EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3093-3108 ISSN 1307-5543 – ejpam.com Published by New York Business Global Multivalued Almost JS-Contractions, Related Fixed Point Results in Complete b-Metric Spaces and Applications Maroua Meneceur1, Haitham Qawaqneh2,∗, Habes Alsamir3, Gawhara Al-Musannef4 1 Department of Mathematics, Exact Sciences Faculty, University of El Oued, P.O.Box 789, El Oued 39000, Algeria 2 Department of Mathematics,Faculty of Science and Information Technology, Al-Zaytoonah University of Jordan, Amman 11733, Jordan 3 Finance and Banking Department, Business Administration College, Dar Aluloom University, riyadh, Saudi Arabia 3 Faculty of business studies,Arab Open University, Jeddah, Saudi Arabia Abstract. In this paper, we introduce a new class of multivalued contractions and prove the existence of a fixed point for such contractions. Some consequences are presented in b-metric spaces endowed with partial order or with graph. To illustrate the applicability of our results, we offer an example and an application to the existence of solutions of an integral inclusions. 2020 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: ϑ-contraction, fixed point, αs-admissible, Volterra integral inclusions. 1. Introduction and preliminaries One of the key generalizations for metric spaces is the idea of a b-metric space. Bakhtin [4] first proposed the idea of building such spaces, and Czerwik[6] refined it. Several fixed- point results were provided in this way for single or set valued mappings, for instance, see[7, 13, 23, 24, 26]. One of the key generalizations for metric spaces is the idea of a b-metric space. Bakhtin [4] first proposed the idea of building such spaces, and Czerwik citesc1 refined it. Several fixed-point results were provided in this way for single or set valued mappings, for instance, see [7, 13, 23, 24, 26]. However, Samet et al. [27] presented the idea of α-admissible, and they established some results. Some results were reached by using this notion in conjunction with non-linear contractions; see [11, 12, 18, 23]. This ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5466 Email addresses: chaimanoor9@gmail.com (M. Meneceur), h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), habes@dau.edu.sa (H. Alsamir), G.almusannef@arabou.edu.sa (J.M. Al-musannef) https://www.ejpam.com 3093 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3094 idea was later extended to αs-admissible in the setting of b-metric spaces by Ali et al. [14]. Jleli and Samet [10] established a novel idea known as the ϑ-contraction and proved the existence of fixed points. Here, it is