@ Appl. Gen. Topol. 23, no. 2 (2022), 377-390 doi:10.4995/agt.2022.17359 © AGT, UPV, 2022 Fixed point theorems for a new class of nonexpansive mappings Rajendra Pant a and Rahul Shukla b a Department of Mathematics & Applied Mathematics, University of Johannesburg Kingsway Campus, Auckland Park 2006, South Africa (pant.rajendra@gmail.com, rpant@uj.ac.za) b Department of Mathematical Sciences and Computing, Walter Sisulu University, Mthatha 5117, South Africa (rshukla.vnit@gmail.com, rshukla@wsu.ac.za) Communicated by S. Romaguera Abstract We consider a new class of nonlinear mappings that generalizes two well-known classes of nonexpansive type mappings and extends some other classes of mappings. We present some existence and convergence results for this class of mappings. Some illustrative examples presented herein show the generality of the obtained results. 2020 MSC: 47H10; 54H25. Keywords: α-nonexpansive; Opial property; condition (C). 1. Introduction Let K be a nonempty subset of X of a Banach space (X ,‖.‖). A self-mapping Ψ : K→K is 1-Lipschitz or nonexpansive if ‖Ψ(σ) − Ψ(υ)‖≤‖σ −υ‖ for all σ,υ ∈ K. A fixed point σ of the mapping Ψ is the point at which the mapping is invariant, that is, Ψ(σ) = σ. In 1965, Browder [6, 7], Göhde [9] and Kirk [10] initiated the existence theory for fixed points of nonexpansive mapping, independently (cf. [8]). In general nonexpansive mapping are uni- formly continuous on their domains. To generalize, extend and accommodate discontinuous nonexpansive type mappings, many authors considered various Received 16 March 2022 – Accepted 12 May 2022 http://dx.doi.org/10.4995/agt.2022.17359 R. Pant and R. Shukla classes of mappings [18, 11, 19, 2, 3, 14, 1, 8] for more details, see [15]. In 2008, Suzuki [18] considered a more general class of nonexpansive mappings (also known as Suzuki type generalized nonexpansive mapping) and presented some interesting results for these mappings: Definition 1.1 ([18]). Assume that K is a nonempty subset of a Banach space X . A mapping Ψ : K→K is said to satisfy condition (C) if 1 2 ‖σ − Ψ(σ)‖≤‖σ −υ‖ implies ‖Ψ(σ) − Ψ(υ)‖≤‖σ −υ‖ for all σ,υ ∈K. In 2011, Aoyama and Kohsaka [3] introduced another class of nonexpansive type mappings (called as α-nonexpansive mappings). This class of mappings generalizes several classes of mappings including λ-hybrid and nonspreading mappings. For more details one may refer to [11, 19, 2]. Definition 1.2. Let K be a nonempty subset of a Banach space X and Ψ : K →K a self-mapping. Then Ψ is an α-nonexpansive if there exists an α < 1 such that ‖Ψ(σ) − Ψ(υ)‖2 ≤ α‖Ψ(σ) −υ‖2 + α‖Ψ(υ) −σ‖2 +(1 − 2α)‖σ −υ‖2(1.1) for all σ,υ ∈K. Remark 1.3. Even though, the class of α-nonexpansive mappings was consid- ered in [3] for any real number α < 1, but Ariza-Ruiz et al. [4] pointed out that for α < 0, this concept is trivial (see