A New Four-Step Iterative Procedure for Approximating Fixed Points with Application to 2D Volterra Integral Equations

  Citation: Hammad, H.A.; Rehman, H.u.; De la Sen, M. A New Four-Step Iterative Procedure for Approximating Fixed Points with Application to 2D Volterra Integral Equations. Mathematics 2022,10, 4257. https://doi.org/10.3390/math 10224257 Academic Editor: Paul Bracken Received: 21 October 2022 Accepted: 9 November 2022 Published: 14 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article A New Four-Step Iterative Procedure for Approximating Fixed Points with Application to 2D Volterra Integral Equations Hasanen A. Hammad 1,2,* , Habib ur Rehman 3and Manuel De la Sen 4 1Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia 2Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt 3Department of Mathematics, Monglkut’s University of Technology, Bangkok 10140, Thailand 4Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, 48940 Leioa, Bizkaia, Spain *Correspondence: [email protected] or [email protected] Abstract: This work is devoted to presenting a new four-step iterative scheme for approximating fixed points under almost contraction mappings and Reich–Suzuki-type nonexpansive mappings (RSTN mappings, for short). Additionally, we demonstrate that for almost contraction mappings, the proposed algorithm converges faster than a variety of other current iterative schemes. Furthermore, the new iterative scheme’s ω2− stability result is established and a corroborating example is given to clarify the concept of ω2− stability. Moreover, weak as well as a number of strong convergence results are demonstrated for our new iterative approach for fixed points of RSTN mappings. Further, to demonstrate the effectiveness of our new iterative strategy, we also conduct a numerical experiment. Our major finding is applied to demonstrate that the two-dimensional (2D) Volterra integral equation has a solution. Additionally, a comprehensive example for validating the outcome of our application is provided. Our results expand and generalize a number of relevant results in the literature. Keywords: RSTN mapping; almost contraction mapping; ω2− stability; fixed point methodology; nonlinear integral problem MSC: 47H05; 39B82; 47H09 1. Prelude and Basic Notions Nowadays, after the huge amount of valuable papers that include the fixed point (FP) method, these points have become the mainstay for nonlinear analysis due to the ease and smoothness of this method, in addition to the numerous and exciting applications in economics, biology, chemistry, game theory, engineering, physics, etc. [1–5]. A very important branch is the involvement of FPs in approximation by algorithms. Numerous problems such as convex feasibility problems, convex optimization problems, monotone variational inequalities, and image restoration problems can be thought of as FP problems of nonexpansive mappings, hence approximating them has a range of specialized applications, see [ 6 – 12 ]. Iteration approaches for FP issues of nonexpansive mappings have received a lot of attention in the literature, for example, see [13–17]. From now on, the symbols R , N , Ξ(=) , ∆ , and Π , denote the set of real numbers, natural numbers, FPs of the mapping = , and a nonempty subset of a Banach space (BS) Π, respectively. Assume that =:∆→∆is a self-mapping, then for each v,υ∈∆, •=is called a contraction if there is `∈[0, 1)so that k=v− =υk≤`kv−υk. •=is called nonexpansive if k=v− =υk≤kv−υk, i.e., it is a contraction with `=1. •=owns an FP v, if v==v. Mathematics 2022,10, 4257. https://doi.org/10.3390/math10224257 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 4257 2 of 26 There are two main categories that can be used to group