EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5994 ISSN 1307-5543 – ejpam.com Published by New York Business Global Unique Fixed Point Results via Geraghty Type Contraction with Applications Kamaleldin Abodayeh1, Badshah-e-Rome2, Sidra Nizam3, Muhammad Sarwar1,3,∗ 1 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia 2 Department of Mathematics, Government Degree College Kabal Swat, Khyber Pakhtunkhwa, Pakistan 3 Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan Abstract. In this research work, using the concept of Banach contraction principle and Geraghty type contractions, some unique fixed point results are established in the context of µ-extended fuzzy b-metric spaces. The developed results are applied to ensure the existence of solution to integral equation and non-linear fractional order differential equation. 2020 Mathematics Subject Classifications: 34A08, 34A12, 34B15 Key Words and Phrases: Fixed point results, Geraghty type contraction, fuzzy b-metric spaces, integral equation, fractional differential equation 1. Introduction L. Zadeh [1] was first to propose the concept of fuzzy set in 1965. Fuzzy set theory includes the ordinary set theory as a special case. Membership degree of an element in fuzzy set is described by a real number from the unit interval as opposed to conventional set theory where it is 1 or 0. There is evidence that the concept of distance measurement is fuzzy instead of clas- sical. Improving this idea, Kramosil and Michalek [2] introduced the notion of fuzzy metric. Fuzzy metric is generalization of probabilistic metric. George and Veeramani [3] modified Kramosil’s definition of fuzzy metric and proved that every metric induces a fuzzy metric. The idea of fixed point in fuzzy metric space (FWS̃) has been studied and explored by several authors. In 1988, Grabiec [4] proved fuzzy version of the well known Banach contraction principle. Gregori and Sapena [5] presented the concept of fuzzy contraction and proved a fixed point theorem in FWS̃ in the sense of George and ˚Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5994 Email addresses: kamal@psu.edu.sa (K. Abodayeh), baadeshah1@gmail.com (B. Rome), sidranizam2244@gmail.com (S. Nizam), sarwarswati@gmail.com (M. Sarwar) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 2 of 22 Veeramani. In [6] existence and uniqueness of fixed points in modified intuitionistic fuzzy metric spaces is studied. V. Gupta at al.[7] established coupled fixed point theorems on V -fuzzy metric spaces, while in [8] authors studied new results on modified intuitionis- tic generalized fuzzy metric spaces. Nădăban [9] initiated fuzzy b-metric space (FbWS̃) extending FWS̃. Mehmood etal.[10] generalized fuzzy b-metric by coming up with the concepts of extended fuzzy b-metric space(EFbWS̃) and fuzzy rectangular b-metric [11]. Rome et al.[12] introduced µ-extended fuzzy b-metric space (µEFbWS̃) and generalized several fixed point results to this newly introduced fuzzy space. It was M. Geraghty [13] who first replaced the condition of Cauchy convergence for a contraction, in a complete metric space, by an equivalent functional condition. J.Mart́inez-Morenoet al.[14] obtained some new common fixed point theorems for Geraghty’s type contraction mappings using the monotone property with two metrics. Ashraf et al.[15] established some fixed point results for Geraghty-type contraction in fuzzy b-metric space. The aim of this paper is to explore Geraghty-type contractions in the setting of µEFbWS̃. Besides Banach contraction principle some well known fixed point results are generalized to µEFbWS̃. To demonstrate validity of the main work, a supporting example and a couple of applications to the existence of solution to non-linear fractional order differential equation and integral equation are provided. 