EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 996-1008 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Common Fixed Point for Contractive Mappings in p-Pompeiu-Hausdorff Metric Spaces Arta Ekayanti1,2,∗, Marjono Marjono1, Mohamad Muslikh1, Sa’adatul Fitri1 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Veteran Road, Malang 65145, Indonesia 2 Department of Mathematics Educations, Faculty of Teacher Training and Education, Universitas Muhammadiyah Ponorogo, Ponorogo, East Java 63471, Indonesia Abstract. In this paper we establish the existence of a common fixed point from a pair of set- valued mappings. By utilizing the concept of convergence of set-valued mappings’ sequences, both ordinary and pointwise convergence, we establish a common fixed point theorem. This our newly result is a generalization of common fixed point theorem of set-valued mappings on partial metric spaces. Further, we establish newly common fixed point theorem under ϕ-contraction on partial metric spaces. 2020 Mathematics Subject Classifications: 47H10, 26E25 Key Words and Phrases: common fixed point, set-valued mappings, contractive mappings, partial metric spaces, p-Pompeiu-Hausdorff Metric Spaces 1. Introduction Discussions regarding Banach’s principle of contraction often appear in various refer- ences. Many generalizations are also given for which a comparative study of these gener- alizations is given by Rhoades [18]. One of the generalizations of the Banach contraction principle that is also quite widely discussed is in the set-valued mapping. Various results of the generalization of Banach’s contraction principle can be found in [8, 13, 14, 17, 20] and reference therein. Further results on the general fixed point of the set-valued map- ping of the contractive type may be found in Kubiak [12] and Singh [19]. On the other hand a generalization of the principle of Banach contraction for single-valued mapping on partial metric spaces can be seen in [2, 5, 10, 11] and reference therein. Furthermore, a generalization of the Banach contraction principle for set-valued mappings in partial ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5152 Email addresses: arta ekayanti@ub.ac.id (A. Ekayanti), marjono@ub.ac.id (Marjono), mslk@ub.ac.id (M. Muslikh), saadatulfitri@ub.ac.id (S. Fitri) https://www.ejpam.com 996 © 2024 EJPAM All rights reserved. A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 997 metric spaces can be found in [1, 4, 6]. This generalization builds upon the Banach con- traction principle for set-valued mappings, which was initially introduced by Nadler [16]. And further results on the general fixed point of the set-valued mapping on partial metric space be found in Aydi et. al. [7] and Ahmad et. al. [3]. In this paper, we will generalize some results of Aydi et. al.