings and fixed point theorem,Journal of Mathematical Analysis and Applications83(2) (1981),566–569.
12.
HitzlerP. and SedaA.K.,Dislocated Topologies,Journal of Electrical Engineering51(12/s) (2000),3–7.
13.
HussainN.,AhmadJ. and AzamA.,Generalized fixed point theorems for multi-valued α - ψ -contractive mappings,J Inequalities and Appl2014 (2014),348.
14.
HussainN.,Al-MezelS.,SalimiP., Fixed points for ψ - graphic contractions with application to integral equations, Abstract and Applied Analysis, Volume 2013, Article ID 575869.
15.
JachymskiJ.,The contraction principle for mappings on a metric space with a graph,Proc Amer Math Soc1(136) (2008),1359–1373.
16.
KarapnarE.,PiriH.,AlsulamiH.H., Fixed points of modified F-contractive mappings in complete metric-like spaces, Journal of Function Spaces, Volume 2015, Article ID 270971, pp. 9.
17.
MatthewsS.G.,Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applicationn,New York Acad Sci728 (1994),183–197.
RashamT.,ShoaibA.,HussainN. and ArshadM.,Fixed point results for a pair of α*-dominated multivalued mappings with applications,U.P.B. Sci Bull, Series A81(3),2019.
20.
RashamT.,ArshadM. and KhanS.U.,Fixed point results on closed ball for a new rational type contraction mapping in complete dislocated metric spaces,Turkish Journal of Analysis and Number Theory5 (2017),86–92.
21.
RashamT.,ShoaibA.,ArshadM. and KhanS.U.,Fixed point results for a pair of multivalued mappings on closed ball for new rational type contraction in dislocated metric spaces,J Inequal Spec Functs8 (2017),74–85.
22.
SenapatiT. and DeyL.K.,Common fixed point theorems for multivalued β* - ψ -contractive mappings,Thai J Math15(3) (2017),747–759.
23.
ShahzadA.,ShoaibA.,KhammahawongK.,KumamP., New Ciric Type Rational Fuzzy F-Contraction for Common Fixed Points, Springer Nature Switzerland AG, ECONVN (2019), SCI 809, pp. 215–229.
24.
ShahzadA.,ShoaibA. and MahmoodQ.,Fixed Point Theorems for Fuzzy Mappings in b- Metric Space,Italian Journal of Pure and Applied Mathematics38 (2017),419–427.
25.
ShoaibA.,KumamP.,ShahzadA.,PhiangsungnoenS. and MahmoodQ.,Fixed point results for fuzzy mappings in a b-metric space,Fixed Point Theory and Applications2018 (2018),2.
26.
ShoaibA.,α - η Dominated Mappings and Related Common Fixed Point Results in Closed Ball,J Concr Appl Math13(1-2) (2015),152–170.
27.
ShoaibA.,HussainA.,ArshadM. and AzamA.,Fixed point results for α* - ψ - Ćirić type multivalued mappings on an intersection of a closed ball and a sequence with graph,J Math Anal7 (2016),41–50.
28.
ShoaibA.,Fixed Point Results for α* - ψ -multivalued Mappings,Bull Math Anal Appl8(4) (2016),43–55.
29.
ShoaibA.,ArshadM.,RashamT. and AbbasM.,Unique fixed point results on closed ball for dislocated quasi G-metric spaces,Transactions of A. Razmadze Mathematical Institute171(2) (2017),221–230.
30.
ShoaibA.,RashamT.,HussainN. and ArshadM.,α*-dominated set-valued mappings and some generalised fixed point results,Journal of the National Science Foundation of Sri Lanka47(2),2019.
31.
TiammeeJ. and SuantaiS.,Coincidence point theorems for graph-preserving multi-valued mappings,Fixed Point Theory Appl2014 (2014),70.
32.
WeissM.D.,Fixed points and induced fuzzy topologies for fuzzy sets,J Math Anal Appl50 (1975),142–150.
33.
ZadehL.A.,Fuzzy Sets,Information and Control8(3) (1965),338–353.