2. Preliminaries In the following some specific definitions and terms are recalled which are essential for main work of this paper. R and N will be used to represent sets of real numbers and positive integers respectively. S will represent a non-empty set. Definition 1. [16] A binary operation *, mapping r0, 1s2 onto r0, 1s, is called continuous t-norm, if for all κ, χ and z̃ in r0, 1s, it satisfies: (t1)κ ˚ y “ y ˚ κ; (t2)pκ ˚ χq ˚ z̃ “ κ ˚ pχ ˚ z̃q; (t3) ˚ is continuous; (t4) κ ˚ 1 “ κ for all κ P r0, 1s; (t5) κ ˚ z̃ ď χ ˚ w̃ whenever κ ď χ and z̃ ď w̃. Below are some instances of the continuous t-norm in usage: κ˚Lχ “ maxtκ`χ´1, 0u, κ ˚P χ “ κχ and κ ˚M χ “ mintκ, χu, are respectively called Lukasievicz, product and minimum t-norm. The relation among these t-norms is: ˚M ě ˚P ě ˚L. In the sequel, ˚ will stand for an arbitrary continuous t-norm. Definition 2. [2] A 3-tuplepS,W, ˚q is called a FWS̃, if W is a fuzzy set on S2xr0,`8q satisfying the following conditions for all κ, ỹ, z̃ P X̃ and ζ, s ą 0: (Fm1) Wpκ, ỹ, 0q “ 0; (Fm2) Wpκ, ỹ, ζq “ 1 if and only if κ “ ỹ; (Fm3) Wpκ, ỹ, ζq “ Wpỹ,κ, ζq; K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 3 of 22 (Fm4) Wpκ, ỹ, ζq ˚ Wpỹ, z̃, s̃q ď Wpκ, z̃, ζ ` s̃q; (Fm5) Wpκ, ỹ, ¨q : r0,`8q Ñ r0, 1s is left continuous. Definition 3. [9] For a given b P r0,`8q, the fuzzy set W: S2 ˆ r0,`8q Ñ r0, 1s is called FbW if for all κ, ỹ, z̃ P S, ζ, s̃ ą 0 it satisfies: pFbm1q Wbpκ, ỹ, 0q “ 0; pFbm2q Wbpκ, ỹ, ζq “ 1 if and only if κ “ ỹ; pFbm3q Wbpκ, ỹ, ζq “ Wpỹ,κ, ζq; pFbm4q Wbpκ, ỹ, ζb q ˚ Wbpỹ, z̃, s̃ b q ď Wbpκ, z̃, ζ ` s̃q; pFbm5q Wbpκ, ỹ, ¨q : p0,`8q Ñ r0, 1s is continuous and lim nÑ`8 Wµpκ, ỹ, ζq “ 1. Remark 1. FbWS̃ becomes FWS̃ for b = 1. Remark 2. [17] Generally speaking, a FbWS̃ is not continuous. Example 1 demonstrates that it is not necessary for a FbW on S to be a FW on S. Definition 4. [10] The ordered triple pS,We, ˚q is called EFbWS̃ by function α : SˆS Ñ r1,`8q, where We : S ˆ S Ñ r0,`8q is fuzzy set which for all κ, ỹ, z̃ P S satisfies the conditons: pFebm1q Wepκ, ỹ, 0q “ 0; pFebm2q Wepκ, ỹ, ζq “ 1, for all ζ ą 0 if and only if ỹ “ κ; pFebm3q Wepκ, ỹ, ζq “ Wepỹ,κ, ζq; pFebm4q Wepκ, z̃, αpκ, z̃qpζ ` s̃q ě Wepκ, ỹ, ζq ˚ Wepỹ, z̃, s̃q, for all s, ζ ą 0; pFebm5q Wepκ, ỹ, .q : p0,`8q Ñ r0, 1s is continuous and lim ζÑ`8 Wepκ, ỹ, ζq “ 1. Definition 5. [12] Let α, µ : S ˆ S Ñ r1,`8q be bounded functions. A fuzzy set Wµ : S ˆ S ˆ r0,`8q is said to be µEFbW if for all κ, ỹ, z̃ P S, the following con- ditions are satisfied: pµe1q Wµpκ, ỹ, 0q “ 0; pµe2q Wµpκ, ỹ, ζq “ 1, for all ζ ą 0 if and only if κ “ ỹ; pµe3q Wµpκ, ỹ, ζq “ Wµpỹ,κ, ζq; pµe4q Wµpκ, z̃, αpκ, z̃qζ ` µpκ, z̃qȷq ě Wµpκ, ỹ, ζq ˚ Wµpỹ, z̃, ȷq, for all ȷ, ζ ą 0; pµe5q Wµpκ, ỹ, .q : p0,`8q Ñ r0, 1s is continuous and lim ζÑ`8 Wµpκ, ỹ, ζq “ 1. Then pS,Wµ, ˚, α, µq is called µEFbWS̃, Remark 3. It is important to note that FbWS̃ and EFbWS̃ are specific forms of µEFbWS̃ when αpκ, ỹq “ µpκ, ỹq “ b, for some b ě 1 αpκ, ỹq “ µpκ, ỹq, respectively. Example 1. [18] Let Wpκ, ỹ, ζq “ e´|κ´ỹ|p{ζ , where p P p1,`8q. Then W is a FbWS̃ with b “ 2p´1. pS,W, ˚q in the above example is not a FWS̃ for p “ 2. K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 4 of 22 Example 2. [12] Let S “ r0, 1s and Wµpκ, ỹ, ζq “ e ´|κ´ỹ| ζ , for all κ, ỹ P S. It can be readily confirmed that pS,Wµ, ˚, α, µq is a G-complete µEFbWS̃ with mappings α, µ : S ˆ S Ñ r1,`8q defined by αpκ, ỹq “ 1 ` κỹ and µpκ, ỹq “ 1 ` κ ` ỹ, respectively and