EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1602-1617 ISSN 1307-5543 – ejpam.com Published by New York Business Global Modified Inertial Krasnosel’skii-Mann type Method for Solving Fixed Point Problems in Real Uniformly Convex Banach Spaces Besheng George Akuchu1, Maggie Aphane2, Godwin Chidi Ugwunnadi2,3,∗, George Emeka Okereke4 1 Department of Mathematics, University of Nigeria Nsukka, Enugu State, Nigeria 2 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa, P.O. Box 94, Pretoria 0204, South Africa 3 Department of Mathematics, Faculty of Science and Engineering, University of Eswatini, Private Bag 4, Kwaluseni M201, Eswatini 4 Department of Computer Science, University of Nigeria Nsukka, Enugu State, Nigeria Abstract. We present an altered version of the inertial Krasnosel’skii-Mann algorithm and demon- strate convergence outcomes for mappings that are asymptotically nonexpansive within real, uni- formly convex Banach spaces. To achieve our results, we skillfully construct the inequality in equation (6) and apply it accordingly. Our findings support and broadly generalize a number of significant findings from the literature. We demonstrate, as an application, the generation of max- imal monotone operators’ zeros via fixed point methods in Hilbert spaces. Additionally, we solve convex minimization issues using our fixed-point techniques. 2020 Mathematics Subject Classifications: 47H09 Key Words and Phrases: Asymptotically Nonexpansive Mappings, Modified Inertial Krasnosel’skii- Mann, Continuous Mappings, Convergence, Fixed Points, Banach Spaces 1. Introduction We consider a Banach space X and any self map of X to be T : X → X in this paper. The fixed points set of T is denoted by F (T ) and may be found with the formula F (T ) := {x ∈ X : T a = a}). We refer to the mapping T as follows: (i) Nonexpansive, if ||T a− T b|| ≤ ||a− b|| for all a, b ∈ X , ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5187 Email addresses: george.akuchu@unn.edu.ng (B. G. Akuchu), maggie.aphane@smu.ac.za (M. Aphane), gcugwunnadi@uniswa.sz (G. C. Ugwunnadi), george.okereke@unn.edu.ng (G. E. Okereke) https://www.ejpam.com 1602 © 2024 EJPAM All rights reserved. B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1603 (ii) Asymptotically Nonexpansive, if ∀a, b ∈ X , there exists a sequence {κn} ⊂ [1,+∞), with limn→∞ κn = 1, such that ||T na− T nb|| ≤ κn||a− b|| ∀n ≥ 1. (1) (iii) Uniformly L− Lipschitzian (see for example [19]), if there exists a real constant L > 0, such that for all a, b ∈ X , n ≥ 1, the following holds ||T na− T nb|| ≤ L||a− b||. We can easily observe that any nonexpansive maps with sequence κn = 1∀ n ≥ 1 are asymptotically nonexpansive maps. It is also bounded because κn is convergent. All asymptotically nonexpansive mappings are therefore uniformly L-Lipschitzian, and so con- tinuous. Goebel and Kirk (see [9]) introduced the class of asymptotically nonexpansive mappings as a natural extension of the nonexpansive mappings class, and the construction of fixed points of nonexpansive mappings and their generalizations have historically attracted a great deal of research interest. This is due to the fact that nonexpansive mappings are strongly associated with numerous other mapping classes, such as the accretive operators, and the practical applications of their fixed point construction in image recovery, computer tomography, signal processing, and other fields are numerous. In 1967, Browder [5] and Kato [12] independently