[7] and Ahmad et. al. [3]. Referring to Kubiak [12], we will use the common fixed point existence of a sequence of set-valued mappings to derived on a pair of set-valued mappings so that the existence of a common fixed point is guaranted. Furthermore, referring to Singh [19] we will use some functions that he has defined to give a new generalization of the contraction of Banach’s principle for set-valued mappings on partial metric spaces. By using this contraction we obtain the common fixed points of a pair of set-valued mappings. 2. Preliminaries Let (X, p) be a partial metric spaces. Suppose that CBp(X) be class of all nonempty, closed and bounded subsets of X. Let mapping Hp : X → CBp(X) define Hp(A,B) = max{sup{p(x,B) : x ∈ A}, sup{p(y,A) : y ∈ B}}, for each A,B ∈ X and p(x,B) = inf{p(x, y) : y ∈ B}. The mapping Hp is p-Pompeiu- Hausdorff (partial Pompeiu Hausdorff) metric, and the pairs (CBp(X), Hp) is called p- Pompeiu-Hausdorff metric spaces. (The use of the term Pompeiu-Hausdorf refers to [9]). Some properties of metric Hp can be found in [6, 7, 15]. Definition 1. [15] Let (CBp(X), Hp) be a p-Pompeiu-Hausdorff metric spaces. A se- quence (Fn) in CBp(X) converges to set F ∈ CBp(X) if lim n→∞ Hp(Fn, F ) = Hp(F, F ). Definition 2. [15] Let (CBp(X), Hp) be a p-Pompeiu-Hausdorff metric spaces. A se- quence (Fn) in CBp(X) is said to a Cauchy sequence if lim n,m→∞ Hp(Fn, Fm) exists and finite. Sequence (Fn) is Cauchy sequence if the sequence Hp(Fn, Fm) tends to some λ ∈ R as n,m approach to infinity, that is, limn,m→∞Hp(Fn, Fm) = λ < ∞, i.e. for each ε > 0 there exists N ∈ N such that |Hp(Fn, Fm)− λ| < ε, for all n,m ≥ N. Furthermore, lets consider the properties of Cauchy sequence (Fn) in (CBp(X), Hp). A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 998 Theorem 1. [15] A sequence (Fn) in p-Pompeiu-Hausdorff metric spaces (CBp(X), Hp) is Cauchy if and only if for all ε > 0 there exists N ∈ N such that Hp(Fn, Fm)−Hp(Fm, Fm) < ε, for every n,m ≥ N . Definition 3. [15] A p-Pompeiu-Hausdorff metric spaces (CBp(X), Hp) is called complete if every Cauchy sequences Fn ∈ CBp(X) converges to F ∈ CBp(X) and lim n→∞ Hp(Fn, F ) = Hp(F, F ). One of the relationships between the partial metric space (X, p) and the p-Pompeiu- Hausdorff metric space (CBp(X), Hp) can be seen in its completeness. This is shown in the following Theorem 2. Theorem 2. [15] If (CBp(X), Hp) be a complete partial metric spaces then (CBp(X), Hp) is complete. For set-valued mapping F : X → CBp(X), a point x ∈ X is called a fixed point of F if x ∈ F (x). Analogously, for F and G set-valued mappings from X into CBp(X), a point x ∈ X is called as a common fixed point of F and G if x ∈ F (x) and x ∈ G(x). 