the main concepts of FP theory. Finding the prerequisites and requirements necessary for an operator to admit fixed points is the first step. Another option is to locate these fixed points using certain schematic methods. The first category is known formally as the existence part, while the second category is known as the computation or approximation part. Studying the behaviors of FPs, such as stability and data dependence, is an essential but less well-known topic of FP theory. The class of weak contractions that appropriately covers the class of Zamfirescu operators [ 18 ] was supplied by Berinde in [ 19 ]. Many authors also refer to this class of mappings as “almost contraction mappings (ACM)”. Definition 1. If there are `∈[0, 1)and δ≥0, the inequality below holds k=v− =υk≤`kv−υk+δkv− =vk,for all v,υ∈∆. (1) Then =:∆→∆is called ACM. Via the concept of strictly increasing continuous functions (SIC functions), the condition (1) generalized by Imoru and Olantiwo [20] as follows: Definition 2. If there is a constant `∈[ 0, 1 ) and a SIC function ξ:[ 0, ∞)→[ 0, ∞) with ξ(0) = 0such that k=v− =υk≤`kv−υk+ξ(kv− =vk),for all v,υ∈∆. (2) Then =:∆→∆is called contractive-like. Clearly, the inequality (2) reduces to (1), if ξ(τ) = δτ. Due to its significance in terms of applications, numerous writers have studied nonexpansive mappings extensions and generalizations in recent years. Suzuki [ 20 ] presented an intriguing generalization of nonexpansive mappings and attained some results for existence and convergence. These mappings are frequently referred to as mappings satisfying condition (C). Definition 3. If the inequality below is true 1 2kv− =vk≤kv−υk⇒k=v− =υk≤kv−υk,for all v,υ∈∆. (3) Then =:∆→∆is said to satisfy condition (C). In 2019, the class of RSTN mappings was considered by Pant and Pandey [ 21 ] as the following: Definition 4. If there is a constant `∈[0, 1)so that 1 2kv− =vk≤kv−υk⇒k=v− =υk≤`kv− =vk+`kυ− =υk+ (1−2`)kv−υk, (4) for all v,υ∈∆.Then =:∆→∆is called an RSTN mapping. Surely, every mapping satisfying condition (C) is an RSTN mapping with `= 0. The converse, however, is false, as demonstrated in [21]. The analysis of the performance and behavior of algorithms that make significant contributions to real-world applications is one of the key trends in FP techniques. Therefore, in order to enhance the functionality and convergence behavior of algorithms for nonexpansive mappings, several authors tended to develop numerous iterative schemes for approximating FPs, for example Mann [ 22 ], Ishikawa [ 23 ], Noor [ 24 ], Argawal et al. [25] , Mathematics 2022,10, 4257 3 of 26 Abbas and Nazir [ 26 ], CR [ 27 ], Normal-S [ 28 ], PicardS [ 29 ], Thakur et al. [ 30 ], and Miterative [31] schemes. Recently, Ahmad et al. [ 32 ] presented a good iterative method known as the JKiterative procedure:        z1∈∆, vr= (1−ηr)zr+ηr=zr, ϑr==vr, zr+1==((1−γr)=vr+γr=ϑr), for all r≥1, (5) where ηr and γr are sequences in ( 0, 1 ) . For the mappings satisfying condition (C) , the authors generated several weak and strong convergence results and also showed numerically that the iterative method (5) converges quicker than the iteration [25,30]. Very recently, Hasanen et al. [ 33 ] presented a novel four-step iterative scheme known as the HR-iteration:            z0∈∆, vr= (1−ηr)zr+ηr=zr, ωr==((1−αr)vr+αr=vr), ϑr==((1−γr)=ωr+γr=ωr) zr+1==ϑr, for all r≥1, (6) where αi , ηi , and γi are sequences in [ 0, 1 ] . Additionally, the authors proved that this algorithm converges faster than the methods presented in [27,29–31] numerically. According to the above works, we build a new four-step iterative procedure called HR*-iteration for obtaining a novel approximation to FPs of ACMs and RSTN mappings as