Open AccessArticle Fixed Point Results for Large Contraction Mappings in b-Metric Spaces with Applications to Fractional Differential Equations by Mouataz Billah Mesmouli Mouataz Billah Mesmouli SciProfiles Scilit Preprints.org Google Scholar 1, Doha A. Abulhamil Doha A. Abulhamil SciProfiles Scilit Preprints.org Google Scholar 2, Loredana Florentina Iambor Loredana Florentina Iambor SciProfiles Scilit Preprints.org Google Scholar 3,* and Taher S. Hassan Taher S. Hassan SciProfiles Scilit Preprints.org Google Scholar 1,4,5 1 Department of Mathematics, College of Science, University of Ha’il, Ha’il 55476, Saudi Arabia 2 Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 21493, Saudi Arabia 3 Department of Mathematics and Computer Science, University of Oradea, Universitatii nr. 1, 410087 Oradea, Romania 4 Jadara University Research Center, Jadara University, Irbid 21110, Jordan 5 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt * Author to whom correspondence should be addressed. Mathematics 2026, 14(11), 1991; https://doi.org/10.3390/math14111991 (registering DOI) Submission received: 19 April 2026 / Revised: 25 May 2026 / Accepted: 2 June 2026 / Published: 4 June 2026 Abstract In this paper, we establish new fixed point results for Burton-type large contractions in complete b-metric spaces and introduce a Rakotch-type generalization in this setting. We establish existence and uniqueness results for fixed points, with an example to illustrate the applicability. Furthermore, an application to a fractional differential equation is presented. Our results generalize classical fixed point theorems and contribute to the theory of nonuniform contractions in generalized metric spaces. Keywords: fixed point theorems; fractional boundary value problems; b-metric spaces MSC: 47H10; 45G10; 34A08 1. Introduction Fixed point theory plays a fundamental role in nonlinear analysis and has wide applications in differential and integral equations. One of the most celebrated results in this area is the Banach contraction principle, which guarantees the existence and uniqueness of fixed points for contractive mappings in complete metric spaces [ 1]. This classical result has become a cornerstone of modern analysis and has inspired numerous generalizations. To extend this framework, Czerwik introduced the concept of b-metric spaces, which generalize classical metric spaces by relaxing the triangle inequality [ 7]. This generalization has opened the door to studying fixed point theory in more flexible and nonstandard settings. Since then, considerable research has been devoted to the development of fixed point results in b-metric spaces. For instance, Koleva and Zlatanov [ 8] established fixed point results for Chatterjea-type mappings, showing that classical contraction principles can be extended to this generalized framework. Berinde and Păcurar [ 9] provided a comprehensive survey of early developments in fixed point theory on b-metric spaces, highlighting the evolution and significance of this area. Moreover, Aydi et al. [ 10] extended fixed point theory to set-valued quasi-contractions, demonstrating the richness of the structure of b-metric spaces and their applicability to more general mappings. George et al. [ 11] introduced and studied rectangular b-metric spaces, further generalizing the concept and establishing corresponding contraction principles. In addition, Miculescu and Mihail [ 12] developed new fixed point