EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6441 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Existence of Solutions for Second-Order Differential Inclusions under Almost Fisher-Type Multivalued F -Contractions in Metric Spaces Mustafa Mudhesh1,∗, Muhammad Arshad1, Aftab Hussain2, Hamed Alsulami2 1 Department of Mathematics, International Islamic University, H-10, Islamabad - 44000, Pakistan 2 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia Abstract. This paper presents novel fixed point (FP) theorems for a specific class of multival- ued contractions, referred to as ”Almost Fisher-type multivalued F -contractions” within complete metric spaces (MSs) endowed with a Γ-transitive binary relation ℜ. These theorems establish the existence of FPs for such contractions and explore their intrinsic properties. Illustrative exam- ples are provided to demonstrate the applicability and effectiveness of the proposed results. An application to second-order differential inclusions (SODIs) is given under these contractions. 2020 Mathematics Subject Classifications: 47H10, 47H09 Key Words and Phrases: Fixed point, metric space, almost Fisher-type contraction, binary relation, second-order differential inclusion 1. Introduction and Preliminaries In recent years, fixed point (FP) theory has undergone substantial development with various generalizations and refinements of the classical Banach contraction principle (BCP). In 1977, Jaggi [1] generalized the BCP in complete MSs with the condition: ∀ς1, ς2 ∈ ∆ℜ, ∃λ1, λ2 ∈ [0,∞) with λ1+λ2 < 1 such that d (Γς1,Γς2) ≤ λ1d (ς1, ς2)+λ2 d(ς1,Γς1).d(ς2,Γς2) d(ς1,ς2) , where the map Γ : ∆ℜ → ∆ℜ has a unique fixed point (UFP) ς∗ ∈ ∆ℜ. Karapinar [2] further enriched FP theory by introducing interpolative-type contractions, establishing connections with interpolation theory, as explored in related works, which was also dis- cussed in [3, 4]. Karapinar and Fulga [5] introduced a hybrid contraction by combining Jaggi-type and interpolative-type contractions. In 2012, Wardowski [6] introduced the concept of F -contraction as an extension of the BCP. Subsequent works in [7, 8] applied ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6441 Email addresses: mustafa.phdma115@iiu.edu.pk (M. Mudhesh), marshadzia@iiu.edu.pk (M. Arshad), aniassuirathka@kau.edu.sa (A. Hussain), hhaalsalmi@kau.edu.sa (H. Alsulami) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 2 of 21 these ideas to combine the results of Wardowski’s cyclic contraction operators and admis- sible mappings of Geraghty F -contraction, producing new FP theorems. Ali et al. [9] discussed new FP results of dynamic process of integral Ciric-type F -contractions set- valued mappings in MSs. Nadler [10] earlier initiated the study of FPs for multivalued mappings. Notably, recent articles in the field of FP theory for multivalued mappings have been published, providing valuable assistance to researchers. Acar and Altun [11] extended multivalued F -contractions with δ-Distance and established FP results in com- plete MSs, (see [12, 13]). In 1980, Fisher [14] explored new results that generalized the BCP by employing a new rational inequality, ∀ς1, ς2 ∈ ∆ℜ, ∃λ1, λ2 ∈ [0,∞) such that d (Γς1,Γς2) ≤ λ1d (ς1, ς2) + λ2 d(ς1,Γς1).d(ς2,Γς2) 1+d(ς1,ς2) , broadening fixed point theory and inspir- ing further research on generalized contractive mappings. Consequently, Fisher-type F- contractions and Jaggi-type F-contractions are important generalizations of classical BCs within FP theory, particularly for multivalued mappings. By incorporating functional inequalities through auxiliary functions (F-functions), they provide greater flexibility in defining contraction conditions, making them applicable to a broader class of problems. These contractions have proven effective in establishing the existence of solutions to nonlin- ear integral equations (IEs), differential inclusions, and equilibrium