EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5744 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Four-Step Semi-Implicit Midpoint Approximation Scheme for Fixed Point with Applications Mohammad Akram Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351 Saudi Arabia Abstract. This paper aims to put forward and design a four-step semi-implicit approximation scheme to work out the fixed point of a contractive mapping. Convergence analysis and stability of the proposed scheme is incorporated under some mild assumptions. Finally, the significance and applications of the proposed scheme and theoretical findings are proven by exploring a general quasi-variational inequality and a nonlinear fractional differential equation. 2020 Mathematics Subject Classifications: 47H09, 47H10, 47H22, 47H25, 49J40 Key Words and Phrases: Semi-implicit midpoint approximation, fixed point, convergence and stability, quasi-variational inequality, nonlinear fractional differential equation 1. Introduction Throughout this paper, we presume that Ω ̸= ϕ is a subset of a Banach space X,R signifies the set of real numbers and Ξ(Ψ) = {ϱ ∈ Ω : Ψϱ = ϱ}, the set of fixed points of the mapping Ψ. A mapping Ψ : Ω → Ω is referred to as contraction if ∃κ ∈ [0, 1) such that ∥Ψϱ − Ψς∥ ≤ κ∥ϱ − ς∥,∀ϱ, ς ∈ Ω and non-expansive for κ = 1. Non-expansive mappings are crucial generalized notion of contraction mappings and fundamental tools in the theory of fixed points, see, [48]. Clearly, Ξ(Ψ) for a non-expansive self mapping Ψ on a bounded, closed and convex subset Ω is non-empty, see, [10]. Detailed information on non-expansive mappings and related results can be found in [16, 38]. Non-expansive mappings play vital role in the journey of nonlinear analysis and have been employed to deal various problems of nonlinear analysis such as variational inequality, optimization, equilibrium and initial value problems. In fact, a non-expansive self-mapping on a complete metric space not necessarily owns a fixed point. Example 1. [37] Consider a closed and bounded subset Ω = {ϱ = (ϱ1, ϱ2, · · · ) : ϱk ≥ 0, ∀k, ∞∑ k=1 ϱk = 1} of a Banach space X of all real absolutely summable sequences (l1, ∥ ·∥1). Then the non-expansive mapping Ψ : Ω → Ω described by Ψ(ϱ) = (0, ϱ1, ϱ2, ·) does not admit a fixed point. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5744 Email address: akramkhan 20@rediffmail.com (M. Akram) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 2 of 19 Also, unlike the contraction mappings, the Picard sequence may not converge to a fixed point of a non-expansive mapping. These facts motivated the researchers to explore the mappings which own fixed points over such spaces. A class of weak contractions also known as almost contraction mappings (ACM) was brought into existence by Berinde [8] which is defined below: Definition 1. A mapping Ψ : Ω → Ω is called ACM if for some κ ≥ 0,∃τ ∈ (0, 1) so that ∥Ψ(ϱ)−Ψ(ς)∥ ≤ τ∥ϱ− ς∥+ κ∥ϱ−Ψ(ϱ)∥,∀ϱ, ς ∈ Ω. (1) Osilike obtained contractive condition (1) by extending the work of Rhoades [39] and the author proved numerous stability results for (1). If κ = 2δ, τ = δ, then ACM coincides with Zamfirescu contraction [7, 19], where δ = max { α, β 1− β , γ 1− γ } , α ∈ [0, 1), β, γ ∈ [0, 0.5]. Further, Imoru and Olantiwo [22] generalized the mapping defined in (1) by involving monotonic increasing function and defined as under: Definition 2. A mapping Ψ : Ω → Ω is referred to as contractive-like if there exists a strictly increasing continuous function g : [0,∞) → [0,∞) with g(0) = 0 and τ ∈ [0, 1) so that ∥Ψ(ϱ)−Ψ(ς)∥ ≤ g(∥ϱ−Ψ(ϱ)∥) + τ∥ϱ− ς∥,∀ϱ, ς ∈ Ω. (2) The contractive condition in (2) is much broader which include several contractive conditions, see, [7, 19, 35, 39, 40]. If gu = κu, where κ ≥ 0 then (2) coincides with (1). Further, for κ = mτ,m = (1 − τ)−1, 0 ≤ τ < 1, we acquire the contractive condition due to Rhoades [40]. Further, if κu = 0, then (2) becomes ∥Ψ(ϱ)−Ψ(ς)∥ ≤ τ∥ϱ− ς∥, τ ∈ [0, 1),∀ϱ, ς ∈ Ω, (3) which is considered by Berinde [7], and Harder and Hicks [19]. In past few years, a tremendous interest has been shown to the fixed point theory which has become most versatile and applicable area of research. Several problems which we encounter in real-world including zeros of monotone operators, ODEs, PDEs, integral equations, VIs, etc., can be reformulated as a fixed point problem. Owing to the sig- nificance of fixed point theory, numerous approaches have been carried out to deal with fixed point problems. Among these approaches, iterative approximation is one of the most handy and applicable tools for exploring nonlinear problems. In recent time, several new iterative schemes have been designed and employed. One of the most common schemes for investigating fixed points is named as Mann iterative scheme [28]:{ ϱ0 ∈ Ω, ϱk+1 = (1− αk)ϱk + αkΨ(ϱk), k ∈ N, (4) M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 3 of 19 where {αk} ∈ [0, 1] and Ψ : Ω → Ω is a non-expansive mapping. In 1974, Ishikawa [23] approximated the fixed points by designing the scheme as under: ϱ0 ∈ Ω, σk = (1− βk)ϱk + βkΨ(ϱk), ϱk+1 = (1− αk)ϱk + αkΨ(σk), k ∈ N, (5) where {αk}, {βk} ∈ [0, 1]. Further, Noor[33] posed a three-step scheme which comprises Mann [28] and Ishikawa [23] schemes and expressed as under: ϱ0 ∈ Ω, ρk = (1− γk)ϱk + γkΨ(ϱk), σk = (1− βk)ϱk + βkΨ(ρk), ϱk+1 = (1− αk)ϱk + αkΨ(σk), k ∈ N, (6) where {αk}, {βk}, {γk} ∈ [0, 1]. Among the numerous iterative methods posed so far, a few common and intensively used schemes include S-iteration [41], M -iteration [49], Normal-S [43], Picard-Ishikawa scheme [34], etc.. Recently, Okeke et al. [15] contrived an efficient four step iterative scheme: ϱ0 ∈ Ω, ϱk+1 = Ψ(ςk), ςk = Ψ[(1− αk)ϑk + αkΨ(ϑk)], ϑk = (1− βk)Ψ(ϱk) + βkΨ(εk), εk = (1− γk)ϱk + γkΨ(ϱk), k ∈ N, (7) where {αk}, {βk}, {γk} ⊂ [0, 1]. The authors approximated the fixed point of a contrac- tion mapping in a uniformly convex Banach space and proved the stability of the proposed scheme. Additionally, the weak convergence for Suzuki’s generalized non-expansive map- ping was analyzed. The efficiency of the scheme was demonstrated by illustrative example and comparing some known schemes. A mapping Ψ in a Banach space X with domain D(Ψ) and range R(Ψ) is referred to as accretive, if ⟨Ψϱ−Ψς, J(ϱ− ς)⟩ ≥ 0, ∀ϱ, ς ∈ D(Ψ), where J : X → X ∗ is the duality mapping and Ψ is referred to as monotone, if ⟨Ψϱ−Ψς, ϱ− ς⟩ ≥ 0,∀ϱ, ς ∈ D(Ψ). If X = H , a Hilbert space, then both the concepts are identical in the sense of Minty [30] and Browder [10]. On the contrary, number of real-life problems appearing in science and engineering can be studied by formulating as a model of the following initial-value problem (IVP): dϱ dt = Ψ(ϱ); ϱ(0) = ϱ0. (8) M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 4 of 19 Since the accretive and monotone mappings are connected to the evolution model (8) and this relation makes these mappings quite fruitful and applicable. A fundamental result documented by [10] affirms that (8) admits a solution when Ψ is locally Lipshitzian and accretive on X . Additionally, estimating a zero of nonlinear mapping Ψ, i.e., 0 ∈ Ψϱ is a significant and powerful tool in approximation theory because solutions of elliptic differential equations, optimization problems, inclusion problems, fixed point problems can be obtained as a model of inclusion problem 0 ∈ Ψϱ which is identical to the equilibrium state: dϱ dt = 0,Ψ(ϱ) = 0, see, [10, 11]. To obtain a numerical solution is challenging task when the involved mapping Ψ is not continuous. Several researchers obtained numerical solutions of (8) by approximation approaches, see, Mustafa [47], Duffull and Hegarty [13], Khorasani and Adibi [25]. One of the fundamental and impressive techniques is implicit midpoint rule (IMR): 1 µ (ϱk+1 − ϱk) = Ψ (ϱk+1 + ϱk 2 ) , (9) where µ > 0 is a step-size. The sequence {ϱk} induced by (9) converges to the exact solution of (8) under modest assumptions, see, [3, 5]. If Ψ is expressed as Ψ(ϱ) := Γ(ϱ)−ϱ, then the IVP (8) transformed into ϱ ′ = ϱ− Γ(ϱ), ϱ(0) = ϱ0 (10) and the IMR (9) becomes: 1 µ (ϱk+1 − ϱk) = [ϱk+1 + ϱk 2 − Γ (ϱk+1 + ϱk 2 )] , (11) In [26], the authors deployed the fact that equilibrium associated to (10) is identical to the fixed point ϱ = Γ(ϱ), which compelled the authors to design the following fixed point implicit iterative scheme: ϱk+1 = (1− αk)ϱk + αkΓ (ϱk+1 + ϱk 2 ) , (12) where {αk} ⊂ (0, 1) and Γ : H → H is nonexpansive. The authors carried out weak convergence results by taking some modest assumptions into consideration. Same fact motivated, Xu et al. [20] to design the following implicit midpoint method using viscosity technique for non-expansive mapping: ϱk+1 = αkψ(ϱk) + (1− αk)Γ (ϱk+1 + ϱk 2 ) , (13) where, Γ is non-expansive and ψ is contraction mapping. More precisely, following result was proved. Theorem 1. Let Ω ̸= ∅ be a closed convex set in a Hilbert space H . Suppose that Γ : Ω → Ω is a non-expansive and ψ : Ω → Ω is a contraction mapping. If {αk} complies with the following preassumptions: M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 5 of 19 (P1) lim k→∞ αk = 0; (P2) ∞∑ k=0 αk = ∞; (P3) ∞∑ k=0 |αk+1 − αk| <∞. Then {ϱk}∞k=1 produced by (13) converges to ϱ ∈ Fix(Γ) and ϱ solves the following varia- tional inequality: ⟨(I − τ)ψ, ϱ− τ⟩ ≥ 0, ∀ϱ ∈ Fix(Γ). Luo et al. [36] obtained the results of Xu et al. [20] in uniformly smooth Banach space. For more details on implicit schemes, we refer, [12, 21, 31, 51]. Motivated and encouraged by the earlier revealed results and iterative process (7), we propose and design a four-step semi-implicit approximation scheme (14) to work out the fixed point of a contractive mapping. The accomplishment of the task is performed as mentioned herein: Second section begins with the designing of a semi-implicit mid point scheme followed by some basic results. Convergence of the planned is analyzed to explore a fixed point of a contractive mapping and the uniqueness of the solution is established. Further, the stability of the designed scheme is discussed. In the third section, we discuss the significance and applicability of our designed scheme. A general quasi-variational inequality and a fractional differential equation are investigated by employing our designed scheme. The concluding comments and expected future research plans are outlined in the last section. 