Fixed point theorems for multivalued mappings in G- and G-cone metric spaces using G-Hausdorff contractions

Introduction

Fixed-point theory is a core component of nonlinear analysis, which has been found to be a useful tool in various applications, including differential equations and integral equations, control systems, economics, and computer science. The classical Banach contraction principle is an important finding in this field that ensures that the unique fixed point of self-mappings of complete metric spaces is a contraction. Since its formulation, this principle has spawned an enormous literature to generalize fixed-point results in terms of weakening the assumption made on either the metric structure or the contractive condition or the character of the mappings one makes. One significant advance in this direction was achieved by Nadler, who generalized the contraction principle of Banach to multivalued mappings using the Hausdorff metric. This extension was feasible for studying the fixed points of set-valued operators, which naturally occur in optimization problems, differential inclusions, and equilibrium models. After this, it has been noted that much interest has been given to fixed point theory in generalized metric spaces, in which the classical metric axioms are relaxed to facilitate wider classes of nonlinear problems. Among these generalizations is the definition of G-metric spaces, which was proposed by Mustafa and Sims, in which the distance function takes three variables rather than two. It is more flexible and may be interpreted geometrically richer than standard metric spaces. Many fixed-point findings, such as Banach-type and cyclic contraction theorems, have been successfully extended to G-metric spaces and their variants [7,17,21] . Huang and Zhang independently proposed cone metric spaces, where the distance is a value of an ordered Banach space, with a cone. Cone metric spaces have also been applied to solve problems and systems in the form of vectors controlled by partial orderings [5,18,22] . The intersection of the two structures results in the presentation of G-cone metric spaces, which brings together the ideas of G-metric spaces and cone metric spaces on one platform. The distance of a G-cone metric space is a function of three variables and is valued in a cone, which means that both scalar-valued and concurrently vector-valued fixed-point problems can be treated. This single structure has become the focus of increasing attention because of its capacity to make generalizations about a variety of known results and offers a natural framework for partially ordered nonlinear systems [1,3,23] . The fixed-point theorems of multivalued mappings have been widely studied in recent years in many generalized metric-type spaces, such as b-metric spaces, partial metric spaces, and cone metric spaces. Other authors have generalized Nadler-type contraction results using generalized Hausdorff and graph-based methods [6,8,15,24] . However, many of these studies are confined only to scalar-valued spaces or are heavily based on abstract set-theoretic arguments, which tend to mask the convergence mechanism at work. Furthermore, only a small number of studies have discussed multivalued contractions in the common context of G-metric and G-cone metric spaces in terms of constructive methods designed through overt recursive inequalities. Based on these findings, this study attempts to develop new fixed point theorems for multivalued contractive-type mappings in both Gmetric and G-cone metric spaces through a generalized G-Hausdorff contraction condition. We generalize some classical findings of Nadler, Mustafa-Sims, and Huang-Zhang [1,5,7,25] to compact-valued multivalued operators and cone induced partial orderings. In contrast to the use of purely abstract methods, the suggested algorithm builds sequences as a result of multivalued mapping and examines the convergence of the latter using explicit contractive estimates, producing transparent and verifiable proofs.

Let M be a nonempty setM and the function d: M × M × M → [0, ∞) be defined such that for all p, q, r, s ∈ M, The following circ and umstathes are fulfilled:  d(p, p, q) = 0 if and only if p = q.  d(p, p, q) > 0 whenever p ≠ q,  d(p, p, q) ≤ d(p, q, r),  d(p, q, r) remains unchanged under any permutation of p, q, r ,  d(p, q, r) ≤ d(p, s, s) + d(s, q, r).

A structure (M, d) satisfying the above is termecalledmetric space.

The space (M, d) is said to be complete if every G-Cauchy sequence converges to M point.

Definition 2.3 (Cone in a Normed Vector Space)

Let V ibera eal Banach space and lC ⊂ Vbe a nontrivial closed subset. Then C is a cone if:  C is closed, nonempty, and C ≠ {0},  λx + μy ∈ C for all x, y ∈ C and λ, μ ≥ 0,  If x ∈ C and -x ∈ C, then x = 0.

A partial ordering on V is then described by: x ⪯ y ⟺ y -x ∈ C.

Definition 2.4 (Multivalued Mappings)

A mapping F: M → 2 M ∖ {∅} is termed multivalued if for each p ∈ M, F(p) is a nonempty subset of M. With reference to fixedpoint theory using the G-Hausdorff distance, values of F are usually assumed to be compact.