introduced the class of accretive op- erators. Browder [5] proved a basic result in the theory of accretive operators: if A is Lipschitzian and accretive, then dz dt +T Az = 0, z(0) = z0 is a solved initial value problem. Assume we have a Hilbert space H. It is well known (see, for example, [1]) that if A : H → H is an accretive operator, then the resolvent of A, given by Jλ A := (I + λA)−1, is a nonexpansive operator for any real constant λ > 0. The zeros of A are obviously the fixed points of Jλ A. Thus, several application domains united by the theory of accretive operators are brought together by studying fixed points of nonexpansive mappings and their generalizations. This makes the study current. The approximation of fixed points of nonexpansive mappings has been studied by a number of authors in various ways. An iteration approach for the production of fixed points of nonexpansive mappings was introduced by W. R. Mann in [16], to be precise: Given a real Hilbert space H, let C be a nonempty convex subset of it. Starting from any random x0 ∈ C, the Mann’s sequence is produced by an+1 = (1− ξn)an + ξnT an, (2) where the real sequence {ξn} ⊂ (0, 1). The author demonstrated that the sequence {an} converges weakly to a fixed point of T with the constraint ∑ ξn(1− ξn) = +∞. Studying convergence results to its fixed points becomes relevant when one realizes that asymptotically nonxepansive mappings are generalizations of nonexpansive mappings. Nu- merous scholars have examined convergence outcomes for fixed points of asymptotically B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1604 nonexpansive mappings (see, for instance, to [9, 10]). Some authors used the modified Mann iteration sequence, which is defined as follows, to achieve this: The modified Mann iteration sequence (see, for example, [19]) is formed from an arbitrary a0 ∈ C if C is a nonempty convex subset of a Banach space, E , and T : C → C is any map. an+1 = (1− ξn)an + ξnT nan, (3) where a real sequence {ξn} ⊂ (0, 1). The authors (see, for example, [9]) demonstrated con- vergence results to fixed points of T under specific restrictions on the iteration parameter. 2. Inertial Iteration Schemes Many authors (see, for example, [1, 4, 6–8, 13, 14, 17, 21, 23, 24]) have recently investigated iteration schemes known as ’inertial iteration schemes’ since the rate of convergence of iteration sequences is equally highly significant. The characteristic of these schemes is that they are known to be faster than well-known convergent iteration schemes because of the addition of a term called the inertial term. A few examples of inertial schemes are given in [1], where the authors presented the inertial Krasnosel’skii-Mann iteration as follows for a self nonexpansive mapping of a real Hilbert space H: a0, a1 ∈ H bk = ak + tk(ak − ak−1) ak+1 = (1− ξk)ak + ξkTbk, k = 0, 1, ... (4) The authors demonstrated the weak convergence of the scheme to fixed points of non- expansive mappings under the criteria that 0 ≤ tk ≤ t < 1, for some t ∈ (0, 1), and∑ tk∥ak − ak−1∥2 < +∞. The inertial term is denoted by tk(ak − ak−1). The interested reader might refer to [1, 4, 7, 15], etc. for additional arguments on the insertion of inertial terms to iteration schemes. The concept of inertial technique was merged with the Halpern viscosity algorithms in [25], whereby modified inertial Mann algorithms were introduced. The following theorems were used by the authors to examine convergence