continuous t-norm ˚ as usual product. Definition 6. Let tκmu be a sequence in pS,Wµ, ˚, α, µq, then: p1q tκmu is called G-convergent to κ0 P S, if lim mÑ`8 Wµpκm,κ0, ζq “ 1 for all ζ ą 0. p2q tκmu is G-Cauchy if lim mÑ`8 Wµpκm,κm`p, ζq “ 1 for all ζ, p ą 0. p3q S is G-complete, if every Cauchy sequence converges to some point of S. 3. Main results Let Fb denote the collection of all mappings β : r0,`8q Ñ r0, 1b q , that satisfy the following condition: βpζnq Ñ 1 b implies ζn ÝÑ 0 as n Ñ `8, for a given real numberb ą 1. (3.1) Our main results make use of the following Lemma. Lemma 1. [19] Let pS,W, ˚q be a complete FWS̃ and Wpκ, ỹ, kζq ě Wpκ, ỹ, ζq for all κ, ỹ P S. With k P p0, 1q and ζ ą 0, then κ “ ỹ. Theorem 1. Let pS,W, ˚, α, µq be a G-complete µEFbWS̃ with mappings α, µ : SˆS Ñ r1,`8q. Let g be a self mapping satisfying Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq ě Wµpκ1,κ2, ζq, for all κ1,κ2 P S with β P Fb and max " sup pě1 lim mÑ`8 αpκm,κn`pq, sup pě1 lim mÑ`8 µpκm,κm`pq * ă b, m, p P N. Then g has a fixed point. Proof. Set κm`1 “ gκm, where m “ 0, 1, 2, ¨ ¨ ¨ , and κ0 P S is fixed. Wµpκm,κm`1, ζq (3.2) “ Wµ ˆ gκm´1, gκm, ζ ˙ ě Wµpκm´1,κm, ζ βpWµpκm´1,κm, ζqq q K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 5 of 22 ě Wµ ˆ κm´2,κm´1, ζ βpWµpκm´1,κm, ζqqβpWµpκm´2,κm´1, ζqq ˙ ě Wµ ˆ κm´3,κm´2, ζ ś3 i“1 βpWµpκm´i,κm`1´i, ζqq ˙ ... ě Wµ ˆ κ0,κ1, ζ śm i“1 βpWµpκm´i,κm`1´i, ζqq ˙ . implies Wµpκm,κm`1, ζq ě Wµ ˆ κ0,κ1, ζ śm i“1 βpWµpκm´i,κm`1´i, ζqq ˙ . (3.3) For any p P N, writing ζ “ pζ p “ ζ p ` ζ p ` ¨ ¨ ¨ ` ζ p , and applying pµ4q repeatedly, we have Wµpκm,κm`p, ζq ě Wµ ˆ κm,κm`1, ζ pαpκm,κm`pq ˙ ˚ Wµ ˆ κm`1,κm`2, ζ pµpκm,κm`pqαpκm`1,κm`pq ˙ ˚ ¨ ¨ ¨ ˚ Wµ ˆ κm`pp´1q,κm`p, ζ pµpκm,κm`pqµpκm`1,κm`pq ¨ ¨ ¨µpκm`pp´3q,κm`pqαpκm`pp´2q,κm`pq ˙ using inequality (3.2), we get Wµpκm,κm`p, ζq ě Wµ ˆ κ0,κ1, ζ pαpκm,κm`pq śm i“1 βpWµpκm´i,κm`1´i, ζqq ˙ ˚ Wµ ˆ κ0,κ1, ζ pµpκm,κm`pqαpκm`1,κm`pq śm`1 i“1 βpWµpκm´i,κm`1´i, ζqq ˙ ˚ Wµ ˆ κ0,κ1, ζ pµpκm,κm`pqµpκm`1,κm`pqαpκn`2,κm`pq śm`2 i“1 βpWµpκm´i,κm`1´i, ζqq ˙ ˚ ¨ ¨ ¨ ˚ Wµ ˆ κ0,κ1, ζ p śp´2 i“1 µpκn`i´1,κm`pqαpκn`pp´2q,κm`pq śm`pp´1q i“1 βpWµpκm´i,κm`1´i, ζqq ˙ . The hypothesismax " sup pě1 lim mÑ`8 αpκm,κm`pq, sup pě1 lim mÑ`8 µpκm,κm`pq * ă b, along with p3.1q, gives lim mÑ`8 Wµpκm,κm`p, ζq ě Wµ ˆ κ0,κ1, ζbm´1 p ˙ ˚ Wµ ˆ κ0,κ1, ζbm´1 p ˙ ˚ ¨ ¨ ¨ ˚ Wµ ˆ κ0,κ1, ζbm´1 p ˙ . Therefore lim nÑ`8 Wµpκm,κm`p, ζq “ 1 ˚ 1 ˚ ... ˚ 1 “ 1. Therefore tκmu is Cauchy. The com- pleteness of matric space pS,Wu, ˚, α, µq guarantees the existence of some z̃ in pS,Wu, ˚, α, µq, such that lim mÑ`8 κm “ z̃. Thus we have Wµpgz̃, z̃, ζq ě Wµ ˆ gz̃, gκm, ζ 2αpgz̃, z̃q ˙ ˚ Wµ ˆ gκm, z̃, ζ 2µpgz̃, z̃q ˙ K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 6 of 22 ě lim mÑ`8 Wµ ˆ z̃,κm, ζ 2αpgz̃, z̃qβpWµpz̃,κmqq ˙ ˚ Wµ ˆ κm`1,κm, ζ 2µpgz̃, z̃q ˙ ÝÑ 1 ˚ 1 “ 1. Hence gz̃ “ z̃ is fixed point. For uniqueness assume that gs̃ “ s̃ ‰ z̃ “ gz̃, for some s̃ϵS, then Wµps̃, z̃, ζq “ Wµpgs̃, gz̃, ζq ě Wµps̃, z̃, ζ βpWµps̃, z̃, ζqq q “ Wµpgs̃, gz̃, ζ βpWµps̃, z̃, ζqq q ě Wµps̃, z̃, ζ βpWµps̃, z̃, ζqq2 q ... ě Wµps̃, z̃, ζ βpWµps̃, z̃, ζqqm q “ Wµps̃, z̃, bmζq. That is Wµps̃, z̃, ζq Ñ 1 as m Ñ `8. Thus s̃ “ z̃ and hence fixed point is unique. Letting βpWµpκ1,κ2, ζqq “ k for some k, ϵr0, 1b q. The following Theorem’s corol- lary, which generalizes the widely known Banach contraction theorem in the context of µEFbWS̃, is obtained. Corollary 1. Let pS,Wu, ˚, α, µq be a G-complete µEFbWS̃ with mappings α, µ : SˆS Ñ r1,`8q. Let g be a self mapping satisfying Wµpgκ1, gκ2, kζq ě Wµpκ1,κ2, ζq, for all κ1,κ2 P Swhere β P Fb. Then g has a fixed point. An example will be provided