Let A, B ⊂ M be compact sets that are nonempty in a G-metric (or G-cone metric) space. Then the G-Hausdorff distance H d (A, B) is defined as: In cone-based structures, the distance d is evaluated with respect to the cone-induced partial order in V.

Suppose that V is a Banach space and that the C ⊂ V satisfying:  C + C ⊂ C, and αC ⊂ C for all α ≥ 0,  C ∩ (-C) = {0}, where 0 is the zero element in V.

Define the order

 It satisfies all G-metric conditions (adapted with respect to the cone order),  The image of d lies in the cone C.

Completeness and convergence are defined analogously, using the norm and partial ordering from V. In the context of cone metric structures, this expression takes values in a cone Q ⊂ E, and comparisons use the induced partial order ⪯.

Lemma 2.8 (Convergence of Geometrically Contractive Sequences)

Let {y n } ⊂ Y be a sequence satisfying the inequality: G(y n , y n+1 , y n+1 ) ≤ λ n G(y 0 , y 1 , y 1 ) for some 0 < λ < 1

Then, the sequence {y n } is G-Cauchy.

Suppose {y n } is a G-Cauchy sequence that converges to a point y * ∈ Y. Then, any convergent subsequence of {y n } also converges to y * , and

Let S:Y→Y be a single-valued mapping such that, given (Y,G) as a full G-metric space,

Then S possesses a unique fixed point in Y.

Theorem 2.11 (Huang-Zhang Fixed Point in Cone Metric Spaces)

Let S:Y→Y meet the following conditions, and let (Y,d) be a full cone metric space equipped with a normal cone Q⊂E.

Then S admits a unique fixed point in Y.

Theorem 2.12 (The Fixed Point Theorem for Cone Metric Spaces with Multivalued Contractions)

Consider a complete cone metric space (Y, d), and let F: Y → 𝒞ℬ(Y) be a multivalued operator such that:

Then F has at least one fixed point, i.e., there exists y ∈ Y with y ∈ F(y). (This result generalizes Nadler's theorem to cone metric spaces.)

Lemma 2.13 (Fixed Point Containment via Upper Semi-Continuity)

If a multivalued operator F: Y → 𝒞ℬ(Y) in a G -metric or G -cone metric space has compact values and is upper semi-continuous, and if a fixed point exists, then that point belongs to its image: y * ∈ F(y * ). (This ensures the constructed sequence's limit is a valid fuxed point.)

Lemma 2.14 (Compact Multivalued Mappings)

Let F: Y → 𝒞ℬ(Y) be a multivalued operator with compact values. If Y is complete under a G-metric (or G-cone metric), then the graph of F, defined as:

is closed in the topology induced by the G-metric.

The first part of this paper prepares a solid background for the fixed point results by introducing and investigating fundamental concepts in G-metric and G-cone metric theories. Definition 2.1 The notion of a G-metric space introduced by Mustafa and Sims by a metric with three arguments. This is an extension of the classical metrics in which richer geometric interpretations are allowed, and is essential from the point of view of nonlinear and multivalued analysis. Definition 2.2 defines the concepts of a G-Cauchy sequence and G-convergence that generalize the traditional notions of convergence and completeness. These are crucial for the fixed point analysis where convergence of sequences arising from PMEP contractive mappings constitutes a major proof technique. Summarizing this concept, Definition 2.3 introduces cones in Banach spaces which then induces a partial ordering, and this partial ordering will play a crucial role in the structure of G-cone metric spaces. Definition 2.7 then represents a synthesis of these ideas which defines G-cone metric spaces in terms of the metric being a G-metric with its image in a cone. This two-sided system is particularly handy to study vector-valued or partially ordered systems. Nevertheless, as we demonstrate in the following sections, the multivalued mappings and the G-Hausdorff distance provided in Definitions 2.5 and 2.6 serve to extend the fixed point results to set-valued functions. The convergence results are strengthened by Lemma 2.9, which states that geographically contractive sequences are G-Cauchy, and Lemma 2.10, which guarantees the uniqueness of limits in such sequences. These estimates guarantee stability in the convergence analysis. Theorems 2.11, 2.12 and 2.13 give essential fixed point theorems for single valued and multivalued mappings in G-metric and G-cone metric spaces. Theorem 2.13 is a generalization of Nadler's result on cone metric spaces. Lastly, we have the following Lemma 2.14, which enforces a fixed point to be in the same set, given by compactness of a graph in Lemma 2.15. Combined these classical facts provide a theoretical foundation for the advanced fixed point theorems presented later in the paper.