results for set-valued contractions, while Kir and Kiziltunc [ 13] revisited several classical fixed point theorems and showed that they remain valid under the weaker assumptions of b-metric spaces. Also, in [ 14], some fixed points in complete extended b-metric spaces for single-valued and multi-valued mappings were studied. These contributions collectively confirm that many classical fixed point results can be successfully extended to b-metric spaces, thereby enriching the theory and expanding its range of applications. In parallel, Burton introduced the notion of large contraction mappings as a nonuniform generalization of classical contraction mappings [ 15]. Unlike Banach contractions, where the contractive behavior is governed by a fixed constant, large contractions allow the contraction rate to depend on the distance between points, making the framework more flexible and suitable for nonlinear analysis. This concept has since attracted considerable attention. In particular, necessary and sufficient conditions for large contractions were further investigated in later works, providing a deeper understanding of their structural properties [ 16]. Moreover, several extensions and applications have been developed in different directions. For instance, Dehici et al. [ 17] studied large Kannan contraction mappings and their applications to fixed point problems. Mesmouli et al. [ 18] studied large Chatterjea contraction mappings and their applications to delay fractional differential equations, highlighting the relevance of large contractions in applied analysis. More recently, further generalizations have been proposed, including large triangle perimeter contractions in metric spaces [ 19], coupled large Kannan contractions with applications to integral equations [ 20], and cyclic large contractions under perturbation frameworks [ 21]. These developments demonstrate that the theory of large contractions is rapidly evolving and has become an active area of research with significant applications in nonlinear analysis and differential equations [ 22]. However, the study of large contraction mappings in the context of b-metric spaces remains limited. Motivated by this gap, we extend Burton’s concept of large contractions to complete b-metric spaces and establish new fixed point theorems for this class of mappings. Unlike the classical metric setting, the proof in b-metric spaces requires the additional control of the weighted geometric terms generated by the coefficient s in the generalized triangle inequality. This creates technical difficulties that do not appear in the standard metric setting and shows that the extension is not merely formal. The main contributions of this paper are threefold. First, we prove a Burton-type fixed point theorem for large contractions in complete b-metric spaces. Second, we introduce a Rakotch-type version of large contractions adapted to the coefficient s of the b-metric space. Third, we provide a nontrivial example and apply the obtained results to a Caputo-type fractional boundary value problem. Compared with existing results on generalized contractions in b-metric spaces, the present work focuses on nonuniform contractive conditions depending on distance thresholds rather than global Lipschitz constants. Therefore, the obtained results extend and complement several classical Banach-type, Burton-type, and Rakotch-type fixed point theorems. The remainder of this paper is organized as follows. In Section 2, we recall some preliminaries on b-metric spaces and large contractions. Section 3 contains the main fixed point results for large contraction mappings in b-metric spaces and related generalizations. In Section 4, we provide an application to fractional differential equations. Finally, the Section 5 concludes this paper with some remarks and possible future directions. 