important to note that a contraction in in the sense of Banach, is a particular case of ϑ-contraction, while there are some ϑ-contractions which do not satisfy Banach contractive condition. Subsequently, several authors studied different variations of ϑ-contractions and other different contractions in single and set valued cases, for example, see [2, 3, 15–17, 19–22, 25, 28] In this work, we prove the existence of a fixed point for such a novel contraction type in complete b-metric spaces spaces by combining the notion of αs-admissible mapping with ϑ-contraction in the case of multivalued mappings. In this work, we prove the existence of a fixed point for such a novel contraction type in complete b-metric spaces by combining the notion of αs-admissible mapping with ϑ- contraction in the case of multivalued mappings. Using our major findings, we also infer the existence of fixed points in partially ordered metric spaces. Lastly, to demonstrate the applicability of our results, we offer an example and an application pertaining to an existence problem of solutions for a Volterra integral inclusion and for applications in factional equtions, see [1, 8] Definition 1. [14] Let X be a non-empty set and s be a real number with s ≥ 1. A function d : X × X → [0,∞) is a b-metric on X if for all ν, µ, η ∈ X, it satisfies the following conditions: (b1) d(ν, µ) = 0 iff ν = µ, (b2) d(ν, µ) = d(µ, ν), (b3) d(ν, η) ≤ s[d(ν, µ) + d(µ, η)]. A triplet (X, d, s) is called a b-metric space. Every metric space is a b-metric space with s = 1. Denote the family of non-empty, closed and bounded subsets of X by CB(X). For A,B ∈ CB(X), define H : CB(X)× CB(X) → [0,+∞) by H(A,B) = max { sup a∈A d(a,B), sup b∈B d(b,A) } where d(a,B) = inf {d(b, ν) : ν ∈ B}. Such a function H is called the Pompeiu-Hausdorff metric induced by d, for more details, see [5]. Also, denote the family of non-empty and closed subsets of X by CL(X). Lemma 1. [14] Let (X, d, s) be a b-metric space. The following properties are satisfied: 1) d(ν,B) ≤ d(ν, b) for all ν ∈ X, b ∈ B and B ∈ CB(X). 2) d(ν,B) ≤ H(A,B) for all ν ∈ X and A,B ∈ CB(X). 3) d(ν,A) ≤ s(d(ν, µ) + d(µ,B)) for all ν, µ ∈ X and A,B ∈ CB(X). Lemma 2. [6] Let (X, d, s) be a b-metric space and A,B ∈ CL(X) with H(A,B) > 0. Then, for each b ∈ B, there exists a = a(b) ∈ A such that d(a, b) ≤ sH(A,B). H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3095 Definition 2. [26] Consider a non-empty set X and two mappings T : X → X and α : X ×X → [0,+∞). For a given real number s ≥ 1, T is weak α-admissible of type S if for ν ∈ X and α(ν, T ν) ≥ s, we have α(T ν, T T ν) ≥ s. Definition 3. [14] Let (X, d, s) be