Demiclosedness and weak convergence of supper hybrid mappings in Banach spaces

 Corresponding author: Ikechukwu Godwin Ezugorie Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0. Demiclosedness and weak convergence of supper hybrid mappings in Banach spaces Ikechukwu Godwin Ezugorie * Department of Mathematics, Enugu State University of Science and Technology, Enugu, Enugu State, Nigeria. World Journal of Advanced Research and Reviews, 2025, 27(02), 1564-1570 Publication history: Received on 09 July 2025; revised on 19 August; accepted on 22 August 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.27.2.2987 Abstract We introduce and study a new class of mapping in Banach Spaces, termed (𝛼 , 𝛽,𝛾) - supper hybrid mappings, which generalize the well – known ( 𝛼 , 𝛽 ) - generalized hybrid mappings. This extended framework encompasses a broader spectrum of nonlinear of nonlinear operators and allows for refined control via an additional parameter 𝛾 ≥ 0. We establish several foundational properties of supper hybrid mappings, including quasi – nonexpansivenes and the demiclosedness principle at zero. Furthermore, we prove a nonlinear ergodic theorem of Baillon’s type in Hilbert spaces for supper hybrid mappings, demonstrated weak convergence of the Cesàro means to a fixed point. Our approach leverages metric projections and techniques inspired by Takahashi, thereby extending classical fixed point theory to this new operator class. Keywords: Supper hybrid mapping; Nonlinear ergodic theorem; Quasi – nonexpansive mapping; Fixed point; Banach space; Demiclosedness principle; Cesàro mean; Weak convergence 1. Introduction Fixed point theory for nonlinear mappings in Banach and Hilbert spaces has seen extensive development, particularly through the study of nonexpensive, quasi – nonexpansive, hybrid mappings. Among these generalized hybrid mappings, introduced to interpolate between contractive and nonexpansive behaviours. Have proven instrumental in analyzing iterative algorithms and variational inequalities. In this paper, we propose a new class of mappings, termed (𝛼 ,𝛽 ,𝛾)- supper hybrid mappings, which extend the classical (𝛼 ,𝛽) - generalized hybrid mappings by incorporating an additional nonnegative parameters 𝛾. This extension allows for greater flexibility in modeling nonlinear phenomenon and unifies several operator classes under one single framework. Our primary contributions are threefold. First, we show that supper hybrid mappings with fixed points are quasi – nonexpansive, thereby inheriting a key stability property. Second, we establish the demiclosednes principle for (𝐼−𝑇) at zero under mild assumptions on the duality mapping, both for supper hybrid and generalized hybrid mappings. Third, we prove a nonlinear ergodic theorem of Baillon’s type for supper hybrid mappings in Hilbert spaces, demonstrating weak convergence of the Cesàro means to a fixed point. The techniques employed draw inspiration from Takahashi’s work on on ergodic theorems and fixed point approximations, and our results contribute to the ongoing effort to generalize and refine convergence principle in nonlinear analysis. The structure of the paper is as follows. In section 2, we present the definition of supper hybrid mappings and establish their basic properties. Section 3 contains the main results, including the demiclosedness principle and the ergodic theorem. We conclude with remarks on potential extensions and applications. World Journal of Advanced Research and Reviews, 2025, 27(02), 1564-1570 1565 2. Preliminaries Let 𝐸 be a real Banach space with dual space 𝐸∗ and let 〈 .,.〉 denote the duality pairing between 𝐸 and 𝐸∗. A subset 𝐶 ⊂ 𝐸 is said to be convex if for all 𝑥 ,𝑦 ∈ 𝐶 and ∈ [ 0 ,1 ] , the point 𝑡𝑥 + (1 − 𝑡)𝑦 ∈ 𝐶 . A mapping 𝐽 ∶ 𝐸 ⟶ 2𝐸∗ is called a duality mapping if 𝐽(𝑥):={ 𝑥∗∈ 𝐸∗∶ 〈 𝑥 ,𝑥∗ 〉 = ‖𝑥∗‖2 = ‖𝑥‖2 },∀ 𝑥 ∈ 𝐸 . We say that 𝐽 is weakly continuous if 𝑥𝑛 ⇀ 𝑥 in 𝐸 implies 𝐽(𝑥𝑛) ⟶ 𝐽(𝑥) in the weak topology of 𝐸∗ Let 𝐻 be a real Hilbert space. The metric projection 𝑃𝐶 : 𝐻 ⟶ 𝐶 onto a nonempty closed convex subset 𝐶 ⊂ 𝐸 is defined by 𝑃𝐶 𝑥 ∶ = argmin 𝑦 ∈ 𝐶‖𝑥−𝑦‖ ,∀ 𝑥 ∈ 𝐻 It is well known that 𝑃𝐶 is nonexpansive and satisfies the variational inequality 〈 𝑥 − 𝑃𝐶 𝑥 ,𝑦 − 𝑃𝐶 𝑥 〉 ≤0 , ∀ 𝑦 ∈ 𝐶 . A mapping 𝑇 ∶ 𝐶 ⟶ 𝐶 is called: - nonexpansive if ‖𝑇𝑥−𝑇𝑦‖ ≤ ‖𝑥−𝑦‖ for all 𝑥 ,𝑦 ∈ 𝐶 ; quasi nonexpansive if ‖𝑇𝑥−𝑝‖ ≤‖𝑥−𝑝‖ , for all 𝑥 ∈ 𝐶 and 𝑝 ∈𝐹 (𝑇) , where 𝐹 (𝑇)∶={ 𝑥 ∈ 𝐶 ∶ 𝑇𝑥 = 𝑥 }, denotes the set of fixed points of 𝑇. We recall the demiclosedness principle, which plays a central role in fixed point theory: 2.1. Lemma 2.1 (Demiclosedness Principle) Let 𝐸 be a Banach space with a weakly continuous duality mapping, and let 𝑇 ∶ 𝐶 ⟶ 𝐸 be a mapping. If 𝑇 is quasi – nonexpansive and 𝑥𝑛⇀ 𝑥 in 𝐸 with (𝐼−𝑇)𝑥𝑛 ⟶ 0 , then 𝑥 ∈ 𝐹(𝑇). We also recall the classical Cesàro means used in ergodic theory. For a mapping ∶ 𝐶 ⟶ 𝐶 , the sequence { 𝑆𝑛 𝑥 } defined by 𝑆𝑛 𝑥 ∶ = 1 𝑛 ∑𝑇𝑘 𝑛−1 𝑘=0 𝑥 is called the Cesàro mean of the iterates of 𝑇. In Hilbert space, such sequence often converge weakly to a fixed point under suitable conditions. Throughout this paper, we use the notation 𝑇𝑛𝑥 to denote the 𝑛− fold composition of 𝑇 applied to , and we assume that all mappings act on nonempty closed convex subsets of Banach or Hilbert spaces unless stated otherwise. 2.2. Lemma 2.2 ([2]) Assuming that E is a Banach space has a weakly continuous duality mapping with guage 𝜑. Then for any sequences {𝑥𝑛} that converges weakly to 𝑥 , we have for any 𝑦 ∈ 𝐸, lim sup 𝑛→∞ Φ (‖𝑥𝑛−𝑦‖) = lim sup 𝑛→∞ Φ (‖𝑥𝑛−𝑥‖) + lim sup 𝑛→∞ Φ (‖𝑥−𝑦‖) Definition 2.2 Let 𝐾 be a nonempty closed subset of a Banach space . A mapping 𝑇 ∶ 𝐾 ⟶ 𝐸 is called supper hybrid if there are 𝛼,𝛽,𝛾 ∈ ℝ with 𝛾 ≥0 such that for all 𝑥 ,𝑦 ∈𝐾, we have 𝛼‖𝑇𝑥−𝑇𝑦‖2 + (1−𝛼+𝛾)‖𝑥−𝑇𝑦‖2≤ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑥−𝑦‖2 World Journal of Advanced Research and Reviews, 2025, 27(02), 1564-1570 1566 + (1 − 𝛽− (𝛽− 𝛼− 1)𝛾) ‖𝑥−𝑦‖2+(𝛼−𝛽)𝛾)‖𝑥−𝑇𝑦‖2+ 𝛾 ‖𝑦−𝑇𝑦‖2, ….. (1) We call such a mapping an (𝛼,𝛽,𝛾) - supper hybrid mapping (see [3]) . Notice that an (𝛼,𝛽,0)- supper hybrid mapping is (𝛼,𝛽)- generalized hybrid mapping, that is 𝛼‖𝑇𝑥−𝑇𝑦‖2 + (1−𝛼)‖𝑥−𝑇𝑦‖2≤ 𝛽‖𝑇𝑥−𝑦‖2 + (1 − 𝛽) ‖𝑥−𝑦‖2 (2) So, the class of supper hybrid mappings contains the class of generalized hybrid mappings. 