2. Preliminaries In this section, we recall some basic definitions and fundamental results that will be used throughout this paper. These notions play a central role in the study of fixed point theory in b-metric spaces and contractive mappings. Definition 1.Let ℵ be a nonempty set and s ≥ 1 . A function ℘ : ℵ ୍ଠ ℵ → [ 0 , ∞ ) is called a b-metric if for all ρ , σ , z ∈ ℵ , the following conditions hold: 1. ℘ ( ρ , σ ) = 0 if and only if ρ = σ . 2. ℘ ( ρ , σ ) = ℘ ( σ , ρ ) . 3. ℘ ( ρ , z ) ≤ s ( ℘ ( ρ , σ ) + ℘ ( σ , z ) ) . Then ( ℵ , ℘ ) is called a b-metric space. Remark 1.It is worth noting that every metric space is a b-metric space with s = 1 ; however the converse is not necessarily true. The relaxation introduced by the coefficient s ≥ 1 allows for a wider class of spaces, which is particularly useful in applications where the classical triangle inequality is too restrictive. This generalization has led to significant developments in fixed point theory. Example 1.Let ℵ = R , and define℘ ( ρ , σ ) = | ρ − σ | p , p > 1 . Then ( ℵ , ℘ ) is a b-metric space with constant s = 2 p − 1 . Indeed, for all ρ , σ , z ∈ R , using the inequality | ρ − z | p ≤ ( | ρ − σ | + | σ − z | ) p ≤ 2 p − 1 ( | ρ − σ | p + | σ − z | p ) , we obtain℘ ( ρ , z ) ≤ s ( ℘ ( ρ , σ ) + ℘ ( σ , z ) ) . Thus all conditions of a b-metric are satisfied. We now recall a fundamental result that extends the classical Banach contraction principle to the setting of b-metric spaces, see [ 7, 13]. Theorem 1([ 7, 13]) . Let ( ℵ , ℘ ) be a complete b-metric space, and let Υ : ℵ → ℵ satisfy℘ ( Υ ρ , Υ σ ) ≤ k ℘ ( ρ , σ ) , 0 0 , there exists δ ( ε ) ∈ ( 0 , 1 ) such that℘ ( ρ , σ ) ≥ ε ⇒ ℘ ( Υ ρ , Υ σ ) ≤ δ ( ε ) ℘ ( ρ , σ ) . The following theorem is given by Burton in [ 15] for a large contraction mapping in a metric space. Theorem 2([ 15]) . Let ℵ , ℘ be a complete metric space and Υ : ℵ → ℵ be a large contraction mapping. Suppose there exist ρ 0 ∈ ℵ and L > 0 such that℘ ( ρ 0 , Υ n ρ 0 ) ≤ L , ∀ n ≥ 1 . Then Υ has a unique fixed point in ℵ. Example 2.Let ℵ = R equipped with the usual metric ℘ ( ρ , σ ) = | ρ − σ | , and defineℏ ( ρ ) = ρ − ρ 3 . Then ℏ exhibits a large contraction. The following remarks have been noted in [ 17]. Remark 2.Observe that if ℵ , ℘ is a compact metric space, then the assumption that there exist ρ 0 ∈ ℵ and L > 0 such that℘ ρ 0 , Υ n ρ 0 ≤ L for all n ≥ 1 can be omitted. Indeed, in this case, the existence and uniqueness of a fixed point follow directly from Edelstein’s theorem. Remark 3.If ℵ , ℘ is a bounded complete metric space, then for any ρ 0 ∈ ℵ and for all integers n ≥ 1 , we have℘ ρ 0 , Υ n ρ 0 ≤ diam ( ℵ ) , where diam ( ℵ ) denotes the diameter of ℵ. Hence, the boundedness condition on the orbit is automatically satisfied in this setting. 