for the aforementioned Theorem in order to validate it Example 3. Let S “ r0, 1s, Let Wµ : SˆSˆr1,`8q Ñ r0, 1s, defined by Wµpκ1,κ2, ζq “ e ´|κ1´κ2| ζ . pS,Wu, ˚, α, µq is µEFbWS̃ with b=2. Consider the mapping g : S Ñ S defined by gpκq “ 1´ 1 6κ, and β : r0, 1s Ñ r0, 12q as βpζq “ 1 4 . obviously β P F2. For all κ1,κ2 P S, ζ ą 0, we have Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq “ e ´|gκ1´gκ2| βpWµpκ1,κ2,ζqζq “ e ´|1´ 1 6κ1´p1´ 1 6κ2q| 1 4 ζ “ e ´|´ 1 6κ1` 1 6κ2| 1 4 ζ K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 7 of 22 “ e ´ 1 6 |κ1´κ2| 1 4 ζ “ e ´ 2 3 |κ1´κ2| ζ ě e ´|κ1´κ2| ζ . This implies that Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq ě Wµpκ1,κ2, ζq. And gp67q “ 6 7 . Consequently, 6 7 is an unique fixed point of g. Since Theorem 1 criteria are all fulfilled. Theorem 2. Consider a G-complete µEFbWS̃, pS,Wu, ˚, α, µq with mappings α, µ : Sˆ S Ñ r1,`8q. Let g be a self mapping on S such that Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq ě min " Wµpκ1,κ2, ζq,Wµpκ1, gκ1, ζq,Wµpκ2, gκ2, ζq, Wµpκ1, gκ2, pαpκ1,κ2q ` µpκ1,κ2qqζq ˚ Wµpκ2, gκ1, pαpκ1,κ2q ` µpκ1, x2qqζq * , for all κ1,κ2 P S. Then there is unique fixed point of g in S. Here ˚ “ ˚m. Proof. Beginning in the same manner as in Theorem 1, we have Wµpκm,κm`1, ζq “ Wµpgκm´1, gκm, ζq ě min " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1, gκm´1, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm, gκm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1, gκm, pαpκm´1,κmq ` µpκm´1,κmqqζ βpWµpκm´1,κm, ζqq ˙ ˚ Wµ ˆ κm, gκm´1, pαpκm´1,κmq ` µpκm´1,κmqqζq βpWµpam´1, am, ζqq ˙* ě min " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm`1, pαpκm´1,κmq ` µpκm´1,κmqqζ βpWµpκm´1,κm, ζqq ˙ ˚ Wµ ˆ κm,κm, pαpκm´1,κmq ` µpκm´1,κmqqζq βpWµpκm´1,κm, ζqq ˙* ě min " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 8 of 22 Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm`1, pαpκm´1,κmq ` µpκm´1,κmqq βpWµpκm´1,κm, ζqq ˙ ˚ 1 * ě min " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm´1,κm`1, pαpκm´1,κmq ` µpκm´1,κmqqζ βpWµpκm´1,κm, ζqq ˙* ě min " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ˚ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙* . (3.4) If min " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙* “ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , then inequality (3.4) implies that Wµ ˆ κm,κm`1, ζ ˙ ě Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , the remaining part of the proof is the consequence of the Lemma 1. If min " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙* “ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ then from (3.4) we have Wµ ˆ κm,κm`1, ζ ˙ ě Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , continuing in this way, we get Wµ ˆ κm,κm`1, ζ ˙ ě ˆ κ0,κ1, ζ śm i“1 βpWµpκm´i,κm`1´i, ζqq ˙ . The same method employed after inequality (3.2) in Theorem 1 can be used to finish the proof of this result. Letting min " Wµpκ1,κ2, ζq,Wµpκ1, gκ1, ζq,Wµpκ2, gκ2, ζq, K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 9 of 22 ˆ Wµpκ1, gκ2, pαpκ1,κ2q ` µpκ1,κ2qqζq ˚ Wµpκ2, gκ1, pαpκ1,κ2q ` µpκ1,κ2qqζq ˙* “ Wµpκ1,κ2, ζq. We have deduced the following corollary from Theorem 2, which reduce the above theorem to Theorem 1. Corollary 2. Let pS,Wu, ˚, α, µq be a G-complete µEFbWS̃ with mappings α, µ : SˆS Ñ r1,`8q. Let g be a self mapping satisfying Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq ě Wµpκ1,κ2, ζq, for all κ1,κ2 P S whereβ P Fb. Then g has a fixed point. Theorem 3. Let pS,Wu, ˚, α, µq be a G-completeµEFbWS̃ with mappings α, µ : SˆS Ñ r1,`8q and g : S Ñ S be a mapping satisfying Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq ě min " Wµpgκ1, gκ2, ζq,Wµpκ1, gκ1, ζq,Wµpκ2, gκ2, ζq,Wµpκ1,κ2, ζq * , for all κ1,κ2 P S and β P Fb. Then there is a unique κ0 in S such that gpκ0q “ κ0. Proof. Beginning in the same manner as in Theorem 1, we have Wµpκm,κm`1, ζq “ Wµpgκm´1, gκm, ζq ě min " Wµ ˆ gκm´1, gκm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1, gκm´1, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm, gκm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* ě min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* ě min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* , employing the identical method as applied in Theorem 2 subsequent to the inequality (3.4), the existence and uniqueness of fixed point can be shown. Theorem 4. Let pS,Wu, ˚, α, µq be a G-complete µEFbWS̃ with mappings α, µ : SˆS Ñ r1,`8q. Let g : S Ñ S be a mapping satisfying Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq ě η ` κ1,κ2, ζ ˘ max " Wµpκ1, gκ1, ζq,Wµpκ2, gκ2, ζq * , K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 10 of 22 where η ` κ1,κ2, t ˘ “ min " Wµpgκ1, gκ2, ζqWµpκ1,κ2, ζq,Wµpκ1, gκ1, ζqWµpκ2, gκ2, ζq * , for all κ1, x2 P S, and β P Fb. Then there is a unique κ0 in S such that gpκ0q “ κ0.. Proof. Arguing as in Theorem 1, we have Wµpκm,κm`1, ζq “ Wµpgκm´1, gκm, ζq ě η ` κn,κm`1, ζ βpWµpκm´1,κm,ζqq ˘ max " Wµ ` κm´1, gκm´1, ζ βpWµpκm´1,κm,ζqq ˘ ,Wµ ` κm, gκm, ζ βpWµpκm´1,κm,ζqq ˘ * .(3.5) Now, η ˆ κn,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ “ min " Wµ ˆ gκm´1, gκm, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm´1, gκm´1, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm, gκm, ζ βpWµpκm´1,κm, ζqq ˙* “ min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙* . η ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ “ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ . (3.6) By substituting (3.6) in (3.5), we get Wµpκm,κm`1, ζq ě Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm,ζqq ˙ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm,ζqq ˙ max " Wµ ` κm´1,κm, ζ βpWµpκm´1,κm,ζqq ˘ ,Wµ ` κm,κm`1, ζ βpWµpκm´1,κn,ζqq ˘ * . (3.7) If max " Wµ ` κm´1,κm, ζ βpWµpxm´1,κm, ζqq ˘ ,Wµ ` κm,κm`1, ζ βpWµpκm´1,κn,ζqq ˘ * “ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 11 of 22 as inequality (3.7) implies that Wµpκm,κm`1, ζq ě Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , then there is nothing to prove by the assertion stated in lemma 1. If max " Wµ ` κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˘ ,Wµ ` κm,κm`1, ζ βpWµpκm´1,κn,ζqq ˘ * “ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , then from inequality (3.7) , we have Wµpκm,κm`1, ζq ě Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ě Wµ ˆ κm´2,κm´1, ζ βpWµpκm´1,κm, ζqq.βpWµpκm´2,κm´1, ζqq ˙ ... ě Wµ ˆ κ0,κ1, ζ βpWµpκm´1,κm, ζqq.βpWµpκm´2,κm´1, ζqq . . . βpWµpκ0,κ1, ζqq ˙ . The theorem can be proven using the same method as demonstrated in Theorem 1. After considering inequality (3.2), we are able to demonstrate the fixed point’s existence and uniqueness. Theorem 5. Let pS,Wu, ˚, α, µq be a G-complete µEFbWS̃ with mappings α, µ : SˆS Ñ r1,`8q. Let g be a self mapping satisfying Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq ě λpκ1,κ2, ζq ˚ γpκ1,κ2, ζq, where λ ` κ1,κ2, t ˘ “ min " Wµpgκ1, gκ2, ζq,Wµpκ1, gκ1, ζq,Wµpκ2, gκ2, ζq,Wµpκ1,κ2, ζq * γ ` κ1,κ2, ζ ˘ “ max " Wµpκ1, gκ2, ζq,Wµpgκ1,κ2, ζq * , for all κ1,κ2 P S, and β P Fb. Then there will be a unique fixed point of g in S. Proof. Using similar arguments as used in the proof of Theorem 1, we have Wµpxm,κm`1, ζq K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 12 of 22 “ Wµpgκm´1, gκm, ζq ě λ ˆ κm´1,κm, ζ βpWµpκm´1,κn,ζqq ˙ ˚ γ ˆ κm´1,κm, ζ βpWµpκm´1,κn,ζqq ˙ . (3.8) Now, λ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ “ min " Wµ ˆ gκm´1, gκm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1, gκm´1, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm, gκm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* . “ min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* . λ ˆ κm´1,κm, ζ βpWµpκm´1,κn,ζqq ˙ “ min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* .