problems. They play a vital role in optimization theory, control systems, fuzzy dynamics, game theory, and fractal-based image compression. Moreover, they are instrumental in best approximation problems in Banach spaces and in the study of non-expansive multivalued mappings in geodesic spaces. Extending FP theory to multivalued settings through Jaggi-type con- tractions equips researchers with powerful tools for analyzing complex systems beyond the scope of single-valued mappings. The development of new FP theorems with sophisti- cated contractive conditions on different spaces is essential for filling gaps in the existing literature. By introducing new FP theorems based on this contractive approach, our study intends to address the gaps in the literature and provide further insights into the theory of FPs for multivalued mappings. This research has the potential to enhance our un- derstanding of the subject and pave the way for future developments in the field. This approach likely incorporates the almost Fisher-type multivalued F -contractions discussed earlier, within the framework of complete MSs endowed with a Γ-transitive binary relation ℜ. Alam and Imdad [15] introduced the relation-theoretic contraction principle on MS endowed with an arbitrary binary relation. Recently, Tomar and Joshi [16] discussed the relation-theoretic contractions in F -MSs to demonstrate the existence of FP and solved a two-point boundary value problem arising in a hanging cable problem. Alam and Imdad [17, 18] generalized some metrical notions related to relation-theoretic setting and locally T -transitive binary relation with utilizing these notions to prove coincidence points and FP theorems for self-mappings on an MS. Authors in [19–21] extended the above results for set-valued mappings in MSs. Negi and Gairola [22] introduced the notion of general- ized multivalued (ψ-Fℜ)-contraction in PMS endowed with an arbitrary binary relation and established a new FP theorem, see [23, 24]. In line with these advancements, we touch on some basics and concepts that are far famed in the literature: M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 3 of 21 Definition 1. [6] Let (∆ℜ, d) be a MS. A map Γ : ∆ℜ → ∆ℜ is said to be an F -contraction if there exists τ ∈ R+ such that ∀ς1, ς2 ∈ ∆ℜ, d(Γς1,Γς2) > 0 ⇒ τ + F (Γς1,Γς2) ≤ F (d (ς1, ς2)), where ∆w is the family of functions F : R+ → R satisfying the assumptions below: (F1) F is strictly increasing, i.e., ∀ς1, ς2 ∈ (0,∞), so that ς1 < ς2, then F (ς1) < F (ς2) ; (F2) for {ςȷ}∞ȷ=1 ⊆ R+, lim ȷ→∞ ςȷ = 0 ⇔ lim ȷ→∞ F (ςȷ) = −∞; (F3) ∃ ℓ ∈ (0, 1) , so that lim ȷ→∞ ςℓF (ςȷ) = 0. Theorem 1. [6] Let (∆ℜ, d) be a complete MS and Γ : ∆ℜ → ∆ℜ be an F -contraction map. If ∃ F ∈ ∆w and τ ∈ (0,∞). Thus, Γ has a UFP. Remark 1. [3] If F is right continuous and satisfies (F2) , then F (inf A) = inf F (A) ,∀A ⊂ (0,∞)with inf (A) > 0. Now, we introduce some primary debates and terminology about MS that are well known in the literature. Definition 2. [6] Let (∆ℜ, d) be a MS. Then, we have (1) A sequence {ςn} converges to a point ς iff lim n→∞ d (ςn, ς) = 0. (2) A sequence {ςn} in ∆ℜ is said to be a Cauchy sequence iff lim n,m→∞ d (ςn, ςm) = 0. (3) An MS is said to be complete if every Cauchy sequence {ςn} converges to a point ς such that lim n→∞ d (ςn, ς) = 0. If (ς1, ς2) ∈ ℜ, then it is said that ς1 is related to ς2. Here, we take ℜ as a binary relation on a nonempty subset ∆ℜ and (∆ℜ, d) is a MS equipped with a binary relation ℜ. Definition 3. A binary relation ℜ on ∆ℜ ̸= 0 is a subset of ∆ℜ ×∆ℜ, we say that ς1 is related to ς2 (i.e.