2. Iterative Scheme and Convergence Let ∅ ̸= Ω be a closed convex subset of a Banach space X equipped with norm ∥ · ∥. Suppose the mapping Ψ : Ω → Ω satisfies contractive condition (2). Based on the iterative scheme (7), we are interested to suggest and analyze the following semi-implicit midpoint scheme (SIMPS) as under: ϱk+1 = Ψ(σk), σk = Ψ [ (1− αk) (σk + ϑk 2 ) + αkΨ (σk + ϑk 2 )] , ϑk = (1− βk)Ψ (ϑk + ϱk 2 ) + βkΨ (ϑk + θk 2 ) , θk = (1− γk) (ϱk + θk 2 ) + γkΨ (ϱk + θk 2 )] , (14) where {αk}, {βk}, {γk} ⊆ (0, 1). Definition 3. [9] Let {φk} ⊂ Ω be an arbitrary sequence. An iterative scheme ϱk+1 = Λ(Ψ, ϱk) so as {ϱk} → ϱ ∈ Ξ(Ψ) is said to be Ψ-stable. If for µk = ∥φk+1 − Λ(Ψ, φk)∥, lim k→∞ µk = 0 if and only if lim k→∞ φk = ϱ. Lemma 1. [50] Suppose the nonnegative real sequences {ϱk}∞k=1 and {ςk}∞k=1 satisfy ϱk+1 ≤ (1− pk)ϱk + ςk, where pk ∈ (0, 1), ∞∑ k=1 pk = ∞ and lim k→∞ ςk pk = 0. Then lim k→∞ ϱk = 0. M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 6 of 19 Theorem 2. Let ∅ ≠ Ω ⊆ X be a closed convex bounded set and Ψ : Ω → Ω satisfies (2). If Ξ(Ψ) ̸= ∅, then {ϱk}∞k=1 initiated by SIMPS (14) converges strongly to ϱ ∈ Ξ(Ψ). Proof. Suppose that ϱ ∈ Ξ(Ψ). Then, it results from the last formulation of (14) that ∥θk − ϱ∥ = ∥∥∥(1− γk) (ϱk + θk 2 ) + γkΨ (ϱk + θk 2 )] − ϱ ∥∥∥ ≤ (1− γk) ∥∥∥ϱk + θk 2 − ϱ ∥∥∥+ γk ∥∥∥Ψ(ϱk + θk 2 ) − ϱ ∥∥∥ = (1− γk) ∥∥∥ϱk + θk 2 − ϱ ∥∥∥+ γk ∥∥∥Ψ(ϱ)−Ψ (ϱk + θk 2 )∥∥∥ ≤ (1− γk) ∥∥∥ϱk + θk 2 − ϱ ∥∥∥+ γk [ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥(ϱk + θk 2 ) − ϱ ∥∥∥] ≤ εk 2 (∥ϱk − ϱ∥+ ∥θk − ϱ∥), where εk = (1− γk + τγk), which turns after simplification into ∥θk − ϱ∥ ≤ εk (2− εk) ∥ϱk − ϱ∥. (15) Again it yields from scheme (14) that ∥ϑk − ϱ∥ = ∥∥∥(1− βk)Ψ (ϑk + ϱk 2 ) + βkΨ (ϑk + θk 2 ) − ϱ ∥∥∥ ≤ (1− βk) ∥∥∥Ψ(ϑk + ϱk 2 ) − ϱ ∥∥∥+ βk ∥∥∥Ψ(ϑk + θk 2 ) − ϱ ∥∥∥ ≤ (1− βk) ∥∥∥Ψ(ϱ)−Ψ (ϑk + ϱk 2 )∥∥∥+ βk ∥∥∥Ψ(ϱ)−Ψ (ϑk + θk 2 )∥∥∥ ≤ (1− βk) [ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥(ϑk + ϱk 2 ) − ϱ ∥∥∥] + βk [ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥(ϑk + θk 2 ) − ϱ ∥∥∥] ≤ (1− βk)τ ∥∥∥(ϑk + ϱk 2 ) − ϱ ∥∥∥+ βkτ ∥∥∥(ϑk + θk 2 ) − ϱ ∥∥∥ ≤ τ 2 ∥ϑk − ϱ∥+ τ 2 [(1− βk)∥ϱk − ϱ∥+ βk∥θk − ϱ∥], which yields into ∥ϑk − ϱ∥ ≤ τ (2− τ) [(1− βk)∥ϱk − ϱ∥+ βk∥θk − ϱ∥]. (16) Combining (15) and (16), one gets ∥ϑk − ϱ∥ ≤ τ (2− τ) [ 1− βk ( 1− εk (2− εk) )] ∥ϱk − ϱ∥. (17) M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 7 of 19 Further, the second formulation of SIMPS (14) yields ∥σk − ϱ∥ = ∥∥∥Ψ[ (1− αk) (σk + ϑk 2 ) + αkΨ (σk + ϑk 2 )] − ϱ ∥∥∥ = ∥∥∥Ψ(ϱ)−Ψ [ (1− αk) (σk + ϑk 2 ) + αkΨ (σk + ϑk 2 )]∥∥∥ ≤ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥[(1− αk) (σk + ϑk 2 ) + αkΨ (σk + ϑk 2 )] − ϱ ∥∥∥ ≤ τ(1− αk) ∥∥∥(σk + ϑk 2 ) − ϱ ∥∥∥+ ταk ∥∥∥Ψ(ϱ)−Ψ (σk + ϑk 2 )∥∥∥ ≤ τ [1− αk(1− τ)] ∥∥∥(σk + ϑk 2 ) − ϱ ∥∥∥ = πk 2 [∥σk − ϱ∥+ ∥ϑk − ϱ∥]. Thus, we acquire ∥σk − ϱ∥ ≤ πk (2− πk) ∥ϑk − ϱ∥, (18) where πk = τ [1−αk(1− τ)]. The straightforward calculation after taking the assumptions {αk}∞k=1 ∈ (0, 1), τ ∈ [0, 1) and (17) into play leads εk ∈ [0, 1). Thus, we acquire 1 − βk ( 1− εk 2− εk ) ≤ 1 and hence, ∥σk − ϱ∥ ≤ πk (2− πk) τ (2− τ) ∥ϱk − ϱ∥. (19) Finally, the first formulation of SIMPS (14) along with (19) turns into ∥ϱk+1 − ϱ∥ = ∥Ψ(σk)− ϱ∥ = ∥Ψ(ϱ)−Ψ(σk)∥ ≤ g(∥ϱ−Ψ(ϱ)∥) + τ∥σk − ϱ∥ ≤ (1− ℓ̂k)∥ϱk − ϱ∥, (20) where, ℓ̂k = (2− τ)(2− πk)− τ2πk (2− τ)(2− πk) . (21) Since πk = τ(1−αk+ ταk), τ ∈ [0, 1) and {αk}∞k=1 ⊆ (0, 1) yields πk ≤ τ . Thus, we obtain ℓ̂k ≥ 1 4 [(2− τ)(2− πk)− τ2πk] ≥ 1 4 [1 + (1− τ)][1 + (1− τ)]− τ3 > 0. Further, 1 − ℓ̂k = τ2πk (2− τ)(2− πk) ≥ 0 and ∞∑ k=0 ℓ̂k = ∞. Utilizing Lemma 1, it follows from (20) that lim k→∞ ∥ϱk − ϱ∥ = 0. Next, we manifest the uniqueness of ϱ, suppose that M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 8 of 19 ϱ1, ϱ2 ∈ Ω so that ϱ1 ̸= ϱ2 and ϱ1, ϱ2 ∈ Ξ(Ψ). Then ∥ϱ1 − ϱ2∥ = ∥Ψ(ϱ1)−Ψ(ϱ2)∥ ≤ g(∥ϱ1 −Ψ(ϱ1)∥) + τ∥ϱ1 − ϱ2∥ = τ∥ϱ1 − ϱ2∥. (22) Since τ ∈ [0, 1), then (22) gives ∥ϱ1 − ϱ2∥ = 0 and consequently ϱ1 = ϱ2. Theorem 3. Suppose that ∅ ≠ Ω ⊆ X is a closed convex bounded set and the mapping Ψ : Ω → Ω satisfies (2). If ϱ ∈ Ξ(Ψ), then {ϱk}∞k=1 initiated by SIMPS (14) is Ψ-stable. Proof. Let {φk} ⊂ Ω be an arbitrary sequence and {ϱk}∞k=1 initiated by SIMPS (14) is ϱk+1 = Λ(Ψ, ϱk) such as {ϱk} → ϱ ∈ Ξ(Ψ). Suppose that µk = ∥φk+1 − Λ(Ψ, φk)∥, where {φk} is initiated as under: φk+1 = Ψ(ζk), ζk = Ψ [ (1− αk) (ζk + ξk 2 ) + αkΨ (ζk + ξk 2 )] , ξk = (1− βk)Ψ (ξk + φk 2 ) + βkΨ (ξk + ωk 2 ) , ωk = (1− γk) (φk + ωk 2 ) + γkΨ (φk + ωk 2 )] . (23) To establish the Ψ-stability of the scheme (14), we corroborate lim k→∞ µk = 0 if and only if lim k→∞ φk = ϱ. Assume that lim k→∞ µk = 0. By utilizing the triangle inequality, we acquire ∥φk+1 − ϱ∥ = ∥φk+1 − Λ(Ψ, φk) + Λ(Ψ, φk)− ϱ∥ ≤ ∥φk+1 − Λ(Ψ, φk)∥+ ∥Λ(Ψ, φk)− ϱ∥ ≤ µk + ∥φk+1 − ϱ∥ = µk + ∥Ψ(ζk)− ϱ∥ ≤ µk + g(∥ϱ−Ψ(ϱ)∥) + τ∥ζk − ϱ∥ = µk + τ∥ζk − ϱ∥. (24) M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 9 of 19 Again, from second equation of (23), we estimate ∥ζk − ϱ∥ = ∥∥∥Ψ[ (1− αk) (ζk + ξk 2 ) + αkΨ (ζk + ξk 2 )] − ϱ ∥∥∥ = ∥∥∥Φ(ϱ)−Ψ [ (1− αk) (ζk + ξk 2 ) + αkΨ (ζk + ξk 2 )]∥∥∥ ≤ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥(1− αk) (ζk + ξk 2 ) + αkΨ (ζk + ξk 2 ) − ϱ ∥∥∥ ≤ τ(1− αk) ∥∥∥(ζk + ξk 2 ) − ϱ ∥∥∥+ ταk ∥∥∥Ψ(ζk + ξk 2 ) − ϱ ∥∥∥ ≤ τ(1− αk) ∥∥∥(ζk + ξk 2 ) − ϱ ∥∥∥+ ταk [ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥(ζk + ξk 2 ) − ϱ ∥∥∥] ≤ τ 2 [1− αk(1− τ)][∥ζk − ϱ∥+ ∥ξk − ϱ∥] = πk 2 [∥ζk − ϱ∥+ ∥ξk − ϱ∥] which turns into ∥ζk − ϱ∥ ≤ πk (2− πk) ∥ξk − ϱ∥. (25) ∥ξk − ϱ∥ = ∥∥∥(1− βk)Ψ (ξk + φk 2 ) + βkΨ (ξk + ωk 2 ) − ϱ ∥∥∥ ≤ (1− βk) ∥∥∥Ψ(ξk + φk 2 ) − ϱ ∥∥∥+ βk ∥∥∥Ψ(ξk + ωk 2 ) − ϱ ∥∥∥ ≤ (1− βk) [ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥(ξk + φk 2 ) − ϱ ∥∥∥] + βk [ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥(ξk + ωk 2 ) − ϱ ∥∥∥] ≤ τ 2 ∥ξk − ρ∥+ τ 2 [(1− βk)∥φk − ϱ∥+ βk∥ωk − ϱ∥] which turns into ∥ξk − ϱ∥ ≤ τ (2− τ) [(1− βk)∥φk − ϱ∥+ βk∥ωk − ϱ∥], (26) and ∥ωk − ϱ∥ = ∥∥∥(1− γk) (φk + ωk 2 ) + γkΨ (φk + ωk 2 ) − ϱ ∥∥∥ ≤ (1− γk) ∥∥∥(φk + ωk 2 ) − ϱ ∥∥∥+ γk ∥∥∥Ψ(φk + ωk 2 ) − ϱ ∥∥∥ ≤ (1− γk) ∥∥∥(φk + ωk 2 ) − ϱ ∥∥∥+ γk [ g(∥ϱ−Ψ(ϱ)∥) + τ ∥∥∥(φk + ωk 2 ) − ϱ ∥∥∥] ≤ εk 2 [∥φk − ϱ∥+ ∥ωk − ϱ∥] which turns into ∥ωk − ϱ∥ ≤ εk (2− εk) ∥φk − ϱ∥, (27) M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 10 of 19 where εk = (1−γk+τγk). By implementing back substitution from (25)-(27), (24) becomes ∥φk+1 − ϱ∥ ≤ µk + (1− ℓ̂k)∥φk − ϱ∥, (28) where ℓ̂k is identical as given in (21). By availing the assumption lim k→∞ µk = 0, Lemma 1 yeilds ∥φk − ϱ∥ → 0 as k → ∞, i.e., lim k→∞ φk = ϱ. Conversely, assume that lim k→∞ φk = ϱ, and following the same procedure, we achieve µk = ∥φk+1 − Λ(Ψ, φk)∥ = ∥φk+1 − ϱ+ ϱ− Λ(Ψ, φk)∥ ≤ ∥φk+1 − ϱ∥+ ∥Λ(Ψ, φk)− ϱ∥ ≤ ∥φk+1 − ϱ∥+ (1− ℓ̂k)∥φk − ϱ∥. Appealing to the assumption lim k→∞ φk = ϱ, it follows that lim k→∞ µk = 0. Hence, SIMPS (14) is Ψ-stable. 3. Applications In this section, we shall explore and examine a general quasi-variational inequality and a nonlinear fractional differential equation by employing our outlined semi-implicit midpoint scheme. 3.1. General quasi-variational inequality Let H be a Hilbert space over R and C(H ), the collection of non empty closed convex subsets of H . We contemplate the problem to observe an element ϱ ∈ H : ψ(ϱ) ∈ C(ϱ) so that ⟨Ψ(ϱ), ψ(ς)− ψ(ϱ)⟩ ≥ 0,∀ς ∈ H , ψ(ς) ∈ C(ϱ), (29) where Ψ, ψ : H → H be (not necessarily) linear mappings, and the set-valued mapping C : H → 2H assigns each element ϱ ∈ H , a closed convex subset C(ϱ) of H . The inequality (29) is called the generalized quasi variational inequality (GQV I) and we signify its solution set by ✠(C(ϱ),Ψ, ψ). In fact, a quasi-variational inequality (QV I) is a kind of modified variational inequality in which the constraint set varies with the variable. Numerous economic and engineering problems, such as Nash equilibrium problems, control and optimization, operations research, etc., are recognized to be well suited for modeling and analysis using QV Is. GQV I (29) can be seen as a unified problem and consists several considerably significant problems as special cases which are listed as under. (i) For C(ϱ) = C, GQV I (29) is identical to the following general variational inequality which was set forth by Noor [32]. ⟨Ψ(ϱ), ψ(ς)− ψ(ϱ)⟩ ≥ 0,∀ς ∈ H , ψ(ς) ∈ C. (30) M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 11 of 19 (ii) Further for ψ = I, Problem (30) becomes the classical variational inequality intro- duced by Stampacchia[45]. (iii) If ψ = I, GQVI (29) turns into the following classical quasi-variational inequality introduced in [6]: ⟨Ψ(ϱ), ς − ϱ⟩ ≥ 0, ∀ς ∈ C(ϱ). (31) (iv) For ϱ0 ∈ H , the dual cone of C(ϱ0) ⊂ H is described by C̄(ϱ0) = {ϱ ∈ H : ⟨ϱ, ς⟩ ≥ 0, ∀ς ∈ C(ϱ0)}. Then problem (30) turns into a general complementarity problem of discerning an element ϱ ∈ H so that ⟨Ψ(ϱ), ψ(ϱ)⟩ ≥ 0, ψ(ϱ) ∈ C(ϱ) and Ψ(ϱ) ∈ C̄(ϱ). (32) Now, to bring off the required goal, we accumulate a few supplementary results and definitions below. Lemma 2. [4] If for any ς ∈ H , ϱ ∈ C(ϱ), the implicit projection PC(ϱ) : H → C(ϱ) ⊂ H obeys the inequality ⟨κ− ω,ϖ − κ⟩ ≥ 0 if and only if PC(ϱ)(ω) = κ,∀ϖ ∈ C(ϱ). Next, we shall design the following fixed point problem associated to GQV I (29) by imposing the Lemma 2. Lemma 3. An element ϱ ∈ H : ψ(ϱ) ∈ C(ϱ) solves GQVI (29) if and only if ϱ ∈ Ξ(Π), where Π(ϱ) = ϱ− ψ(ϱ) + PC(ϱ)[ψ(ϱ)− λΨ(ϱ)] and