a b-metric space. For a given function α : X ×X → [0,+∞), a multivalued mapping T : X → CL(X) is (1) αs-admissible, if for each ν ∈ X and µ ∈ T ν with α(ν, µ) ≥ s2, we have α(µ, η) ≥ s2 for each η ∈ T µ. (2) α∗ s -admissible, if for ν, µ ∈ X with α(ν, µ) ≥ s2 we have α∗(T ν, T µ) ≥ s2, where α∗(T ν, T µ) = inf {α(a, b) : a ∈ T ν, b ∈ T µ} . Definition 4. [9, 18] Let (X, d) be a metric space, and T : X → CL(X) and α : X×X → [0,+∞) be given maps. Then T is called an αs-lower semi-continuous if for ν ∈ X and a sequence {νn} in X with limn→∞ d(νn, ν) = 0 and α(νn, νn+1) ≥ s2 for all n ∈ N, implies lim inf n→∞ d(νn, T νn) ≥ d(ν, T ν). Definition 5. [10] Let Θs be the set of all functions ϑ : (0,+∞) → (1,+∞) such that (ϑ1) ϑ is a strictly increasing function; (ϑ2) for each sequence {ωn} of positive real numbers limn→∞ ϑ(ωn) = 1 iff limn→∞ ωn = 0; (ϑ3) there exist ρ ∈ (0, 1) and χ ∈ (0,+∞] such that limω→0+ ϑ(ω)−1 ωρ = χ; (ϑ4) for each sequence {ωn} in R+ such that ϑ(sωn) ≤ [ ϑ(ωn−1) ]ρ , where ρ ∈ (0, 1), then ϑ(snωn) ≤ [ ϑ(sn−1ωn) ]ρ . Example 1. The following functions ϑi : (0,+∞) → (1,+∞) for i ∈ {1, 2, 3, 4} , are the elements of Θs. (i) ϑ1(ω) = eω; (ii) ϑ2(ω) = eωe ω ; (iii) ϑ3(ω) = e √ ω; (iv) ϑ4(ω) = e √ ωeω . 2. Main results We begin this section with the following definition. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3096 Definition 6. Let (X, d, s) be a b-metric space and α : X×X → [0,+∞) be given. A map T : X → CL(X) is called multivalued almost (αs, ϑ, κ)-contraction of Hardy-Rogers type if there exist ϑ ∈ Θs, L ≥ 0 and κ : (0,+∞) → [0, 1) satisfies limω→z+ supκ(ω) < 1 for all z ∈ (0,+∞) and non-negative real numbers a1, a2, a3, a4, a5 with a1 + a2 + a3 + 2sa4 = 1, and a3 ̸= 1 such that ϑ(s3H(T ν, T µ)) ≤ [ ϑ(Ns(ν, µ)) ]κ(d(ν,µ) + Lmin{d(ν, T µ), d(µ, T ν)}, (2.1) for all ν, µ ∈ X with α(ν, µ) ≥ s2 and H(T ν, T µ) > 0 where Ns(ν, µ) = a1d(ν, µ) + a2d(ν, T ν) + a3d(µ, T µ) + a4d(ν, T µ) + a5d(µ, T ν). If α(ν, µ) = s2, T is said to be an almost (ϑ, κ)-contraction of Hardy-Rogers type. Theorem 1. Let (X, d, s) be a complete b-metric space and T : X → CB(X) be a mul- tivalued almost (αs, ϑ, κ)-contraction of Hardy-Rogers type. Assume that the following conditions are satisfied: (i) T is αs-admissible; (ii) there exist ν0 ∈ X and ν1 ∈ T ν0 such that α(ν0, ν1) ≥ s2; (iii) T is αs-lower semi-continuous, or X is αs-regular, that is, for every sequence {νn} in X such that νn → ν∗ ∈ X and α (νn, νn+1) ≥ s2 for all n ∈ N, then α (νn, ν ∗) ≥ s2, for all n ∈ N. Then T has a fixed point. Proof. From the hypothesis (2), there exist ν0 ∈ X and ν1 ∈ T ν0 such that α(ν0, ν1) ≥ s2. If ν0 = ν1 or ν1 ∈ T ν1, then