(3.9) Also γ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ “ max " Wµ ˆ κm´1, gκm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ gκm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* “ max " Wµ ˆ κm´1,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm,κm, ζ βpWµpκm´1,κm, ζqq ˙* “ max " Wµ ˆ κm´1,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , 1 * γ ` κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˘ “ 1. (3.10) Using equation(3.9) and (3.10) in inequality (3.8), we have Wµpκm,κm`1, ζq ě min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* ˚ 1 Wµpκm,κm`1, ζq ě min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* .(3.11) K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 13 of 22 If min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* “ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ . Inequality (3.11) implies that Wµpκm,κm`1, ζq ě Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ . The remaining portion of the proof follows from Lemma 1. If min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙* “ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , then from inequality (3.8), we have Wµpκm,κm`1, ζq ě Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ě Wµ ˆ κm´2,κm´1, ζ βpWµpκm´1,κm, ζqqβpWµpκm´2,κm´1, ζqq ˙ ... ě Wµ ˆ κ0,κ1, ζ βpWµpκm´1,κm, ζqqβpWµpκm´2,κm´1, ζqq . . . βpWµpκ0,κ1, ζqq ˙ . Using the identical process as applied in Theorem 2 following the inequality (3.4), we can show the existence and uniqueness of fixed point. Theorem 6. Let pS,Wu, ˚, α, µq be a G-complete µEFbWS̃ and b ě 1. Let g be a self mapping satisfying Wµpgκ1, gκ2, βpWµpκ1,κ2, ζqqζq ě λ ` κ1,κ2, ζ ˘ ˚ γ ` κ1,κ2, ζ ˘ η ` κ1,κ2, ζ ˘ , where λ ` κ1,κ2, ζ ˘ “ min " Wµpgκ1, gκ2, ζqWµpκ1,κ2, ζq,Wµpκ1, gκ1, ζqWµpκ2, gκ2, ζq * γ ` κ1,κ2, ζ ˘ “ max " Wµpκ1, gκ1, ζqWµpκ1, gκ2, ζq, pWµpκ2, gκ1, ζqq2 * K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 14 of 22 η ` κ1,κ2, ζ ˘ “ max " Wµpκ1, gκ1, ζq,Wµpκ2, gκ2, ζq * . For all κ1,κ2 P S, with β P Fb. Then g has a unique fixed point. Proof. By utilizing the identical method as outlined in the previous Theorems, we arrive at the same result. Wµpκm,κm`1, ζq “ Wµpgκm´1, gκm, ζq ě λ ˆ κm´1,κm, ζ βpWµpκm´1,κm,ζqq ˙ ˚ γ ˆ κm´1,κm, ζ βpWµpκm´1,κm,ζqq ˙ η ˆ κm´1,κm, ζ βpWµpκm´1,κm,ζqq ˙ . (3.12) Now λ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ “ min " Wµ ˆ gxm´1, gκm, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm´1,κm, ζ βpWµpxm´1,κm, ζqq ˙ , Wµ ˆ κm´1, gκm´1, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm, gκm, ζ βpWµpκm´1,κm, ζqq ˙* “ min " Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κn, ζqq ˙ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙* λ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ “ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ . (3.13) γ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ “ max " Wµ ˆ κm´1, gκm´1, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm´1, gκm, ζ βpWµpκm´1,κm, ζqq ˙ , ˆ Wµ ˆ κm, gκm´1, ζ βpWµpκm´1,κm, ζqq ˙˙2* “ max " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm´1,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , ˆ Wµ ˆ κm,κm, ζ βpWµpκm´1,κm, ζqq ˙˙2* K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 15 of 22 “ max " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ Wµ ˆ κm´1,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , 1 * γ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ “ 1. (3.14) η ˆ κm´1,κm, ζ βpWµpxm´1,κm, ζqq ˙ “ max " Wµ ˆ κm´1, gκm´1, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm, gκm, ζ βpWµpκm´1,κm, ζqq ˙* “ max " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙* (3.15) Using equation (3.13),(3.14)and (3.15) in equation(3.12), we get Wµpgκm´1, gκm, ζq ě Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm,ζqq ˙ Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm,ζqq ˙ max " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm,ζqq ˙ ,Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm,ζqq ˙* .(3.16) If max " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙* “ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , in inequality (3.16), implies that Wµ ˆ κm,κm`1, ζ ˙ ě Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙ , then there is nothing to prove by Lemma 1. If max " Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ,Wµ ˆ κm,κm`1, ζ βpWµpκm´1,κm, ζqq ˙* “ Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ , then inequality (3.16), implies that Wµ ˆ κm,κm`1, ζ ˙ ě Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ . K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 16 of 22 Continuing in the same way, we get Wµpκm,κm`1, ζq ě Wµ ˆ κm´1,κm, ζ βpWµpκm´1,κm, ζqq ˙ ě Wµ ˆ κm´2,κm´1, ζ βpWµpκm´1,κm, ζqqβpWµpκm´2,κm´1, ζqq ˙ ... ě Wµ ˆ κ0,κ1, ζ βpWµpκm´1,κm, ζqqβpWµpκm´2,κm´1, ζqq ¨ ¨ ¨βpWµpκ0,κ1, ζqq ˙ using the identical method applied subsequent to Theorem 1 right after the inequality (3.2), we have the capability to demonstrate both the presence and exclusivity of a fixed point. 4. Application to Nonlinear Fractional Differential Equations In this section, existence of solution to a nonlinear fractional order differential equation pNFDEq is investigated via the results established in this manuscript. The set S̃ ” Cpr0, 1s,Rq, furnished with norm }κ} :“ sup sPr0,1s |κpsq| is Banach space. Consider Dξ 0`pũpζqq “ gpζ, ũpζqq, ζ Ps0, 1r, (4.1) with conditions κp0q ` κ́p0q “ 0, κp1q ` κ́p1q “ 0, here κ P S̃, ξ Ps1, 2s and g : r0, 1s ˆ R Ñ R is a continuous mapping. Now κ P S̃ will be a solution to (4.1) if it solves the following equation κps̃q “ 1 Γpξq ż 1 0 p1 ´ r̃qξ´1p1 ´ s̃qgpr̃,κpτ̃qqdr̃` 1 Γpξ ´ 1q ż 1 0 p1 ´ r̃qξ´2p1 ´ s̃qgpr̃,κpr̃qqdr̃ ` 1 Γpξq ż s̃ 0 ps̃ ´ r̃qξ´1gpr̃,κpr̃qqdr̃. (4.2) For further detail and comprehensive discussion of the problem setting, [20–23] should be consulted. Define an integral operator J̃ : S̃ Ñ S̃ by J̃κps̃q “ 1 Γpξq ż 1 0 p1 ´ r̃qξ´1p1 ´ s̃qgpr̃,κpr̃qqdr̃ ` 1 Γpξ ´ 1q ż 1 0 p1 ´ r̃qξ´2p1 ´ s̃qgpr̃,κpr̃qqdr̃ ` 1 Γpξq ż s̃ 0 ps̃ ´ r̃qξ´1gpr̃,κpr̃qqdr̃. (4.3) K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 17 of 22 where S̃ is an µEFbWS̃, Wµ is a µEFbW with functions µ “ 1 ` κps̃q ` ỹps̃q and α “ 1 ` κps̃qỹps̃q, for all ζ ą 0 and ũ, ỹ P S̃. Wµpκ, ỹ, ζq “ e ´ sup s̃Pr0,1s |κps̃q´ỹps̃q| ζ , Theorem 7. Define tgmu Ă S̃, by gm`1 “ J̃pgmq, where m P N Y t0u and g P S̃. Let there be βpWµpκ, ỹ, ζqq P p0, 1 k2 q, with k “ lim sup m,nÑ`8 Ωpgm, gnq, for a bounded function Ω : S̃ ˆ S̃ Ñ r1,`8q and the following conditions are satisfied: • |gpr̃,κpr̃qq ´ gpr̃, ~ypr̃qq| ď |κpr̃q ´ ~ypr̃q|, for all κ, ỹ P S̃ • sup ζPr0,1s " 1´s̃ Γpξ`1q ` 1´s̃ Γpξq ` sξ Γpξ`1q * ď βpWµpκ, ỹ, ζqq ă 1. Then pNFDEq (4.1) has unique solution K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 18 of 22 Proof. ˇ ˇJ̃κps̃q ´ J̃~yps̃q ˇ ˇ “ ˇ ˇ ˇ ˇ 1 ´ s̃ Γpξq ż 1 0 