follows:            $0∈∆, ρr= (1−sr)$r+sr=$r, ωr==((1−tr)ρr+tr=ρr), ϑr==(=(ωr)), $r+1= (1−er)ϑr+er=ϑr, for all r≥1, (7) where sr,tr, and erare sequences in (0, 1). The goal of this manuscript is to show that the iteration (7) converges faster than iterations (5), (6), and Thakur et al.’s [ 30 ] iterative scheme. Hence, it is faster than many sober iterative methods in this direction for ACMs. Additionally, the property of ω2− stability for the proposed algorithm is shown with a supported example. Moreover, weak and strong convergence results of the considered method are obtained for RSTN mappings. Ultimately, we prove that a 2D Volterra integral equation has a solution in BSs using our main findings. 2. Definitions and Auxiliary Lemmas In this part, we provide some basic definitions and concepts that help us in our desired goal and also facilitate the reader to understand our manuscript. Assume that Π∗ is a dual of a BS Π , h ., . i refers to the generalized duality pairing between Π and Π∗ , −→ denotes strong convergence, and * denotes weak convergence. For v∈Π , the normalized duality mapping Θ:Π→ 2 Π∗ is a multivalued mapping defined as Θ(v) = nυ∈Π∗:hv,υi=kvk2=kυk2o. A BS Πis called smooth if the limit below exists for all v,υ∈P lim a→0 kv+aυk−kvk a, (8) where P={υ∈Π:kυk=1} . Here, the norm of Π is called Gâteaux differentiable. Clearly, if Π is smooth, then Θ is a single-valued mapping. Further, if the limit (8) exists and is Mathematics 2022,10, 4257 4 of 26 attained uniformly for υ∈Z , then the norm of Π is called Fréchet differentiable for v∈P and the following inequality is true hv,Θ(v)i+1 2kvk2≤1 2kv+υk2≤ hυ,Θ(v)i+1 2kvk2+z(υ), where z:[0, ∞)→[0, ∞)is an increasing function so that limυ↓0z(υ) υ=0. Definition 5. If for each e∈( 0, 2 ] ,there exists δ> 0so that kυk≤ 1, kυk≤ 1and kυ−vk>e , we get  υ+v 2 <1−δfor υ,v∈Π.Then a BS Πis called a uniformly convex. Definition 6. If for any sequence {νi}in Πso that υi*υ∈Π,implies lim sup r→∞ kυr−υk<lim sup r→∞ kυr−vk,for all v∈Πwith ν6=v. Then a BS Πis said to satisfy Opial’s condition. Definition 7. Assume that {υr}is a bounded sequence in a BS Π.For υ∈∆⊂Π,put <(υ,{υr}) = lim sup r→∞ kυr−υk. •The asymptotic radius of {υi}relative to Πis described as <(Π,{υr}) = inf{<(υ,{υr}):υ∈Π}. •The asymptotic center of {υi}relative to Πis given by Z(Π,{υr}) = {υ∈Π:<(υ,{υr}) = <(Π,{υr})}. Clearly, Z(Π,{υr})consists of exactly one point in a uniformly convex BS. Definition 8. Assume that ∆6=∅ is a closed convex subset of a BS Π .A self-mapping =:∆→∆ is called demiclosed with respect to v∈Π ,if for all a sequence {vr}*∆ and {=vr} −→ υ implies =v=υ. Definition 9 ([ 34 ]) . Suppose that {sr} and {tr} are two sequences of real numbers that, respectively, converge to s and t.If there is α=limr→∞ksr−sk ktr−tk.Then (i) {sr}is converges to s faster than {tr}does to t, if α=0, (ii) the two sequences {sr}and {tr}have the same rate of convergence, if α∈(0, ∞). Definition 10 ([ 34 ]) . Assume that {ϕr} and {φr} are two FP iteration procedures which converge to the same point e υ,the error estimates kϕr−e υk≤srand kφr−e υk≤tr,r∈N are accessible, where {sr} and {tr} are defined in Definition 9and converging to 0. Then, {ϕr} converges faster to e υthan {φr}if {sr}converges faster than {tr}. Definition 11. For a mapping =:∆→∆,if lim r→∞k=υr−υrk=0. (9) Then the sequence {υr}in ∆is called an approximate FP sequence for a mapping =. Mathematics 2022,10, 4257 5 of 26 Definition 12 ([ 35 ]) . Assume that κ:( 0, ∞)→( 0, ∞) is a nondecreasing function with κ( 0 ) = 0 and for each τ> 0, if κ(τ)> 0so that k=v−vk≥κ(d(v,Ξ(=))) ,for all v∈∆ ,where d(v , Ξ(=)) = infv∗∈Ξ(=)kv−v∗k ,then the mapping =:∆→∆ is said to satisfy the condition (I). Lemma 1 ([ 36 ]) . Assume that {ξr} and {ζr} are two non-negative real sequences verifying the inequality below ξr+1≤(1−θr)ξr+ζr,∀r∈N, where θr∈(0, 1), ∞ ∑ r=0 θr=∞and limr→∞ζr θr=0, then limr→∞ξr=0. Lemma 2 ([ 28 ]) . Suppose that {vr} and {υr} are any sequences of a uniformly convex BS Π such that the following inequalities hold lim sup r→∞ kvrk≤h, lim sup r→∞ kυrk≤h and lim sup r→∞ kςrvr+ (1−ςr)υrk=h, for some h≥ 0, where {ςr} is any sequence satisfying 0 <v≤ςr≤υ< 1. Then limr→∞kvr−υrk = 0. Lemma 3 ([ 32 ]) . Assume that =:∆→∆ is a given mapping. If = is an RSTN mapping with Ξ(=)6=∅ ,then for arbitrary point v∈∆ and v∗∈Ξ(=) ,we have k=v− =v∗k≤kv−v∗k . Moreover, if =satisfies condition (C), then =is an RSTN mapping. Lemma 4 ([ 37 ]) . Suppose that =:∆→∆ is an RSTN mapping, then for all v , υ∈∆ and some `∈(0, 1), the inequality below holds kv− =υk≤3+` 1−`kv− =vk+kv−υk. (10) We now provide a numerical example that meets the inequality (10) but does not satisfy condition (C). Example 1. Assume that R endowed with a usual norm k.k is a BS and − 1 ≤∆≤ 1. Define a mapping =:∆→∆by =v=   −v 4,if −1≤v<0, −v,if v∈[0, 1]\{1 4}, 0, if v∈ {1 4}. If we set v=1 4and υ=1, we have 1 2kv− =vk=1 2    1 4− =1 4   =1 8≤3 4=kv−υk. However, k=v− =υk=   =1 4− =(1)   =1>3 4=kv−υk. Therefore, the mapping =:∆→∆does not satisfy condition (C). On the other hand, w prove that = fulfills the inequality (10). To reach this result, we suggest the following positions: (p1)if −1≤v,υ<0, we get |v− =υ|≤|v− =v|+|=v− =υ|=|v− =v|+1 4|v−υ| ≤3+v 1−v|v− =v|+|v−υ|. Mathematics 2022,10, 4257 6 of 26 (p2)if v,υ∈[0, 1]\{1 4},then |v− =υ|≤|v− =v|+|=v− =υ|=|v− =v|+|v−υ|. (p3)if −1≤v<0and υ∈[0, 1]\{1 4},we have |v− =υ|=|v+υ|≤|v|+|υ| ≤5 4|v|+|v−υ|(since v<0and υ≥0) =v−−v 4+|v−υ| =|v− =v|+|v−υ|. (p4)if −1≤v<0and υ=1 4,one can write |v− =υ|=|v|≤5 4|v|+v−1 4=|v− =v|+|v−υ|. (p5)if v∈[0, 1]\{1 4}and υ=1 4,we obtain |v− =υ|=|v|≤2|v|+v−1 4=|v− =v|+|v−υ|. Based on the above cases, we conclude that =fulfills the inequality (10) with 3+v 1−v≥1. 3. Rate of the Convergence In this part, we demonstrate analytically that for ACMs, our iterative method (7) converges faster than the iterative method in (5). Theorem 1. Let ∆6=∅ be a closed convex subset of a BS Π and =:∆→∆ be ACM. If {$r} is a sequence iterated by (7). Then {$r} −→ $,where $is a unique FP of =. Proof. Consider $∈Ξ(=). Based on (1) and (7), we have kρr−$k=k(1−sr)$r+sr=$r− =$k ≤(1−sr)k$r−$k+srk=$r− =$k(11) ≤(1−sr)k$r−$k+sr[`k$r−$k+δk$− =$k] = (1−sr(1−`))k$r−$k. From (7) and (12), we get kωr−$k=k=((1−tr)ρr+er=ρr)− =$k ≤`k(1−tr)ρr+er=ρr−$k ≤`[(1−tr)kρr−$k+erk=ρr− =$k](12) ≤`[(1−tr(1−`))kρr−$k] ≤`[(1−sr(1−`))(1−tr(1−`))]k$r−$k. Using (7) and (13), we obtain that kϑr−$k=k=(=ωr)− =$k ≤`k=ωr−$k(13) ≤`2kωr−$k ≤`3[(1−sr(1−`))(1−tr(1−`))]k$r−$k. Mathematics 2022,10, 4257 7 of 26 Finally, from (7) and (14), one can write k$r+1−$k=k(1−er)ϑr+er=ϑr− =$k ≤(1−er)kϑr−$k+erk=ϑr− =$k(14) ≤(1−er(1−`))kϑr−$k ≤`3(1−er(1−`))(1−sr(1−`))(1−tr(1−`))k$r−$k. As `∈( 0, 1 ) and 0 <er , sr , tr< 1, it follows that (1−er(1−`))< 1, (1−sr(1−`)) < 1 and (1−tr(1−`)) <1, hence (1−er(1−`))(1−sr(1−`))(1−tr(1−`)) <1. Thus, (15) reduces to k$i+1−$k≤`3k$r−$k. By induction, one can write k$r+1−$k≤`3(r+1)k$0−$k→0 as r→∞. (15) Hence, $r−→ $ . The uniqueness $ follows immediately by the definition of = . This finishes the proof. Theorem 2. Let ∆6=∅ be a closed convex subset of a BS Π and =:∆→∆ be ACM. If {$r} is a sequence iterated by (7). Then {$r} converges faster than {zr} , which is made by the iterative scheme (5). Proof. Keeping in mind (15) of Theorem 1, we get k$r+1−$k≤`3(r+1)k$0−$k,r∈N. Additionally, using (5), one can obtain kvr−$k=k(1−ηr)zr+ηr=zr− =$k ≤(1−ηr)kzr−$k+ηrk=zr− =$k(16) ≤(1−ηr(1−`))kzr−$k. From (5) and (17), we have kϑr−$k=k=vr− =$k ≤`kvr−$k(17) ≤`(1−ηr(1−`))kzr−$k. Again, using (5), (17), and (18), one has kzr+1−$k=k=((1−γr)=vr+γr=ϑr)− =$k ≤`k(1−γr)=vr+γr=ϑr−$k ≤`((1−γr)k=vr− =$k+γrk=ϑr− =$k) ≤`2((1−γr)kvr−$k+γrkϑr−$k) ≤`2[(1−γr)(1−ηr(1−`))kzr−$k+γr`(1−ηr(1−`))kzr−$k] ≤`2[(1−γr(1−`))(1−ηr(1−`))kzr−$k] ≤`2kzr−$k. Mathematics 2022,10, 4257 8 of 26 By induction, we have kzr+1−$k≤`2(r+1)kzr−$k. (18) Dividing (15) by (18), we find that k$r+1−$k kzr+1−$k≤`3(r+1)k$0−$k `2(r+1)kzr−$k=`(r+1)k$0−$k kzr−$k→0, as r→∞, which implies that {$r}converges faster than {zr}to $. Example 2. Assume that Π=R3 and ∆=v= (v1,v2,v3):(v1,v2,v3)∈[0, 6]3 ,where [0, 6]3= [0, 6]×[0, 6]×[0, 6] is a subset of Π equipped with the norm kvk=k(v1,v2,v3)k = |v1|+|v2|+|v3|.Define a mapping =:∆→∆by =v=v1 3,v2 3,v3 3,if (v1,v2,v3)∈[0, 3)3, v1 6,v2 6,v3 6,if (v1,v2,v3)∈[3, 6]3. It is clear that = owns a unique FP, it is ( 0, 0, 0 ) . Now, we shall show that = is a contractivelike mapping and, hence, ACM. For this, we define the function ξ:[ 0, ∞)→[ 0, ∞) by ξ(v) = v 4 . Obviously, ξis a SIC function with ξ(0) = 0. If v∈[0, 3)3,we have kv− =vk=  (v1,v2,v3)−v1 3,v2 3,v3 3  =   2v1 3,2v2 3,2v3 3   , and ξ(kv− =vk)=ξ   2v1 3,2v2 3,2v3 3    =  v1 6,v2 6,v3 6  =v1 6+v2 6+v3 6. (19) Analogously, if v∈[3, 6]3,one has kv− =vk=  (v1,v2,v3)−v1 6,v2 6,v3 6  =   5v1 6,5v2 6,5v3 6   , and ξ(kv− =vk)=ξ   5v1 6,5v2 6,5v3 6    =   5v1 24 ,5v2 24 ,5v3 24    = 5v1 24 + 5v2 24 + 5v3 24 . (20) After that, we discuss the cases below: Mathematics 2022,10, 4257 9 of 26 (I) If v,υ∈[0, 3)3,then by (19), we get k=v− =υk=  v1 3,v2 3,v3 3−υ1 3,υ2 3,υ3 3   =v1 3−υ1 3+v2 3−υ2 3+v3 3−υ3 3 =1 3[|v1−υ1|+|v2−υ2|+|v3−υ3|] =1 3k(v1,v2,v3)−(υ1,υ2,υ3)k=1 3kv−υk ≤1 3kv−υk+v1 6+v2 6+v3 6 =1 3kv−υk+ξ(kv− =vk). (II) If v,υ∈[3, 6]3,then by (20), we have k=v− =υk=  v1 6,v2 6,v3 6−υ1 6,υ2 6,υ3 6   =v1 6−υ1 6+v2 6−υ2 6+v3 6−υ3 6 =1 6[|v1−υ1|+|v2−υ2|+|v3−υ3|] =1 6k(v1,v2,v3)−(υ1,υ2,υ3)k=1 6kv−υk ≤1 6kv−υk+ 5v1 24 + 5v2 24 + 5v3 24  ≤1 3kv−υk+ξ(kv− =vk). (III) If v∈[0, 3)3and υ∈[3, 6]3,then by (19), we obtain that k=v− =υk=  v1 3,v2 3,v3 3−υ1 6,υ2 6,υ3 6   =  v1 3−υ1 6,v2 3−υ2 6,v3 3−υ3 6   =  v1 6+v1 6−υ1 6,v2 6+v2 6−υ2 6,v3 6+v3 6−υ3 6   ≤v1 6+v1 6−υ1 6+v2 6+v2 6−υ2 6+v3 6+v3 6−υ3 6 ≤v1 6+v2 6+v3 6+v1 6−υ1 6+v2 6−υ2 6+v3 6−υ3 6 =1 6[|v1−υ1|+|v2−υ2|+|v3−υ3|]+ξ(kv− =vk) ≤1 3k(v1,v2,v3)−(υ1,υ2,υ3)k+ξ(kv− =vk) =1 3kv−υk+ξ(kv− =vk). Mathematics 2022,10, 4257 16 of 26 The strong convergence results that we now establish are as follows: Theorem 5. Let ∆ , = , and Π be as in Lemma 6. The sequence {$r} produced by HR ∗ iterative procedure (7) converges to an element of Ξ(=) iff lim infr→∞d($r , Ξ(=)) = 0, where d($r,Ξ(=)) = inf{k$r−v∗k:v∗∈Ξ(=)}. Proof. Prove the necessity is clear. Contrariwise, assume that lim infr→∞d($r , Ξ(=)) = 0 and v∗∈Ξ(=) . From Lemma 5, limr→∞k$r−v∗k exists for any v∗∈Ξ(=) . It is enough to demonstrate that the sequence {$r} is Cauchy in ∆ . As limr→∞d($r , Ξ(=)) = 0, then for given ε>0, there is θ0∈Nso that d($r,Ξ(=)) <ε 2and inf{k$r−v∗k:v∗∈Ξ(=)}<ε 2, for all r≥θ0. Particularly, inf $θ0−v∗ :v∗∈Ξ(=)<ε 2. Hence, there is v∗∈Ξ(=)so that  $θ0−v∗ <ε 2. Now, for θ,r≥θ0, we get k$θ+r−$rk≤k$θ+r−v∗k+k$r−v∗k ≤ $θ0−v∗ + $θ0−v∗  =2 $θ0−v∗ <ε. This proves that the sequence {$r} is Cauchy in ∆ . The closedness of ∆ implies that there is an element q∈∆ so that limr→∞$r=q . Additionally, limr→∞d($r , Ξ(=)) = 0 leads to d(q,Ξ(=)) = 0, that is q∈Ξ(=). If we take the set ∆ as nonempty compact convex (NCC, for short), we have the following theorem: Theorem 6. Let = and Π be as in Lemma 6. Assume that ∆ is a NCC subset of Π .If {$r} is an iterative sequence generated by HR∗iterative scheme (7), then {$r} −→ q∈Ξ(=). Proof. Based on Lemma 6, limr→∞k=$r−$rk= 0. Because ∆ is a NCC, then there is a convergent