ς1ℜς2) if and only if (ς1, ς2) ∈ ℜ. Definition 4. [15] Let Γ : ∆ℜ → ∆ℜ. Then, ℜ on ∆ℜ is designated as Γ-closed, if (ς1, ς2) ∈ ℜ ⇒ (Γς1,Γς2) ∈ ℜ, ς1, ς2 ∈ ∆ℜ. M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 4 of 21 Definition 5. [15] Let ℜ be a binary relation on ∆ℜ. Then, a sequence {ςn} ⊂ ∆ℜ is designated as ℜ-preserving if ∀n ∈ N, then (ςn, ςn+1) ∈ ℜ. Definition 6. [18] Let Γ : ∆ℜ → ∆ℜ. Then, ℜ on ∆ℜ is designated as Γ-transitive, if for any ς1, ς2, ς3 ∈ ∆ℜ, (Γς1,Γς2) , (Γς2,Γς3) ∈ ℜ ⇒ (Γς1,Γς3) ∈ ℜ. Definition 7. [25] An MS (∆ℜ, d) is ℜ-regular if for every {ςn}n∈N ⊂ ∆ℜ, (ςn, ςn+1) ∈ ℜ, ςn → ς ∈ ∆ℜ ⇒ (ςn, ς) ∈ ℜ. Definition 8. [17] Let ℜ be a binary relation in an MS (∆ℜ, d) and ς∗ ∈ ∆ℜ. A map Γ : ∆ℜ → ∆ℜ is ℜ-continuous at ς∗ if for any ℜ-preserving sequence {ςn} →d ς∗, we have Γςn →d Γς∗. Γ is ℜ-continuous if it is ℜ-continuous at each point of ∆ℜ. Definition 9. [22] Let (∆ℜ, d) be a partial MS with a binary relation ℜ and Γ a multivalued- maps on ∆ℜ. Then, (1) ℜ is designated as Γ-closed, if for any ς1, ς2 ∈ ∆ℜ, (ς1, ς2) ∈ ℜ ⇒ (a, b) ∈ ℜ for some a ∈ Γς1 and b ∈ Γς2. (2) Γ is called ℜ-continuous at ς∗ ∈ ∆ℜ, if for any ℜ-preserving sequence {ςn} ⊂ with {ςn} →d ς∗, we have Γςn →Hd Γς∗, we say that Γ is ℜ-continuous if it is ℜ-continuous at each point of ∆ℜ. (3) ℜ is designated as Γ-transitive, if for any ς1, ς2, ς3 ∈ ∆ℜ, a ∈ Γς1, b ∈ Γς2, c ∈ Γς3, we have (a, b) ∈ ℜ, (b, c) ∈ ℜ ⇒ (a, c) ∈ ℜ. Definition 10. [19] Let ∆ℜ ̸= ∅ and Γ : ∆ℜ → CP (∆ℜ). A binary relation ℜ on an MS ∆ℜ is designated as Γ-transitive, if for any ς1, ς2, ς3 ∈ ∆ℜ, a ∈ Γς1, b ∈ Γς2, c ∈ Γς3, we have (a, b) ∈ ℜ, (b, c) ∈ ℜ ⇒ (a, c) ∈ ℜ. Definition 11. [26] Let (∆ℜ, d) be a MS with a binary relation ℜ and Γ : ∆ℜ → CP (∆ℜ). Then, ℜ is called Γ-d-closed, if (ς1, ς2) ∈ ℜ, r ∈ Γς1, s ∈ Γς2, d(r, s) ≤ d(ς1, ς2) ⇒ (r, s) ∈ ℜ. Let (∆ℜ, d) be a MS. we shall denote CB (∆ℜ) the family of all bounded and closed subsets of ∆ℜ and K (∆ℜ) the class of all compact subsets of ∆ℜ. M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 5 of 21 Definition 12. [10] Let H : CB (∆ℜ) × CB (∆ℜ) → [0,∞) be the Pompeiu-Hausdorff metric induced by d so that H (A,B) = max{ sup ς1∈A D (ς1, B) , sup ς2∈B D (A, ς2)}, where ∀ς1 ∈ ∆ℜ and A,B ∈ CB (∆ℜ) D (ς1, B) = inf ς2∈B d (ς1, ς2) . In addition, CB (∆ℜ) ,H is known as a Pompeiu-Hausdorff MS. Here, we say that an element a ∈ ∆ℜ is an FP of a multivalued map Γ : ∆ℜ → CB (∆ℜ) if a ∈ Γa. Lemma 1. [10] If A,B ∈ CB (∆ℜ) , then D (ς, B) ≤ H (A,B) for every ς ∈ A. Lemma 2. [10] If A,B ∈ CB (∆ℜ) . For µ > 0, a ∈ A there is e ∈ B such that d (a, e) ≤ H (A,B) + µ. Definition 13. [1] Let (∆ℜ, d) be an MS and Γ : ∆ℜ → ∆ℜ be a self-map, then Γ is called a Jaggi contraction if there are λ1, λ2 ∈ [0,∞) with λ1 + λ2 < 1 such that ∀ς1, ς2 ∈ ∆ℜ d (Γς1,Γς2) ≤ λ1d (ς1, ς2) + λ2 d (ς1,Γς1) .d (ς2,Γς2) d (ς1, ς2) . Theorem 2. [1] Let (∆ℜ, d) be a complete MS and Γ : ∆ℜ → ∆ℜ be a Jaggi contraction map. Then, Γ has a UFP in ∆ℜ. Definition 14. [2] Let (∆ℜ, d) be an MS. We say that the self-mapping Γ : ∆ℜ → ∆ℜ is an interpolative Kannan-type contraction, if there exist λ ∈ [0,∞) and α ∈ (0, 1), such that ∀ς1, ς2 ∈ ∆ℜ with ς1 ̸= Γς1 d (Γς1,Γς2) ≤ λ [d (ς1,Γς1)] α . [d (ς2,Γς2)] 1−α . Theorem 3. [2] Let (∆ℜ, d) be a complete MS and Γ : ∆ℜ → ∆ℜ be an interpolative Kannan-type contraction map. Then, Γ has a UFP in ∆ℜ. Definition 15. [5] A self-mapping Γ on an MS (∆ℜ, d) is called a Jaggi-type hybrid contraction if there is ψ ∈ Ψ so that d (Γς1,Γς2) ≤ ψ (MJ (ς1, ς2)) , where for s ≥ 0, ς1, ς2 ∈ ∆ℜ and λi ≥ 0, i = 1, 2, ... with λ1 + λ2 = 1 and MJ (ς1, ς2) =  [ λ1 ( d(ς1,Γς1)d(ς2,Γς2) d(ς1,ς2) )s + λ2 (d (ς1, ς2)) s ] 1 s , for s > 0, ς1 ̸= ς2, (d (ς1,Γς1)) λ1 (d (ς2,Γς2)) λ2 for s = 0, ς1, ς2 ∈ ∆ℜ\FΓ(∆ℜ) , where FΓ(∆ℜ) = {ς ∈ ∆ℜ : Γς = ς}. M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 6 of 21 Theorem 4. [5] Let (∆ℜ, d) be a complete MS and Γ : ∆ℜ → ∆ℜ be a continuous Jaggi- type hybrid contraction. Then, Γ has an FP in ∆ℜ. Moreover, for any ς0 ∈ ∆ℜ, the sequence {Γς0} converges to ς. Definition 16. [27] A self-mapping Γ on a graphical b-MS (∆ℜ, d) with s ≥ 1 is called a Fisher-type graph contraction for HG on (∆ℜ, d) if HG is graph preserving and if there exist nonnegative constants λ1, λ2 with λ1 + λ2 < 1 s so that for every ς1, ς2 ∈ ∆ℜ with (ς1, ς2) ∈ E(HG), we have d (Γς1,Γς2) ≤ λ1d (ς1, ς2) + λ2 d (ς1,Γς1) d (ς2,Γς2) 1 + d (ς1, ς2) . 