Main Results

Theorem 3.1: Let (Y, d G ) be a complete G -metric space where the G -metric is symmetric. Suppose ℱ: Y → 𝒦(Y) is a multivalued function assigning to each point a nonempty compact subset of Y. Assume that for all a, b, c ∈ Y, the following contraction condition holds:

where θ ∈ [0,1) and H G represents the G-Hausdorff metric. Then there exists a point y * ∈ Y such that y * ∈ ℱ(y * ); i.e., y * is a fixed point of ℱ.

Proof

Choose an arbitrary y 0 ∈ Y. Since ℱ(y 0 ) is nonempty and compact, select y 1 ∈ ℱ(y 0 ). Inductively define a sequence {y n } ⊂ Y such that y n+1 ∈ ℱ(y n ) for each n ∈ ℕ. For a small tolerance δ n: = Thus, the contradiction implies:

Hence, y * is a fixed point of ℱ.

The existence of fixed points of multivalued contractive mappings in full symmetric G-metric spaces is demonstrated by Theorem 3.1. With the aid of a generalized G-Hausdorff contraction condition, the theorem establishes a sequence {𝑦 𝑛 } where each point is taken from the image of the next one under the multivalued map F. By comparing to a recursive inequality with an additive error term 𝛿 𝑛 = 1/(𝑛 + 1) it is proved that the sequence satisfies a contractive estimate and then converges to a limit point 𝑦 * . The completeness of the G-metric space and the compactness of the set-valued mapping serve as the foundation for the convergence proof. To confirm that y* is, in fact, the fixed point, it is shown to belong to F(y*) by way of contradiction. Our result generalizes Nadler's fixed-point theorem to multivalued mappings in generalized matric spaces and gives an effective and meticulous method for studying multivalued mappings.

Statement: Let (M, d G ) be a complete G-metric space where the metric is symmetric. Consider a multivalued operator ℱ: M → 𝒦(M), where 𝒦(M) represents the collection of all compact nonempty subsets of M. Suppose that ℱ satisfies the generalized contraction criterion:

for some constant μ ∈ [0,1). Then, there exists a point u * ∈ M such that u * ∈ ℱ(u * ), i.e., u * is a fixed point of ℱ. This result is a direct application of Theorem 3.1. Given the assumptions-namely, the completeness of the G-metric space, the compactness and non-emptiness of ℱ(u) for all u ∈ M, and the Hausdorff-type contractive condition with coefficient μ ∈ [0,1)-all prerequisites of the theorem are met. Thus, by the previously established convergence process, a fixed point u * exists such that u * ∈ ℱ(u * ).

Corollary 3.2 (Convergence of Generated Sequence to a Fixed Point)

Let (𝑋, 𝐺) be a complete symmetric G-metric space, and suppose 𝒯: 𝑋 → 𝒦(𝑋) is a multivalued mapping such that for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, the following contractive condition holds:

𝐻 𝐺 (𝒯(𝑥), 𝒯(𝑦), 𝒯(𝑧)) ≤ 𝛾 ⋅ 𝐺(𝑥, 𝑦, 𝑧)

for some constant 𝛾 ∈ [0,1). Assume an iterative sequence {𝑥 𝑛 } is constructed beginning from an arbitrary point 𝑥 0 ∈ 𝑋, such that each 𝑥 𝑛+1 ∈ 𝒯(𝑥 𝑛 ) satisfies:

Then, the sequence {𝑥 𝑛 } converges to a limit point 𝑥 * ∈ 𝑋 𝑟 and this point satisfies the fixed point condition:

The recursive inequality establishes a decay pattern in the distances 𝐺(𝑥 𝑛+1 , 𝑥 𝑛 , 𝑥 𝑛 ), showing that the sequence contracts over time. Because 𝛾 < 1 and the additive term 1 𝑛+1 → 0, the sequence is proven to be G-Cauchy. The completeness of (𝑋, 𝐺) ensures that the sequence converges to some limit point 𝑥 * ∈ 𝑋. Moreover, since the values of 𝒯 are assumed to be compact, and the sequence {𝑥 𝑛 } is drawn from successive images of 𝒯, the limit 𝑥 * must lie in the closure of 𝒯(𝑥 * ). Hence, 𝑥 * ∈ 𝒯(𝑥 * ), confirming it as a fixed point.