3. Main Results In this section, we will investigate fixed point results for large contractions in the setting of b-metric spaces and establish new extensions of classical fixed point theorems. Theorem 3.Let ( ℵ , ℘ ) be a complete b-metric space with constant s ≥ 1 , and let Υ : ℵ → ℵ be a large contraction mapping. Suppose there exist ρ 0 ∈ ℵ and L > 0 such that℘ ( ρ 0 , Υ n ρ 0 ) ≤ L , ∀ n ≥ 1 . Then Υ has a unique fixed point in ℵ. Proof.Let ρ 0 ∈ ℵ , and define ρ n = Υ n ρ 0 . If there exists n 0 ≥ 1 such that ρ n 0 = ρ n 0 + 1 , then ρ n 0 is a fixed point of Υ . Assume now that ρ n ≠ ρ n + 1 for all n ≥ 0 . Step 1. Since Υ is a large contraction, we have ℘ ( ρ n + 1 , ρ n ) = ℘ ( Υ ρ n , Υ ρ n − 1 ) 0 . Then ζ n ≥ γ for all n. By the large contraction condition, there exists δ ( γ ) ∈ ( 0 , 1 ) such that ζ n = ℘ ( ρ n + 1 , ρ n ) ≤ δ ( γ ) ℘ ( ρ n , ρ n − 1 ) = δ ( γ ) ζ n − 1 . Iterating this inequality gives ζ n ≤ δ ( γ ) n ζ 0 . Letting n → ∞ , we obtain ζ n → 0 , which contradicts γ > 0 . Therefore, lim n → ∞ ℘ ( ρ n + 1 , ρ n ) = 0 . Step 2. We prove that ( ρ n ) is a Cauchy sequence. Assume, to the contrary, that ( ρ n ) is not Cauchy. Then there exist ε > 0 and subsequences ( ρ m j ) and ( ρ n j ) such that m j > n j → ∞ and ℘ ( ρ m j , ρ n j ) ≥ ε for all j . Since Υ is a large contraction, there exists δ = δ ( ε ) ∈ ( 0 , 1 ) such that ℘ ( x , y ) ≥ ε ⟹ ℘ ( Υ x , Υ y ) ≤ δ ℘ ( x , y ) . Moreover, from the strict contractive property, ℘ ( ρ m j , ρ n j ) 0 and σ = 0 , we have | Υ ( ρ ) − Υ ( 0 ) | | ρ − 0 | = 1 1 + ρ → 1 as ρ → 0 + . Hence there is no constant k ∈ ( 0 , 1 ) such that ℘ ( Υ ρ , Υ σ ) ≤ k ℘ ( ρ , σ ) , ∀ ρ , σ ∈ R . Thus the Banach contraction principle is not applicable. Now we prove that Υ is a large contraction. Let ρ , σ ∈ R , ρ ≠ σ . If ρ , σ ≥ 0 , then | Υ ( ρ ) − Υ ( σ ) | = | ρ − σ | ( 1 + ρ ) ( 1 + σ ) 0 , and suppose that ℘ ( ρ , σ ) ≥ ε . Then | ρ − σ | ≥ ε . From the previous estimates, we obtain | Υ ( ρ ) − Υ ( σ ) | ≤ 1 1 + ε 2 | ρ − σ | . Consequently, ℘ ( Υ ρ , Υ σ ) ≤ 1 1 + ε 2 2 ℘ ( ρ , σ ) . Set δ ( ε ) = 1 1 + ε 2 2 . Then δ ( ε ) ∈ ( 0 , 1 ) , and hence Υ is a large contraction. Finally, the fixed point equation is ρ = ρ 1 + | ρ | . If ρ ≠ 0 , then ρ ( 1 + | ρ | ) = ρ , which gives ρ | ρ | = 0 , a contradiction. Hence ρ = 0 . Therefore, the unique fixed point is ρ * = 0 . □ Corollary 1.Let ( ℵ , ℘ ) be a complete b-metric space and Υ : ℵ → ℵ such that Υ m 0 is a large contraction for some m 0 ≥ 1 . Then Υ has a unique fixed point. Proof.By Theorem 3, there exists z 0 ∈ ℵ such that Υ m 0 z 0 = z 0 . Then Υ m 0 ( Υ z 0 ) = Υ m 0 + 1 z 0 = Υ z 0 , so Υ z 0 is also a fixed point of Υ m 0 . By uniqueness, Υ z 0 = z 0 . If z 1 is another fixed point of Υ , then it is also a fixed point of Υ m 0 ; hence z 1 = z 0 . □ To further generalize this setting, we combine the approach of large contractions with the functional control of Rakotch-type mappings [ 23]. This leads to a more flexible class of contractions in which the contractive behavior is governed by a family of control functions depending on distance thresholds. Unlike the classical metric case, the generalized triangle inequality in b-metric spaces generates additional weighted terms involving the coefficient s. Therefore, stronger control conditions are required in order to compensate for the influence of this coefficient. Let Ω s denote the class of real-valued control functions satisfying Ω s = ℏ : ( 0 , ∞ ) → 0 , 1 s : ∀ ε > 0 , sup t ≥ ε ℏ ( t ) 0 , there exists a function ℏ ε ∈ Ω s such that℘ ( ρ , σ ) ≥ ε ⇒ ℘ ( Υ ρ , Υ σ ) ≤ ℏ ε ( ℘ ( ρ , σ ) ) ℘ ( ρ , σ ) . Then Υ has a unique fixed point z 0 ∈ ℵ . Moreover, for any ρ 0 ∈ ℵ , the sequenceρ n = Υ n ρ 0 converges to z 0 . Proof.Let ρ 0 ∈ ℵ , and define ρ n = Υ n ρ 0 , n ≥ 0 . If there exists n 0 ≥ 0 such that ρ n 0 = ρ n 0 + 1 , then ρ n 0 is a fixed point of Υ . Assume now that ρ n ≠ ρ n + 1 for all n ≥ 0 . Step 1. Let ζ n = ℘ ( ρ n , ρ n + 1 ) . Since ζ n + 1 = ℘ ( Υ ρ n , Υ ρ n + 1 ) 0 . Then, for all sufficiently large n, ζ n ≥ γ 2 . Putting ε = γ 2 . By the assumption, there exists ℏ ε ∈ Ω s such that ζ n + 1 ≤ ℏ ε ( ζ n ) ζ n . Let k ε = sup t ≥ ε ℏ ε ( t ) . Since ℏ ε ∈ Ω s , we have k ε 0 . Hence lim n → ∞ ℘ ( ρ n , ρ n + 1 ) = 0 . Step 2. We show that the orbit { ρ n } is bounded. Set a = ℘ ( ρ 0 , ρ 1 ) . If a = 0 , then ρ 0 is a fixed point, and there is nothing to prove. Assume a > 0 . Fix η > 0 , and let k η = sup t ≥ η ℏ η ( t ) 0 , since ℏ ε ∈ Ω s , we have k ε = sup t ≥ ε ℏ ε ( t ) 0 , there exists a function ℏ ε ∈ Ω such that℘ ( ρ , σ ) ≥ ε ⇒ ℘ ( Υ ρ , Υ σ ) ≤ ℏ ε ( ℘ ( ρ , σ ) ) ℘ ( ρ , σ ) . Then Υ has a unique fixed point in ℵ. Proof.The result follows directly from Theorem 4 by taking s = 1 , which reduces the b-metric space to a classical metric space. □ Remark 6.The obtained results extend classical Banach-type and Rakotch-type contraction principles in several directions. In the Banach contraction principle, the contractive condition depends on a uniform Lipschitz constant k ∈ ( 0 , 1 ) , while in the present work, the contractive behavior depends on the distance scale through large contraction conditions. Moreover, Rakotch contractions involve variable contractive factors depending on the distance between points, but the current approach combines this idea with Burton-type large contractions in the setting of complete b-metric spaces. This provides greater flexibility in the study of nonlinear operators, especially in situations where global uniform contractions are not available. In particular, the obtained results remain applicable to certain nonlinear problems where classical Banach contractions fail, as illustrated in Example 3 and in the application to fractional differential equations. 4. Application to Fractional Differential Equation In this section, we apply the obtained fixed point results for large contractions in b-metric spaces to establish the existence and uniqueness of solutions for a fractional differential equation. Fractional differential equations have attracted significant attention due to their ability to model memory and hereditary properties in various physical and engineering systems. In particular, Caputo-type fractional derivatives are widely used because they allow for classical boundary conditions. Let ℵ = C ( [ 0 , 1 ] , R ) , and define ℘ ( ρ , σ ) = ∥ ρ − σ ∥ ∞ 2 . Then ( ℵ , ℘ ) is a complete b-metric space with coefficient s = 2 . Consider the fractional boundary value problem D α C ρ ( t ) + ℏ ( t , ρ ( t ) ) = 0 , t ∈ [ 0 , 1 ] , 1 0 , there exists δ ( ε ) ∈ ( 0 , 1 ) such that r ≥ ε ⇒ ϕ ( r ) ≤ δ ( ε ) r ; If M = sup t ∈ [ 0 , 1 ] ∫ 0 1 G ( t , s ) d s , then M δ ( ε ) 0 . Accordingly, solutions of the fractional boundary value problem correspond to fixed points of the operator Υ : ℵ → ℵ defined by ( Υ ρ ) ( t ) = ∫ 0 1 G ( t , s ) ℏ ( s , ρ ( s ) ) d s . Proof.Let ρ , σ ∈ ℵ . Then | ( Υ ρ ) ( t ) − ( Υ σ ) ( t ) | ≤ ∫ 0 1 G ( t , s ) | ℏ ( s , ρ ( s ) ) − ℏ ( s , σ ( s ) ) | d s . Using the assumption on ℏ and the monotonicity of ϕ , we get | ℏ ( s , ρ ( s ) ) − ℏ ( s , σ ( s ) ) | ≤ ϕ ( | ρ ( s ) − σ ( s ) | ) ≤ ϕ ( ∥ ρ − σ ∥ ∞ ) . Hence | ( Υ ρ ) ( t ) − ( Υ σ ) ( t ) | ≤ ϕ ( ∥ ρ − σ ∥ ∞ ) ∫ 0 1 G ( t , s ) d s . Taking the supremum over t ∈ [ 0 , 1 ] , we obtain ∥ Υ ρ − Υ σ ∥ ∞ ≤ M ϕ ( ∥ ρ − σ ∥ ∞ ) . Let ρ ≠ σ , and set r = ∥ ρ − σ ∥ ∞ > 0 . By the large contraction condition applied with ε = r , we have ϕ ( r ) ≤ δ ( r ) r . Therefore, ∥ Υ ρ − Υ σ ∥ ∞ ≤ M δ ( r ) ∥ ρ − σ ∥ ∞ . Since M δ ( r ) 0 , and suppose that ℘ ( ρ , σ ) ≥ ε . Then ∥ ρ − σ ∥ ∞ ≥ ε . Using the large contraction condition for ϕ , we get ϕ ( ∥ ρ − σ ∥ ∞ ) ≤ δ ( ε ) ∥ ρ − σ ∥ ∞ . Thus, ∥ Υ ρ − Υ σ ∥ ∞ ≤ M δ ( ε ) ∥ ρ − σ ∥ ∞ . Set δ 1 ( ε ) = M δ ( ε ) 2 . By assumption, δ 1 ( ε ) ∈ ( 0 , 1 ) . Therefore, ℘ ( Υ ρ , Υ σ ) ≤ δ 1 ( ε ) ℘ ( ρ , σ ) , whenever ℘ ( ρ , σ ) ≥ ε . Hence Υ is a large contraction on the complete b-metric space ( ℵ , ℘ ) . By Theorem 3, Υ has a unique fixed point in ℵ. Therefore, the fractional boundary value problem admits a unique solution on [ 0 , 1 ] . □ 5. Conclusions In this paper, we studied fixed point results for a class of mappings known as large contractions in b-metric spaces. This class extends the classical Banach contraction principle by allowing the contractive condition to depend on the distance scale rather than requiring a uniform Lipschitz constant. We also derived theorems and corollaries using the functional control of Rakotch, showing that our results generalize classical fixed point theorems. As an application, we studied a Caputo-type fractional boundary value problem by transforming it into an equivalent integral equation. We showed that the associated operator satisfies the large contraction condition, which guarantees the existence and uniqueness of solutions. Finally, future work may focus on extending these results to more general spaces, such as partial b-metric spaces or generalized b-metric spaces, and to more complex fractional systems with delay or impulsive effects. Author Contributions Conceptualization, M.B.M. and L.F.I.; methodology, D.A.A. and T.S.H.; investigation, M.B.M.; writing—original draft preparation, M.B.M.; writing—review and editing, D.A.A., L.F.I. and T.S.H.; supervision, T.S.H.; funding acquisition, L.F.I. All authors have read and agreed to the published version of the manuscript. Funding This research was funded by the University of Oradea. Data Availability Statement The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. Conflicts of Interest The authors declare no conflicts of interest. 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Fixed Point Results for Large Contraction Mappings in b-Metric Spaces with Applications to Fractional Differential Equations. Mathematics. 2026; 14(11):1991. https://doi.org/10.3390/math14111991 Chicago/Turabian Style Mesmouli, Mouataz Billah, Doha A. Abulhamil, Loredana Florentina Iambor, and Taher S. Hassan. 2026. "Fixed Point Results for Large Contraction Mappings in b-Metric Spaces with Applications to Fractional Differential Equations" Mathematics 14, no. 11: 1991. https://doi.org/10.3390/math14111991 APA Style Mesmouli, M. B., Abulhamil, D. A., Iambor, L. F., & Hassan, T. S. (2026). Fixed Point Results for Large Contraction Mappings in b-Metric Spaces with Applications to Fractional Differential Equations. Mathematics, 14(11), 1991. https://doi.org/10.3390/math14111991 Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here. 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Fixed Point Results for Large Contraction Mappings in b-Metric Spaces with Applications to Fractional Differential Equations