ν1 is a fixed point of T and the proof is completed. Assume that ν0 ̸= ν1 and ν1 /∈ T ν1, then H(T ν0, T ν1) ≥ d(ν1, T ν1) > 0. From Lemma 2, there exists ν2 ∈ T ν1 such that d(ν1, ν2) ≤ sH(T ν0, T ν1) ≤ s2H(T ν0, T ν1), which implies sd(ν1, ν2) ≤ s3H(T ν0, T ν1). Since ϑ is strictly increasing, we get ϑ(sd(ν1, ν2)) ≤ ϑ(s3H(T ν0, T ν1)). Then by using (2.1) we get ϑ(sd(ν1, ν2)) ≤ ϑ(s3H(T ν0, T ν1)) ≤ [ ϑ(Ns(ν0, ν1)) ]κ(d(ν0,ν1)) + Lmin{d(ν0, T ν1), d(ν1, T ν0)} H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3097 < [ϑ(Ns(ν0, ν1))] κ(d(ν0,ν1)) < ϑ(Ns(ν0, ν1)), which gives ϑ(sd(ν1, ν2)) < ϑ(Ns(ν0, ν1)). Since ϑ is increasing, we get sd(ν1, ν2) < Ns(ν0, ν1), where Ns(ν0, ν1) = a1d(ν0, ν1) + a2d(ν0, T ν0) + a3d(ν1, T ν1) + a4d(ν0, T ν1) + a5d(ν1, T ν0) ≤ a1d(ν0, ν1) + a2d(ν0, ν1) + a3d(ν1, ν2) + a4d(ν0, ν2) ≤ a1d(ν0, ν1) + a2d(ν0, ν1) + a3d(ν1, ν2) + sa4(d(ν0, ν1) + d(ν1, ν2)) ≤ (a1 + a2 + sa4)d(ν0, ν1) + (a3 + sa4)d(ν1, ν2), which implies that d(ν1, ν2) ≤ sd(ν1, ν2) ≤ (a1 + a2 + sa4)d(ν0, ν1) + (a3 + sa4)d(ν1, ν2). Then, d(ν1, ν2) ≤ a1 + a2 + sa4 1− a3 − sa4 d(ν0, ν1). Since a1 + a2 + a3 + 2sa4 = 1, we get d(ν1, ν2) < d(ν0, ν1). Thus, sd(ν1, ν2) < (a1 + a2 + sa4)d(ν0, ν1) + (a3 + sa4)d(ν0, ν1) = d(ν0, ν1), and so ϑ(sd(ν1, ν2)) ≤ [ ϑ(d(ν0, ν1)) ]κ(d(ν0,ν1)) . Assume that ν1 ̸= ν2, then ν2 /∈ T ν2 and d(ν2, T ν2) > 0 so H(T ν1, T ν2) > 0. From Lemma 2, there exists ν3 ∈ T ν2 such that ϑ(sd(ν2, ν3)) ≤ ϑ(s3H(T ν1, T ν2)) ≤ [ ϑ(Ns(ν1, ν2)) ]κ(d(ν1,ν2)) + Lmin{d(ν1, T ν2), d(ν2, T ν1)} < [ϑ(Ns(ν1, ν2))] κ(d(ν1,ν2)) < ϑ(Ns(ν1, ν2)). Then, ϑ(sd(ν2, ν3)) ≤ ϑ(Ns(ν1, ν2)), H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3098 which gives sd(ν2, ν3) < Ns(ν1, ν2), where Ns(ν1, ν2) = a1d(ν1, ν2) + a2d(ν1, T ν1) + a3d(ν2, T ν2) + a4d(ν1, T ν2) + a5d(ν2, T ν1). ≤ a1d(ν1, ν2) + a2d(ν1, ν2) + a3d(ν2, ν3) + sa4(d(ν1, ν2) + d(ν2, ν3)) ≤ (a1 + a2 + sa4)d(ν1, ν2) + (a3 + sa4)d(ν2, ν3). Hence, d(ν2, ν3) ≤ sd(ν2, ν3) ≤ (a1 + a2 + sa4)d(ν1, ν2) + (a3 + sa4)d(ν2, ν3), and so d(ν2, ν3) ≤ a1 + a2 + sa4 1− a3 − sa4 d(ν1, ν2). Since a1 + a2 + a3 + 2sa4 = 1, we get d(ν2, ν3) < d(ν1, ν2). Then, we infer that ϑ(sd(ν2, ν3) ≤ [ ϑ(d(ν1, ν2)) ]κ(d(ν1,ν2)) . By continuing in this manner, we construct a sequence {νn} in X, if there exists n0 such that νn0 = νn0+1, or νn0+1 ∈ T νn0+1 then νn0+1 is fixed point. If νn ̸= νn+1 and νn+1 /∈ T νn+1, then H(T νn, T νn+1) > 0. From Lemma 2, there exists νn+1 ∈ T νn such that θ(sd(νn, νn+1)) ≤ [ θ(d(νn−1, νn)) ]κ(d(νn−1,νn)) , for all n ∈ N. (2.2) It follows by (2.2) and (ϑ4) that θ(snd(νn, νn+1)) ≤ [ θ(sn−1d(νn−1, νn)) ]κ(d(νn−1,νn)) , for all n ∈ N. (2.3) Since ϑ is increasing, then the sequence {d(νn, νn+1)} is decreasing and so convergent. By the property of κ, there exist δ ∈ (0, 1) and n0 ∈ N such that κ(d(νn, νn+1)) < δ, for all n ≥ n0. Thus, from (2.3), we deduce 1 < ϑ(snd(νn, νn+1)) ≤ [ ϑ(sn−1d(νn−1, νn)) ]κ(d(νn−1,νn) ≤ [ ϑ(sn−2d(νn−2, νn−1)) ]κ(d(νn−2,νn−1)κ(d(νn−1,νn) ... ≤ [ ϑ(d(ν0, ν1)) ]δn−n0 , (2.4) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3099 for all n ≥ n0. On taking the limit as n→ ∞, we get lim n→∞ ϑ(snd(νn, νn+1)) = 1, and from (ϑ2), lim n→∞ snd(νn, νn+1) = 0. Now, we prove {νn} is a Cauchy sequence, by (ϑ3) there exist ρ ∈ (0, 1) and χ ∈ (0,+∞] such that lim n→∞ ϑ(snd(νn, νn+1))− 1 (snd(νn, νn+1))ρ = χ. Take δ ∈ (0, χ). By the definition of limit, there exists n1 ∈ N such that (snd(νn, νn+1)) ρ ≤ δ−1[θ(snd(νn, νn+1))− 1], for all n ≥ n1. Using (2.4) and the above inequality, we deduce n(snd(νn, νn+1)) ρ ≤ δ−1n([ϑ(d(ν0, ν1))] δn−n0 − 1), for all n ≥ n1. This implies that lim n→∞ n(snd(νn, νn+1)) ρ = 0. Thence, there exists n2 ∈ N such that snd(νn, νn+1) ≤ 1 n 1 ρ , for all n ≥ n2. (2.5) Let m > n ≥ max{n0, n1, n2}. Then, using the triangular inequality and (2.5), we have d(νn, νm) ≤ m−1∑ j=n d(νj , νj+1) ≤ m−1∑ j=n snd(νj , νj+1) ≤ m−1∑ j=n 1 j 1 p ≤ ∞∑ j=n 1 j 1 p <∞, and so {νn} is a Cauchy sequence. Since (X, d, s) is complete, so {νn} converges to some ν∗ ∈ X. If T is αs-lower semi-continuous, then for all n ∈ N, we have d(νn, T νn) ≤ d(νn, νn+1). Passing to the limit, we get lim n→∞ d(νn, T νn) = 0. Then taken in the account T is αs-lower semi-continuous, we obtain 0 < d(ν∗, T ν∗) ≤ lim inf n→∞ d(νn, T νn) = 0, H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3100 which gives d(ν∗, T ν∗) = 0. Hence, ν∗ is a fixed point of T . If X is αs-regular, so for every sequence {νn} converges to ν∗ with α(νn, νn+1) ≥ s2, then α(νn, ν ∗) ≥ s2 so d(νn+1, T ν∗) > 0, which implies H(T νn, T ν∗) > 0, then by (2.1), we get 1 < ϑ(sd(νn+1, T ν∗)) ≤ ϑ(s3H(T νn, T ν∗) ≤ [ϑ(Ns(d(ν0, ν1)))] δn−n0 < [ϑ(d(ν0, ν1))] δn−n0 . Passing to the limit, we have lim n→∞ ϑ(d(νn+1, T ν∗)) = 1, then (ϑ2) gives lim n→∞ d(νn+1, T ν∗) = 0, which implies d(ν∗, T ν∗) = 0. Hence ν∗ is a fixed point of T . Since each α∗ s-admissible mapping is also αs-admissible, we obtain the following result. Corollary 1. Let (X, d, s) be a complete b-metric space and T : X → CB(X) be a mul- tivalued almost (αs, ϑ, κ)-contraction of Hardy-Rogers type. Assume that the following conditions are satisfied: (i) T is an α∗ s-admissible; (ii) there exist ν0 ∈ X and ν1 ∈ T ν0 such that α(ν0, ν1) ≥ s2; (iii) T is αs-lower semi-continuous, or for every sequence {νn} ⊂ X converges to some ν∗ in X and α∗(νn, νn+1) ≥ s2, for all n ∈ N. Then α∗(νn, ν ∗) ≥ s2, for all n ∈ N. Then T has a fixed point. Corollary 2. Let (X, d, s) be a complete b-metric space, α : X×X → [0,+∞) be a function and T : X → CB(X) be a multivalued mapping. Assume that the following conditions are satisfied: (i) T is an αs-admissible; (ii) there exist ν0 ∈ X and ν1 ∈ T ν0 such that α (ν0, ν1) ≥ s2; (iii) T is αs-lower semi-continuous, or X is αs-regular; (iv) there exist ϑ ∈ Θs, L ≥ 0 and κ : (0,+∞) → [0, 1) satisfies limω→z+ supκ(ω) < 1 for all z ∈ (0,+∞) and nonnegative real numbers a1, a2, a3, a4, a5 with a1 + a2 + a3 + 2sa4 = 1, and a3 ̸= 1 such that ϑ(s3α(ν, µ)H(T ν, T µ)) ≤ [ ϑ(Ns(ν, µ))))] κ(d(ν,µ) + Lmin{d(ν, T µ), d(µ, T ν)}, for all ν, µ ∈ X with H(T ν, T µ) > 0, where Ns(ν, µ) = a1d(ν, µ) + a2d(ν, T ν) + a3d(µ, T µ) + a4d(ν, T µ) + a5d(µ, T ν). H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3101 Then T has a fixed point. Proof. For all ν, µ ∈ X, we have H(T ν, T µ) ≤ α(ν, µ)H(T ν, T µ), since ϑ is increasing function, we get ϑ(s3H(T ν, T µ)) ≤ ϑ(s3α(ν, µ)H(T ν, T µ)) ≤ [ ϑ(Ns(ν, µ))))] κ(d(ν,µ) + Lmin{d(ν, T µ), d(µ, T ν)}. So this result is a consequence of Theorem 1. Corollary 3. Let (X, d, s) be a complete b-metric space, α : X×X → [0,+∞) be a function and T : X → CB(X) be a multivalued mapping. Assume that the following conditions hold: (i) T is almost (ϑ, κ)-contraction of Hardy Rogers type. (i) T is lower semi continuous. Then T has a fixed point. Proof. It suffices to take α(ν, µ) = s2 for all ν, µ ∈ X in Theorem1. Example 2. Let X = [0, 2] be a set endowed with a b-metric d(ν1, ν2) = |ν1−ν2|2. Define T : X → CB(X) and α : X×X → [0,∞) by T ν = { [0, ν4 ], ν ∈ [0, 2) {2}, ν = 2 and α(ν, µ) = { 4, (ν, µ) ∈ [0, 2) 0, otherwise. Taking ϑ(ω) = eω, κ = 3/4, s = 2, a1 = 4 5 , a2 = a4 = a5 = 0 and a3 = 1/8. For all ν, µ ∈ (0, 2), we have α(ν, µ) = 4, H(T ν, T µ) > |ν−µ 4 |2 > 0 and d(ν, µ) = |ν − µ|2. Then 8H(T ν, T µ) = 1 2 |ν − µ|2 ≤ 3 4 |ν − µ|2 ≤ 3 4 Ns(ν, µ), which implies that e8H(T ν,T µ) ≤ e 9 16 d(ν,µ) ≤ e 9 16 Ns(ν,µ). T is α-continuous, since if (νn) is a sequence in X converges to ν∗ with α(νn, νn+1) ≥ 4, then (νn) ⊂ [0, 2) which implies Tνn = [0, νn4 ] and lim n→∞ T νn = [0, ν 4 ] = T ν∗. Consequently, all conditions of Theorem 1 are satisfied. Then T has a fixed point which is 2. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3102 Now, we give some consequences concerning, two existence theorems of fixed point in metric space endowed with a graph and other in partially order metric spaces. Theorem 2. Let (X,⪯, d) be a complete ordered b-metric space and T : X → CB(X) be a multivalued mapping. Assume that the following assertions hold. (i) For each ν ∈ X and µ ∈ T ν with ν ⪯ µ, we have µ ⪯ η for all ν3 ∈ T µ. (ii) There exist ν0 ∈ X and ν0 ∈ T ν0 such that ν0 ⪯ ν1; (iii) For ν∗ ∈ X and a sequence {νn} in X with limn→∞ d(νn, ν ∗) = 0 and νn ⪯ νn+1 for all n ∈ N, implies lim inf n→∞ d(νn, T νn) ≥ d(ν∗, T ν∗) or, for every sequence {νn} in X such that νn → ν∗ ∈ X and νn ⪯ νn+1 for all n ∈ N, we have νn ⪯ ν∗ for all n ∈ N. (iv) There exist ϑ ∈ Θs, L ≥ 0 and κ : (0,∞) → [0, 1) satisfies lim ω→z+ supκ(ω) < 1 for all z ∈ (0,∞) such that ϑ(s3H(T ν, T µ)) ≤ [ ϑ(Ns(ν, µ)) ]κ(d(ν,µ)) + Lmin{d(ν, T µ), d(µ, T ν)}, where Ns(ν, µ) = a1d(ν, µ) + a2d(ν, T ν) + a3⌈(µ, T µ) + a4d(ν, T µ) + a5d(µ, T ν). Then T has a fixed point. Proof. Define α : X×X → [0,+∞), α (ν, µ) = { s2, if ν ⪯ µ, 0, otherwise. The rest of proof is like the proof of Theorem 1. Nextly, we present an existence theorem of a fixed point for multivalued ϑ-contractions in a b-metric space X, endowed with a graph, into the space of nonempty closed and bounded subsets of the metric space. Consider a graph G̃ such that the set V ( G̃ ) of its vertices coincides with X and the set E ( G̃ ) of its edges contains all loops; that is, E ( G̃ ) ⊇ ∆, where ∆̃ = {(ν, ν) , ν ∈ X}. We assume G̃ has no parallel edges, so we can identify G̃ with the pair ( V ( G̃ ) , E ( G̃ )) . We define the function α : X×X → [0,+∞), α (ν, µ) = { s2, if (ν, µ) ∈ E ( G̃ ) , 0, otherwise. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3103 Theorem 3. Let (X, d, s) be a complete b-metric space endowed with a graph G̃ and T : X → CB(X) be a multivalued mapping. Assume that the following conditions hold: (i) For each ν ∈ X and µ ∈ T ν with (ν, µ) ∈ E(G̃), we have (µ, η) ∈ E(G̃) for all η ∈ Tµ; (ii) There exist ν0 ∈ X and ν1 ∈ T ν0 such that (ν0, ν1) ∈ E(G̃); (iii) For every sequence {νn} in X such that νn → ν∗ ∈ X and (xn, xn+1) ∈ E(G̃) for all n ∈ N, we have (νn, ν ∗) ∈ E(G̃) for all n ∈ N; (iv) There exist ϑ ∈ Θs and κ : (0,∞) → [0, 1) satisfies lim ω→z+ supκ(ω) < 1 such that ϑ(s3H(T ν, T µ)) ≤ [ ϑ(Ns(ν, µ)) ]κ(d(ν,µ)) + Lmin{d(ν, T µ), d(µ, T ν)}, (2.6) where Ns(ν, µ) = a1d(ν, µ) + a2d(ν, Tµ) + a3d(µ, T µ) + a4d(ν, T µ) + a5d(µ, T ν). Then T has a fixed point. Proof. It suffices to consider α : X×X → [0,+∞), α (ν, µ) = { s2, if (ν, µ) ∈ E ( G̃ ) , 0, otherwise. . 