3. Main Results In the following discussion, we assume that (X, p) is a complete partial metric space. Theorem 3. Let (CBp(X), Hp) be a p-Pompeiu-Hausdorff metric spaces. Suppose that Fn, Gn : X → CBp(X), n ∈ N be sequence of set-valued mappings on CBp(X), there exists κ where 0 ≤ κ < 1 such that Hp(Fm(x), Gn(y)) ≤ κmax { p(x, y), p(x, Fm(x)), p(y,Gn(y)), 1 2 (p(x,Gn(y)) + p(y, Fm(x))) } , for each m,n ∈ N and x, y ∈ X, then (Fn) and (Gn) have a common fixed point, i.e. there exist a point x ∈ X such that x ∈ Fm(x) and x ∈ Gn(x) for each m,n ∈ N. Proof. Let we consider that 0 ≤ κ < 1. For the first we assume that κ = 0. Suppose that x0 ∈ X and x1 ∈ F1(x0), then for all n ∈ N we have p(x1, Gn(x1) ≤ Hp(F1(x0), Gn(x1)) = 0. It means p(x1, Gn(x1)) = 0. Since Gn are closed for each n then x1 ∈ Gn(x1). In the similar way, we can obtain that for x0 ∈ X and x1 ∈ G1(x0), then for all n ∈ N we have p(x1, Fn(x1)) ≤ Hp(G1(x0), Fn(x1)) = 0, A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 999 i.e., p(x1, Fn(x1)) = 0, then x1 ∈ Fn(x1). From this result, it can be seen that x1 is the common fixed point of Fn and Gn. Next we assume that κ ̸= 0. Suppose that x0 ∈ X and x1 ∈ F1(x0). Furthermore, define the sequence (xn) where x2n ∈ Gn(x2n−1) and x2n−1 ∈ Fn(x2n−2) are such that p(x2n−1, x2n) ≤ 1√ κ Hp(Fn(x2n−2), Gn(x2n−1)) p(x2n, x2n+1) ≤ 1√ κ Hp(Fn(x2n), Gn(x2n−1)), for n = 1, 2, 3, . . . . Suppose that xn ̸= xn+1 for all n ∈ N. For n being even, we have x2n ∈ Fn+1(x2n) thus for each m ∈ N p(x2n, Gm(x2n)) ≤ Hp(Fn+1(x2n), Gm(x2n)) ≤ κmax{p(x2n, x2n), p(x2n, Fn+1(x2n)), p(x2n, Gm(x2n)), 1 2(p(x2n, Fn+1(x2n)) + p(x2n, Gm(x2n))} ≤ κp(x2n, Gm(x2n)). Since 0 < κ < 1 then p(x2n, Gm(x2n) = 0. Therefore, we have x2n ∈ Gm(x2n) for each m ∈ N. Similarly, for n being odd numbers, we have x2n+1 ∈ Gn+1(x2n+1), and for every m implies p(x2n+1, Fm(x2n+1)) ≤ Hp(Gn+1(x2n+1), Fm(x2n+1)) ≤ κmax{p(x2n+1, x2n+1), p(x2n+1, Gn+1(x2n+1)), p(x2n+1, Fm(x2n+1)), 1 2(p(x2n+1, Fm(x2n+1)) +p(x2n+1, Gn+1(x2n+1))} ≤ κp(x2n+1, Fm(x2n+1)). Analogous to n is even, it can be concluded that p(x2n+1, Fm(x2n+1)) = 0, it means x2n+1 ∈ Fm(x2n+1). For the next step, we will show that (xn) is Cauchy sequence in (X, p). Let we consider p(x2n, x2n+1)) ≤ 1√ κ Hp(Fn+1(x2n), Gn(x2n−1)) ≤ 1√ κ κmax{p(x2n, x2n−1), p(x2n, Fn+1(x2n)), p(x2n−1, Gn(x2n−1)), 1 2(p(x2n, Gn(x2n−1)) + p(x2n−1, Fn+1(x2n))} ≤ √ κmax{p(x2n, x2n−1), p(x2n, x2n+1), p(x2n−1, x2n), 1 2(p(x2n, x2n+1) + p(x2n−1, x2n))} ≤ √ κmax{p(x2n−1, x2n), p(x2n, x2n+1)}, when p(x2n, x2n+1) is the maximum then we have x2n = x2n+1. Since xn ̸= xx+1 for each n thus we get a contradiction. Therefore, we have the maximum is p(x2n−1, x2n). It implies p(x2n, x2n+1) ≤ √ κp(x2n−1, x2n). A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 1000 In the similar way, we have p(x2n+1, x2n+2) ≤ √ κp(x2n, x2n+1). Therefore, we obtain p(x2n, x2n+1)) ≤ √ κp(x2n−1, x2n) ≤ √ κ √ κp(x2n−2, x2n−1) = ( √ κ)2p(x2n−2, x2n−1) ≤ ( √ κ)2 √ κp(x2n−3, x2n−2) = ( √ κ)3p(x2n−3, x2n−2) ... ≤ ( √ κ)2np(x0, x1) = κnp(x0, x1). And also we have p(x2n+1, x2n+2)) ≤ κnp(x1, x2). Let t(x0) := max{p(x0, x1), p(x1, x2)}, then for m > n w have p(xm, xn)) ≤ m−(n+1)∑ i=0 p(xn+i, xn+1+i) ≤ m−(n+1)∑ i=0 hn+it(x0) = t(x0) m−(n+1)∑ i=0 hn+i = thn m−(n+1)∑ i=0 hi ≤ t(x0)hn 