results to fixed points of nonexpansive mappings using these algorithms: Theorem 1. [25] Let T : C → C be a nonexpansive mapping with F (T ) ̸= ∅ and C a nonempty closed convex subset of a real Hilbert space H. The following criteria are met given a point z ∈ C and two sequences {ψn} and {vn} in (0, 1): (D1) ∑ vn = ∞ and lim vn = 0; (D2) lim δn vn ∥an − an−1∥ = 0. B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1605 Assume that a−1, a0 ∈ C are arbitrary. Use the following algorithm to define a sequence {an}:  wn = an + δn(an − an−1), bn = ψnwn + (1− ψn)T wn an+1 = vnz + (1− vn)bn, n ≥ 0. The generated iterative sequence {an} then strongly converges to a∗ = PF (T )z. Theorem 2. [25] Let T : C → C be a nonexpansive mapping with F (T ) ̸= ∅. Assume that C is a nonempty closed convex subset of a real Hilbert space H. Define a ρ-contraction h : C → C such that ∥h(a) − h(b)∥ ≤ ρ∥a − b∥, ∀a, b ∈ C. Given two sequences in (0, 1), {ψn} and {vn}, the following conditions hold: (D1) ∑ vn = ∞ and lim vn = 0; (D2) lim δn vn ∥an − an−1∥ = 0. Assume that a−1, a0 ∈ C are arbitrary. Generate a sequence {an} using the procedure below:  wn = an + δn(an − an−1), bn = ψnwn + (1− ψn)T wn an+1 = vnh(an) + (1− vn)bn, n ≥ 0. The resultant iterative sequence {an} then converges strongly to a∗ = PF (T )h(a ∗). Observation 1: Given condition (D1), condition (D2) sug gests that δn∥an − an−1∥ approaches zero at a quicker rate than vn. This suggests that lim δn∥an − an−1∥ = 0. This further suggests that lim δn∥an − an−1∥2 = (lim δn∥an − an−1∥)(lim ∥an − an−1∥) = 0(lim ∥an − an−1∥) = 0. Therefore, δn∥an − an−1∥2 is bounded. Consequently, it is weaker to impose a boundedness constraint on δn∥an − an−1∥2. This will be helpful for our out- comes in the follow-up. Observation 2: It is possible that {wn} will not belong in C since C is a convex subset. This suggests that the schemes in [25] need precise definitions. A well-defined scheme can only exist if C is either the entire space or an affine subset of it. An inertial accelerated algorithm for obtaining a fixed point in the fixed points set of an asymptotically nonexpansive mapping in a real uniformly convex Banach space that satisfies Opial criteria was recently explored by Murtala et al. (see [11]). More specifically, the authors suggested the following outcomes: Assumption 1. [11] Let X be a real uniformly convex Banach space. (i) Choose sequences {ξn} ⊂ (0, 1), {βn}, {δn} ⊂ [0,+∞) and ∑∞ n=1 δn < +∞ with δn = o(βn) which means limn→∞ δn βn = 0. (ii) Let a0, a1 ∈ X be arbitrary points, for the iterates an−1 and an for each n ≥ 1, choose θn such that 0 ≤ θn ≤ θ̄n where, for η ≥ 3 θ̄n :=  min{ n−1 n+η−1 , δn ∥an−an−1∥}, if an ̸= an−1 n−1 n+η−1 , otherwise B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1606 According to the authors, the extrapolation phase described in [3] provides the concept of Assumption 1. Along with it, the authors added this: It is easy to see from Assumption 1 that for each n ≥ 1, we have θn∥an − an−1∥ ≤ δn, which together with ∑ δn < +∞ and limn→∞ δn βn = 0, we obtain∑ θn∥an − an−1∥ < +∞ and lim n→∞ θn βn ∥an − an−1∥ ≤ lim n→∞ δn βn = 0. Using Assumption 1, the authors stated and proved the following theorem: Theorem 3. [11] Let X be a Banach space that is real and uniformly convex, possessing Opial’s property. With sequence {κn} ⊂ [0,∞), let T : X → X be an asymptotically nonexpansive mapping such that ∑∞ n=0 κn < ∞ and F (T ) ̸= ∅. Let {an} be the sequence that is produced in