2. Main Result In this part, we study the existence of FPs for an almost Fisher-type multivalued F -contraction mappings endowed with a Γ-transitive binary relation ℜ into a MS. Definition 17. Let (∆ℜ, d) be a MS. A map Γ : ∆ℜ → CB(∆ℜ) is called an almost Fisher-type multivalued F -contraction endowed with a Γ-transitive binary relation ℜ, If there exist τ ∈ R+, F ∈ ∆w and λ1, λ2, L ≥ 0 with λ1 + λ2 ≤ 1 such that for all (ς1, ς2) ∈ ℜ∗ = {(ς1, ς2) ∈ ℜ : ς1, ς2 ∈ ∆ℜ\Fix (Γ)} , we have τ + F (H(Γς1,Γς2)) ≤ F (Mℜ(ς1, ς2)) + LNℜ (ς1, ς2) , (1) where Mℜ(ς1, ς2) =  [ λ1 ( D(ς1,Γς1)D(ς2,Γς2) 1+D(ς1,ς2) )β + λ2 (d(ς1, ς2)) β ] 1 β if β > 0; (D (ς1,Γς1)) λ1 (D (ς2,Γς2)) λ2 if β = 0, (2) and Nℜ(ς1, ς2) = min {D (ς1,Γς1) , D (ς2,Γς2) , D (ς1,Γς2) , D (ς2,Γς1)} . (3) Theorem 5. Let (∆ℜ, d) be an ℜ-complete MS, and let Γ : ∆ℜ → CB(∆ℜ) be an al- most Fisher-type multivalued F -contraction mapping endowed with a Γ-transitive binary relation ℜ. Assume that (T1) ∆ℜ (Γ,ℜ) = {ς ∈ ∆ℜ : (ς,Γς) ∈ ℜ} ̸= ∅; (T2) ℜ is Γ-closed; (T3) Γ is ℜ-continuous or (∆ℜ, d) is ℜ-regular space. Then, Γ has a FP ς∗ ∈ ∆ℜ. Proof. From (T1) ∃ ς0 ∈ ∆ℜ such that (ς0,Γς0) ∈ ℜ and from (T2) we have ( Γς0,Γ 2ς0 ) ∈ ℜ. Since Γς0 ̸= ∅ and closed ∃ ς1 ∈ ∆ℜ such that ς1 ∈ Γς0 ⊂ ∆ℜ such that (ς1,Γς1) ∈ ℜ and from (T2) , we have ( Γ2ς0,Γ 3ς0 ) ∈ ℜ. Continuing in this way, we construct a sequence {ςn} by ςn ∈ Γςn−1 = Γnς0 ∀n ∈ N0. If ςn ∈ Γςn for some n ∈ N0 then ςn becomes a FP of Γ M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 7 of 21 and the proof is done. So, we assume that ςn /∈ Γςn ∀ n ∈ N0 then D (ςn,Γςn) > 0 and by Lemma 1.15, we have 0 < D (ςn,Γςn) ≤ H (Γςn−1,Γςn) ∀ n ∈ N0 (4) Since (ς0,Γς0) ∈ ℜ and from (T2) , we conclude that ( Γnς0,Γ n+1ς0 ) ∈ ℜ ∀n ∈ N0. That is (ςn, ςn+1) ∈ ℜ ∀n ∈ N0. (5) Now, from (4) and (5), we have (ςn−1, ςn) ∈ ℜ∗ ∀n ∈ N0.Utilizing (F1) in (4) and applying Remark 1.3 together with (1), we get F (d(ςn, ςn+1)) = F (D(ςn,Γςn)) ≤ F (H(Γςn−1,Γςn)) (6) ≤ F (Mℜ(ςn−1, ςn)) + LNℜ(ςn−1, ςn)− τ, where if β > 0, then Mℜ(ςn−1, ςn) = [ λ1 ( D (ςn−1,Γςn−1)D (ςn,Γςn) 1 + d (ςn−1, ςn) )β + λ2 (d(ςn−1, ςn)) β ] 1 β = [ λ1 ( d (ςn−1, ςn) d (ςn, ςn+1) 1 + d (ςn−1, ςn) )β + λ2 (d(ςn−1, ςn)) β ] 1 β ≤ [ λ1 ( d (ςn−1, ςn) d (ςn, ςn+1) d (ςn−1, ςn) )β + λ2 (d(ςn−1, ςn)) β ] 1 β = [ λ1 (d (ςn, ςn+1)) β + λ2 (d(ςn−1, ςn)) β ] 1 β . Assume that d (ςn−1, ςn) ≤ d (ςn, ςn+1) , then we get Mℜ(ςn−1, ςn) ≤ [ λ1 (d (ςn, ςn+1)) β + λ2 (d (ςn, ςn+1)) β ] 1 β (7) = [ (λ1 + λ2) (d (ςn, ςn+1)) β ] 1 β < d (ςn, ςn+1) and Nℜ(ςn−1, ςn) = min {d (xn−1, xn) , d (ςn, ςn+1) , d (ςn−1, ςn+1) , d (ςn, ςn)} = 0. Then from (6), we have F (d(ςn, ςn+1)) ≤ F (d (ςn, ςn+1))− τ < F (d (ςn, ςn+1)) . M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 8 of 21 A contradiction. Therefore, d (ςn−1, ςn) > d (xn, xn+1) , which implies that Mℜ(ςn−1, ςn) ≤ [ λ1 (d (ςn−1, ςn)) β + λ2 (d (ςn−1, ςn)) β ] 1 β = [ (λ1 + λ2) (d (ςn−1, ςn)) β ] 1 β < d (ςn−1, ςn) . Then from (6), we have F (d(ςn, ςn+1)) ≤ F (d (ςn−1, ςn))− τ (8) < F (d (ςn−1, ςn)) . Thus, the sequence {d(ςn, ςn+1)} is decreasing and convergent. By induction on n, we obtain F (d(ςn, ςn+1)) ≤ F (d (ςn−1, ςn))− τ < ... < F (d (ς0, ς1))− nτ. Taking limit as n→ ∞ above, we get lim n→∞ F (d(ςn, ςn+1)) = −∞. By (F2) , we have lim n→∞ d(ςn, ςn+1) = 0. (9) We claim that {ςn} is a Cauchy sequence, by supposing on the contrary that it is not. Then ∃ ε > 0 and subsequences {ςln} and {ςqn} so that for ln > qn > n, we have d(ςln , ςqn) ≥ ε and d(ςln−1, ςqn) < ε ∀ n ∈ N. (10) By triangle inequality, we have ε ≤ d(ςln , ςqn) ≤ d(ςln , ςln−1) + d(ςln−1 , ςqn). Taking limit as n→ ∞ and using (9) and (10), we get lim n→∞ d(ςln , ςqn) = ε. (11) Using triangle inequality again, we get d(ςlm , ςqm+1) ≤ d(ςlm , ςqm) + d(ςqm , ςqm+1). Taking limit as m→ ∞ above and using (9) and (11), we get lim m→∞ d(ςlm , ςqm+1) ≤ ε. (12) Similarly, we have ε ≤ d(ςlm , ςqm) ≤ d(ςlm , xqm+1) + d(ςqm+1 , ςqm). M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 9 of 21 Taking limit as m→ ∞ above and using (9) and (11), we get lim m→∞ d(ςlm , ςqm+1) ≥ ε. (13) Therefore, from (12) and (13), we get lim m→∞ d(ςlm , ςqm+1) = ε. By similar way, we conclude lim m→∞ d(ςqm , ςlm+1) = ε. Next, we claim that d(ςln+1, ςqn+1) > 0 ∀ n ∈ N. (14) Arguing by contradiction, there exists m ∈ N, such that d(ςlm+1, ςqm+1) = 0. (15) Using triangle inequality, we have ε ≤ d(ςlm , ςqm) ≤ d(ςlm , ςlm+1) + d(ςlm+1, ςqm) ≤ d(ςlm , ςlm+1) + d(ςlm+1, ςqm+1) + d(ςqm+1, ςqm). Taking limit as m→ ∞ above and from (9) and (15), we get a contradiction. Hence (14) hold true. Now, we have 0 < d(ςlm+1, ςqm+1) = d(ςlm+1,Γςqm) ≤ H(Γςlm ,Γςqm). (16) Since {ςn} is ℜ-preserving sequence, then by Γ-transitivity of ℜ we have (ςln , ςqn) ∈ ℜ and from (16), we have (ςln , ςqn) ∈ ℜ∗. Utilizing (F1) in (16) and applying (1), we obtain F (d(ςlm+1, ςqm+1)) ≤ F (H(Γςlm ,Γςqm)) (17) ≤ F (Mℜ(ςlm , ςqm)) + LNℜ(ςlm , ςqm)− τ, where Mℜ(ςlm , ςqm) = [ λ1 ( D(ςlm ,Γςlm)D(ςqm ,Γςqm) 1 + d(ςlm , ςqm) )β + λ2 (d(ςlm , ςqm)) β ] 1 β = [ λ1 ( d(ςlm , ςlm+1)d(ςqm , ςqm+1) 1 + d(ςlm , ςqm) )β + λ2 (d(ςlm , ςqm)) β ] 1 β and Nℜ(ςlm , ςqm) = min {D(ςlm ,Γςlm), D(ςqm ,Γςqm), D(ςlm ,Γςqm), D(ςqm ,Γςlm)} M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 10 of 21 = min {d(ςlm , ςlm+1), d(ςqm , ςqm+1), d(ςlm , ςqm+1), d(ςqm , ςlm+1)} . As F is continuous then taking limit as m→ ∞ in (17), we get F (ε) ≤ F ( β √ λ2ε ) + L (0)− τ < F (ε) , which gives a contradiction. So, {ςn} is ℜ-Cauchy sequence in an ℜ-complete MS such that ∃ ς∗ ∈ ∆ℜ implies that lim n→∞ ςn = ς∗. To proof that ς∗ ∈ Γς∗, we claim that D (ς∗,Γς∗) > 0 and by using (1), Lemma 1.15 and (T3), we write F (D (ς∗,Γς∗)) ≤ lim n→∞ F (D (ςn,Γς ∗)) (18) ≤ lim n→∞ F (H (Γςn−1,Γς ∗)) ≤ lim n→∞ F (Mℜ (ςn−1, ς ∗)) + lim n→∞ LNℜ (ςn−1, ς ∗)− τ, where Mℜ (ςn−1, ς ∗) = [ λ1 ( D (ςn−1,Γς ∗)D (ς∗,Γςn−1) 1 +D (ςn−1, ς∗) )β + λ2d (ςn−1, ς ∗)β ] 1 β = [ λ1 ( D (ςn−1,Γς ∗) d (ς∗, ςn) 1 + d (ςn−1, ς∗) )β + λ2D (Γxn−2, x ∗)β ] 1 β , and Nℜ (ςn−1, ς ∗) = min {D (ςn−1,Γςn−1) , D (ς∗,Γς∗) , D (ςn−1,Γς ∗) , D (ς∗,Γςn−1)} = min {d (ςn−1, ςn) , D (ς∗,Γς∗) , D (ςn−1,Γς ∗) , D (ς∗, ςn)} . Taking limit as n→ ∞ in (18) with continuity of F , we have F (D (ς∗,Γς∗)) ≤ F ( β √ λ2D (ς∗,Γς∗) ) + L (0)− τ < F (D (ς∗,Γς∗)) . A contradiction, thence D (ς∗,Γς∗) = 0 and ς∗ ∈ Γς∗ is a FP of Γ. Now, if β = 0, then Mℜ(ςn−1, ςn) = D (ςn−1,Γςn−1) λ1 D (ςn,Γςn) λ2 (19) = d (ςn−1, ςn) λ1 d (ςn, ςn+1) λ2 . Assume that d (ςn−1, ςn) ≤ d (ςn, ςn+1) , then (19) becomes Mℜ(ςn−1, ςn) ≤ d (ςn, ςn+1) λ1 d (ςn, ςn+1) λ2 (20) = d (ςn, ςn+1) λ1+λ2 M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 11 of 21 = d (ςn, ςn+1) . Thereafter, from (6) and (20), we get a contradiction. Thus, d (ςn−1, ςn) > d (ςn, ςn+1) and (19) becomes Mℜ(ςn−1, ςn) < d (ςn−1, ςn) (21) Applying (21) in (6) and following the same steps from (8) till (17), we get Mℜ(ςlm , ςqm) = d(ςlm ,Γςlm) λ1d(ςqm ,Γςqm) λ2 = d(ςlm , ςlm+1) λ1d(ςqm , ςqm+1) λ2 and Nℜ(ςlm , ςqm) = min {d(ςlm ,Γςlm), d(ςqm ,Γςqm), d(ςlm ,Γςqm), d(ςqm ,Γςlm)} = min {d(ςlm , ςlm+1), d(ςqm , ςqm+1), d(ςlm , ςqm+1), d(ςqm , ςlm+1)} . As F is continuous then taking limit as m→ ∞ in (17), we get F (ε) < F (0) + L (0)− τ which gives a contradiction again. So, {ςn} is ℜ-Cauchy sequence in an ℜ-complete MS such that ∃ ς∗ ∈ ∆ℜ implies that lim n→∞ ςn = ς∗. To proof that ς∗ ∈ Γς∗, we claim that D (ς∗,Γς∗) > 0 and by using (1), Lemma 1.15 and (T3), we write F (D (ς∗,Γς∗)) ≤ lim n→∞ F (D (ςn,Γς ∗)) (22) ≤ lim n→∞ F (H (Γςn−1,Γς ∗)) ≤ lim n→∞ F (Mℜ (ςn−1, ς ∗)) + lim n→∞ LNℜ (ςn−1, ς ∗)− τ, where Mℜ (ςn−1, ς ∗) = D (ςn−1,Γς ∗)λ1 D (ς∗,Γςn−1) λ2 = D (ςn−1,Γς ∗)λ1 d (ς∗, ςn) λ2 , and Nℜ (ςn−1, ς ∗) = min {D (ςn−1,Γςn−1) , D (ς∗,Γς∗) , D (ςn−1,Γς ∗) , D (ς∗,Γςn−1)} = min {d (ςn−1, ςn) , D (ς∗,Γς∗) , D (ςn−1,Γς ∗) , d (ς∗, ςn)} . Taking limit as n→ ∞ in (22) with continuity of F , we have F (D (ς∗,Γς∗)) ≤ F (0) + L (0)− τ, this implies that D (ς∗,Γς∗) = 0 and ς∗ ∈ Γς∗ is a FP of Γ. M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 12 of 21 Example 1. Let ∆ℜ = [0,∞) equipped with a usual metric. Take a sequence {ςn} ⊂ ∆ℜ given by ςn = n2(n+1)2 4 ∀ n ≥ 1. Set ℜ = {(ςn, ςn) , (ςn, ςn+1) , (ςn, ςn+2) : n = 1, 2, ...