λ > 0 is a constant. Proof. Assume that ϱ ∈ ✠(C(ϱ),Ψ, ψ) then ⟨Ψ(ϱ), ψ(ς) − ψ(ϱ)⟩ ≥ 0, ∀ς ∈ H : ψ(ς) ∈ C(ϱ). By making use of Lemma 2, we acquire ΠC(ϱ)[ψ(ϱ)− λΨ(ϱ)] = ψ(ϱ). So, ϱ ∈ Ξ(Π). On the other side, assume that ϱ ∈ Ξ(Π) then for all ϱ ∈ H : ψ(ϱ) ∈ C(ϱ), we obtain Π(ϱ) = ϱ, thus, one can write ψ(ϱ) = ΠC(ϱ)[ψ(ϱ)−λΨ(ϱ)]. Again, by the virtue of Lemma 2, we get ⟨Ψ(ϱ), ψ(ς)− ψ(ϱ)⟩ ≥ 0, ∀ς ∈ H : ψ(ς) ∈ C(ϱ), i.e., ϱ ∈ ✠(C(ϱ),Ψ, ψ). Now, we take the following assumption into account to accomplish the required goal. Assumption A: For given elements ϱ, τ, ς ∈ H and κ > 0, PC obeys the following inequality ∥PC(τ)(ς)− PC(ϱ)(ς)∥ ≤ κ∥τ − ϱ∥. Definition 4. A mapping Ψ : H → H is called • ρ1-strongly monotone if ∃ρ1 > 0 so that ⟨Ψ(ϱ)−Ψ(ς), ϱ− ς⟩ ≥ ρ1∥ϱ− ς∥2,∀ϱ, ς ∈ H ; • relaxed (u, v)-cocoercive, if ∃u, v > 0 so that ⟨Ψ(ϱ)−Ψ(ς), ϱ− ς⟩ ≥ (−u)∥Ψ(ϱ)−Ψ(ς)∥2 + v∥ϱ− ς∥2, ∀ϱ, ς ∈ H ; • ρ2-Lipschitz continuous, if ∃ρ2 > 0 so that ∥Ψ(ϱ)−Ψ(ς)∥ ≤ ρ2∥ϱ− ς∥,∀ϱ, ς ∈ H . M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 12 of 19 Now, as an application of SIMPS (14), we shall re-structure the following semi- implicit midpoint scheme to find the common solution of GQV I (29) and fixed point of the mapping defined in (2). Algorithm 1. For given initial point ϱ0, estimate the sequence {ϱk}∞k=1 by the following implicit iterative scheme: ϱk+1 = Ψ[σk − ψ(σk) + PC(σk)[ψ(σk)− λΨ(σk)]], σk = Ψ [ (1− αk) (σk + ϑk 2 ) + αkΨ (σk + ϑk 2 )] , ϑk = (1− βk)Ψ (ϑk + ϱk 2 ) + βkΨ (ϑk + θk 2 ) , θk = (1− γk) (ϱk + θk 2 ) + γkΨ (ϱk + θk 2 )] , (33) where {αk}, {βk}, {γk} ⊆ (0, 1). Theorem 4. Let PC(ϱ) : H → C(ϱ) be a projection mapping and Ψ, ψ : H → H be non-linear mappings so that Ψ satisfies (2) and Ξ(Ψ) ∩ ✠(C(ϱ),Ψ, ψ) ̸= ∅. Assume that the assumption A and the following relations hold: (R1) Ψ is l-Lipschitz continuous and relaxed (u, v)-cocoercive and ψ is t-Lipschitz contin- uous and r-strongly monotone. (R2) The constant λ > 0 obeys the following relation: λl2 ≤ 2λv +∆(∆− 2) λ+ 2u ,∆ = 2 √ 1− 2r + t2 + κ. (34) Then {ϱk}∞k=1 approximated by (33) converges strongly to ϱ ∈ Ξ(Ψ) ∩✠(C(ϱ),Ψ, ψ). Proof. Invoking the l-Lipschitz continuity and relaxed (u, v)-cocoercivity of Ψ yields ∥(σk − ϱ)− λ[Ψ(σk)−Ψ(ϱ)]∥2 = ∥σk − ϱ∥2 − 2λ⟨Ψ(σk)−Ψ(ϱ), σk − ϱ⟩+ λ2∥Ψ(σk)−Ψ(ϱ)∥2 ≤ ∥σk − ϱ∥2 + 2λu∥Ψ(σk)−Ψ(ϱ)∥2 − 2λv∥σk − ϱ∥2 + λ2l2∥σk − ϱ∥2 ≤ ∥σk − ϱ∥2 + 2λul2∥σk − ϱ∥2 − 2λv∥σk − ϱ∥2 + λ2l2∥σk − ϱ∥2 = [1− 2λ(v − ul2) + λ2l2]∥σk − ϱ∥2 = B2∥σk − ϱ∥2. (35) Employing the t-Lipschitz continuity and r-strongly monotone property of ψ provide with the relation ∥σk − ϱ− [ψ(σk)− ψ(ϱ)]∥2 = ∥σk − ϱ∥2 − 2⟨ψ(σk)− ψ(ϱ), σk − ϱ⟩+ ∥ψ(σk)− ψ(ϱ)∥2 ≤ ∥σk − ϱ∥2 − 2r∥σk − ϱ∥2 + t2∥σk − ϱ∥2 = (1− 2r + t2)∥σk − ϱ∥2 = A2∥σk − ϱ∥2. (36) M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 13 of 19 ∥ϱk+1 − ϱ∥ = ∥Ψ[σk − ψ(σk) + PC(σk)[ψ(σk)− λΨ(σk)]]− ϱ∥ = ∥Ψ(ϱ)−Ψ[σk − ψ(σk) + PC(σk)[ψ(σk)− λΨ(σk)]]∥ ≤ g(∥ϱ−Ψ(ϱ)∥) + τ∥σk − ψ(σk) + PC(σk)[ψ(σk)− λΨ(σk)]− ϱ∥ = τ∥σk − ψ(σk) + PC(σk)[ψ(σk)− λΨ(σk)]− [ϱ− ψ(ϱ) + PC(ϱ)[ψ(ϱ)− λΨ(ϱ)]∥ ≤ τ ( ∥σk − ϱ− [ψ(σk)− ψ(ϱ)]∥+ ∥ψ(σk)− ψ(ϱ)− λ[Ψ(σk)−Ψ(ϱ)]∥+ κ∥σk − ϱ∥ ) ≤ 2τ∥σk − ϱ− [ψ(σk)− ψ(ϱ)]∥+ τ∥(σk − ϱ)− λ[Ψ(σk)−Ψ(ϱ)]∥+ τκ∥σk − ϱ∥ ≤ τ(2A+ B+ κ)∥σk − ϱ∥. (37) Replicating the process as from (15)-(19) and combining with (37) yields ∥ϱk+1 − ϱ∥ ≤ (1− ℓ̂n)(2A+ B+ κ)∥ϱk − ϱ∥, (38) where ℓ̂k is described in (21). Evidently, (2A + B + κ) < 1 from the assumption (R2). Then (38) turns into ∥ϱk+1 − ϱ∥ ≤ (1− ℓ̂n)∥ϱk − ϱ∥. (39) Thus, from (39) and implementing Lemma 1, we acquire lim k→∞ ∥ϱk − ϱ∥ = 0. By taking C(ϱ) =: C, we deduce the following corollary to estimate the common solution of the GV I (30) and the contractive mapping (2). Corollary 1. Let PC : H → C be a projection mapping and Ψ, ψ : H → H be non-linear mappings so that Ψ satisfies (2) and Ξ(Ψ)∩✠(C,Ψ, ψ) ̸= ∅, where ✠(C,Ψ, ψ) signifies the solution set of the GV I (30). Assume that the following relations hold: (v1) Ψ is l-Lipschitz continuous and relaxed (u, v)-cocoercive and ψ is t-Lipschitz contin- uous and r-strongly monotone. (v2) The constant λ > 0 obeys the following relation: λl2 ≤ 2λv +∆(∆− 2) λ+ 2u ,∆ = 2 √ 1− 2r + t2. (40) Then {ϱk}∞k=1 approximated by (33) converges strongly to ϱ ∈ Ξ(Ψ) ∩✠(C,Ψ, ψ). Example 2. Let l2 = {ϱ = (ϱ0, ϱ1, ϱ2, · · · ) : ∑∞ k=0 |ϱ2k| < ∞, ϱk ∈ R, ∀n = 0, 1, 2, · · · } be a Hilbert space with norm ∥ϱ∥2 = √∑∞ k=0 |ϱ2k|. Define Ψ, ψ : l2 → l2 by Ψ(ϱ) = (ϱ0 3 , 0, 0, · · · ) , and ψ(ϱ) = (3ϱ0 4 , 0, 0, · · · ) ,∀ϱ ∈ l2. Then, for all ϱ, ω ∈ l2, we calculate ⟨Ψ(ϱ)−Ψ(ω), ϱ− ω⟩ = 〈(ϱ0 3 − ω0 3 , 0, 0, · · · ) , (ϱ0 − ω0, ϱ1 − ω1, ϱ2 − ω2, · · · ) 〉 ≥ −1 3 ∥Ψ(ϱ)−Ψ(ω)∥22 + 1 3 ∥ϱ− ω∥22, ∥Ψ(ϱ)−Ψ(ω)∥2 = ∥∥∥ϱ 3 − ω 3 ∥∥∥ 2 = 1 3 ∥ϱ− ω∥2, M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 14 of 19 i.e., Ψ is relaxed (13 , 1 3)-cocoercive and 1 3 -Lipschitz continuous and ⟨ψ(ϱ)− ψ(ω), ϱ− ω⟩ = 〈(3ϱ0 4 − 3ω0 4 , 0, 0, · · · ) , (ϱ0 − ω0, ϱ1 − ω1, ϱ2 − ω2, · · · ) 〉 = 3 4 ∥ϱ− ω∥22, ∥ψ(ϱ)− ψ(ω)∥2 = ∥∥∥3ϱ 4 − 3ω 4 ∥∥∥ 2 = 3 4 ∥ϱ− ω∥2. Thus, ψ is 3 4 -strongly monotone and 3 4 -Lipschitz continuous. Also, for τ = 1 3 and strictly continuous function g : [0,∞) → [0,∞) with g(0) = 0, we have ∥Ψ(ϱ)−Ψ(ω)∥ − τ∥ϱ− ω∥ − g(∥ϱ−Ψ(ϱ)∥) = 1 3 |ϱ− ω| − 1 3 |ϱ− ω| − g(|ϱ− ϱ 3 |) = −g (2ϱ 3 ) ≤ 0, i.e., ∥Ψ(ϱ) − Ψ(ω)∥ ≤ τ∥ϱ − ω∥ + g(∥ϱ − Ψ(ϱ)∥). Thus, Ψ satisfies (2) and also ϱ∗ = (0, 0, 0, · · · ) ∈ Ξ(Ψ). Now, define C : H → H by C(ϱ) = C({ϱn}) = {a = {ak} : a0 ≥ 9 16ϱ0, ak = 0,∀n = 1, 2, · · · }. Now, for any α ∈ [0, 1] and a0, b0 ∈ C(ϱ) gives αa0 + (1 − α)b0 ≥ 9 16ϱ0, thus, C(ϱ) is convex set. Now, we shall verify that C(ϱ) is closed. Define Q : [ 916ϱ0,∞) → C(ϱ) by Q(s) = (s, 0, 0, · · · ). Then Q is well defined and for distinct a0, b0 ∈ [ 916ϱ0,∞), we acquire (a0, 0, 0, · · · ) ̸= (b0, 0, 0, · · · ), i.e., Q is one-to-one and there exists an a0 ∈ [ 916a0,∞), such that Q(a0) = (a0, 0, 0, · · · ),∀a = (a0, 0, 0, · · · ) ∈ C(ϱ), i.e., Q is onto. Consider the usual metric spaces (R, d) and (l2, d ′ ), then for all a, b ∈ [ 916a0,∞), we acquire d ′ (Q(a), Q(b)) = d ′ ((a0, 0, 0, · · · ), (b0, 0, 0, · · · )) = |a− b| = d(a, b). Thus, Q is continuous. Q−1 is also continuous and one-to-one and onto, so Q is home- omorphism. Since C(ϱ) is the homeomorphic image of a closed set [ 916a0,∞), and hence closed. Define PC(ϱ) : H → C(ϱ) as under: PC(ϱ)(l0, l1, l2, · · · ) =  (l0, l1, l2, · · · ), if (l0, l1, l2, · · · ) ∈ C(ϱ) ( 9 16ϱ0, 0, 0, · · · ), if (l0, l1, l2, · · · ) /∈ C(ϱ), l0 < 9 16ϱ0 (l0, 0, 0, · · · ), if(l0, l1, l2, · · · ) /∈ C(ϱ), l0 ≥ 9 16ϱ0. Then ∥PC(a)(l) − PC(b)(l)∥ ≤ 9 16∥a − b∥, i.e., PC fulfills assumption A. Next, we shall explore an element ϱ∗ such that ϱ∗ ∈ Ξ(Ψ) ∩ Ξ(Π). Consider ϱ∗ = (ϱ∗0, 0, 0, · · · ) : ϱ∗0 ≥ 0. If ϱ∗ > 0, then ⟨Ψ(ϱ∗), ψ(ω∗)− ψ(ϱ∗)⟩ = 〈ϱ∗ 3 , 3ω∗ 4 − 3ϱ∗ 4 〉 = 1 4 ⟨(ϱ∗0, 0, 0, · · · ), (ω∗ 0 − ϱ∗0, 0, 0, · · · )⟩ < 0, ∀ω∗ = (ω∗ 0, 0, 0, · · · ) ∈ C(ϱ∗). M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 15 of 19 On the other hand, for ϱ∗ = (0, 0, 0, · · · ), we acquire ⟨Ψ(ϱ∗), ψ(ω∗)−ψ(ϱ∗)⟩ = ⟨(0, 0, 0, · · · ), (ω∗ 0−ϱ∗0, 0, 0, · · · )⟩ = 0, ∀ω∗ = (ω∗ 0, 0, 0, · · · ) ∈ C(ϱ∗). Thus, for ϱ∗ = (0, 0, 0, · · · ) ∈ Ξ(Ψ) ∩ Ξ(Π). 3.2. Fractional differential equation The history of fractional calculus can be traced back to the middle of the 19th century from the pure mathematics. But a century later, its substantial and significant applica- tions have been drawn by engineers and physicists in their respective fields. Fractional derivatives are generalization of ordinary derivatives which simultaneously set out the be- havior of several physical phenomena. A very much attention have been paid to Fractional differential equations (FDEs) due to their worthy applications in several physical phenom- ena appearing in engineering, mechanics, economics, biology, etc., see, [27, 29, 46, 52, 53] and references therein and so these equations are widely used in many different domains. Now a days, fixed point theory has become a crucial tool to handle nonlinear problems arising in multi-disciplinary sciences. Its capacity and aptitude to demonstrate the exis- tence and uniqueness of solutions, provides researchers to construct fixed point iterative methods to research and examine FDEs. In recent time, researchers have been explored different classes of FDEs by implementing fundamental tools of fixed point theory, for more details, we refer, [1, 2, 14, 17, 18, 24, 42, 44]. Now, we take SIMPS (14) into account to examine the following Caputo-type non- linear fractional differential equation (C-NFDE):{ γDξϱ(u) + Φ(u, ϱ(u)) = 0, ϱ(0) = ϱ(1) = 0, 1 < ξ < 2, u ∈ [0, 1], (41) here, γDξ signifies a Caputo-fractional derivative of order ξ and Φ : [0, 1] × R → R is a continuous function. Let X = {ℑ : ℑ : [0, 1] → R} is a real continuous function equipped with supremum norm. The Green’s function related to (41) is expressed as under: G(u, v) = { 1 Γ(ξ)(u(1− v)(ξ−1) − (u− v)(ξ−1)), if 0 ≤ v ≤ u ≤ 1, u(1−v)(ξ−1) Γ(ξ) , if 0 ≤ u ≤ v ≤ 1. Now, we proceed to accomplish the goal of this sub-section. Theorem 5. Let X = C[0, 1] and the operator ℑ : X → X is defined by ℑ(ϱ(u)) = ∫ 1 0 G(u, v)Φ(v, h(v))dv,∀ϱ ∈ X . If, |Φ(v, ϱ(v))− Φ(v, j(v))| ≤ g(ϱ−ℑ(ϱ)) + τ |ϱ− j|,∀v ∈ [0, 1], ϱ, j ∈ X . (42) Then the scheme (14) associated to ℑ converges to the solution of C-NFDE (41). M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 16 of 19 Proof. Evidently, if ϱ ∈ X solves (41) iff ϱ solves: ϱ(u) = ∫ 1 0 G(u, v)Φ(v, h(v))dv. Then for all ϱ, j ∈ X and u ∈ [0, 1], imposing the assumption (42) and employing the definition of operator ℑ, we acquire ∥ℑ(ϱ(u))−ℑ(j(u))∥ = ∣∣∣ ∫ 1 0 G(u, v)Φ(v, ϱ(v))dv − ∫ 1 0 G(u, v)Φ(v, j(v))dv ∣∣∣ = ∣∣∣ ∫ 1 0 G(u, v)[Φ(v, ϱ(v))− Φ(v, j(v))]dv ∣∣∣ ≤ ∫ 1 0 G(u, v)|Φ(v, ϱ(v))− Φ(v, j(v))|dv ≤ ∫ 1 0 G(u, v)[g(|ϱ(v)−ℑ(ϱ(v))|) + τ |ϱ(v)− j(v)|]dv ≤ sup u∈[0,1] ∫ 1 0 G(u, v)[g(|ϱ(v)−ℑ(ϱ(v))|) + τ |ϱ(v)− j(v)|]dv ≤ g(∥ϱ(v)−ℑ(ϱ(v))∥) + τ∥ϱ(v)− j(v)∥, (43) which yields ∥ℑ(ϱ)−ℑ(j)∥ ≤ g(∥ϱ−ℑ(ϱ)∥) + τ∥ϱ− j∥. So, ℑ satisfies (2) and by the virtue of the Theorem 2, the sequence initiated by SIMPS (14) converges to an element in Ξ(ℑ) which solves C-NFDE (41). 