subsequence {$ri} of {$r} so that {$ri} −→ q∈Ξ(=) . Setting $ri=υ in Lemma 4, we have k$ri− =qk≤3+` 1−`k$ri− =$rik+k$ri−qk. As i→∞ , one can find that $ri→ =q , this implies that q==q , i.e., q∈Ξ(=) . We conclude from Lemma 5that limr→∞k$r−qkexists, hence {$r} −→ q∈Ξ(=). The following theorem is obtained in the strong convergence for the sequence {$r} if the operator =meets condition (I): Theorem 7. Let ∆ , = , and Π be as in Lemma 6. If {$r} is an iterative sequence generated by HR ∗ iterative scheme (7), then {$r} −→ q∈Ξ(=)if =satisfies condition (I). Proof. According to Lemma 6, limr→∞k=$r−$rk=0. Using Definition 12, we get 0≤lim r→∞κ(d($r,Ξ(=)))≤lim r→∞k$r− =$rkimplies lim r→∞κ(d($r,Ξ(=)))=0. Since κ:( 0, ∞)→( 0, ∞) is a nondecreasing function with κ( 0 ) = 0 and for all w> 0, κ(w)> 0, we get limr→∞d($r , Ξ(=)) = 0. Because all of the prerequisites of Theorem 5 have been demonstrated, then one can infer that the sequence {$r} −→ q∈Ξ(=). Mathematics 2022,10, 4257 17 of 26 6. Numerical Example In this part, we provide an illustrative example of an RSTN mapping that does not meet condition (C) . We also assess the convergence of the HR ∗ iterative scheme in comparison to some of the most popular iterative schemes in the literature. Example 4. Consider (R,k.k) as a BS equipped with the usual norm and ∆= [ 3, 5 ] .Define a mapping =:∆→∆by =v=v+6 3,if v<5, 2, if v=5. In order to prove that =does not satisfy condition (C), we take v=4and υ=5, hence 1 2|v− =v|=1 2|4− =4|=1 3<1=|v−υ|. However, |=v− =υ|≤|=4− =5|= 10 3−6 3=4 3>1=|v−υ|. Now, to show that =is an RSTN mapping, we consider the cases below: (I) If v,υ<5, we get `|v− =v|+`|υ− =υ|+ (1−2`)|v−υ| =1 2v−v+6 3+1 2υ−υ+6 3 =1 2 2v−6 3+1 2 2υ−6 3 ≥1 22v−6 3−2υ−6 3 =1 2 2v 3−2υ 3=1 3|v−υ|=|=v− =υ|. (II) If v<5and υ=5, we obtain `|v− =v|+`|υ− =υ|+ (1−2`)|v−υ| =1 2v−v+6 3+1 2|5−2| =1 2 2v−6 3+3 2=v 3+1 2 ≥v 3=|=v− =υ|. (III) If υ<5and v=5, we have `|v− =v|+`|υ− =υ|+ (1−2`)|v−υ| =1 2|5−2|+1 2υ−υ+6 3 =3 2+1 2 2υ−6 3=1 2+υ 3 ≥υ 3=|=v− =υ|. Mathematics 2022,10, 4257 18 of 26 •If υ=v=5, we can write `|v− =v|+`|υ− =υ|+ (1−2`)|v−υ| =3>0=|2− =υ|=|=v− =υ|. Hence, =is RSTN mapping and has a unique FP 3. Numerically, by using MATLAB R2015a, we found that our iterative scheme converges faster than both iterations (5) and (6) according to Tables 1and 2and Figures 1–6as follows: Table 1. Numerical comparison of results of Algorithms (5)–(7). Number of Iterations Initial Point (z1) Algorithm (5) Algorithms (6) Algorithms (7) 3.00 16 13 7 3.82 23 18 10 4.44 25 20 10 Table 2. Numerical comparison of results of Algorithms (5)–(7). Execution Time in Seconds Initial Point (z1) Algorithm (5) Algorithms (6) Algorithms (6) 3.00 0.00483290000000000 0.00595750000000000 0.000157200000000000 3.82 0.00236760000000000 0.00755520000000000 0.00779860000000000 4.44 0.00705930000000000 0.00946030000000000 0.00744400000000000 0 2 4 6 8 10 12 14 16 Number of Iterations 10-12 10-10 10-8 10-6 10-4 10-2 100 102 Figure 1. A graphical comparison of Algorithms (5)–(7), where z1=3.00. Mathematics 2022,10, 4257 19 of 26 0123456 Elapsed time [sec] 10-3 10-12 10-10 10-8 10-6 10-4 10-2 100 102 Figure 2. A graphical comparison of Algorithms (5)–(7), where z1=3.00. 0 5 10 15 20 25 Number of Iterations 10-12 10-10 10-8 10-6 10-4 10-2 100 102 Figure 3. A graphical comparison of Algorithms (5)–(7), where z1=3.82. Mathematics 2022,10, 4257 20 of 26 12345678 Elapsed time [sec] 10-3 10-12 10-10 10-8 10-6 10-4 10-2 100 102 Figure 4. A graphical comparison of Algorithms (5)–(7), where z1=3.82. 0 5 10 15 20 25 Number of Iterations 10-12 10-10 10-8 10-6 10-4 10-2 100 102 Figure 5. A graphical comparison of Algorithms (5)–(7), where z1=4.44. Mathematics 2022,10, 4257 21 of 26 1 2 3 4 5 6 7 8 9 10 Elapsed time [sec] 10-3 10-12 10-10 10-8 10-6 10-4 10-2 100 102 Figure 6. A graphical comparison of Algorithms (5)–(7), where z1=4.44. 