} , and Γ : ∆ℜ → CB (∆ℜ) by Γx =  {3x} if 0 ≤ x ≤ ς1 {0, ς1} if ς1 ≤ x ≤ ς2{ ςn−1 + ( ςn−ςn−1 ςn+1−ςn ) (x− ςn) } if ςn ≤ x ≤ ςn+1, n = 2, ... . Then, (∆ℜ, d) is a complete MS, ∆ℜ (Γ,ℜ) ̸= ∅ as ς1 = 1 ∈ ∆ℜ, 1 ∈ Γς1 = Γ1 = {0, 1} and (1, 1) ∈ ℜ. also it is easy to check that ℜ is Γ-transitive and Γ is ℜ-continuous or (∆ℜ, d) is ℜ-regular space. Now, for n = 2, ..., we have H (Γςn,Γςn+1) = max { sup r∈Γςn D (r,Γςn+1) , sup s∈Γξn+1 D (Γςn, s) } = max {d (ςn−1, ςn) , d (ςn−1, ςn)} = d (ςn, ςn−1) . From (2) and (3) with β ≥ 0, we get Mℜ (ςn, ςn+1) = d (ςn+1, ςn) and Nℜ (ςn, ςn+1) = 0. Then, utilizing (1), we have τ + F (H (Γςn,Γςn+1)) = τ + F (d (ςn, ςn−1)) (23) = τ + ln (d (ςn, ςn−1)) , and F (Mℜ (ςn, ςn+1)) + LNℜ (ςn, ςn+1) (24) ≤ F (d (ςn+1, ςn)) + L (0) = ln (d (ςn+1, ςn)) . From (23) and (24), we deduce that τ + ln (d (ςn, ςn−1)) ≤ ln (d (ςn+1, ςn)) , implies that τ ≤ ln ( d (ςn+1, ςn) d (ςn, ςn−1) ) . (25) Let f (n) = ln ( |ςn+1 − ςn| |ςn − ςn−1| ) . (26) M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 13 of 21 Table 1: Iteration and f(n) Iter f(n) Iter f(n) n=2 1.21 n=11 0.26 n=3 0.86 n=12 0.24 n=4 0.66 n=13 0.22 n=5 0.55 n=14 0.20 n=6 0.46 n=15 0.19 n=7 0.40 n=16 0.18 n=8 0.35 n=17 0.17 n=9 0.31 ... ... n=10 0.28 n=50 0.059 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 1.2 n f (n ) Figure 1: Behavior of f(n) for n ∈ [2, 50]. In view of Table1 and Figure1, since the sequence {f (n)}n≥2 is decreasing and discon- tinuous, the smallest value in (26) is 0.059. Therefore, the Eq (25) holds for 0 < ς < 0.059. So, the contraction (1) is satisfied for all ς1, ς2 ∈ ∆ℜ such that (ς1, ς2) ∈ ℜ∗. Hence, Γ has infinite FPs. Example 2. Let ∆ℜ = {1, 2, 3, 4} and d : ∆ℜ ×∆ℜ → [0,∞) be a metric on ∆ℜ defined as d (1, 2) = 4, d (1, 3) = 6, d (1, 4) = 3, d (2, 3) = 4, d (2, 4) = 3, d (3, 4) = 5, d (ς1, ς2) = d (ς2, ς1) and d (ς1, ς1) = 0, ∀ς1, ς2 ∈ ∆ℜ. We define a binary relation on ∆ℜ as ℜ ={ (1, 1) , (1, 2) , (1, 3) , (1, 4) , (4, 1) , (2, 1) , (2, 2) , (2, 4) , (3, 1) , (4, 3) } . M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 14 of 21 Consider a mapping Γ : ∆ℜ → CB (∆ℜ) as Γς =  {3, 4} , ς ∈ {1, 4} ; {3} , ς = 2; {4} , ς = 3. Then, ∆ℜ (Γ,ℜ) ̸= ∅ as 4 ∈ ∆ℜ, 4 ∈ Γ3 = {4} and (4, 3) ∈ ℜ. also it is easy to check that ℜ is Γ-transitive and Γ is ℜ-continuous or (∆ℜ, d) is ℜ-regular space. Now, for all λ1 = 1 5 , λ2 = 3 5 , β = 2, L = 1 5 and (ς1, ς2) ∈ ℜ∗ = {(1, 1) , (1, 2) , (1, 3) , (3, 1) , (2, 2) , (2, 1)} , we have for (ς1, ς2) = (1, 2) H (Γ1,Γ2) = max { sup a∈Γ1 D (a,Γ2) , sup b∈Γ2 D (Γ1, b) } = max {d (4, 3) , d (3, 3)} = max {5, 0} = 5, where Mℜ (1, 2) = [ λ1 ( D (1,Γ1)D (2,Γ2) 1 + d (1, 2) )β + λ2 (d (1, 2)) β ] 1 β = [ λ1 ( d (1, 3) d (2, 3) 1 + d (1, 2) )β + λ2 (d (1, 2)) β ] 1 β = [ λ1 ( 6× 4 1 + 4 )β + λ2 (4) β ] 1 β ≤ [ 1 3 ( 24 5 )2 + 2 3 (4)2 ] 1 2 = ( 86 75 ) 1 2 × 4 = 4.283300908, and Nℜ (1, 2) = min {D (1,Γ1) , D (2,Γ2) , D (1,Γ2) , D (2,Γ1)} = min {d (1, 3) , d (2, 3) , d (1, 3) , d (2, 3)} = min {6, 4, 6, 4} = 4. For β = 0, we get Mℜ (1, 2) = D (1,Γ1) 1 5 D (2,Γ2) 3 5 = d (1, 3) 1 5 d (2, 3) 3 5 M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 15 of 21 = 6 1 5 × 4 3 5 ≈ 3.3. Therefore, τ + F (5) ≤ F (4) + 4 5 and For β = 0, we get τ + F (5) ≤ F (3.3) + 4 5 . For (ς1, ς2) = (1, 3) , we have H (Γ1,Γ3) = 5, Mℜ (1, 3) = 5.488, and Nℜ (1, 3) = 0. Therefore τ + F (5) ≤ F (5.488) + 0 For (ς1, ς2) = (1, 1) , we have H (Γ1,Γ1) = 5, Mℜ (1, 1) = 12 √ 3 and Nℜ (1, 1) = 6. Therefore τ + F (5) ≤ F ( 12 √ 3 ) + 6 5 . For (ς1, ς2) = (2, 2) ,we have H (Γ2,Γ2) = 0, where Mℜ (2, 2) = 