4. Conclusions A four-step semi-implicit midpoint scheme is designed to investigate the fixed point of a contractive mapping under some mild assumptions. Convergence analysis and stability results of the implied scheme are exhibited. Furthermore, we applied our scheme to ex- amine a general quasi-variational inequality. By redesigning the proposed mid-point rule, we inspected a common solution of a contractive mapping and a general quasi-variational inequality. Finally, a nonlinear fractional differential equation is studied by employing SIMPS (14). In future, semi-implicit type schemes could be implemented to explore fixed points of some generalized nonexpansive mappings including Suzuki’s generalized nonexpansive mapping, asymptotically and total asymptotically non-expansive mappings. Some nonlinear problems such as variational inequalities, quasi-variational inequalities and inclusions are some worthy future research directions. Acknowledgements The author would like to thank the referees for their valuable comments and DSR, Islamic University of Madinah. M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 17 of 19 References [1] AAN Abdou. Fixed point theorems: Exploring applications in fractional differential equations for economic growth. Fractal Fract., 8, 2024. [2] A. Ahmadkhanlu. Existence and uniquensess for a class of fractional differential equations with an integral fractional boundary condition. Filomat, 31(5):1241–1246, 2017. [3] G. Bader and P. Deuflhard. A semi-implicit midpoint rule for stiff systems of ordinary differential equations. Numer. Math., 41:373–398, 1983. [4] C. Baiocchi and A. Capelo. Variational and Quasi Variational Inequalities. Wiley, New York, 1984. [5] A. Bayreuth. The implicit midpoint rule applied to discontinuous differential equa- tions. Computing., 49:45–62, 1992. [6] A. Bensoussan and J.L. Lions. VApplication des Inequalities Variationnelles en Con- trol Eten Stochastique. Dunod, Paris, France, 1978. [7] V. Berinde. On the stability of some fixed point procedures. Bul. Ştiinţ. Univ. Baia Mare, Ser. B. Matematică Informatică, 18(1):7–14, 2002. [8] V. Berinde. On the approximation of fixed points of weak contractive mapping. Carpath J. Math., 17:7–22, 2003. [9] V. Berinde. Picard iteration converges faster than mann iteration for a class of quasi- contractive operators. Fixed Point Theory Appl., 2:1–9, 2004. [10] F.E. Browder. Nonlinear mappings of nonexpansive and accretive-type in banach spaces. Bull. Amer. Math. Soc., 73:875–882, 1967. [11] C.E. Chidume. Geometric Properties of Banach Spaces and Nonlinear Iterations., volume 1965. Springer, London, 2009. [12] M. Dilshad D. Filali, M. Akram and A.A. Khidir. General semi-implicit mid- point approximation for fixed point of almost contraction mapping and appli- cations. Applied Mathematics in Science and Engineering, 32(1), 2024. doi: 10.1080/27690911.2024.2365687. [13] S.B. Duffull and G. Hegarty. An inductive approximation to the solution of systems of nonlinear ordinary differential equations in pharmacokinetics-pharmacodynamics. J. Theor. Comput. Sci., 1:4(1), 2014. doi: 10.4172/jtco.1000119. [14] M. Turkyilmazoglu G. H. Ibraheem and M.A. AL-Jawary. Novel approximate solution for fractional differential equations by the optimal variational iteration method. J. Comput. Sci., 64, 2022. doi: 10.1016/j.jocs.2022.101841. [15] R.T. Alqahtani G.A. Okeke, A.V. Udo and N.H. Alharthi. A faster iterative scheme for solving nonlinear fractional differential equations of the caputo type. AIMS Math- ematics, 8:28488–28516, 2023. [16] K. Goebel and S. Reich. Uniform Convexity, Hyperbolic Geometry, and Non-expansive Mappings. Marcel Dekker, New York and Basel, 1984. [17] H. Aydi H.A. Hammad and D.A. Kattan. New contributions to fixed point techniques with applications for solving fractional and differential equations. Qual. Theory Dyn. Syst., 23, 2024. doi: 10.1007/s12346-023-00932-7. M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 18 of 19 [18] H.A. Hammad and S.F. Aljurbua. Solving fractional random differential equations by using fixed point methodologies under mild boundary conditions. Fractal Fract., 8, 2024. doi: 10.3390/fractalfract8070384. [19] A.M. Harder and T.L. Hicks. Stability results for fixed point iteration procedures. Math. Japonica, 33:693–706, 1998. [20] M.A. Alghamdi H.K. Xu and N. Shahzad. The viscosity technique for the implicit midpoint rule of nonexpansive mappings in hilbert spaces. Fixed Point Theory Appl., 2015. doi: 10.1186/s13663-015-0282-9. [21] H.Y. Lan H.Y. Xu and F. Zhang. General semi-implicit approximations with errors for common fixed points of nonexpansive-type operators and applications to stampacchia variational