3. Main Results 3.1. Proposition 3.1 Let 𝐸 be a Banach space, let 𝐶 be a nonempty subset of 𝐸, then a supper hybrid mappings with a fixed point is quasi – nonexpansive. Proof: Since 𝑇 ∶ 𝐶 ⟶ 𝐶 is a supper hybrid mapping for 𝛼,𝛽,𝛾 ∈ ℝ with 𝛾 ≥0 and 𝑥 ,𝑦 ∈𝐶, as in (1). Let 𝑣 ∈ 𝐹 (𝑇), then we have that for any 𝑥 ∈𝐶, from (1), that 𝛼‖𝑇𝑥−𝑣‖2 ≤ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑥−𝑣‖2+ (1−𝛽−(𝛽−𝛼−1)𝛾)‖𝑥−𝑣‖2 +(𝛼−𝛽)𝛾)‖𝑥−𝑣‖2+ 𝛾 ‖𝑣−𝑣‖2−(1−𝛼+𝛾)‖𝑥−𝑣‖2 Which implies that [𝛼−𝛽−(𝛽−𝛼)𝛾]‖𝑇𝑥−𝑣‖2 ≤[𝛼− 𝛽+ (𝛼−𝛽)𝛾)‖𝑥−𝑣‖2 and hence ‖𝑇𝑥−𝑣‖2≤ ‖𝑥−𝑣‖2. This implies that 𝑇 is quasi–nonexpansive. 3.2. Proposition 3.2 Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸, with a weakly continuous duality mapping and let 𝑇 ∶ 𝐶 ⟶ 𝐸 be (𝛼,𝛽,𝛾) –supper hybrid mappings with 𝛼,𝛽,𝛾 ∈ ℝ with 𝛾 ≥0. Then with (𝐼−𝑇) is demiclosed at with 0. Proof: Let {𝑥𝑛}𝑛=1 ∞ C be a sequence in 𝐶 which converges weakly to 𝑝 and {𝑥𝑛− 𝑇𝑥𝑛}𝑛=1 ∞ Converges strongly to 0. We show that 𝑝 is a fixed point of 𝑇. Since {𝑥𝑛}𝑛=1 ∞ converges weakly, it is bounded. Clearly, {𝑇𝑥𝑛}𝑛=1 ∞ is also bounded sequence. Since 𝑇 ∶ 𝐶 ⟶ 𝐸 is supper–hybrid mapping, implies that from (1), since {𝑥𝑛}𝑛=1 ∞ converges weakly, it is bounded. For each 𝑥 ∈𝐸 Define by 𝑓 ∶ 𝐸 ⟶[0,∞) by 𝑓(𝑥)∶= lim sup 𝑛→∞ ‖𝑥𝑛−𝑥‖2 Then from Lemma 2.2, taking Φ(‖𝑥‖) = 1 2‖𝑥‖2, we obtain, 𝑓(𝑥) = lim sup 𝑛→∞ ‖𝑥𝑛−𝑝‖2+‖𝑝−𝑥‖2, ∀ 𝑥 ∈𝐸 Thus, 𝑓(𝑥) = 𝑓(𝑝) + ‖𝑝−𝑥‖2 , ∀ 𝑥 ∈𝐸 and 𝑓(𝑇𝑝) = 𝑓(𝑝) + ‖𝑝−𝑇𝑝‖2 ……… (3) World Journal of Advanced Research and Reviews, 2025, 27(02), 1564-1570 1567 Observe also that from (1) and (3) 𝛼𝑓(𝑇𝑝) = 𝛼 lim sup 𝑛→∞ ‖𝑥𝑛−𝑇𝑝‖2 = 𝛼 lim sup 𝑛→∞ ‖𝑥𝑛−𝑇𝑥𝑛+𝑇𝑥𝑛−𝑇𝑝‖2 = 𝛼limsup 𝑛→∞ ‖𝑇𝑥𝑛− 𝑇𝑝‖2 ≤ limsup 𝑛→∞ [ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑥𝑛−𝑝‖2+ (1 −𝛽− (𝛽−𝛼−1)) ‖𝑥𝑛−𝑝‖2 + (𝛼−𝛽)‖𝑥𝑛−𝑇𝑥𝑛‖2+ 𝛾 ‖𝑝−𝑇𝑝‖2− (1 − 𝛼+ 𝛾)‖𝑥𝑛−𝑇𝑝‖2] = (𝛽+(𝛽−𝛼)𝛾) 𝑓(𝑝)+(1−𝛽−(𝛽−𝛼−1) 𝛾 )𝑓(𝑝) + 𝛾 ‖𝑝−𝑇𝑝‖2− (1 − 𝛼+ 𝛾) 𝑓(𝑇𝑝) = 𝑓(𝑝)+ 𝛾 [ 𝑓(𝑝) + ‖𝑝−𝑇𝑝‖2] − (1 − 𝛼+ 𝛾) 𝑓(𝑇𝑝) = 𝑓(𝑝)+ 𝛾 𝑓(𝑝) − (1 − 𝛼+ 𝛾) 𝑓(𝑇𝑝) = 𝑓(𝑝) − (1−𝛼) 𝑓(𝑇𝑝) Therefore, 𝑓(𝑇𝑝) ≤ 𝑓(𝑝) …………. (4) Hence it follows from (3) and (4) that ‖𝑝−𝑇𝑝‖ = 0. 