this way: a0, a1 ∈ X wn = an + θn(an − an−1), dn+1 = 1 λ(T n(wn)− wn) + βndn, bn = wn + λdn+1, an+1 = µξnwn + (1− µξn)yn, n ≥ 1, where µ ∈ (0, 1], λ > 0, assuming that Assumption 1 holds and set d1 = 1 λ(T nw0 − w0). Then the sequence {an} converges weakly to a point a∗ ∈ F (T ), provided that the following conditions hold: (C1) ∑∞ n=0 βn < +∞ (C2) lim infn→∞ µξn(1− µξn) > 0 Moreover, {wn} satisfies (C3) {T nwn − wn} is bounded. Observation 3: The discussion in Observation 1 also holds for θn∥an − an−1∥2, based on Assumption 1 and the fact that ∑ θn∥an − an−1∥ <∞. Moreover, lim θn∥an − an−1∥2p = (lim θn∥an−an−1∥) holds for every positive integer p > 1,. When (lim∥an−an−1∥2p−1) = 0, Limit∥an − an−1∥2p−1) = 0. It is therefore weaker to impose a boundedness constraint on θn∥an − an−1∥2p. This will help with the outcomes we get in the follow-up. Observation 4: The computations and analysis performed in [11] are negatively impacted by the inequality that characterizes uniformly convex Banach spaces, which is unfortu- nately economically quoted in [11] (only for p = 2). For real uniformly convex Banach spaces, therefore, the conclusions in [11] are not applicable in general. B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1607 In this article, we modify the inertial iteration scheme introduced in [1] and prove con- vergence results for fixed points of asymptotically nonexpansive mappings in some real uniformly convex Banach spaces. We do this by imposing different sets of conditions, some of which are weaker than those imposed in [11]. Our motivation comes from the aforementioned works and observations. Compared to the class examined in [11], our class of spaces is more general (just for p = 2). Our improved inertial strategy for a real uniformly convex Banach X is as follows: a0, a1 ∈ X bn = an + νn(an − an−1) an+1 = (1− ξn)bn + ξnT nbn, n = 1, 2, ... (5) 3. Preliminaries Assume that the Banach space X is real. It’s common knowledge that if D is a nonempty convex subset of X and h : X → ℜ̄ := ℜ∪{+∞} is a suitable functional, then h is convex on D if h(λa+ (1− λ)b) ≤ λh(a) + (1− λ)h(b) for all 0 ≤ λ ≤ 1 and a, b ∈ D. In D, h is considered uniformly convex (refer to [27]) if and only if there is a function µ : ℜ+ := [0,+∞) → ℜ+ with µ(t) = 0, such that h(λa+ (1− λ)b) ≤ λh(a) + (1− λ)h(b)− λ(1− λ)µ(∥a− b∥) for all 0 ≤ λ ≤ 1 and a, b ∈ D. A definition and a few lemmas that will be helpful in the sequel are provided before we express and demonstrate our primary findings: Definition 1. (see e.g [20]) Consider the Banach space E. When {an} is a sequence in D(T ) such that {an} converges weakly to z ∈ D(T ) and {T an} converges strongly to z, then T z = z. This mapping T : D(T ) ⊆ E → E is said to be demiclosed at a point z ∈ D(T ). Lemma 1. (see e.g [10]) Assume that E is a uniformly convex Banach space, D is a nonempty closed convex subset of E, and T : D → D is an asymptotically nonexpan- sive mapping with a sequence {κn} ⊂ [1,∞), where limn→∞ κn = 1.. At zero, I − T is demiclosed. Lemma 2. (see e.g [19]) For all n ≥ 1, let {ξn}, {βn}, and {δn} be sequences of nonneg- ative real numbers that fulfill the inequality ξn+1 ≤ (1 + δn)ξn + βn, lim ξn exists if and only if ∑ δn = +∞ and ∑ βn = +∞. Additionally, if {ξn} has a subsequence that strongly converges to zero, then lim ξn = 0. B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1608 Lemma 3. (see [26]) Assume that p > 1 is a fixed real value. If X is p-uniformly convex, then the functional ∥.