inequality. Comput. Appl. Math., 31, 2022. doi: 10.1007/s40314-022- 01890-7. [22] C.O. Imoru and M.O. Olatinwo. On the stability of picard and mann iteration pro- cesses. Carpathian J. Math., 19:155–160, 2003. [23] S. Ishikawa. Fixed points by a new iteration method. Proc. Amer. Math. Soc., 44: 147–150, 1974. [24] H.A. Hammad J. Ahmad, K. Ullah and R. George. A solution of a fractional differen- tial equation via novel fixed-point approaches in banach spaces. AIMS Mathematics, 8(6):12657–12670, 2023. [25] S. Khorasani and A. Adibi. Analytical solution of linear ordinary differential equations by differential transfer matric method. Electron. J. Differ. Equ., 79:1–8, 2003. [26] N. Shahzad M.A. Alghamdi, A.M. Alghamdi and H.K. Xu. The implicit midpoint rule for nonexpansive mappings. Fixed Point Theory Appl., 96, 2014. doi: 10.1186/1687- 1812-2014-96. [27] R.L. Magin. Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl., 59:1586–1593, 2010. [28] W.R. Mann. Mean value methods in iteration. Proc. Amer. Math. Soc., 4:506–510, 1953. [29] K.S. Miller and B. Ross. An Introduction to Fractional Calculus and Fractional Dif- ferential Equations. Wiley, New York, 1993. [30] G.J. Minty. Monotone (nonlinear) operators in hilbert space. Duke Math. J., 29(4): 341–346, 1962. [31] S.C. Thakur M.O. Aibinu and S. Moyo. The implicit midpoint procedures for asymp- totically nonexpansive mappings. Journal of Mathematics, 2020. [32] M.A. Noor. General variational inequalities. Appl. Math. Lett., 1:119–122, 1988. [33] M.A. Noor. New approximation scheme for general variational inequalities. J. Math. Anal. Appl., 251:217–229, 2000. [34] G.A. Okeke. Convergence analysis of the picard-ishikawa hybrid iterative process with applications. Afr. Mat., 30:817–835, 2019. [35] M.O. Osilike. Some stability results for fixed point iteration procedures. J. Nigerian Math. Soc., 14-15:17–29, 1995. [36] G. Cai P. Luo and Y. Shehu. The viscosity iterative algorithms for the implicit midpoint rule of nonexpansive mappings in uniformly smooth banach spaces. J. M. Akram / Eur. J. Pure Appl. Math, 18 (1) (2025), 5744 19 of 19 Inequal. Appl., 2017. doi: 10.1186/s13660-017-1426-8. [37] R. Pant, R. Shukla, and P. Patel. Nonexpansive mappings, their extensions and generalizations in banach spaces. In P. Debnath, N. Konwar, and S. Radenović, edi- tors, Metric Fixed Point Theory, Forum for Interdisciplinary Mathematics. Springer, Singapore, 2021. [38] S. Reich. Some remarks concerning contraction mappings. Canad. Math. Bull., 14 (1):121–124, 1971. [39] B.E. Rhoades. Some fixed point iteration procedures. Int. J. Math. Math. Sci., 14: 1–16, 1991. [40] B.E. Rhoades. Fixed point theorems and stability results for fixed point iteration procedures ii. Indian J. Pure Appl. Math., 24(11):691–703, 1993. [41] D.O’ Regan R.P. Agarwal and D.R.Sahu. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal., 8(1):61– 79, 2007. [42] I. Uddin S. Khatoon and D. Baleanu. Approximation of fixed point and its application to fractional differential equation. J. Appl. Math. Comput., 66:507–525, 2021. [43] D.R. Sahu and A. Petrusel. Strong convergence of iterative methods by strictly pseu- docontractive mappings in banach spaces. Nonlinear Anal. Theory Methods Appl., 74:6012–6023, 2011. [44] A.O. Nazek S.R. Mahmoud and A. Hala. New class of nonlinear fractional integro- differential equations with theoretical analysis via fixed point approach: Numerical and exact solutions. J. Appl. Anal. Comput., 13(5):2767–2787, 2023. [45] G. Stampacchia. Formes bilineaires coercivites sur les ensembles convexes. Comptes Rendus de l’Academie des Sciences, 258:4413–4416, 1964. [46] X. Su. Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett., 22:64–69, 2009. [47] M. Turkyilmazoglu. Approximate analytical solution of the nonlinear system of differ- ential equations having asymptotically stable equilibrium. Filomat, 31(9):2633–2641, 2017. [48] M. Turkyilmazoglu. Optimization by the convergence control parameter in iterative methods. Appl. Math. Lett., 23(2):105–116, 2024. [49] K. Ullah and M. Arshad. Numerical reckoning fixed points for suzuki’s generalized nonexpansive mappings via new iteration process. Filomat, 32:187–196, 2018. [50] X. Weng. Fixed point iteration for local strictly pseudo-contractive mappings. Proc. Amer. Math. Soc., 113:727–731, 1991. [51] N. Shahzad Y. Yao and Y.C. Liou. Modified semi-implicit midpoint rule for nonex- pansive mappings. Fixed Point Theory Appl., 2015. doi: 10.1186/s13663-015-0414-2. [52] J. Yu and Y. Feng. Symmetry analysis, optimal system, conservation laws and exact solutions of time fractional diffusion-type equation. Int. J. Geom. Methods Mod. Phys., 2024. doi: 10.1142/S0219887824502864. [53] J. Yu and Y. Feng. On the generalized time fractional reaction-diffusion equation: Lie symmetries, exact solutions and conservation laws. Chaos Solit. Fractals, 182, 2024. doi: 10.1016/j.chaos.2024.114855.