also [17]). We note that α-nonexpansive and mappings satisfying the condition (C) are independent, and need not be continuous on their domains of definitions, unlike nonexpansive mappings. A couple of examples below illustrate these facts. Example 1.4. Let K = [0, 5] ⊂ R with the usual norm on R. Assume that Ψ : K→K is a self-mapping defined as: Ψ(σ) =   1 −σ, if σ ∈ [0, 1] 0, if σ ∈ (1, 5) 1, if σ = 5. If σ < υ and (σ,υ) ∈ ([0, 5] × [0, 5])\((4, 5) ×{5}), then it can be easily seen that ‖Ψ(σ) − Ψ(υ)‖≤‖σ −υ‖ holds. If σ ∈ (4, 5) and υ = 5, then 1 2 ‖σ − Ψ(σ)‖ = σ 2 > 1 > ‖σ −υ‖ and 1 2 ‖υ − Ψ(υ)‖ = 2 > ‖σ −υ‖. Hence Ψ satisfies condition (C). Contrarily, at σ = 0 and υ = 1 α‖Ψ(σ) −υ‖2 + α‖Ψ(υ) −σ‖2 + (1 − 2α)‖σ −υ‖2 = 1 − 2α ≤ 1 = ‖Ψ(σ) − Ψ(υ)‖2 © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 378 Fixed point theorems for a new class of nonexpansive mappings in Banach spaces holds only if α = 0. But for α = 0, and σ = 9 2 , υ = 5, we get ‖Ψ(σ) − Ψ(υ)‖ = 1 > 1 2 = ‖σ −υ‖. Therefore, Ψ is not an α-nonexpansive mapping for any α ∈ [0, 1). Example 1.5. [14]. Let K = [0, 4] ⊂ R endowed with the usual norm. Define Ψ : K→K as follows: Ψ(σ) = { 0, if σ 6= 4, 2, if σ = 4. Then it can be easily verified that Ψ is α-nonexpansive mapping for α ≥ 1 2 . However Ψ is not a mpping satisfying the condition (C) for σ ∈ (2, 3] and υ = 4. In [14], we introduced the following class of mappings: Definition 1.6. Suppose K is a nonempty subset of a Banach space X , and Ψ : K → K a self-mapping. Then Ψ is called a generalized α-nonexpansive mapping if there exists an α ∈ [0, 1) such that 1 2 ‖σ − Ψ(σ)‖ ≤ ‖σ −υ‖ implies ‖Ψ(σ) − Ψ(υ)‖ ≤ α‖Ψ(σ) −υ‖ + α‖Ψ(υ) −σ‖ + (1 − 2α)‖σ −υ‖(1.2) for all σ,υ ∈K. The implication in inequality (1.2) is more restrictive than in (1.1), and therefore the above mapping does not contain α-nonexpansive mapping, prop- erly. The present paper deals with this problem. Indeed, we consider a class of mappings which properly contains the class of α-nonexpansive mappings. To show the generality of the class of mappings considered herein, we present some illustrative examples. We also obtain the Demi-closedness principle in Banach spaces. Further, we employ a three step iterative method to approximate the fixed point of mapping considered herein. 2. Preliminaries Now onwards, R denotes the set of real numbers and N the set of natural numbers. Definition 2.1. Assume that K is a nonempty subset of a Banach space X . A self-mapping Ψ : K→K is a quasinonexpansive mapping if ‖Ψ(σ) −w†‖≤‖σ −w†‖ for all σ ∈K and w† ∈ F(Ψ). A Banach space X is said to be uniformly convex, for each ε ∈ (0, 2], there exists δ > 0 such that the following holds: for each σ,υ ∈X ‖σ‖≤ 1 ‖υ‖≤ 1 ‖σ −υ‖≥ ε   ⇒ ∥∥∥∥σ + υ2 ∥∥∥∥ ≤ 1 − δ. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 379 R. Pant and R. Shukla Definition 2.2 ([16]). Let X be a normed space and K nonempty subset of X . A mapping Ψ : K→K is said to satisfy Condition (I) if there exists a function f : [0,∞) → [0,∞) with the following properties: • f is nondecreasing; • f(r) > 0 for all r ∈ (0,∞) and f(0) = 0; • ‖σ − Ψ(σ)‖≥ f(d(σ,F(Ψ))) for all σ ∈K, where d(x,F(Ψ)) denotes distance of x from F(Ψ). A Banach X satisfies the Opial conditions [13] if for each weakly convergent sequence {σn}⊂X having weak limit σ, we have lim inf n→∞ ‖σn −σ‖ < lim inf n→∞ ‖σn −υ‖ for all υ ∈ X , σ 6= υ. It can be easily seen that on passing through appropri- ate subsequences, the lower limit can be replaced with upper limits in Opial property. The sequence {σn} is an approximate fixed point sequence for Ψ (in short, a.f.p.s.) if lim n→∞ ‖σn − Ψ(σn)‖ = 0. 3. C-α nonexpansive mapping We introduce the following notion of C-α nonexpansive mapping Definition 3.1. Suppose K is a nonempty subset of a Banach space X and Ψ : K→K a self-mapping. We say Ψ is a C-α nonexpansive mapping if 1 2 ‖σ − Ψ(σ)‖ ≤ ‖σ −υ‖ implies ‖Ψ(σ) − Ψ(υ)‖2 ≤ α‖Ψ(σ) −υ‖2 + α‖Ψ(υ) −σ‖2 + (1 − 2α)‖σ −υ‖2(3.1) for all σ,υ ∈K, where α ∈ [0, 1). We discuss some fundamental properties of C-α nonexpansive mapping. Proposition 3.2. Let Ψ : K → K be a mapping satisfying the condition (C). Then Ψ is a C-α nonexpansive mapping. In the next example we show that the reverse implication is not true, in general. Example 3.3. Let (`2,‖.‖2) be the Banach space of square-summable se- quences endowed with its standard norm. Assume that {en} is the canonical basis of `2. Define K := conv{e1,e2} = {µe1 + (1 −µ)e2 : µ ∈ [0, 1]}, where conv{e1,e2} denotes the convex closure of {e1,e2}. Now, define Ψ : K→ K as follows: Ψ(µe1 + (1 −µ)e2) =  e1, if µ = 0,(e1 + e2) 2 , otherwise. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 380 Fixed point theorems for a new class of nonexpansive mappings in Banach spaces Then Ψ is C- 1 3 nonexpansive mapping. Indeed, if σ := e2 and υ := µe1 + (1 − µ)e2, µ ∈ (0, 1], we have ‖Ψ(σ) − Ψ(υ)‖2 = ∥∥∥∥e1 − ( e1 + e2 2 )∥∥∥∥ 2 = ∥∥∥∥e1 −e22 ∥∥∥∥ 2 = √ 2 2 , ‖Ψ(υ) −σ‖2 = ∥∥∥∥e1 + e22 −e2 ∥∥∥∥ 2 = ∥∥∥∥e1 −e22 ∥∥∥∥ 2 = √ 2 2 , ‖Ψ(σ) −υ‖2 = ‖e1 − (µe1 + (1 −µ)e2)‖2 = (1 −µ) √ 2 ‖σ −υ‖2 = ‖e2 − (µe1 + (1 −µ)e2)‖2 = µ √ 2. Therefore, for α = 1 3 1 3 ‖Ψ(σ) −υ‖2 + 1 3 ‖Ψ(υ) −σ‖2 + ( 1 − 2 3 ) ‖σ −υ‖2 = 1 3 (1 −µ) √ 2 + 1 3 √ 2 2 + 1 3 µ √ 2 = √ 2 2 = ‖Ψ(σ) − Ψ(υ)‖2. By the convexity of function t 7→ t2, we obtain (‖Ψ(σ)−Ψ(υ)‖2)2 ≤ 1 3 (‖Ψ(σ)−υ‖2)2 + 1 3 (‖Ψ(υ)−σ‖2)2 + ( 1 − 2 3 ) (‖σ−υ‖2)2. Contrarily, if σ := e2 and υ := 1 3 e1 + 2 3 e2, then 1 2 ‖υ − Ψ(υ)‖2 = 1 2 ∥∥∥∥13e1 + 23e2 − (e1 + e2)2 ∥∥∥∥ 2 = 1 12 ‖e1 −e2‖2 = 1 12 √ 2 ≤ 1 3 √ 2 = ‖σ −υ‖2 and ‖Ψ(σ) − Ψ(υ)‖2 = √ 2 2 > 1 3 √ 2 = ‖σ − υ‖2. Therefore, Ψ does not satisfy the criterion of condition (C). Note that (e1 + e2) 2 is a fixed point of Ψ. A generalized α-nonexpansive