7. Solving 2D Volterra Integral Equation In this section, we investigate how our main results can be applied to the nonlinear 2D Volterra integral equation of the form: κ(λ,δ) = β(λ,δ) + λ Z0 δ Z0 Ω1(r,u,κ(r,u))drdu +η λ Z0 Ω2(δ,u,κ(λ,u))du +γ δ Z0 Ω3(λ,r,κ(δ,r))dr, (34) for all λ , δ , r , u∈[ 0, 1 ] , where κ∈Λ×Λ , β:[ 0, 1 ]×[ 0, 1 ]→R2 , Ωi(i= 1, 2, 3 ):[ 0, 1 ]× [0, 1]×R2→R2,η,γ≥0 and Λ=C([0, 1])is a BS with the maximum norm kv−υk∞=max τ∈[0,1]|v(τ)−υ(τ)|, for all v,υ∈C([0, 1]). Now, our main theorem here is as follows: Theorem 8. Assume that f is a nonempty closed convex subset of Λ and =:f→f described as =κ(λ,δ) = β(λ,δ) + λ Z0 δ Z0 Ω1(r,u,κ(r,u))drdu +η λ Z0 Ω2(δ,u,κ(λ,u))du +γ δ Z0 Ω3(λ,r,κ(δ,r))dr. Assume also the assertions below are true (A1)the function κ:Λ×Λ→R2is continuous; Mathematics 2022,10, 4257 22 of 26 (A2) the functions Ωi(i= 1, 2, 3 ):[ 0, 1 ]×[ 0, 1 ]×R2→R2 are continuous and there are the constants `1,`2,`3>0so that |Ω1(r,u,v1(r,u))−Ω1(r,u,v2(r,u))|≤`1|v1−v2|, |Ω2(r,u,v1(r,u))−Ω2(r,u,v2(r,u))|≤`2|v1−v2|, |Ω3(r,u,v1(r,u))−Ω3(r,u,v2(r,u))|≤`3|v1−v2|, for v1,v2∈R2; (A3)for η,γ≥0, `1+η`2+γ`3≤ξ,where ξ∈(0, 1). Then, the 2D Volterra integral Equation (34) has a solution in f×f provided that = has an FP. Proof. Let κ,κ∗∈Λ×Λ, then kκ− =κ∗k∞=max τ∈[0,1]|κ(λ,δ)(τ)− =κ∗(λ,δ)| =max τ∈[0,1] κ(λ,δ)(τ)−β(λ,δ)(τ)− λ Z0 δ Z0 Ω1(r,u,κ∗(r,u))drdu −η λ Z0 Ω2(δ,u,κ∗(λ,u))du −γ δ Z0 Ω3(λ,r,κ∗(δ,r))dr ≤max τ∈[0,1]   κ(λ,δ)(τ)−β(λ,δ)(τ)− λ Z0 δ Z0 Ω1(r,u,κ(r,u))drdu −η λ Z0 Ω2(δ,u,κ(λ,u))du −γ δ Z0 Ω3(λ,r,κ(δ,r))dr + λ Z0 δ Z0 Ω1(r,u,κ(r,u))drdu − λ Z0 δ Z0 Ω1(r,u,κ∗(r,u))drdu +η λ Z0 Ω2(δ,u,κ(λ,u))du − λ Z0 Ω2(δ,u,κ∗(λ,u))du +γ δ Z0 Ω3(λ,r,κ(δ,r))dr −γ δ Z0 Ω3(λ,r,κ∗(δ,r))dr   ≤max τ∈[0,1]|κ(λ,δ)(τ)− =κ(λ,δ)| +`1max τ∈[0,1] λ Z0 δ Z0 |κ(r,u)−κ∗(r,u)|drdu +η`2max τ∈[0,1] λ Z0 |κ(r,u)−κ∗(r,u)|du +γ`3max τ∈[0,1] δ Z0 |κ(r,u)−κ∗(r,u)|dr, Mathematics 2022,10, 4257 23 of 26 which implies that kκ− =κ∗k∞≤max τ∈[0,1]|κ(λ,δ)(τ)− =κ(λ,δ)| +max τ∈[0,1]`1|κ(r,u)−κ∗(r,u)|+η`2max τ∈[0,1]|κ(r,u)−κ∗(r,u)| +γ`3max τ∈[0,1]|κ(r,u)−κ∗(r,u)| ≤kκ− =κ∗k∞+ (`1+η`2+γ`3)max τ∈[0,1]|κ(r,u)−κ∗(r,u)| ≤kκ− =κ∗k∞+ξkκ−κ∗k∞ ≤kκ− =κ∗k∞+kκ−κ∗k∞. Hence, by Lemma 4, = is an RSTN mapping because it fulfills the condition (10) on f with 3+` 1−`= 1. Set f=∆ and Λ=Π , we find that all requirements of Lemma 6are satisfied. Therefore, = has at least one FP. Thus, problem (33) has a solution on f×f . The following example support Theorem 8: Example 5. Consider the following 2D Volterra integral equation κ(λ,δ) = π 2λ−δ2 7π+ λ Z0 δ Z0 cos κ(ru) 2drdu +2 7 λ Z0 cos κ(λu) 2du +1 7 δ Z0 cos κ(δr) 2dr. (35) It is clear that problem (35) is a special case of (34) with β(λ,δ) = π 2λ−δ2 7π,Ω1(r,u,κ(r,u))=cos κ(ru) 2, Ω2(δ,u,κ(λ,u))=cos κ(λu) 2,Ω3(λ,r,κ(δ,r))=cos κ(δr) 2,η=2 7and γ=1 7. Then, for any r,u∈[0, 1]and v1,v2∈R2, we find that |Ω1(r,u,v1(r,u))−Ω1(r,u,v2(r,u))|≤1 2|cos v1−cos v2|, |Ω2(r,u,v1(r,u))−Ω2(r,u,v2(r,u))|≤1 2|cos v1−cos v2|, (36) |Ω3(r,u,v1(r,u))−Ω3(r,u,v2(r,u))|≤1 2|cos v1−cos v2|, According to the mean-value theorem, for any v1 , v2∈R2 with v1<v2 there is b∈[v1,v2]so that cos v1−cos v2 v1−v2 =−sin(b), implies |cos v1−cos v2| |v1−v2|=|−sin(b)|≤1. Hence, |cos v1−cos v2|≤|v1−v2|and (36) reduces to |Ω1(r,u,v1(r,u))−Ω1(r,u,v2(r,u))|≤1 2|v1−v2|, |Ω2(r,u,v1(r,u))−Ω2(r,u,v2(r,u))|≤1 2|v1−v2|, |Ω3(r,u,v1(r,u))−Ω3(r,u,v2(r,u))|≤1 2|v1−v2|, where `1=`2=`3=1 2 and `1+η`2+γ`3=ξ=5 7< 1. It is easy to see that β(λ , δ) is continuous on [0, 1]. Mathematics 2022,10, 4257 24 of 26 Consequently, all conditions of Theorem 8are satisfied. Therefore, there exists a solution to the problem (36). 