∥p is uniformly convex on the entire Banach space X . That is, X is p−uniformly convex if and only if h : ℜ+ := [0,+∞) → ℜ+, where h(0) = 0, exists and such a function ∥λa+(1−λ)a∥p ≤ λ∥a∥p+(1−λ)∥b∥p−λ(1−λ)g(∥a− b∥) for all 0 ≤ λ ≤ 1 and a, b ∈ X . According to [26], this lemma’s result yields the inequality below: A real constant c > 0 exists such that ∥λa+ (1− λ)b∥p ≤ λ∥a∥p + (1− λ)∥b∥p −wp(λ)c∥a− b∥p, where wp(λ) = λ(1− λ)p + λp(1− λ)For every 0 ≤ λ ≤ 1 and a, b ∈ X. 4. Main Results We now state and prove our main results. Theorem 4. Let X be a p−uniformly convex Banach space that satisfies Opial’s condition, and let p > 1 be any positive integer. Given a non-empty fixed points set F (T ) and a sequence {κn} ⊂ [1,∞), let T : X → X be an asymptotically nonexpansive mapping such that ∑ n→∞ κn − 1 < ∞. Let {an} be the modified inertial Krasnosel’skii-Mann sequence for a0, a1 ∈ X , where {ξn} and {νn} are real sequences in (0, 1), satisfying: (i) νn∥an − an−1∥2p ≤ D, for some real positive constant D, (ii) lim inf ξn(1− ξn) > 0, (iii) ∑ ν 1 2 n < +∞. Then {an} converges weakly to a fixed point of T . Proof. Condition (i) in our theorem is motivated by Observation 3 above. For any positive integer p > 1 and since νn ∈ (0, 1) we have (1 + νn)ν p n < νn(1 + νn) p, implying that (1 + νn)ν p n − νn(1 + νn) p < 0. Letting wp(.) to be the functional in Lemma 3 (consequence inequality), we have wp(1+νn) = (1+νn)(−νn)p+(1+νn) p(−νn) =  −[(1 + νn)ν p n + (1 + νn) pνn], if p is odd (1 + νn)ν p n − (1 + νn) pνn, if p is even So, for all positive integers p > 1, we have wp(1+νn) < 0 (since (1+νn)ν p n−νn(1+νn)p < 0,) so that −wp(ν + νn) > 0. Also, observe that −wp(1 + νn) = −[(1 + νn)(−νn)p + (1 + νn) p(−νn)] = (1 + νn) p(νn)− (1 + νn)(−νn)p ≤ (1 + νn) pνn + (1 + νn)(νn) p (6) Since −wp(1 + νn) > 0, the inequality (consequent inequality) in Lemma 3 holds for λ = 1 + νn, for some nonnegative real constant c = c1 and for all a, b in any real p- uniformly convex Banach space with p > 1 being any positive integer. B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1609 Since {κn} ⊂ [1,+∞) converges to 1, there exists a positive real constant D1, such that κpn ≤ D1. By applying the Lagrange mean value theorem, it is easily verifiable that for r > 1, we have rp − 1 ≤ prp−1(r − 1). Let c be a positive real constant and a∗ ∈ F (T ). Using these, (5), (6) and Lemma 3 (consequence inequality), we have ||an+1 − a∗||p = ||(1− ξn)bn + ξnT nbn − a∗||p = ||(1− ξn)(bn − a∗) + ξn(T nbn − a∗)||p ≤ (1− ξn)||bn − a∗||p + ξn||T nbn − a∗||p − wp(ξn)c||T nbn − bn||p ≤ (1− ξn)||bn − a∗||p + ξnκ p n||bn − a∗||p − wp(ξn)c||T nbn − bn||p = [1 + ξn(κ p n − 1)]||bn − a∗||p − wp(ξn)c||T nbn − bn||p = [1 + ξn(κ p n − 1)]||(1 + νn)an− ∋n an−1 − a∗||p − wp(ξn)c||T nbn − bn||p = [1 + ξn(κ p n − 1)]∥(1 + νn)(an − a∗)− νn(an−1 − a∗)||p − wp(ξn)c||T nbn − bn||p ≤ [1 + ξn(κ p n − 1)][(1 + νn)∥an − a∗∥p − νn∥an−1 − a∗∥p − wp(1 + νn)c1∥an − an−1∥p] −wp(ξn)c||T nbn − bn||p ≤ [1 + ξn(κ p n − 1)](1 + νn)∥an − a∗∥p − [1 + ξn(κ p n − 1)]wp(1 + νn)c1∥an − an−1∥p −wp(ξn)c||T nbn − bn||p ≤ ∥an − a∗∥p + [νn + ξn(κ p n − 1) + ξnνn(κ p n − 1)]∥an − a∗∥p − κpnwp(1 + νn)c1∥an − an−1∥p −wp(ξn)c||T nbn − bn||p since wp(1 + νn) < 0 ≤ ∥an − a∗∥p + [νn + 2ξn(κ p n − 1)]∥an − a∗∥p − κpnwp(1 + νn)c1∥an − an−1∥p −wp(ξn)c||T nbn − bn||p ≤ ∥an − a∗∥p + [νn + 2ξnpκ p−1 n (κn − 1)]∥an − a∗∥p − κpnwp(1 + νn)c1∥an − an−1∥p −wp(ξn)c||T nbn − bn||p ≤ ∥an − a∗∥p + [νn + 2pD1(κn − 1)]∥an − a∗∥p − κpnwp(1 + νn)c1∥an − an−1∥p −wp(ξn)c||T nbn − bn||p ≤ [1 + δn]∥an − a∗∥p − κpnwp(1 + νn)c1∥an − an−1∥p − wp(ξn)c||T nbn − bn||p ≤ [1 + δn]∥an − a∗∥p −D1wp(1 + νn)c1∥an − an−1∥p −wp(ξn)c||T nbn − bn||p since wp(1 + νn) < 0 ≤ [1 + δn]∥an − a∗∥p +D1[νn(1 + νn) p + νpn(1 + νn)]c1∥an − an−1∥p −wp(ξn)c||T nbn − bn||p ≤ [1 + δn]∥an − a∗∥p +D1νn[(1 + νn) p + νp−1 n (1 + νn)]c1∥an − an−1∥p −wp(ξn)c||T nbn − bn||p ≤ [1 + δn]∥an − a∗∥p +D1νn[2 p + 2]c1∥an − an−1∥p − wp(ξn)c||T nbn − bn||p = [1 + δn]∥an − a∗∥p + 2D1νn[2 p−1 + 1]c1∥an − an−1∥p − wp(ξn)c||T nbn − bn||p ≤ [1 + δn]∥an − a∗∥p + 2D1[2 p−1 + 1]c1ν 1 2 n [νn∥an − an−1∥2p] 1 2 − wp(ξn)c||T nbn − bn||p ≤ [1 + δn]∥an − a∗∥p + 2D1[2 p−1 + 1]c1ν 1 2 nD 1 2 − wp(ξn)c||T nbn − bn||p B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1610 = [1 + δn]∥an − a∗∥p +Mν 1 2 n − wp(ξn)c||T nbn − bn||p (7) whereM = 2c1D1[2 p−1+1]D 1 2 and δn = νn+2pD1(κn−1) is such that ∑ δn < +∞, since∑ (κn − 1) < +∞ and condition (iii) holds. Using this, (7), Lemma 2 and condition (iii), we have that lim ∥an − a∗∥p exists. This implies {an − a∗} and {an} are norm bounded. Hence, there exists a real constant D2 > 0, such that ∥an − a∗∥p ≤ D2. Using this in 7 and ∀n ≥ 0, we have ∥an+1 − a∗∥p ≤ ∥an − a∗∥p + δnD2 +Mν 1 2 n − wp(ξn)c||T nbn − bn||p From this, the fact that ∑ δn < +∞ and condition (iii), we have∑ n≥0 2[ξn(1− ξn)] pc||T nbn − bn||p ≤ ∑ n≥0 wp(ξn)c||T nbn − bn||p ≤ ∑ n≥0 [||an − a∗||p − ||an+1 − a∗||p] +D2 ∑ n≥0 δn +M ∑ n≥0 ν 1 2 n <∞. This implies from conditions (ii) that lim ||T nbn − bn||p = 0. Hence lim ||T nbn − bn|| = 0. (8) From (5), (iii) and the fact that {an} is norm bounded, we have ∥bn − an∥ = νn∥an − an−1∥ ≤ νn[∥an∥+ ∥an−1∥] → 0. (9) Furthermore, we have from (5), (8) and (9) that ||an+1 − an|| = ||(1− ξn)(bn − an) + ξn(T nbn − an)|| ≤ (1− ξn)∥bn − an∥+ ξn∥T nbn − an∥ = (1− ξn)∥bn − an∥+ ξn∥T nbn − bn + bn − an∥ ≤ (1− ξn)∥bn − an∥+ ξn[∥T nbn − bn∥+ ∥bn − an∥] = ∥bn − an∥+ ξn∥T nbn − bn∥ ≤ ∥bn − an∥+ ∥T nbn − bn∥ → 0. (10) We also have from (5) that ξnT n(an + νn(an − an−1))− ξnan = an+1 − an − (1− ξn)νn(an − an−1) ⇔ ξn[T n(an + νn(an − an−1))− an] = an+1 − an − (1− ξn)νn(an − an−1) ⇔ [T n(an + νn(an − an−1))− an] = 1 ξn [an+1 − an − (1− ξn)νn(an − an−1)]. B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1611 This implies ∥T n(an + νn(an − an−1))− an∥ = 1 ξn ∥[an+1 − an − (1− ξn)νn(an − an−1)]∥ ≤ 1 ξn [∥an+1 − an∥+ ∥(1− ξn)νn(an − an−1)∥] = 1 ξn [∥an+1 − an∥+ (1− ξn)νn∥an − an−1∥] ≤ 1 ξn [∥an+1 − an∥+ νn∥an − an−1∥]. Since T is uniformly L-Lipschitzian(and hence T n is continuous), {an} is norm bounded, using (10), conditions (ii) and (iii), this yields lim ∥T nan − an∥ = 0. (11) Using (10) and (11), we now have ||an − T an|| = ∥an − an+1 + an+1 − T n+1an+1 + T n+1an+1 − T an∥ ≤ ∥an − an+1∥+ ∥T n+1an+1 − an+1∥+ L∥T nan+1 − an∥ = ∥an − an+1∥+ ∥T n+1an+1 − an+1∥+ L∥T nan+1 − T nan + T nan − an∥ ≤ ∥an − an+1∥+ ∥T n+1an+1 − an+1∥+ L[∥T nan+1 − T nan∥+ ∥T nan − an∥] ≤ ∥an − an+1∥+ ∥T n+1an+1 − an+1∥+ L2∥an+1 − an∥+ L∥T nan − an∥ = (1 + L2)∥an+1 − an∥+ ∥T n+1an+1 − an+1∥+ L∥T nan − an∥ → 0. (12) Since {an} is norm bounded, it possesses a subsequence {ank } which converges weakly to a point u ∈ X . Since X satisfies the Opial condition, a standard argument (see e.g [18]) yields that {an} converges weakly to ∈ X . The demiclosedness property of T (see Lemma 1 now yields that u ∈ F (T ). Setting u = a∗ above, our proof is complete. Theorem 5. Let H be a real Hilbert space and let T : H → H be