16√ 3 , and Nℜ (2, 2) = 4. Therefore τ + F (0) ≤ F ( 16√ 3 ) + 4 5 . Hence, for all (ς1, ς2) ∈ ℜ∗, we find that (1) is achieved and all the conditions of Theorem 5 are satisfied so that Γ has an FP 4 ∈ ∆ℜ. 3. Corollaries Corollary 1. Let (∆ℜ, d) be a MS. A map Γ : ∆ℜ → ∆ℜ is called an almost Fisher- type F -contraction endowed with a Γ-transitive binary relation ℜ, If there exist τ ∈ R+, F ∈ ∆w and λ1, λ2, L ≥ 0 with λ1 + λ2 ≤ 1 such that for all (ς1, ς2) ∈ ℜ∗ = {(ς1, ς2) ∈ ℜ : ς1, ς2 ∈ ∆ℜ\Fix (Γ)} , we have τ + F (d(Γς1,Γς2)) ≤ F (Mℜ(ς1, ς2)) + LNℜ (ς1, ς2) , (27) where Mℜ(ς1, ς2) and Nℜ (ς1, ς2) are defined as in (2) and (3) respectively. Hence, Γ has an FP ς∗ ∈ ∆ℜ . Corollary 2. Let (∆ℜ, d) be a MS. A map Γ : ∆ℜ → CB(∆ℜ) is called an almost Fisher- type multivalued F -contraction endowed with a Γ-transitive binary relation ℜ, If there exist τ ∈ R+, F ∈ ∆w and λ1, λ2, L ≥ 0 with λ1 + λ2 ≤ 1 such that for all (ς1, ς2) ∈ ℜ∗ = {(ς1, ς2) ∈ ℜ : ς1, ς2 ∈ ∆ℜ\Fix (Γ)} , we have τ + F (H(Γς1,Γς2)) ≤ F (Mℜ(ς1, ς2)) + Ld (ς2,Γς1) , (28) Where Mℜ(ς1, ς2) is defind as in (2). Hence, Γ has an FP in ∆ℜ. Corollary 3. Let (∆ℜ, d) be a MS. A map Γ : ∆ℜ → ∆ℜ is called an almost Fisher- type F -contraction endowed with a Γ-transitive binary relation ℜ, If there exist τ ∈ R+, F ∈ ∆w and λ1, λ2, L ≥ 0 with λ1 + λ2 ≤ 1 such that for all (ς1, ς2) ∈ ℜ∗ = {(ς1, ς2) ∈ ℜ : ς1, ς2 ∈ ∆ℜ\Fix (Γ)} , we have τ + F (d(Γς1,Γς2)) ≤ F (Mℜ(ς1, ς2)) + Ld (ς2,Γς1) , (29) Where Mℜ(ς1, ς2) is defined as in (2). M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 16 of 21 Hence, Γ has an FP in ∆ℜ. Corollary 4. Let (∆ℜ, d) be a MS. A map Γ : ∆ℜ → CB(∆ℜ) is called an almost multivalued F -contraction endowed with a Γ-transitive binary relation ℜ, If there ex- ist τ ∈ R+, F ∈ ∆w and λ,L ≥ 0 with λ + L ≤ 1 such that for all (ς1, ς2) ∈ ℜ∗ = {(ς1, ς2) ∈ ℜ : ς1, ς2 ∈ ∆ℜ\Fix (Γ)} , we have τ + F (H(Γς1,Γς2)) ≤ F (λd(ς1, ς2)) + Ld (ς1, ς2) . (30) Then, Γ has an FP in ∆ℜ. Corollary 5. Let (∆ℜ, d) be a MS. A map Γ : ∆ℜ → ∆ℜ is called Fisher-type F - contraction, if there exist τ ∈ R+, F ∈ ∆w and λ1, λ2 ≥ 0 with λ1 + λ2 ≤ 1 such that for all ς1, ς2 ∈ ∆ℜ\Fix (Γ) , we have τ + F (d(Γς1,Γς2)) ≤ F (Mℜ(ς1, ς2)) , where Mℜ(ς1, ς2) =  [ λ1 ( d(ς1,Γς1)d(ς2,Γς2) 1+d(ς1,ς2) )β + λ2 (d(ς1, ς2)) β ] 1 β if β > 0; (d (ς1,Γς1)) λ1 (d (ς2,Γς2)) λ2 if β = 0. Hence, Γ has an FP in ∆ℜ. 4. An application to second-order differential inclusions In this section, we apply the previous theoretical results to study the existence of solutions for the following SODI. In line with [28–31], we consider the boundary value problem (BVP) on [0, 1]: { ς ′′(t) ∈ Π(t, ς(t), ς ′(t)), t ∈ [0, 1], ς(0) = 0, ς(1) = 0, (31) where Π : [0, 1] × R2 → CB(R) has nonempty, closed and compact values and satisfies measurability hypotheses to ensure selections. Let ∆ℜ = C1([0, 1],R) with norm ∥ς∥ = supt |ς(t)|+ supt |ς ′(t)| and metric d(ς1, ς2) = ∥ς1 − ς2∥. Denote by G(t, s) the Green function for the homogeneous Dirichlet problem: G(t, s) = { t(1− s), t ≤ s, s(1− t), t > s. Define the multivalued operator Γ : ∆ℜ → CB(∆ℜ) by Γ(ς) = { Υ ∈ C1([0, 1]) : Υ(t) = ∫ 1 0 G(t, s)f(s) ds, f(s) ∈ Π(s, ς(s), ς ′(s))} } . (32) M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 17 of 21 A fixed point ς∗ ∈ Γς∗ is a solution of (31). Compute K = sup t∈[0,1] ∫ 1 0 G(t, s) ds = 1 8 , M = sup t∈[0,1] ∫ 1 0 ∣∣∂tG(t, s)∣∣ ds = 1 2 , so K +M ≤ 5/8. We assume Π satisfies the following pointwise control: (P1) There exist measurable functions a(s), b(s) ≥ 0 and constants λ1, λ2 ≥ 0, 0 ≤ λ1 + λ2 ≤ 1, and β ∈ (0, 1] such that ∀ s ∈ [0, 1] and all ui ∈ Π(s, ςi(s), ς ′ i(s)) (for i = 1, 2), one has |u1 − u2|β ≤ a(s) ( |ς1(s)− ς2(s)|β + |ς ′1(s)− ς ′2(s)|β ) + b(s)Ds(ς1, ς2), (33) where Ds(ς1, ς2) stands for combinations of pointwise “distances-to-value-sets” (these will produce the D(·,Γ(·))-type terms appearing in (2)). This is an abstract but standard assumption, it generalizes the Hausdorff–Lipschitz