3.3. Proposition 3.3 Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸 with a weakly continuous duality mapping, and let 𝑇 ∶ 𝐶 ⟶ 𝐶 be (𝛼,𝛽)–generalized hybrid mappings with 𝛼,𝛽 ∈ ℝ. Then (𝐼−𝑇) is demiclosed at 0. Proof: From Lemma 3.2 if then we obtain the desired result. We now prove the following Nonlinear ergodic theorem of Baillon’s type [1] by using the technique developed by Takahashi [4]. Theorem 3.1 Let 𝐻 be a Hilbert space and let 𝐶 be closed convex subset of 𝐻, let 𝑇∶ 𝐶 ⟶ 𝐶 be a supper hybrid mapping, with 𝐹(𝑇) ≠ 0 and let 𝐶 be a metric projection of 𝐻 onto 𝐹(𝑇). Then for 𝑥 ∈𝐶, 𝑆𝑛 𝑥 ∶ = 1 𝑛 ∑𝑇𝑘 𝑛−1 𝑘=0 𝑥………(5) converges weakly to an element 𝑝 of 𝐹(𝑇), where 𝑝= 𝑙𝑖𝑚𝑃𝑇𝑛𝑥 𝑛→∞ . Proof: Let 𝑇 ∶ 𝐶 ⟶ 𝐶 be (𝛼,𝛽,𝛾) supper-hybrid with with 𝛾 ≥0, then from Proposition 3.1 𝑇 is quasi-nonexpansive, we have that 𝐹(𝑇) is closed and convex. Let 𝑥 ∈𝐶 and let 𝑃 be the metric projection of 𝐻 onto 𝐹(𝑇). Then, we have ‖𝑃𝑇𝑛𝑥−𝑇𝑛𝑥‖ ≤ ‖𝑃𝑇𝑛−1𝑥−𝑇𝑛𝑥‖ …….. (6) ≤ ‖𝑃𝑇𝑛−1𝑥−𝑇𝑛−1𝑥‖ ……….. (7) This implies that {‖𝑃𝑇𝑛𝑥−𝑇𝑛𝑥‖} non increasing. We also know that for any 𝑣 ∈𝐶 and 𝑢 ∈𝐹(𝑇) 〈 𝑣 − 𝑃𝑣 ,𝑃𝑣 − 𝑢 〉 ≥ 0 and hence ‖𝑣−𝑃𝑣‖2 ≤ 〈 𝑣 − 𝑃𝑣 ,𝑃𝑣 − 𝑢 〉. World Journal of Advanced Research and Reviews, 2025, 27(02), 1564-1570 1568 So, we get ‖𝑃𝑣−𝑢‖2= ‖𝑃𝑣−𝑣+𝑣−𝑢‖2 = ‖𝑃𝑣−𝑣‖2−2〈 𝑃𝑣 − 𝑣 ,𝑢 − 𝑣 〉 + ‖𝑣−𝑢‖2 = ‖𝑣−𝑢‖2−‖𝑃𝑣−𝑣‖2 Let 𝑚 ,𝑛 ∈ℕ with 𝑚 ≥ 𝑛. Putting 𝑣 = 𝑇𝑚𝑥 and 𝑢 = 𝑇𝑛𝑥, we have ‖𝑃𝑇𝑚𝑥−𝑃𝑇𝑛𝑥‖2 ≤ ‖𝑇𝑚𝑥−𝑃𝑇𝑛𝑥‖2− ‖𝑃𝑇𝑚𝑥−𝑇𝑚𝑥‖2 ≤ ‖𝑇𝑛𝑥−𝑃𝑇𝑛𝑥‖2− ‖𝑃𝑇𝑚𝑥−𝑇𝑚𝑥‖2 So, {𝑃𝑇𝑛𝑥} is a Cauchy sequence. Since 𝐹(𝑇), is closed, {𝑃𝑇𝑛𝑥} converges strongly to an element 𝑝 of 𝐹(𝑇). Then we obtain, for any 𝑛 ∈ℕ, ‖𝑆𝑛 𝑥 − 𝑢‖ ≤ 1 𝑛 ∑‖ 𝑇𝑘𝑥−𝑢‖ 𝑛−1 𝑘=0 ≤ ‖𝑥−𝑢‖ So, {𝑆𝑛𝑥} is bounded and hence there exists a weakly convergent subsequence {𝑆𝑛𝑖𝑥} of 𝑆𝑛𝑥}. If 𝑆𝑛𝑖𝑥 ⇀𝑣, then we have 𝑣 ∈𝐹(𝑇). In fact, for any 𝑦 ∈𝐶 and 𝑘 ∈ ℕ ⋃ {0}, we have 0 ≤ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑘+1𝑥−𝑦‖2+ 1 − 𝛽− (𝛽−𝛼−1)𝛾)‖𝑇𝑘𝑥−𝑦‖2 + (𝛼−𝛽)𝛾‖𝑇𝑘𝑥−𝑇𝑦‖2+ 𝛾 ‖𝑦−𝑇𝑦‖2 −𝛼‖𝑇𝑘+1𝑥−𝑇𝑦‖2 − (1 − 𝛼+ 𝛾)‖𝑇𝑘𝑥−𝑇𝑦‖2] = (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑘+1𝑥−𝑦‖2+ (1 − 𝛽−(𝛽−𝛼−1)𝛾)‖𝑇𝑘𝑥−𝑦‖2 + (𝛼−𝛽)𝛾‖𝑇𝑘𝑥−𝑇𝑦‖2+ 𝛾 ‖𝑦−𝑇𝑦‖2−𝛼[‖𝑇𝑘+1𝑥−𝑦‖2 + ‖𝑦−𝑇𝑦‖2+2 〈 𝑇𝑘+1𝑥− 𝑦 ,𝑦− 𝑇𝑦 〉] − (1−𝛼+𝛾)[‖𝑇𝑘𝑥−𝑦‖2+‖𝑦−𝑇𝑦‖2+ 2 〈 𝑇𝑘𝑥− 𝑦 ,𝑦− 𝑇𝑦 〉] = (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑘+1𝑥−𝑦‖2+ (1 − 𝛽−(𝛽−𝛼−1)𝛾)‖𝑇𝑘𝑥−𝑦‖2 + (𝛼−𝛽)𝛾‖𝑇𝑘𝑥−𝑇𝑦‖2−‖𝑦−𝑇𝑦‖2 − 𝛼[‖𝑇𝑘+1𝑥−𝑦‖2+ 2 〈 𝑇𝑘+1𝑥− 𝑦 ,𝑦 − 𝑇𝑦 〉]−(1−𝛼+𝛾)[‖𝑇𝑘𝑥−𝑦‖2 + 2 〈 𝑇𝑘𝑥− 𝑦 ,𝑦− 𝑇𝑦 〉] ……… (8) Summing up the inequality (8) with respect to 𝑘 =0,1,2,3,...,𝑛−1, we get 0 ≤ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑛𝑥−𝑦‖2+ (1 − 𝛽− (𝛽−𝛼−1)𝛾)‖𝑥−𝑦‖2 + (𝛼−𝛽)𝛾‖𝑥−𝑇𝑦‖2− 𝑛 ‖𝑦−𝑇𝑦‖2−𝛼[‖𝑇𝑛𝑥−𝑦‖2 + 2 〈 (𝑛+1)𝑆(𝑛+1)𝑥−𝑥−𝑛𝑦 ,𝑦− 𝑇𝑦 〉] − (1−𝛼+𝛾)[‖𝑥−𝑦‖2+ 2 〈 𝑥−𝑦 ,𝑦− 𝑇𝑦 〉 ] ………… (9) Dividing the inequality (9) by 𝑛, we have World Journal of Advanced Research and Reviews, 2025, 27(02), 1564-1570 1569 0 ≤ (𝛽+(𝛽−𝛼)𝛾) 𝑛‖𝑇𝑛𝑥−𝑦‖2+ (1−𝛽−(𝛽−𝛼−1)𝛾) 𝑛‖𝑥−𝑦‖2 + (𝛼−𝛽)𝛾 𝑛‖𝑥−𝑇𝑦‖2− ‖𝑦−𝑇𝑦‖2−𝛼[1 𝑛‖𝑇𝑛𝑥−𝑦‖2 ……… (10) + 2 〈 (𝑛+1) 𝑛𝑆(𝑛+1)𝑥−𝑥 𝑛− 𝑦 ,𝑦− 𝑇𝑦 〉] − (1−𝛼+𝛾)[1 𝑛‖𝑥−𝑦‖2+ 2 𝑛 〈 𝑥− 𝑦 ,𝑦− 𝑇𝑦 〉 ] , ………… (11) where ∑𝑇𝑘+1 𝑛 𝑘=0 𝑥 =(𝑛+1)𝑆(𝑛+1)𝑥−𝑥 from (5). Replacing 𝑛 by 𝑛𝑖 and letting by 𝑛𝑖→∞, we obtain from 𝑆(𝑛𝑖+1)𝑥 ⇀ 𝑣 that 0 ≤ − ‖𝑦−𝑇𝑦‖2 …………. (12) Putting 𝑦= 𝑣 in (12) we get 0 ≤ − ‖𝑣−𝑇𝑣‖2, that is ‖𝑣−𝑇𝑣‖2 ≤0 Hence, 𝑇𝑣 =𝑣. To complete the proof, it is sufficient to show that if 𝑆(𝑛𝑖+1)𝑥 ⇀ 𝑣 then 𝑣 =𝑝. We have that 〈 𝑇𝑘𝑥−𝑃𝑇𝑘𝑥 ,𝑃𝑇𝑘𝑥− 𝑢 〉 ≥0 for all 𝑢 ∈𝐹(𝑇). Since {‖𝑇𝑘𝑥−𝑃𝑇𝑘𝑥‖} is nonincreasing, we have 〈 𝑢− 𝑝 ,𝑇𝑘𝑥−𝑃𝑇𝑘𝑥 〉 ≤ 〈 𝑃𝑇𝑘𝑥−𝑝 ,𝑇𝑘𝑥−𝑃𝑇𝑘𝑥 〉 ≤ ‖𝑃𝑇𝑘𝑥−𝑝‖ .‖𝑇𝑘𝑥−𝑃𝑇𝑘𝑥‖ ≤ ‖𝑃𝑇𝑘𝑥−𝑝‖ .‖𝑥−𝑃𝑥‖ Adding these inequalities from 𝑘 =0 to 𝑘 =𝑛−1 and dividing by 𝑛, we have 〈 𝑢− 𝑝 ,𝑆𝑛 𝑥− 1 𝑛 ∑ 𝑃𝑇𝑘𝑥 𝑛−1 𝑘=0 〉 ≤‖𝑥−𝑃𝑥‖ 𝑛∑‖𝑃𝑇𝑘𝑥−𝑝‖. 𝑛−1 𝑘=0 Since, 𝑆(𝑛𝑖+1)𝑥 ⇀ 𝑣 and 𝑃𝑇𝑘𝑥 →𝑝, we have 〈 𝑢− 𝑝 ,𝑣− 𝑝 〉≤ 0. We know 𝑣 ∈ 𝐹(𝑇). So, putting 𝑢 =𝑣, we have 〈 𝑣− 𝑝 ,𝑣− 𝑝 〉≤0 and hence ‖𝑣−𝑝‖2≤0. So, we obtain 𝑣 =𝑝. This completes the proof. Corollary 3.2 Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇 ∶ 𝐶 ⟶ 𝐶 be a supper hybrid mapping, with nonempty fixed point set 𝐹(𝑇). Then, for any 𝑥 ∈𝐶, the Cesaro means 𝑆𝑛 𝑥 ∶ = 1 𝑛 ∑𝑇𝑘 𝑛−1 𝑘=0 𝑥 converges weakly to a point 𝑝∈𝐹(𝑇). World Journal of Advanced Research and Reviews, 2025, 27(02), 1564-1570 1570 Proof: This follows directly from the nonlinear ergodic theorem established in Theorem 3.1, together with the demiclosedness principle and the weak compactness of closed convex subsets in Hilbert spaces. Applications Variational Inequalities: The convergence of Cesaro means for supper hybrid mappings can be used to approximate solutions of variational inequality problems of the form: find 𝑥∗∈𝐶 such that 〈 𝐴𝑥∗ ,𝑦 − 𝑥∗ 〉≥0, ∀ 𝑦 ∈𝐶, where 𝐴 ∶ 𝐶 ⟶ 𝐻 is a monotone operator. By constructing suitable supper hybrid mappings associated with the resolvent of 𝐴, one