an asymptotically non- expansive mapping with a non-empty fixed points set F (T ) and sequence {κn} ⊂ [1,+∞), such that ∑ n→∞ κn − 1 < +∞. Then the modified inertial Krasnosel’skii-Mann sequence {an} generated from a0, a1 ∈ H by{ bn = an + νn(an − an−1) an+1 = (1− ξn)bn + ξnT nbn, n = 1, 2, ... where {ξn} and {nun} are real sequences in (0, 1), satisfying: (i) 0 < α1 ≤ ξn ≤ α2 < 1 for some real constants α1, α2 ∈ (0, 1), (ii) ∑ νn < +∞ (iii) ∑ νn∥an − an−1∥2 < +∞ B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1612 converges weakly to a fixed point of T . Proof. Since κn → 1, there exists a real constantM1 > 0 such that κn+1 ≤M1, ∀ n ≥ 1. Let z ∈ F (T ). Using (5) and the well-known identity ||(1− λ)a+ λb||2 = (1− λ)||a||2 + λ||b||2 − λ(1− λ)||a− b||2 which holds in Hilbert spaces H, ∀a, b ∈ H and ∀λ ∈ [0, 1], we have ||an+1 − z||2 = ||(1− ξn)bn + ξnT nbn − z||2 = ||(1− ξn)(bn − z) + ξn(T nbn − z)||2 = (1− ξn)||(bn − z)||2 + ξn||T nbn − z||2 − ξn(1− ξn)||T nbn − bn||2 ≤ (1− ξn)||bn − z||2 + ξnκ 2 n||bn − z||2 − ξn(1− ξn)||T nbn − bn||2 = [1 + ξn(κ 2 n − 1)]||bn − z||2 − ξn(1− ξn)||T nbn − bn||2 = [1 + ξn(κ 2 n − 1)]||(1 + νn)an − νnan−1 − z||2 − ξn(1− ξn)||T nbn − bn||2 = [1 + ξn(κ 2 n − 1)]∥(1 + ξn)(an − z)− νn(an−1 − z)||2 − ξn(1− ξn)||xinbn − bn||2 = [1 + ξn(κ 2 n − 1)][(1 + νn)∥an − z∥2 − νn∥an−1 − z∥2 + νn(1 + νn)∥an − an−1∥2] −ξn(1− ξn)||T nbn − bn||2 ≤ [1 + ξn(κ 2 n − 1)](1 + νn)∥an − z∥2 + [1 + ξn(κ 2 n − 1)]νn(1 + νn)∥an − an−1∥2] −ξn(1− ξn)||T nbn − bn||2 ≤ ||an − z||2 + [νn + ξn(κ 2 n − 1) + ξnνn(κ 2 n − 1)]∥an − z∥2 +2(1 +M2 1 )νn∥an − an−1∥2 −ξn(1− ξn)||T nbn − bn||2 ≤ [1 + δn]∥an − z∥2 + 2(1 +M2 1 )νn∥an − an−1∥2 −ξn(1− ξn)||T nbn − bn||2 (13) ≤ [1 + δn]∥an − z∥2 + 2(1 +M2 1 )νn∥an − an−1∥2, where δn = νn + 2M1(κn − 1) is such that ∑ δn < +∞, since ∑ (κn − 1) < +∞ and condition (ii) holds. Using this, Lemma 2 and condition (iii), we have that lim ∥an − a∗∥2 exists. This implies {an − a∗} and {an} are norm bounded. Hence, there exists a real constant D3 > 0, such that ∥an − a∗∥2 ≤ D3. Using this in (13), we have that ∥an+1 − a∗∥2 ≤ ∥an − a∗∥2 + δnD3 + 2(1 +M2 1 )νn∥an − an−1∥2 − ξn(1− ξn)||T nbn − bn||2 From this and condition and (iii), we have∑ n≥0 α1(1− α2)||T nbn − bn||2 ≤ ∑ n≥0 ξn(1− ξn)||T nbn − bn||2 ≤ ∑ n≥0 [||an − a∗||2 − ||an+1 − a∗||2] +D3 ∑ n≥0 δn +2(1 +M2 1 ) ∑ n≥0 νn∥an − an−1∥2 <∞. B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1613 This implies from condition (i) that lim ||T nbn − bn||2 = 0. Hence lim ||T nbn − bn|| = 0. The rest of the proof now follows easily as in that of Theorem 4 above. Theorem 6. Let H be a real Hilbert space and let T : H → H be an asymptotically nonexpansive mapping with a non-empty fixed points set F (T ) and sequence {κn} ⊂ [1,∞), such that ∑ n→∞ κn − 1 < +∞. Then the modified inertial Krasnosel’skii-Mann sequence {an} generated from a0, a1 ∈ H by{ bn = an + νn(an − an−1) an+1 = (1− ξn)bn + ξnT nbn, n = 1, 2, ... where {ξn} and {νn} are real sequences in (0, 1), satisfying: (i) lim inf ξn(1− ξn) > 0, (ii) ∑ νn 1 2 < +∞ (iii) νn∥an − an−1∥4 ≤ D4 for some positive real constant D4, converges weakly to a fixed point of T . Proof. Computing as in Theorem 5 above, we arrive at ||an+1 − z||2 ≤ [1 + δn]∥an − z∥2 + 2(1 +M2)νn∥an − an−1∥2 − ξn(1− ξn)||T nbn − bn||2. This implies ||an+1 − z||2 ≤ [1 + δn]∥an − p∥2 + 2(1 +M2)ν 1 2 n √ νn∥an − an−1∥4 − ξn(1− ξn)||T nbn − bn||2 ≤ [1 + δn]∥an − z∥2 + 2(1 +M2)ν 1 2 n √ D4 − ξn(1− ξn)||T nbn − bn||2 where δn = νn + 2M(κn − 1) is