8. Conclusions and Future Works In this study, a four-step iterative scheme known as the HR∗− iterative scheme (7) is presented for approximating the fixed points of contractive-like mappings and RSTN mappings. Analytically, it has been demonstrated that the new iterative scheme converges faster than the iterative method (5) for contractive-like mappings. Furthermore, we have shown numerically that for contractive-like mappings, our novel iterative method converges faster than several popular iterative schemes in the literature. Additionally, the ω2− stability result of the HR∗− iterative scheme (7) has also been obtained. To clarify the idea of ω2− stability of the considered algorithm with regard to = , we have given an example. Additionally, we have demonstrated a number of weak and strong convergence theorems for RSTN mappings in uniformly convex BSs. In order to compare the convergence behavior of the proposed algorithm (7) with certain well-known iterative schemes, a novel example of RSTN mappings has been supplied. As a practical application, we proved that a 2D Volterra integral equation has a solution. Additionally, we provided an engaging example to explain the outcome of our application. Finally, as future work for this paper, we suggest the following: (1) If we define a mapping = in a Hilbert space ∆ endowed with inner product space, we can find a common solution to the variational inequality problem by using our iteration (7). This problem can be stated as follows: find ℘∗∈∆such that h=℘∗,℘−℘∗i ≥ 0 for all ℘∈∆, where =:∆→∆ is a nonlinear mapping. Variational inequalities are an important and essential modeling tool in many fields such as engineering mechanics, transportation, economics, and mathematical programming, see [45–47]. (2) We can generalize our algorithm to gradient and extra-gradient projection methods, these methods are very important for finding saddle points and solving many problems in optimization, see [6]. (3) We can accelerate the convergence of the proposed algorithm by adding shrinking projection and CQ terms. These methods stimulate algorithms and improve their performance to obtain strong convergence, for more details, see [7]. (4) If we consider the mapping = as an α− inverse strongly monotone and the inertial term is added to our algorithm, then we have the inertial proximal point algorithm. This algorithm is used in many applications such as monotone variational inequalities, image restoration problems, convex optimization problems, and split convex feasibility problems, see [ 48 – 50 ]. For more accuracy, these problems can be expressed as mathematical models such as machine learning and the linear inverse problem. (5) We can try to determine the error of our present iteration. Author Contributions: H.A.H. contributed in conceptualization, investigation, methodology, validation and writing the theoretical results; H.u.R. contributed in conceptualization, investigation and writing the numerical results; M.D.l.S. contributed in funding acquisition, methodology, project administration, supervision, validation, visualization, writing and editing. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported in part by the Basque Government under Grant IT1555-22. Data Availability Statement: The data used to support the findings of this study are available from the corresponding author upon request. Acknowledgments: The authors thank the Basque Government for Grant IT1555-22. Conflicts of Interest: The authors declare that they have no competing interests. Mathematics 2022,10, 4257 25 of 26 References 1. Arias, A.; Gheondea, A.; Gudder, S. Fixed points of quantum operations. J. Math. Phys. 2002,43, 5872. 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