1−h Since t(x0)hn 1−h → 0 as n → ∞, it means we are already shown that (xn) is a Cauchy sequence in X. Since (X, p) is complete partial metric space then there exists x ∈ X such that xn → x whereas n → ∞. Let we observe the following condition p(x2n−1, Gm(x)) ≤ Hp(Fn(x2n−2), Gm(x)) ≤ κmax{p(x2n−2, x), p(x2n−2, Fn(x2n−2)), p(x,Gm(x)), 1 2(p(x2n−2, Gm(x)) + p(x, Fn(x2n−2)))} ≤ κmax{p(x2n−2, x), p(x2n−2, x2n−1), p(x,Gm(x)), 1 2(p(x2n−2, Gm(x)) + p(x, x2n−1)))} by taking n → ∞ we obtain p(x,Gm(x)) ≤ κmax { p(x, x), p(x, x), p(x,Gm(x)), 1 2 (p(x,Gm(x)) + p(x, x)) } , for each m. Therefore, we have p(x,Gm(x)) ≤ κp(x,Gm(x)). A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 1001 Since 0 ≤ κ < 1 then p(x,Gm(x)) = 0. It implies that x ∈ Gm(x) because Gm(x) is closed. Similarly, we can show that x ∈ Fn(x) for each n. So, we obtain x ∈ Gm(x) and x ∈ Fn(x) for each m,n. It means x is common fixed point of Gm and Fn for every m and n. This complete the proof. In Theorem 3 above, we have the principle of contraction on set-valued mapping se- quences as follows: Hp(Fn(x), Gn(y)) ≤ κmax { p(x, y), p(x, Fn(x)), p(y,Gn(y)), 1 2 (p(x,Gn(y)) + p(y, Fn(x))) } , for each n ∈ N and x, y ∈ X and κ ∈ [0, 1). By looking at the sequences Fn and Gn in Theorem 3 respectively as constant se- quences, Corollary 1 can be obtained as follows. This Corollary 1 is a generalization of the result [12] on the partial metric space. Corollary 1. Let (CBp(X), Hp) be a p-Pompeiu-Hausdorff metric spaces. Suppose that F,G : X → CBp(X) with the following condition Hp(F (x), G(y)) ≤ κmax { p(x, y), p(x, F (x)), p(y,G(y)), 1 2 (p(x,G(y)) + p(y, F (x))) } , (1) for each x, y ∈ X where 0 ≤ κ < 1, then F and G have a common fixed point. By utilizing the concept of pointwise convergence of set-valued sequences, we can also investigate the existence of a common fixed point of set-valued mappings. Let see on the following theorem. Theorem 4. Let (CBp(X), Hp) be a p-Pompeiu-Hausdorff metric spaces. Suppose that Fn, Gn : X → CBp(X) sequences in CBp(X). Sequences Fn, Gn converging pointwise to F,G : X → CBp(X) respectively. If the following condition holds Hp(Fn(x), Gn(y)) ≤ κmax { p(x, y), p(x, Fn(x)), p(y,Gn(y)), 1 2 (p(x,Gn(y)) + p(y, Fn(x))) } , (2) for every x, y ∈ X and n ∈ N where 0 ≤ κ < 1 then F and G have a common fixed point. Proof. Take any point x, y ∈ X. Let u ∈ Fn(x) and v ∈ F (x), then we have p(y, u) ≤ p(y, v) + p(v, u)− p(v, v) ≤ p(y, v) + p(v, u) ≤ p(y, F (x)) + p(u, F (x)). Consequently p(y, Fn(x)) ≤ p(y, F (x)) +Hp(Fn(x), F (x)). On the other side we also have the following condition p(y, v) ≤ p(y, u) + p(u, v)− p(u, u) ≤ p(y, u) + p(u, v) ≤ p(y, Fn(x)) + p(v, Fn(x)). A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 1002 It implies p(y, F (x)) ≤ p(y, Fn(x)) +Hp(F (x), Fn(x)). Therefore, we have |p(y, Fn(x))− p(y, F (x))| ≤ Hp(Fn(x), F (x)). (3) On the similar way we can also show that |p(x,Gn(y))− p(x,G(y))| ≤ Hp(Gn(y), G(y)). (4) Furthermore, by using inequality (3) and (4) and also the continuity of Hp then by taking n → ∞ in inequality (2) we obtain Hp(F (x), G(y)) ≤ κmax { p(x, y), p(x, F (x)), p(y,G(y)), 1 2 (p(x,G(y)) + p(y, F (x))) } , These conditions show that the set-valued mapping F and G satisfies the hypothesis on Corollary 1. Thus, based on corollary 1 it can be concluded that F and G have a common fixed point. This complete the proof. Let we consider that for any positive real numbers s and t holds 1 2 (s+ t) ≤ max{s, t}. It implies for any positive real numbers p, q, r, s and t we have max { p, q, r, 1 2 (s+ t) } ≤ max{p, q, r, s, t}. (5) Therefore, we can derive a generalization of contractions that the theorem uses as well as the corollary on the previous discussion. In corollary 1, which indicates the existence of a common fixed point of set-valued mapping, by utilizing inequality (5) we can obtain a generalization of contractions (1) as follows Hp(F (x), G(y)) ≤ κmax {p(x, y), p(x, F (x)), p(y,G(y)), p(x,G(y)), p(y, F (x))} . (6) On the other sides, Singh has given a definition of a function in generalizing the principle of contraction of several references therein (Definition 2.1 in [19]) as follows. Definition 4. Suppose that ϕ : [0,∞) → [0,∞) a function that satisfy the following conditions: (i) ϕ is non-decreasing upper semi-continuous, (ii) ϕ(2u) < u for each u > 0. Using this definition, Singh established the existence of common fixed points of set- valued mappings (Theorem 2.2 in [19]). Referring to these results, we will generalize the theorem to a more general metric space, which is a partial metric space. A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 1003 Theorem 5. Let (CBp(X), Hp) be a p-Pompeiu-Hausdorff metric spaces. Suppose that F,G : X → CBp(X) be set-valued mappings that satisfy the following conditions Hp(F (x), G(y)) ≤ ϕ(max{p(x, y), p(x, F (x)), p(y,G(y)), p(x,G(y)), p(y, F (x))}), (7) for each x, y ∈ X where ϕ : R+ → R+ such that ϕ be a non-decreasing upper semi- continuous and ϕ(2u) < u for u > 0, then set-valued mappings F and G have a unique common fixed point. Proof. Take any x0 ∈ X, but fixed. Let x0 /∈ F (x0) and take x1 ∈ F (x0), then from (7) we obtain p(x1, G(x1)) ≤ Hp(F (x0), G(x1)) ≤ ϕ(max{p(x0, x1), p(x0, F (x0)), p(x1, G(x1), p(x0, G(x1)), p(x1, F (x0))}) ≤ ϕ(max{p(x0, x1), p(x1, G(x1))}) ≤ ϕ(p(x0, x1) + p(x1, G(x1))). Consider that: if p(x0, x1) < p(x1, G(x1)) then p(x1, G(x1)) ≤ ϕ(p(x0, x1) + p(x1, G(x1))) < ϕ(p(x1, G(x1)) + p(x1, G(x1))) = ϕ(2p(x1, G(x1))) < p(x1, G(x1). This condition shows a contradiction, then it must be p(x0, x1) ≥ p(x1, G(x1)). Therefore, we have p(x1, G(x1)) ≤ ϕ(p(x0, x1) + p(x0, x1)) = ϕ(2p(x0, x1)) < p(x0, x1). Furthermore, we can take x2 ∈ G(x1) such that p(x1, x2) ≤ p(x0, x1). Thus, by using inequality (7) we have p(x2, F (x2)) ≤ Hp(G(x1), F (x2)) ≤ ϕ(max{p(x1, x2), p(x1, G(x1)), p(x2, F (x2), p(x1, F (x2)), p(x2, G(x1))}) ≤ ϕ(max{p(x1, x2), p(x2, F (x2))}) ≤ ϕ(p(x1, x2) + p(x2, F (x2))). Let’s observe, when p(x1, x2) < p(x2, F (x2)) then we obtain p(x2, F (x2)) ≤ ϕ(p(x1, x2) + p(x2, F (x2))) < ϕ(p(x2, F (x2)) + p(x2, F (x2))) = ϕ(2p(x2, F (x2))) < p(x2, F (x2). Thus, we found a contradiction. It must holds p(x1, x2) ≥ p(x2, F (x2)). Therefore, we obtain p(x2, F (x2)) ≤ ϕ(p(x1, x2) + p(x1, x2)) = ϕ(2p(x1, x2)) < p(x1, x2). A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 1004 In the similar line, we can choose x3 ∈ F (x2), then we will have p(x2, x3) ≤ p(x1, x2). If this process is continued then a sequence (xn) in X is obtained with the form as follows x2n+1 ∈ F (x2n), and x2n+2 ∈ G(x2n+1), and also p(xn, xn+1) ≤ p(xn−1, xn). (8) Furthermore, we defined pn = p(xn, xn+1). From inequality (8) then we obtain pn ≤ pn+1. This means that pn is a non-decreasing sequences of real numbers and is bounded below by zero. Therefore pn is a convergent sequences. Suppose that limn→∞pn = q. Let q > 0, consider that p(xn, xn+1) ≤ ϕ(2p(xn−1, xn)) < p(xn−1, xn), thus p(xn) ≤ ϕ(2pn−1) < pn−1. (9) Take n → ∞ on inequality (9) then we obtain q ≤ ϕ(2q) < q. Therefore, we have a contradiction. Hence, q = 0, i.e., limn→∞pn = limn→∞p(xn, xn+1) = 0. Furthermore, we will show that (xn) is a Cauchy sequences. Based on the construction of sequence (xn), in showing that sequence (xn) is a Cauchy can be done by showing that (x2n) is a Cauchy sequence. As for the proof using contradiction, that is, if (x2n) is not a Cauchy sequence then there exist ε > 0 such that for every positive integer 2t there is sequence (2mt) and (2nt) where t < nt < mt and we have p(x2nt , x2mt) > ε, t = 1, 2, 3, . . . (10) Suppose that 2mt is the smallest integer that greater than 2nt and satisfies the inequality (10) then we have p(x2nt , x2mt−2) ≤ ε. Hence ε ≤ p(x2nt , x2mt) ≤ p(x2nt , x2mt−2) + p(x2mt−2, x2mt)− p(x2mt−2, x2mt−2) ≤ p(x2nt , x2mt−2) + p(x2mt−2, x2mt) ≤ ε+ p(x2mt−2, x2mt). A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 1005 Let we consider that p(x2mt−2, x2mt) → 0 as mt → ∞, then we have ε ≤ p(x2nt , x2mt) ≤ ε. Consequently, limnt,mt→∞p(x2nt , x2mt) = ε. It is noted that p(x2nt+1, x2mt) ≤ Hp(F (x2nt), G(x2mt−1)) ≤ ϕ(max{p(x2nt , x2mt−1), p(x2nt , F (x2nt)), p(x2mt−1, G(x2mt−1)), p(x2nt , G(x2mt−1)), p(x2mt−1, F (x2nt))}) ≤ ϕ(max{p(x2nt , x2mt−1), p(x2mt−1, G(x2mt−1))}) ≤ ϕ(p(x2nt , x2mt−1) + p(x2mt−1, G(x2mt−1))) ≤ ϕ(p(x2nt , x2mt−1) + p(x2mt−1, x2nt) + p(x2nt , G(x2mt−1)) −p(x2nt , x2nt)) ≤ ϕ(p(x2nt , x2mt−1) + p(x2mt−1, x2nt) + p(x2nt , G(x2mt−1)) ≤ ϕ(p(x2nt , x2mt−1) + p(x2mt−1, x2nt) ≤ ϕ(2p(x2nt , x2mt−1)) ≤ ϕ(2ε) Therefore, we have p(x2nt , x2mt) ≤ p(x2nt , x2nt+1) + p(x2nt+1, x2mt) ≤ p(x2nt , x2nt+1) +Hp(F (x2nt), G(x2mt−1)) ≤ p(x2nt , x2nt+1) + ϕ(2ε) Thus for nt,mt → ∞ we have ε ≤ ϕ(2ε). Since ϕ(2ε) < ε then we have a contradiction. Therefore, it can be concluded that (xn) is Cauchy sequences