3. Application In this section, we apply our obtained results to prove existence theorem of solution for an integral inclusion of Volterra-type. For this purpose, let X := C([a, b],R) be the space of all continuous real valued functions on [a, b]. Note that X is b-complete b-metric space by considering d(ν, µ) = sup ω∈[a,b] |ν(ω)−µ(ω)|2 with s = 2 and define α : X×X → R+ by α(ν, µ) = 4, for all ν, µ ∈ X. Consider now the following problem ν(t) ∈ p(ω) + ∫ ω a F(ω, τ, ν(τ))dτ, ω ∈ J = [a, b]. (3.1) where p ∈ X and F : J × J × R → K(R). Consider the set-valued operator T : X → CL(X) as follows T ν(ω) = { µ ∈ X : µ ∈ p(ω) + ∫ ω a F(ω, τ, ν(τ))dτ, ω ∈ J } . We consider the following hypotheses: H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3104 (A1) : For each ν ∈ X, the multivalued operator Fν : (ω, τ) 7→ F(ω, τ, ν(τ)), is lower semi continuous. (A2) : There exists a continuous function ξ : J × J → [0,+∞) such that |qν(ω, τ)− qµ(ω, τ)| ≤ ξ(ω, τ)|ν(τ)− µ(τ)|. For all ν, µ ∈ X, all qν ∈ Fν , qµ ∈ Fµ and for each (ω, τ) ∈ J × J . (A3) : There exists γ > 0 such that sup ω∈J ∫ ω a |ξ(ω, τ)|dτ ≤ ( e−γ 8 ) 1 2 . Theorem 4. The integral inclusion (3.1) has a solution in X provided the assumptions (A1)− (A3) hold. Proof. The set-valued operator Fν(ω, τ) : J × J → K(R) is lower semi continuous, then from Michael’s selection theorem, for ν ∈ X there exists a continuous function qν : J × J → R such that qν(ω, τ) ∈ Fν(ω, τ) for all ω, τ ∈ J . It follows that p(ω) +∫ ω a qν(ω, τ)ds ∈ T ν, so T ν is non-empty for all ν ∈ X̃. Since p and qν are continuous on J , resp. J 2, their ranges are bounded and closed and hence T ν is bounded, i.e., T : X → K(X). Let ν, µ ∈ X and let ϑ ∈ T ν. Then ϑ(ω) ∈ p(ω) + ∫ ω a F(ω, τ, ν(τ))dτ, ω ∈ J . It follows that there exists qν ∈ F(ω, τ) such that ϑ(ω) = p(ω) + ∫ ω a qν(ω, τ)dτ, (ω, τ) ∈ J × J , From (A2), there exists ς(ω, τ) ∈ Fµ(ω, τ) such that |qν(ω, τ)− ς(ω, τ)| ≤ ξ(ω, τ) · |ν(τ)− µ(τ)|2, for all (ω, τ) ∈ J × J . Let P be a multi valued operator defined by P(ω, τ) = Fµ(ω, τ) ∩ {z ∈ R : |qν(ω, τ)− z| ≤ ξ(ω, τ) · |ν(τ)− µ(τ)|}, for all (ω, τ) ∈ J × J . Since, by (A1), P is lower semi-continuous, there exists a continuous function qµ(ω, τ) ∈ P(ω, τ). Then we have ζ(ω) = p(ω) + ∫ ω a qµ(ω, τ)dτ ∈ p(ω) + ∫ ω a F(ω, τ, µ(τ))dτ, ω ∈ J H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3093-3108 3105 and d(ϑ, T µ) ≤ |ϑ(ω, τ)− ζ(ω, τ)|2 ≤ (∫ ω a |qν(ω, τ)− qµ(ω, τ)|dτ )2 ≤ (∫ ω a ξ(ω, τ)|ν(τ)− µ(τ)|dτ )2 ≤ sup τ∈[a,b] |ν(τ)− µ(τ)|2 (∫ ω a ξ(ω, τ)dτ )2 = d(ν, µ)( ∫ ω a ξ(ω, τ)dτ)2 ≤ e−γ 8 d(ν, µ). Consequently, we have 8d(ϑ, T µ) ≤ e−γd(ν, µ), interchanging the role of ν and µ, we get 8H(T ν, T µ) ≤ e−τd(ν, µ). Taking exponents we get e(8H(T ν,T µ) ≤ [ ed(ν,µ) ]e−γ Then, the mapping T satisfies all the conditions of Corollary 3 with ϑ(ω) = eω, a1 = 1, ai = L = 0, i = 2, 3, 4, 5 and κ = e−γ . So, T has a fixed point, which implies that the integral inclusion (3.1) has a solution in X. 4. 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