can apply the ergodic theorem to obtain weak convergence to a solution. is Convex Feasibility Problems: In the context of finding a point in the intersection of convex sets 𝐶1 ,𝐶2 , 𝐶3 ,..., 𝐶𝑚 ⊂ 𝐻 supper hybrid mapping can be designe to encode projection – based iterative schemes. The ergodic convergence of Cesàro means then provides a mechanism for approximating feasible points when direct projection is computationally expensive or infeasible. 4. Conclusion In this paper, we have introduced and analyzed a nonlinear ergodic theorem for a new class of mappings termed supper hybrid mappings in Banach and Hilbert spaces. By employing the demiclosedness principle and properties of quasinonexpansive mappings, we established the weak convergence of Cesàro means to fixed points under mild assumptions. A key corollary demonstrates that such convergence holds for any initial point in the domain, thereby extending classical ergodic results to a broader class of nonlinear operators. Beyond its theoretical significance, the main result admits applications to variational inequality problems and convex feasibility formulations, where supper hybrid mappings can be used to constructive iterative schemes with guaranteed convergence. These findings offer a unified framework for analyzing nonlinear iterative process in infinite-dimensional settings. Further research may focus on quantitative convergence rates, stability under perturbations, and extensions to more general classes of mappings. Applications to monotone inclusion problems and operator splitting methods also present promising directions. References [1] J.-B. Baillon, Un theorem de type ergodique pour les contractions non lineaire dans un espace de Hilbert, C. R. Acad. Sci. Paris Ser. A-B, 280 (1975), 1511-1514. [2] T. C. Lim, H. K. Xu, Fixed Point theorems for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1994) 1345-1355. [3] P. Kocourek, W. Takahashi, and J.-C. Yao, Fixed Point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese Journal of Mathematics, vol. 14, no. 6, 2497-2511, 2010. [4] W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc., 81 (1981), 253-256.