inX. Since (X, p) is complete partial metric space, then there exist x ∈ X such that limn→∞xn = x. Furthermore, we will establish that x is common fixed point of F and G. Let p(x, F (x)) > 0. Let we consider that p(x2n, F (x)) ≤ Hp(F (x), G(x2n−1)) ≤ ϕ(max{p(x, x2n−1), p(x, F (x)), p(x2n−1, G(x2n−1)), p(x,G(x2n−1)), p(x2n−1, F (x))}) (11) Taking n → ∞ on the inequality (11) above, we obtain p(x, F (x)) ≤ ϕ(max{p(x, F (x)), p(x, F (x))}) ≤ ϕ(p(x, F (x)) + p(x, F (x))) = ϕ(2p(x, F (x))) < p(x, F (x)). Then we have a contradiction. Hence, p(x, F (x)) = 0, i.e., x ∈ F (x). In the similar way it can be shown that p(x,G(x)) = 0, i.e., x ∈ G(x). It means, x is a common fixed point of set-valued mapping F and G. Furthermore, we will show the uniqueness of this common fixed points. A. Ekayanti et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 996-1008 1006 Suppose that v is another common fixed point of set-valued mappings F and G such that v ∈ F (v) and v ∈ G(v). Let p(x, v) > 0 then Hp(F (x,G(v)) ≤ ϕ(max{p(x, v), p(x, F (x)), p(v,G(v)), p(x,G(v)), p(v, F (x))}) ≤ ϕ(p(x, v), p(x,G(v)), p(v, F (x))) ≤ ϕ(p(x, v), p(v, F (x))) ≤ ϕ(p(x, v), p(v, x)) ≤ ϕ(2p(x, v)) Since p(x, v) ≤ Hp(F (x,G(v)) ≤ ϕ(2p(x, v)) < p(x, v), thus we have a contradiction. Hence p(x, v) = 0, i.e., x = v. Therefore, we can conclude that common fixed point x is unique. This complete the proof. Further, we have Corollary 2. Let (CBp(X), Hp) be a p-Pompeiu-Hausdorff metric spaces. Suppose that F : X → CBp(X) be set-valued mappings which satisfy Hp(F (x), F (y)) ≤ ϕ(max{p(x, y), p(x, F (x)), p(y, F (y)), p(x, F (y)), p(y, F (x))}), (12) for each x, y ∈ X and ϕ as defined in Theorem 5, then set-valued mappings F has a unique fixed point. The existence of a common fixed point of set-valued mapping that satisfies the con- traction as in inequality (6) is the consequence of Theorem 5. For ϕ(βu) = βu where β ∈ [0, 12) in Theorem 5 then we have Corollary 3 below. Corollary 3. Let (CBp(X), Hp) be a p-Pompeiu-Hausdorff metric spaces. Suppose that F,G : X → CBp(X) be set-valued mappings which satisfy the contraction as in inequality (6), Hp(F (x), G(y)) ≤ κmax {p(x, y), p(x, F (x)), p(y,G(y)), p(x,G(y)), p(y, F (x))} , for each x, y ∈ X and κ ∈ [0, 1), then set-valued mappings F and G have a unique common fixed point. 4. Conclusion In this manuscript, we have established several theorems concerning common fixed points for set-valued mappings. These theorems introduce novel forms of contraction, which extend the Banach contraction principle to set-valued mappings. Among them are contractions for sequences of set-valued mappings, indicating the existence of a common fixed point for the sequence. This common fixed point is then utilized to infer shared fixed points of the set-valued mappings through sequence convergence. Furthermore, we present a new, more general contraction principle. 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