condition and allows us to produce the more general Mℜ-term. (P2) Fix ς1, ς2 ∈ ∆ℜ. Let f(·) be a measurable selection from Π(·, ς1(·), ς ′1(·)) and g(·) from Π(·, ς2(·), ς ′2(·)). For η = ∫ G(t, s)f(s) ∈ Γ(ς1) and ζ = ∫ G(t, s)g(s) ∈ Γ(ς2) we have, for any t, |η(t)− ζ(t)| ≤ ∫ 1 0 G(t, s) |f(s)− g(s)| ds. (34) Theorem 6. Under the two assumptions above, SODI (31) has at least one solution ς∗ ∈ ∆ℜ iff Γ has an FP. Proof. The set ∆ℜ = C1([0, 1],R) is a complete MS. Define the multivalued operator Γ : ∆ℜ → CB(∆ℜ) as in (32) and from (P2), we raise to the power β ∈ (0, 1] in (34) and use Jensen (or generalized Hölder) to obtain |η(t)− ζ(t)|β ≤ ∫ 1 0 G(t, s)β |f(s)− g(s)|β ds. Taking supremum in t and using C1 = supt ∫ 1 0 G(t, s) βds, C2 = supt ∫ 1 0 |∂tG(t, s)|βds, we get ∥η − ζ∥β ≤ (C1 + C2) ∫ 1 0 |f(s)− g(s)|β ds. From (P2) and applying the structural bound (33) to the integrand and integrate(∫ 1 0 |f − g|β )1/β ≤ Ad(ς1, ς2) +BMℜ(ς1, ς2), M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 18 of 21 for appropriate constants A,B ≥ 0 depending on a(·), b(·), C1, C2, whereMℜ(ς1, ς2) denotes a combination of distances-to-value-sets of the type appearing in (2). Concretely one may choose Mℜ(ς1, ς2) = [ λ1 (D(ς1,Γς1)D(ς2,Γς2) 1 +D(ς1, ς2) )β + λ2 ( d(ς1, ς2) )β]1/β , Combining the previous two displayed bounds yields H(Γς1,Γς2) ≤ C1/β ( Ad(ς1, ς2) +BMℜ(ς1, ς2) ) , where C = C1 + C2. Now choose τ = 0, F (t) = t (t ≥ 0), Nℜ(ς1, ς2) = d(ς1, ς2), and define the constant L = C1/βA and note that F (Mℜ(ς1, ς2)) can absorb the other piece C1/βBMℜ. With these choice the bound becomes exactly of the form τ + F ( H(Γς1,Γς2) ) ≤ F ( Mℜ(ς1, ς2) ) + LNℜ(ς1, ς2), which is equation (1). We now show that the conditions (T1)–(T3) in Theorem 5 can be satisfied by natural choices. • Choose the binary relation ℜ = ∆ℜ × ∆ℜ (the universal relation). It is clearly transitive and closed. • (T1) The set ∆ℜ(Γ,ℜ) = {ς ∈ ∆ℜ : (ς,Γς) ∈ ℜ} is equal to ∆ℜ (hence nonempty) because ℜ is universal. • (T2) ℜ is Γ-closed trivially. • (T3) is satisfied because under the compactness of values and the structural bounds we can show upper semicontinuity of Γ in the Hausdorff metric; hence Γ is ℜ- continuous or the space (∆ℜ, d) is ℜ-regular. Hence, with the above matching of parameters and the structural pointwise control (33), all hypotheses of Theorem 2 are satisfied and yields existence of a fixed point ς∗ ∈ ∆ℜ of Γ, which is a solution of (31). Example 3. Take Π(t, u, v) = [ 1 4(u+ v), 1 2(u+ v) ] , t ∈ [0, 1]. Pointwise one has for any ui ∈ Π(t, u∗i , v ∗ i ) |u1 − u2| ≤ 1 2(|u ∗ 1 − u∗2|+ |v∗1 − v∗2|), so setting β = 1 and integrating the previous estimates yields the linear contraction esti- mate H(Γς1,Γς2) ≤ (K +M)12 d(ς1, ς2), with (K + M)12 = 5 16 < 1. In the view of Definition 1 this corresponds to taking Mℜ negligible (or explicitly zero) and L = (K +M) · 1 2 ; the inequality (1) is thus satisfied and Theorem 2 gives existence of the BVP’s solution. M. Mudhesh et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6441 19 of 21 5. Conclusions In this article, we have introduced and studied a new class of contractions, namely almost Fisher-type multivalued F -contractions equipped with a Γ-transitive binary re- lation. Within this framework, we proved several fixed-point results which extend and unify a number of existing theorems in the literature on multivalued contractions. The obtained results highlight the flexibility of the proposed approach, as they recover well- known outcomes as special cases and, at the same time, yield genuinely new contributions. To illustrate the applicability of our theory, we provided explicit examples and developed an application to second-order differential inclusions. This application shows how the ab- stract fixed point results can be effectively employed to guarantee the existence of solutions to nonlinear problems. 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