mapping is C-α nonexpansive mapping but the reverse implication is not true (see Example 3.5 below). Proposition 3.4. Assume that K is a nonempty subset of a Banach space X and Ψ : K →K a generalized α-nonexpansive mapping for all α ∈ [0, 1 2 ]. Then Ψ is C-α nonexpansive mapping for α ∈ [0, 1 2 ]. Proof. Let σ,υ ∈K and α ∈ [0, 1 2 ]. Note that 1−2α ≥ 0. Since Ψ a generalized α-nonexpansive mapping, by implication in (1.2), we have ‖Ψ(σ) − Ψ(υ)‖≤ α‖Ψ(σ) −υ‖ + α‖Ψ(υ) −σ‖ + (1 − 2α)‖σ −υ‖. Considering the convexity of function t 7→ t2, we conclude that ‖Ψ(σ) − Ψ(υ)‖2 ≤‖Ψ(σ) −υ‖2 + ‖Ψ(υ) −σ‖2 + (1 − 2α)‖σ −υ‖2. That is, Ψ is C-α nonexpansive mapping for all α ∈ [0, 1 2 ]. � © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 381 R. Pant and R. Shukla Example 3.5. Let K = [0, 3] ⊂ R endowed with the usual norm in R. Define a mapping Ψ : K→K as follows: Ψ(σ) = { σ 2 , if σ 6= 3, 5 2 , otherwise. Then Ψ is C-α nonexpansive mapping for α = 3 4 . Indeed, if σ,υ 6= 3, then 3 4 |Ψ(σ) −υ|2 + 3 4 |Ψ(υ) −σ|2 + ( 1 − 2 × 3 4 ) |σ −υ|2 = 3 4 ∣∣∣σ 2 −υ ∣∣∣2 + 3 4 ∣∣∣υ 2 −σ ∣∣∣2 − 1 2 |σ −υ|2 = 7 16 σ2 + 7 16 υ2 − 1 2 συ = ( 1 4 σ2 + 1 4 υ2 − 1 2 συ ) + 3 16 σ2 + 3 16 υ2 ≥ 1 4 σ2 + 1 4 υ2 − 1 2 συ = ∣∣∣σ 2 − υ 2 ∣∣∣2 = |Ψ(σ) − Ψ(υ)|2. Again if σ = 3 and υ 6= 3, then 3 4 |Ψ(σ) −υ|2 + 3 4 |Ψ(υ) −σ|2 + ( 1 − 2 × 3 4 ) |σ −υ|2 = 3 4 ∣∣∣∣52 −υ ∣∣∣∣2 + 34 ∣∣∣υ 2 − 3 ∣∣∣2 − 1 2 |3 −υ|2 = 7 16 υ2 − 12 4 υ + 111 16 = ( 1 4 υ2 − 10 4 υ + 100 16 ) + 3 16 υ2 − 1 2 υ + 11 16 . Since 3 16 υ2 − 1 2 υ + 11 16 ≥ 0 for all υ ∈ [0, 3], we have 3 4 |σ − Ψ(υ)|2 + 3 4 |Ψ(σ) −υ|2 + ( 1 − 2 × 3 4 ) |σ −υ|2 ≥ 1 4 υ2 − 10 4 υ + 100 16 = ∣∣∣∣52 − υ2 ∣∣∣∣2 = |Ψ(σ) − Ψ(υ)|2. Contrarily at σ = 3 and υ = 2, we get 1 2 |σ − Ψ(σ) = 1 2 ∣∣∣∣3 − 52 ∣∣∣∣ = 14 ≤ 1 = |3 − 2| = |σ −υ| © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 382 Fixed point theorems for a new class of nonexpansive mappings in Banach spaces and α|Ψ(3) − 2| + α|Ψ(2) − 3| + (1 − 2α)|3 − 2| = = α ∣∣∣∣52 − 2 ∣∣∣∣ + α|1 − 3| + 1 − 2α = 1 2 α + 2α + 1 − 2α = 1 + 1 2 α < 3 2 = ∣∣∣∣52 − 22 ∣∣∣∣ = |Ψ(σ) − Ψ(υ)|. Hence Ψ is not a generalized α-nonexpansive mapping for any value of α ∈ [0, 1). Proposition 3.6. Every α-nonexpansive is C-α nonexpansive mapping, but the converse is not true. Example 3.7. Let (`∞,‖.‖∞) be the Banach space of all bounded real se- quences endowed with the supremum norm. Assume that {en} is the canonical basis of `∞. Define K := {µe1 : µ ∈ [0, 1]} Define Ψ : K→K as follows: Ψ(µe1) = { 0, if µ 6= 1, e1 3 , if µ = 1. Then Ψ is a C- 1 10 nonexpansive mapping. Indeed, if σ = µ1e1, υ = µ2e1, where µ1,µ2 ∈ [0, 1) then ‖Ψ(σ)−Ψ(υ)‖2∞ = 0 ≤ 1 10 ‖Ψ(σ)−υ‖2∞+ 1 10 ‖Ψ(υ)−σ‖2∞+ ( 1 − 2 × 1 10 ) ‖σ−υ‖2∞. Again if, σ = µ1e1, where µ1 ∈ [0, 23 ] and υ = e1, then 1 10 ‖Ψ(σ) −υ‖2∞ + 1 10 ‖Ψ(υ) −σ‖2∞ + ( 1 − 2 × 1 10 ) ‖σ −υ‖2∞ = 1 10 ‖e1‖2∞ + 1 10 ∥∥∥e1 3 −σ ∥∥∥2 ∞ + 4 5 ‖e1 −σ‖2∞ ≥ 1 10 + 4 5 × 1 9 = 17 90 > 1 9 = ‖Ψ(σ) − Ψ(υ)‖2∞. If σ = µ1e1, where µ1 ∈ ( 2 3 , 1 ) and υ = e1, then 1 2 ‖σ−Ψ(σ)‖∞ = 1 2 ‖σ‖∞ > ‖e1−σ‖∞ and 1 2 ‖υ−Ψ(υ)‖∞ = 1 2 ∥∥∥e1 − e1 3 ∥∥∥ ∞ = 1 3 > ‖e1−σ‖∞. On the other hand, at σ = 9 10 e1 and υ = e1, 1 10 ∥∥∥∥Ψ ( 9 10 e1 ) −e1 ∥∥∥∥2 ∞ + 1 10 ∥∥∥∥Ψ(e1) − 910e1 ∥∥∥∥2 ∞ + ( 1 − 2 × 1 10 )∥∥∥∥ 910e1 −e1 ∥∥∥∥2 ∞ = 1 10 + 289 9000 + 8 1000 = 397 9000 < 1 9 = ∥∥∥∥Ψ ( 9 10 e1 ) − Ψ(e1) ∥∥∥∥2 ∞ . © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 383 R. Pant and R. Shukla Thus Ψ is not 1 10 -nonexpansive mapping. Proposition 3.8. Suppose that K is a nonempty subset a Banach space X and Ψ : K → K a C-α nonexpansive mapping and has a fixed point. Then Ψ is quasinonexpansive. Proof. It follows from the proof of [14, Proposition 3.5]. � Lemma 3.9. Suppose that K is a nonempty subset a Banach space X and Ψ : K → K a C-α nonexpansive mapping. Then F(Ψ) is closed. In addition, if K is convex and X is strictly convex, then F(Ψ) is convex. Proof. The proof is much similar to proof [14, Lemma 3.6] � 4. Main results Proposition 4.1 (Demiclosedness principle). Assume that K is a nonempty subset of a Banach space X which has the Opial property and Ψ : K → K be a C-α nonexpansive mapping. If {σn} converges weakly to a point σ and lim n→∞ ‖Ψ(σn) − σn‖ = 0 then Ψ(σ) = σ. That is, I − Ψ is demiclosed at zero, where I is the identity mapping on X . Proof. Since the sequence {σn} is weakly convergent and lim n→∞ ‖Ψ(σn)−σn‖ = 0, both sequences {σn} and {Ψ(σn)} are bounded. First we assume that lim sup n→∞ ‖σn −σ‖ = 0. Now by the triangle inequality, we get ‖σ − Ψ(σ)‖ ≤ lim sup n→∞ ‖σn −σ‖ + lim sup n→∞ ‖σn − Ψ(σ)‖ = lim sup n→∞ ‖σn − Ψ(σ)‖. Indeed, by Opial property ‖σ − Ψ(σ)‖≤ lim sup n→∞ ‖σn − Ψ(σ)‖ < lim sup n→∞ ‖σn −σ‖ = 0. Thus Ψ(σ) = σ. If we assume that lim sup n→∞ ‖σn −σ‖ = r > 0. Since lim n→∞ ‖Ψ(σn) −σn‖ = 0, for large enough n, there exists a n0 ∈ N such that 1 2 ‖σn − Ψ(σn)‖≤‖σn −σ‖ for all n ≥ n0. By (4.5), we have (4.1) ‖Ψ(σn)−Ψ(σ)‖2 ≤ α‖Ψ(σn)−σ‖2 +α‖Ψ(σ)−σn‖2 + (1−2α)‖σn−σ‖2. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 384 Fixed point theorems for a new class of nonexpansive mappings in Banach spaces Now by the triangle inequality and (4.1), we have ‖σn − Ψ(σ)‖2 ≤ (‖σn − Ψ(σn)‖ + ‖Ψ(σn) − Ψ(σ)‖)2 ≤ ‖σn − Ψ(σn)‖2 + ‖Ψ(σn) − Ψ(σ)‖2 + 2‖σn − Ψ(σn)‖‖Ψ(σn) − Ψ(σ)‖ ≤ ‖σn − Ψ(σn)‖2 + α‖Ψ(σn) −σ‖2 + α‖Ψ(σ) −σn‖2 +(1 − 2α)‖σn −σ‖2 + 2‖σn − Ψ(σn)‖‖Ψ(σn) − Ψ(σ)‖ ≤ ‖σn − Ψ(σn)‖2 + α(‖Ψ(σn) −σn‖ + ‖σn −σ‖)2 + α‖Ψ(σ) −σn‖2 +(1 − 2α)‖σn −σ‖2 + 2‖σn − Ψ(σn)‖‖Ψ(σn) − Ψ(σ)‖ ≤ ‖σn − Ψ(σn)‖2 + α‖Ψ(σn) −σn‖2 + α‖σn −σ‖2 +2α‖Ψ(σn) −σn‖‖σn −σ‖ + α‖Ψ(σ) −σn‖2 +(1 − 2α)‖σn −σ‖2 + 2‖σn − Ψ(σn)‖‖Ψ(σn) − Ψ(σ)‖. This implies that ‖σn − Ψ(σ)‖2 ≤ (1 + α) (1 −α) ‖σn − Ψ(σn)‖2 + 2 (1 −α) (α‖σn −σ‖ + ‖Ψ(σn) − Ψ(σ)‖) ‖Ψ(σn) −σn‖ + ‖σn −σ‖2. Therefore lim sup n→∞ ‖σn − Ψ(σ)‖2 ≤ lim sup n→∞ ‖σn −σ‖2 as an application of Opial property we conclude that Ψ(σ) = σ. � Theorem 4.2. Suppose X is a Banach space having the Opial property. As- sume that K is a nonempty subset of X and Ψ : K → K a C-α nonexpansive mapping such that Ψ admits an a.f.p.s.. Then Ψ has a fixed point. Proof. Demiclosedness principle implies the conclusion. � Noor [12] considered the following iterative process:  σ1 ∈K σn+1 = (1 − ζn)σn + ζnΨ(υn) υn = (1 −γn)σn + γnΨ(wn) wn = (1 −δn)σn + δnΨ(σn), n ∈ N, (4.2) where {ζn}, {γn} and {δn} are sequences in [0, 1]. Lemma 4.3 ([20, p.484]). Assume that 0 < a ≤ ln ≤ b < 1 for all n ∈ N and X is a uniformly convex Banach space. Suppose {σn} and {υn} are sequences such that lim sup n→∞ ‖σn‖≤ r, lim sup n→∞ ‖υn‖≤ r and lim n→∞ ‖lnσn + (1 − ln)υn‖ = r hold for some r ≥ 0. Then lim n→∞ ‖σn −υn‖ = 0. Lemma 4.4. Suppose K is a nonempty closed convex subset of a Banach space X . Let Ψ : K → K be a C-α nonexpansive mapping. Let {σn} be a sequence defined by (4.2). If F(Ψ) 6= ∅, then the following postulation hold: (1): max{‖σn+1 −w†‖,‖υn−w†‖,‖wn−w†‖}≤‖σn−w†‖ for all n ∈ N and w† ∈ F(Ψ); © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 385 R. Pant and R. Shukla (2): lim n→∞ ‖σn −w†‖ exists; (3): lim n→∞ d(σn,F(Ψ)) exists, where d(σ,F(Ψ)) denotes the distance from σ to F(Ψ). Proof. In view (4.2) and Proposition 3.8, we get ‖wn −w†‖ = ‖(1 −δn)σn + δnΨ(σn) −w†‖ ≤ (1 − δn)‖σn −w†‖ + δn‖Ψ(σn) −w†‖ ≤ (1 − δn)‖σn −w†‖ + δn‖σn −w†‖ = ‖σn −w†‖.(4.3) By (4.2), (4.3) and Proposition 3.8, we have ‖υn −w†‖ = ‖(1 −γn)σn + γnΨ(wn) −w†‖ ≤ (1 −γn)‖σn −w†‖ + γn‖Ψ(wn) −w†‖ ≤ (1 −γn)‖σn −w†‖ + γn‖wn −w†‖ ≤ (1 −γn)‖σn −w†‖ + γn‖σn −w†‖ = ‖σn −w†‖.(4.4) Using (4.2), (4.4) and Proposition 3.8, we get ‖σn+1 −w†‖ = ‖(1 − ζn)σn + ζnΨ(vn) −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖Ψ(υn) −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖υn −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖σn −w†‖ = ‖σn −w†‖.(4.5) Combining (4.3), (4.4) and (4.5) proves (1). Also by (4.5) the sequence {‖σn− w†‖} is bounded and hence monotone decreasing. Therefore lim n→∞ ‖σn−w†‖ ex- ists and proves (2). Now, since ‖σn+1−w†‖≤‖σn−w†‖ for each w† ∈ F(Ψ) and for all n ∈ N, d(σn+1,F(Ψ)) ≤ d(σn,F(Ψ)) for all n ∈ N. Thus {d(σn,F(Ψ))} is a bounded sequence and monotone decreasing. Hence, lim n→∞ d(σn,F(Ψ)) ex- ists. � Theorem 4.5. Let K, {σn} and Ψ be same as in Lemma 4.4. If F(Ψ) 6= ∅ and X is a uniformly convex Banach space. Then lim n→∞ ‖Ψ(σn) −σn‖ = 0. Proof. Let w† ∈ F(Ψ). Then from Lemma 4.4, {σn} is bounded and lim n→∞ ‖σn− w†‖ exists. Call it r. That is (4.6) lim n→∞ ‖σn −w†‖ = r. In view of (4.6) and Proposition 3.8 it implies that (4.7) lim sup n→∞ ‖Ψ(σn) −w†‖≤ r. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 386 Fixed point theorems for a new class of nonexpansive mappings in Banach spaces By (4.3) and (4.6), we get (4.8) lim sup n→∞ ‖wn −w†‖≤ lim n→∞ ‖σn −w†‖ = r. From (4.4) and (4.6), we get (4.9) lim sup n→∞ ‖υn −w†‖≤ r. In view of (4.9) and Proposition 3.8 it follows that (4.10) lim sup n→∞ ‖Ψ(υn) −w†‖≤ r. Similarly, (4.11) lim sup n→∞ ‖Ψ(wn) −w†‖≤ r. By (4.2), (4.4) and Proposition 3.8, we have ‖σn+1 −w†‖ = ‖(1 − ζn)σn + ζnΨ(υn) −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖Ψ(υn) −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖υn −w†‖ ≤ (1 − ζn)‖σn −w†‖ + ζn‖σn −w†‖ = ‖σn −w†‖, or ‖σn+1 −w†‖≤‖(1 − ζn)σn + ζnΨ(υn) −w†)‖≤‖σn −w†‖, it implies that r ≤ lim n→∞ ‖(1 − ζn)σn + ζnΨ(υn) −w†)‖≤ r. Then, (4.12) lim n→∞ ‖(1 − ζn)σn + ζnΨ(υn) −w†)‖ = r. By (4.10), (4.11), (4.12) and Lemma 4.3, we get that (4.13) lim n→∞ ‖σn − Ψ(υn)‖ = 0. In view of the triangle inequality and Proposition 3.8, we obtain ‖σn −w†‖ ≤ ‖σn − Ψ(υn)‖ + ‖Ψ(υn) −w†‖ ≤ ‖σn − Ψ(υn)‖ + ‖υn −w†‖ By (4.13), we have lim n→∞ ‖σn −w†‖ ≤ lim n→∞ ‖σn − Ψ(υn)‖ + lim inf n→∞ ‖υn −w†‖ ≤ lim inf n→∞ ‖υn −w†‖. It follows that (4.14) r ≤ lim inf n→∞ ‖υn −w†‖. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 387 R. Pant and R. Shukla Using (4.2), (4.9) and (4.14), we get (4.15) r = lim n→∞ ‖υn −w†‖ = ‖(1 −γn)(σn −w†) + γn(Ψ(wn) −w†)‖. In view of Lemma 4.3, and (4.6), (4.11), (4.15), we have (4.16) lim n→∞ ‖σn − Ψ(wn)‖ = 0. By triangle inequality and Proposition 3.8, we have ‖σn −w†‖ ≤ ‖σn − Ψ(wn)‖ + ‖Ψ(wn) −w†‖ ≤ ‖σn − Ψ(wn)‖ + ‖wn −w†‖, using (4.16) it follows that (4.17) r ≤ lim inf n→∞ ‖wn −w†‖. Combining (4.8) and (4.17) together we get (4.18) lim n→∞ ‖wn −w†‖ = 0. By (4.2) and Proposition 3.8, we have ‖wn −w†‖ = ‖(1 −δn)σn + δnΨ(σn) −w†‖ ≤ (1 − δn)‖σn −w†‖ + δn‖Ψ(σn) −w†‖ ≤ (1 − δn)‖σn −w†‖ + δn‖σn −w†‖ = ‖σn −w†‖. This implies that r ≤ lim n→∞ ‖(1 − δn)(σn −w†) + δn(Ψ(σn) −w†)‖≤ r. Therefore, we get (4.19) lim n→∞ ‖(1 −δn)(σn −w†) + δn(Ψ(σn) −w†)‖ = r. In view of Lemma 4.3 and (4.6), (4.7), (4.19), it follows that lim n→∞ ‖Ψ(σn) − σn‖ = 0. � Theorem 4.6. Let X be a uniformly convex Banach space having the Opial’s property, K, Ψ and {σn} same as in Theorem 4.5. If F(Ψ) 6= ∅ then {σn} weakly converges to a fixed point of Ψ. Proof. This can be completed following [14, Theorem 5.8]. � Theorem 4.7. Suppose that K, X , {σn} and Ψ are same as in Lemma 4.4. Let F(Ψ) 6= ∅ and lim inf n→∞ d(σn,F(Ψ)) = 0. Then the sequence {σn} strongly converges to a fixed point of Ψ. Proof. This can be completed following [14, Theorem 5.9]. � © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 388 Fixed point theorems for a new class of nonexpansive mappings in Banach spaces Theorem 4.8. Assume that K is a subset of a uniformly convex Banach spaceX . Let Ψ and {σn} are same as in Theorem 4.5 and. Let Ψ satisfy condi- tion (I) with F(Ψ) 6= ∅. Then the sequence {σn} strongly converges to a fixed point of Ψ. Proof. This can be completed following [14, Theorem 5.10]. � Acknowledgements. We are very much thankful to the reviewer for his/her constructive comments and suggestions which have been useful for the improve- ment of this paper. References [1] R. P. Agarwal, D. O’Regan and D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, volume 6 Topological Fixed Point Theory and Its Applica- tions, Springer, New York, 2009. [2] K. Aoyama, S. Iemoto, F. Kohsaka and W. Takahashi, Fixed point and ergodic theorems for λ-hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 11 (2010), 335–343. [3] K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Ba- nach spaces, Nonlinear Anal. 74 (2011), 4387–4391. [4] D. Ariza-Ruiz, C. Hernández Linares, E. Llorens-Fuster and E. Moreno-Gálvez, On α- nonexpansive mappings in Banach spaces, Carpathian J. Math. 32 (2016), 13–28. [5] J. Bogin, A generalization of a fixed point theorem of Goebel, Kirk and Shimi, Canad. Math. Bull. 19 (1976), 7–12. [6] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1272–1276. [7] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. [8] K. Goebel and W. A. Kirk, Topics in metric fixed point theory, volume 28 Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990. [9] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258. [10] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. [11] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings re- lated to maximal monotone operators in Banach spaces, Arch. Math. (Basel), 91 (2008), 166–177. [12] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229. [13] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpan- sive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597. [14] R. Pant and R. Shukla, Approximating fixed points of generalized α-nonexpansive map- pings in Banach spaces, Numer. Funct. Anal. Optim. 38, no. 2 (2017), 248–266. [15] R. Pant, R. Shukla and P. Patel, Nonexpansive mappings, their extensions and gen- eralizations in Banach spaces, in: Metric fixed point theory - applications in science, engineering and behavioural sicences, 309–343, Forum Interdiscip. Math., Springer, Sin- gapore, 2021. [16] H. F. Senter and W. G. Dotson Jr, Approximating fixed points of nonexpansive map- pings, Proc. Amer. Math. Soc. 44 (1974), 375–380. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 389 R. Pant and R. Shukla [17] R. Shukla, R. Pant and P. Kumam, On the α-nonexpansive mapping in partially ordered hyperbolic metric spaces, J. Math. Anal. 8 (2017), 1–15. [18] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonex- pansive mappings, J. Math. Anal. Appl. 340 (2008), 1088–1095. [19] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010), 79–88. [20] E. Zeidler, Nonlinear functional analysis and its applications, I, Fixed-point theorems, Springer-Verlag, New York, 1986. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 390