Example (Illustration of the Fixed Point Existence)

Let 𝑀 = [0,1] and define the 𝐺-metric 𝑑 𝐺 as: 𝑑 𝐺 (𝑎, 𝑏, 𝑐) = |𝑎 -𝑏| + |𝑏 -𝑐|, ∀𝑎, 𝑏, 𝑐 ∈ 𝑀 Define a multivalued operator ℱ: 𝑀 → 𝒦(𝑀) as:

That is, for any 𝑥 ∈ [0,1], ℱ(𝑥) is the closed interval from 𝑥/2 to 1. We verify the assumptions:  ℱ(𝑥) is nonempty and compact for every 𝑥 ∈ [0,1].  For all 𝑥, 𝑦, 𝑧 ∈ [0,1], the G -Hausdorff distance between the sets ℱ(𝑥), ℱ(𝑦), ℱ(𝑧) satisfies:

Construct a sequence {𝑥 𝑛 } with 𝑥 0 = 1:

This sequence monotonically decreases and converges to 0. Since ℱ(0) = [0,1], we have:

Thus, 0 is a fixed point of the mapping ℱ.

Let (𝑀, 𝑑 𝐺 ) be a complete 𝐺-cone metric space over a real Banach space 𝑉, equipped with a normal cone 𝐶 ⊂ 𝑉. Consider a multivalued mapping ℱ: 𝑀 → 𝒦(𝑀), where the collection of all nonempty compact subsets of M is represented by 𝒦(𝑀),.

Assume that for all 𝑢, 𝑣, 𝑤 ∈ 𝑀, the following inequality holds:

𝐻 𝐺 (ℱ(𝑢), ℱ(𝑣), ℱ(𝑤)) ⪯ 𝛼 ⋅ 𝑑 𝐺 (𝑢, 𝑣, 𝑤)

For some constant 𝛼 ∈ [0,1), where ⪯ denotes the cone-induced partial ordering and 𝐻 𝐺 is the G-Hausdorff metric. Then, there exists 𝑢 * ∈ 𝑀 such that 𝑢 * ∈ ℱ(𝑢 * ).

Proof Sequence Generation

Let 𝑢 0 ∈ 𝑀 be any initial point. Since ℱ(𝑢 0 ) is compact and nonempty, pick 𝑢 1 ∈ ℱ(𝑢 0 ). Proceed inductively: assume 𝑢 𝑛 ∈ 𝑀 is known. Then, choose 𝑢 𝑛+1 ∈ ℱ(𝑢 𝑛 ) such that:

Applying the contractive property of ℱ:

𝐻 𝐺 (ℱ(𝑢 𝑛-1 ), ℱ(𝑢 𝑛 ), ℱ(𝑢 𝑛 )) ⪯ 𝛼 ⋅ 𝑑 𝐺 (𝑢 𝑛-1 , 𝑢 𝑛 , 𝑢 𝑛 )

𝑑 𝐺 (𝑢 𝑛 , 𝑢 𝑛+1 , 𝑢 𝑛+1 ) ⪯ 𝛼 ⋅ 𝑑 𝐺 (𝑢 𝑛-1 , 𝑢 𝑛 , 𝑢 𝑛 ) for all 𝑛 ≥ 1

Distance Recurrence Relation

Define 𝛿 𝑛: = 𝑑 𝐺 (𝑢 𝑛 , 𝑢 𝑛+1 , 𝑢 𝑛+1 ) ∈ 𝐶. Then, from (3.1), we get the recurrence:

By applying (3.2) recursively:

Since 𝛼 ∈ [0,1) and 𝛿 0 ∈ 𝐶, it follows that 𝛿 𝑛 → 𝜃 𝑉 , the zero element of 𝑉.