such that ∑ δn < +∞, since ∑ (κn − 1) < +∞ and condition (ii) holds. Using this, Lemma 2 and condition (ii), we have that lim ∥an − a∗∥2 exists. This implies {an − a∗} and {an} are norm bounded. Hence, there exists a real constant D5 > 0, such that ∥an − a∗∥2 ≤ D5. Using this in (14), we have that ∥an+1 − a∗∥2 ≤ ∥an − a∗∥2 + δnD5 + 2(1 +M2)ν 1 2 n √ D4 − ξn(1− ξn)||T nbn − bn||2. From this and conditions (ii), we have∑ n≥0 ξn(1− ξn)||T nbn − bn||2 ≤ ∑ n≥0 [||an − a∗||2 − ||an+1 − a∗||2] +D5 ∑ n≥0 δn +2(1 +M2) √ D4 ∑ n≥0 ν 1 2 n <∞. B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1614 This implies from condition (i) that lim ||T nbn − bn||2 = 0. Hence lim ||T nbn − bn|| = 0. From (5), (ii) and the fact that {an} is norm bounded, we have ∥bn − an∥ = νn∥an − an−1∥ ≤ ν 1 2 n [∥an∥+ ∥an−1∥] → 0 The rest of the proof follows easily like in Theorem 4 above. Theorem 7. Let H be a real Hilbert space and let A : H → C ⊆ H be a maximally monotone operator such that Zer(A) ̸= ∅. Let Jλ A := (I + λA)−1 be the resolvent of A. Then the modified inertial Krasnosel’skii-Mann sequence {an} generated from a0, a1 ∈ H by { bn = an + νn(an − an−1) an+1 = (1− ξn)yn + ξnJ λ Abn where {ξn} and {νn} are real sequences in (0, 1), satisfying: (i) 0 < α1 ≤ ξn ≤ α2 < 1 for some real constants α1, α2 ∈ (0, 1), (ii) ∑ νn < +∞ (iii) ∑ νn∥an − an−1∥2 < +∞. converges weakly to an element of F (Jλ A), which is also an element of zer(A). Proof. Since Jλ A is nonexpansive, the proof follows like that of Theorem 5 above, since every nonexpansive mapping is an asymptotically nonexpansive mapping with sequence κn = 1 ∀ n ≥ 1. Theorem 7 can be applied in solving convex optimization problems of the form mina∈H{h(a)}, where h : H → R ∪ {+∞} is a proper, convex and lower semicontinuous function. To do this, we recall the following: If h : H → R ∪ {+∞} is a proper, convex and lower semicontinuous function, then its (convex) subdifferential at a ∈ H is defined by ∂h(a) = {b ∈ H : h(u) ≥ h(a) + ⟨b, u− a⟩∀u ∈ H}, for all a ∈ H, with h(a) = +∞ and ∂h(a) = ∅ otherwise. When the convex subdifferential is seen as a set-valued mapping, then, it is maximally monotone (see [22]) and its resolvent is given by J∂h = proxh (see [2]), where Proxh : H → H is defined by proxh(a) = argminb∈H{h(b) + 1 2 ∥b− a∥2} and is called the proximal operator of h. We now have the following: B. G. Akuchu et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1602-1617 1615 Corollary 1. Let h : H → R ∪ {+∞} be a set-valued, proper, convex and lower semi- continuous function which is such that argmina∈H{h(a)} ≠ ∅. Then the modified inertial Krasnosel’skii-Mann sequence {an} generated from a0, a1 ∈ H by{ bn = an + νn(an − an−1) an+1 = (1− ξn)bn + ξnprox λ hbn where {ξn} and {∋n} are real sequences in (0, 1), satisfying: (i) (i) 0 < α1 ≤ ξn ≤ α2 < 1 for some real constants α1, α2 ∈ (0, 1), (ii) (ii) ∑ νn < +∞ (iii) (iii) ∑ νn∥an − an−1∥2 < +∞ converges weakly to an element of argmina∈H{h(a)}. Proof: Setting ∂h = A, the proof follows as in the proof of Theorem 7 since the zero of ∂h is an element of argmina∈H{h(a). Remark 1. If we let {σn} ⊂ (0, 1) such that ∑ σn <∞ and choose νn ∈ [0, ν̄n] with ν̄n = min{σ2n, 1 n2∥an−an−1∥2p }, then conditions (i) and (iii) of Theorem 4 hold. Remark 2. If we let {σn} ⊂ (0, 1) such that ∑ σn < +∞ and choose νn ∈ [0, ν̄n] with ν̄n = min{σn, 1 n2∥an−an−1∥2 }, then conditions (ii) and (iii) of Theorem 5 hold. Remark 3. 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