p1 ´ r̃qξ´1 rgpr̃,κpr̃qq ´ gpr̃, ~ypr̃qqs dr̃ ` 1 ´ s̃ Γpξ ´ 1q ż 1 0 p1 ´ r̃qξ´2 rgpr̃,κpr̃qq ´ gpr̃, ~ypr̃qqs dr̃ ` 1 Γpξq ż s̃ 0 ps̃ ´ r̃qξ´1 rgpr̃,κpr̃qq ´ gpr̃, ~ypr̃qqs dr̃ ˇ ˇ ˇ ˇ ď 1 ´ s̃ Γpξq ż 1 0 p1 ´ r̃qξ´1 ˇ ˇgpr̃,κpr̃qq ´ gpr̃, ~ypr̃qq ˇ ˇdr̃ ` 1 ´ s̃ Γpξ ´ 1q ż 1 0 p1 ´ r̃qξ´2 ˇ ˇgpr̃,κpr̃qq ´ gpr̃, ~ypr̃qq ˇ ˇdr̃ ` 1 Γpξq ż s̃ 0 ps̃ ´ r̃qξ´1 ˇ ˇgpr̃,κpr̃qq ´ gpr̃, ~ypr̃qq ˇ ˇdr̃ ď 1 ´ s̃ Γpξq ż 1 0 p1 ´ r̃qξ´1 ˇ ˇκpr̃q ´ ~ypr̃q ˇ ˇdr̃ ` 1 ´ s̃ Γpξ ´ 1q ż 1 0 p1 ´ r̃qξ´2 ˇ ˇκpr̃q ´ ~ypr̃q ˇ ˇdr̃ ` 1 Γpξq ż s̃ 0 ps̃ ´ r̃qξ´1 ˇ ˇκpr̃q ´ ~ypr̃q ˇ ˇdr̃ ď sup s̃Pr0,1s ˇ ˇκps̃q ´ ~yps̃q ˇ ˇ ˆ 1 ´ s̃ Γpξq ż 1 0 p1 ´ r̃qξ´1dr̃ ` 1 ´ s̃ Γpξ ´ 1q ż 1 0 p1 ´ r̃qξ´2dr̃ ` 1 Γpξq ż s̃ 0 ps̃ ´ r̃qξ´1dr̃ ˙ “ sup s̃Pr0,1s ˇ ˇκps̃q ´ ~yps̃q ˇ ˇ ˆ 1 ´ s̃ Γpξ ` 1q ` 1 ´ s̃ Γpξq ` sξ Γpξ ` 1q ˙ “ βpWµpκ, ỹ, ζqq sup s̃Pr0,1s ˇ ˇκps̃q ´ ~yps̃q ˇ ˇ, where βpWµpκ, ỹ, ζqq “ 1 ´ s̃ Γpξ ` 1q ` 1 ´ s̃ Γpξq ` sξ Γpξ ` 1q . K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 19 of 22 From the above inequality, it becomes apparent that sup s̃Pr0,1s ˇ ˇJ̃κps̃q ´ J̃~yps̃q ˇ ˇ ď βpWµpκ, ỹ, ζqq sup s̃Pr0,1s ˇ ˇκps̃q ´ ~yps̃q ˇ ˇ ñ ´ sup s̃Pr0,1s ˇ ˇJ̃κps̃q ´ J̃~yps̃q ˇ ˇ βpWµpκ, ỹ, ζqqζ ě ´βpWµpκ, ỹ, ζqq sup s̃Pr0,1s ˇ ˇκps̃q ´ ~yps̃q ˇ ˇ βpWµpκ, ỹ, ζqqζ ñ e ´ sup s̃Pr0,1s ˇ ˇJ̃κps̃q´J̃~yps̃q ˇ ˇ βpWµpκ,ỹ,ζqqζ ě e ´ sup s̃Pr0,1s ˇ ˇκps̃q´~yps̃q ˇ ˇ ζ ñ Wµ ´ J̃κ, J̃~y, βpWµpκ, ỹ, ζqqζ ¯ ě Wµ pκ, ~y, ζq . Theorem 1 implies that J̃ has a fixed point v˚ P S̃, and as a result, the NFDE (4.1) has a unique solution in S̃. 5. Application to Integral Equations In this section, we use our established results to ensure that integral equations of the type upζq “ gpζq ` ż ζ 0 Hpζ, s̃,κps̃qqds, ζ P r0, bs, (5.1) have solution in C ” Cpr0, bs,Rq, being a Banach space, with norm }κ} :“ sup s̃Pr0,bs |κpsq|, where b is some given positive real number. Consider a complete µEFbWS̃, pC,Wµ, ˚p, α, µq where Wµpκ, ỹ, ζq “ e ´ sup s̃Pr0,bs |κps̃q´ỹps̃q|2 ζ , for all κ, ỹ P C and βpWµpκ, ỹ, ζqq ą 0, with t-norm ˚p and bounded functions µ, α : C ˆ C Ñ r1,`8q. The existence of a solution to the integral equation of the form (5.1) is ensured by the following theorem. Theorem 8. Let I : C Ñ C be an integral operator given by rIpuqspζq “ gpζq ` ż ζ 0 Hpζ, s̃,κps̃qqds̃, u P C, ζ P r0, bs. Let tgmu Ă C, be defined by gm`1 “ Ipgmq, m P N Y t0u, for g P C. Suppose there exists βpWµpκ, ỹ, ζqq P p0, 1 k2 q, with κ “ lim sup m,nÑ`8 Ωpgm, gnq, where Ω : C ˆ C Ñ r1,`8q is a bounded mapping and let H P Cpr0, bs ˆ r0, bs ˆ R,Rq holds the following condition: |Hpζ, s̃,κps̃qq ´ Hpζ, s̃, ỹps̃qq| ď b βpWµpκ, ỹ, ζqq|κpr̃q ´ ỹpr̃q|. Then, the integral equation (5.1) has a solution u˚ P C. K. Abodayeh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5994 20 of 22 Proof. For all κ, ỹ P C, and βpWpκ, ỹ, ζqq ą 0, we have WµpIpκq, Ipỹq, βpWµpκ, ỹ, ζqqζq “ e ´ sup r̃Pr0,bs |rIpκqsps̃q´rIpỹqsps̃q|2 βpWµpκ,ỹ,ζqqζ “ e ´ sup r̃Pr0,bs | şs̃ 0 Hps̃,r̃,κpr̃qqdr̃´ şs̃ 0 Hps̃,r̃,ỹpr̃qqdr̃|2 βpWµpκ,ỹ,ζqqζ ě e ´ sup r̃Pr0,bs şs̃ 0 |Hps̃,r̃,κpr̃qq´Hps̃,r̃,ỹpr̃qq|2dr̃ βpWµpκ,ỹ,ζqqζ ě e ´ sup r̃Pr0,bs şs̃ 0p ? βpWµpκ,ỹ,ζqq|κpr̃q´ỹpr̃q|q 2 dr̃ βpWµpκ,ỹ,ζqqζ ě e ´ sup r̃Pr0,bs şs̃ 0 βpWµpκ,ỹ,ζqq|κpr̃q´ỹpr̃q|2dr̃ βpWµpκ,ỹ,ζqqζ ě e ´ sup r̃Pr0,bs βpWµpκ,ỹ,ζqq|κpr̃q´ỹpr̃q|2 βpWµpκ,ỹ,ζqqζ “ e ´ sup r̃Pr0,bs |κpr̃q´ỹpr̃q|2 ζ “ Wµpκ, ỹ, ζq. ñ WµpIpκq, Ipỹq, βpWµpκ, ỹ, ζqqζq ě Wµpκ, ỹ, ζq. As a result, the unique fixed point u˚ P C of I, is unique solution to the integral equa- tion (5.1). Acknowledgements The authors would to thank Prince Sultan University for paying the APC and for the support through the TAS research lab. 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