Showing {𝑢 𝑛 } is G-Cauchy Let 𝜖 ∈ 𝑉 with 𝜖 ≫ 𝜃 𝑉 . Since 𝛿 𝑛 → 𝜃 𝑉 , there exists 𝑁 0 ∈ ℕ such that:

Using the G-cone triangle inequality (property G 5), for 𝑚 > 𝑛, we have:

Existence of the Limit

Because (𝑀, 𝑑 𝐺 ) is complete, the Cauchy sequence {𝑢 𝑛 } converges to some point 𝑢 * ∈ 𝑀, such that:

Using the triangle inequality again:

𝑑 𝐺 (𝑢 𝑛+1 , 𝑢 * , 𝑢 * ) ⪯ 𝑑 𝐺 (𝑢 𝑛+1 , 𝑢 𝑛 , 𝑢 𝑛 ) + 𝑑 𝐺 (𝑢 𝑛 , 𝑢 * , 𝑢 * )

Taking the limit as 𝑛 → ∞, and knowing that both terms on the right approach 𝜃 𝑉 , we get:

Thus, 𝑢 * is a fixed point of the mapping ℱ.

It has been demonstrated that Theorem 3.2 makes it simple to determine the fixed point result for multivalued mappings in symmetric G-cone metric spaces. The theorem generalizes fixed point theorems from the G-metric to the vector-valued case, by imposing generalized G-Hausdorff contraction condition on a cone induced partial order. We recursively define a sequence such that the observables {𝑢 𝑛 } are contained in the image of their successive term. The recurrence relation 𝛿 𝑛 ⪯ 𝛼 𝑛 𝛿 0 guarantees that the sequence verifies a contractive property in the cone. The sequence is G-Cauchy, using the triangle inequality for Gcone. The compactness of the space ensures convergence to a limit point 𝑢 * ∈ 𝑀 The upper semi continuity (or closed graph property) of the mapping rationalizes that 𝑢 * ∈ 𝐹(𝑢 * ). This theorem is not only an extension of scalar-valued fixed-point results but also a rigorous analytical framework for vector-valued multivalued mappings under cone structures.

This mapping assigns to each x ∈ [0,1] a compact interval within [0,1].

We claim that ℱ satisfies the contraction condition:

for all x, y, z ∈ M. The Hausdorff distance between such intervals behaves linearly with respect to their endpoints, and thus scales with the input G-distance. Since < 1, the contraction condition is satisfied, and hence, by the earlier corollary, a fixed point exists.

Corollary 1 ensures uniqueness under contraction for single-valued maps by reducing the problem to a degenerate multivalued case. Corollary 2 extends existence results to compact domains, where the completeness is guaranteed by compactness. The example demonstrates practical applicability using a basic linear mapping over the unit interval, showing how fixed point theory can apply in concrete scenarios even in cone-ordered spaces.

Conclusion

In this article a common and generalized theoretical framework of fixed point theorems for multivalued contractive type mappings is given in G-metric and G-cone metric spaces. It is also shown that some previous main results due to Nadler, Mustafa-Sims and Huang-Zhang can be easily deduced from our results. Contrarily to scalar-oriented or set-theoretic methods, the one developed here provides convergent sequences based on recursive inequalities with stronger convergence properties for the both scalar and vector-valued multivalued mappings. One of the main elements of this work has consisted in showing how the G-cone metric structure contains the G-metric structure as a special case. This discussion allows more flexible analysis, in particular for Banach spaces which have a natural partial ordering in the sense of normal cones. Compactness, usc or contractive properties are such that the fixed-point existence follows, and they will facilitate proofs of convergence and adaptations of algorithms. Our results enrich not only the theory of fixed points but also its applications to various fields of mathematics and computer science. This encompasses operator theory, nonlinear analysis, optimisation, and systems described by multivalued dynamics. The template is a convenient tool in stability, equilibrium, and convergence analysis of these complicated systems. For future studies, there are some promising directions. One possible generalization is to utilize these results for probabilistic, intuitionistic fuzzy, or fuzzy G-cone metric spaces, which can play an important role in modeling uncertainty real problems. Furthermore, the efficient algorithms with the constructed iterative sequences can be useful in practical applications of various problems in applied mathematics and engineering. Another interesting direction is that our approach may be generalized to the equilibrium problem, variational inclusion, and the game theory model with multivalued strategies. In addition, the combination of fractional derivative/integral-type brakes with this G-cone framework may result in powerful tools for modeling memorydependent or hereditary systems. Finally, this paper provides with a solid framework of the theoretical and applied developments of the fixed-point theory in the abstract metric type of and cone-metric type of spaces.

References

1. Azam A, Mehmood N. Fixed point theorems for multivalued mappings in G-cone metric spaces. J Inequal Appl. 2013;2013:354.
2. Ullah K, Katiyar SK. Generalized G-Hausdorff space and applications in fractals. Chaos Solitons Fractals. 2023;174:113819.
3. Chen CM. On set-valued contractions of Nadler type in tυs-G-cone metric spaces. Fixed Point Theory Appl. 2012;2012:52.
4. Hammad HA, Alshehri MG, Shehata A. Control functions in G-metric spaces: novel methods for θ-fixed points and θ-fixed circles with an application. Symmetry (Basel). 2023;15(1):164.
5. Khamsi MA. Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010;2010:315398.
6. Kutbi MA, Sintunavarat W. Fixed point analysis for multi-valued operators with graph approach by the generalized Hausdorff distance. Fixed Point Theory Appl. 2014;2014:142.
7. Mustafa Z, Jaradat MMM, Aydi H, Alrhayyel A. Some common fixed points of six mappings on Gb-metric spaces using (E.A) property. Eur J Pure Appl Math. 2018;11(1):90-109.
8. Karapinar E, Aydi H, Postolache M. Fixed point results for multivalued operators in partial b-metric spaces. Mathematics. 2023;11(17):1768.
9. Rehman SU, Jabeen S, Abbas F, Ullah H, Khan I. Common fixed-point theorems for compatible and weakly compatible maps in fuzzy cone metric spaces. Ann Fuzzy Math Inform. 2020;19(1):1-19.
10. Ige AO, Olaoluwa HO, Olaleru JO. Some fixed points of multivalued maps in multiplicative metric spaces. Heliyon. 2022;8(12):e12453.
11. Rao NS. Coupled fixed points of (φ, ψ, θ)-contractive mappings in partially ordered b-metric spaces. Heliyon. 2022;8(12):e12442.
12. Maheswari JU, Dillibabu K, Mani G, Thabet STM, Kedim I, Vivas-Cortez M. On new common fixed point theorems via bipolar fuzzy b-metric space with their applications. PLoS One. 2024;19(6):e0305316.
13. Okeke GA, Francis D, de la Sen M. Some fixed-point theorems for mappings satisfying rational inequality in modular metric spaces with applications. Heliyon. 2020;6(9):e04785.
14. Saleh HN, Imdad M, Sintunavarat W. Fixed points which belong to the set of unit values of a suitable function on fuzzy metric spaces. Appl Gen Topol. 2023;24(1):9-24.
15. Abd Zaid AI, Hashim AM. Fixed point results for multivalued mappings in partial Hausdorff metric spaces. Basrah J Sci. 2020;38(3):373-386.
16. Jeyaraman M, Aydi H, De La Sen M. New results for multivalued mappings in Hausdorff neutrosophic metric spaces. Axioms. 2022;11(12):724.
17. Bilgili N, Karapınar E. Cyclic contractions via auxiliary functions on G-metric spaces. Fixed Point Theory Appl. 2013;2013:49.
18. Latif A, Shaddad FY. Fixed point results for multivalued maps in cone metric spaces. Fixed Point Theory Appl. 2010;2010:941371.
19. Wangwe L, Kumar S. Common fixed point theorems for F-Kannan-Suzuki type mappings in TVS-valued cone metric space with some applications. J Math. 2022;2022:6504663.
20. Sumitra R, Aruna R, Hemavathy R. A fixed point theorem for non-self G-contractive type mappings in cone metric space endowed with a graph. Nonlinear Funct Anal Appl. 2021;26(5):1105-1114.
21. Diwan SD, Thakur AK, Raja H. Common fixed-point theorems in metric spaces satisfying an implicit relation. Adv Fixed Point Theory. 2016;6(2):167-174.
22. Diwan SD, Thakur AK, Raja H. Some generalization on fixed point theorems related to different types of fuzzy metric spaces. Indian J Appl Res. 2016;6(10):635-644.
23. Diwan SD, Thakur AK, Raja H. Fixed point of expansion mapping in fuzzy Menger space with CLRs property. Int J Innov Sci Math. 2016;4(4):143-145.
24. Diwan SD, Thakur AK, Raja H. Common fixed point theorem of integral type in Menger probabilistic metric spaces. Int J Appl Sci Math. 2016;3(4):153-156.
25. Diwan SD, Thakur AK, Raja H. Common fixed points of four mappings in metric spaces. Int J Sci Res. 2016;5(5):2089-2092.