() @ Appl. Gen. Topol. 19, no. 2 (2018), 291-305doi:10.4995/agt.2018.10213 c© AGT, UPV, 2018 On Reich type λ−α-nonexpansive mapping in Banach spaces with applications to L1([0, 1]) Rabah Belbaki a, E. Karapınar b and Amar Ould-Hammouda a a Laboratory of Physics Mathematics and Applications, ENS, P.O.Box 92, 16050 Kouba, Algiers, Algeria (belbakirab@gmail.com,a.ouldhamouda@yahoo.com) b Atilim University, Department of Mathematics, Incek, 068630 Ankara, Turkey (erdalkarapinar@yahoo.com) Communicated by S. Romaguera Abstract In this manuscript we introduce a new class of monotone generalized nonexpansive mappings and establish some weak and strong conver- gence theorems for Krasnoselskii iteration in the setting of a Banach space with partial order. We consider also an application to the space L1([0, 1]). Our results generalize and unify the several related results in the literature. 2010 MSC: 46T99; 47H10; 54H25. Keywords: fixed point; Krasnoselskii iteration; monotone mapping; Reich type λ − α-nonexpansive mapping; optial property. 1. Introduction and preliminaries The study of the existence of fixed point of nonexpansive mappings, initiated in 1965 independently by Browder [5], Göhde [11] and [16], is one of dynamic research subject in nonlinear functional analysis. In [16], Kirk proved that a self-mapping on a nonempty bounded closed and convex subset of a reflexive Banach space possesses a fixed point if it is nonexpansive and the corresponding subset has a normal structure. In 1992, Veeramani obtained a more general result in this direction by introducing the notion of T−regular set [23]. On the other hand, in 1967, Opial introduced in [18] a class of spaces for which the asymptotic center of a weakly convergent sequence coincides with Received 24 May 2018 – Accepted 09 August 2018 http://dx.doi.org/10.4995/agt.2018.10213 R. Belbaki, E. Karapınar and A. Ould-Hammouda the weak limit point of the sequence. A Banach space X is said to have the Opial property, if for each weakly convergent sequence {xn} in X with limit z, lim inf ‖xn − z‖ < lim inf ‖xn − y‖ for all y ∈ X with y 6= z. In 1972, Gossey and Lami Dozo noticed in [12] that all the spaces of this class have normal structure. It is well known that Hilbert spaces, finite dimensional Banach spaces and lp-spaces, (1 < p < ∞), have the Opial property [8]. In 2008, Suzuki introduced in [21] a new class of mappings satisfying the so-called (C)- condition which also includes nonexpansive mappings and proved that such mappings on a nonempty weakly compact convex set in a Banach space which satisfies Opial’s condition have a fixed point. In 2011, Falset et al. proposed in [8] mappings satisfying (Cλ)-condition, λ ∈ (0, 1), respectively. In [1] Aoyama and Kohsaka introduced a new class of nonexpansive mappings, and obtained a fixed point result for such mappings. Finally, in 2017, in [19] Shukla et al proposed a new generalization and introduce the deneralized α−nonexpansive mapping and obtained a fixed theorem for such mappings. All the results cited above were obtained, in the weak case, with Opial’s condition. In this report, we propose a generalization of the results of Shukla et al. [19] by introducing a class of λ−α- generalized nonexpansive mapping. In addition, we establish some weak and strong convergence theorems for Krasnoselskii iteration in an ordered Banach space with partial order ≤. We also consider an application in the context of L1([0, 1]). The presented results in this report, extend, generalize and unify a number of existing results on the the topic in the literature. Throughout the paper, N denotes the set of natural numbers and R the set of the real numbers. For a non-empty K of a real Banach space X, a mapping T : K → K is said to be nonexpansive if ‖T (x) − T (y)‖ ≤ ‖x − y‖ for all x, y ∈ K. Moreover, a selfmapping T is called quasinonexpansive [7] if ‖T (x) − y‖ ≤ ‖x − y‖ for all x ∈ K and y ∈ F(T ), where F(T ) is the set of fixed points of T . Definition 1.1 ([12, 22]). The norm of a Banach space X is called uniformly convex in every direction, in short, we say that X is UCED, if for ε ∈ (0, 2] and z ∈ X with ‖z‖ = 1, there exists δ(ε, z) > 0 such that for all x, y ∈ X with where ‖x‖≤ 1, ‖y‖≤ 1 and x−y ∈{tz : t ∈ [−2,−ε]∪ [ε, 2]} ‖x + y‖≤ 2(1− δ(ε, z)). Lemma 1.2 ([21]). For a Banach space X, the following are equivalent: (i) X is UCED. (ii) If {xn} is a bounded sequence in X , then the function f on X defined by f(x) = lim sup‖xn −x‖ is strictly quasiconvex , that is, f(λx+(1−λ)y) < max{f(x), f(y)} for all λ ∈ (0, 1) and x, y ∈ X with x 6= y. Lemma 1.3 ([9]). Let (zn) and (wn) be bounded sequences in a Banach space X and let λ belongs to (0, 1). Suppose that zn+1 = λwn +(1−λ)zn and ‖wn+1− wn‖≤‖zn+1 −zn‖ for all n ∈ N. Then lim‖wn −zn‖ = 0. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 292 On Reich type λ − α-nonexpansive mapping Definition 1.4 ([21]). Let K be a nonempty subset of a Banach space X. We say that a mapping T : K → K satisfies (C)-condition on K if for x, y ∈ K we have 1 2 ‖x−T (x)‖≤‖x−y‖⇒‖T (x)−T (y)‖≤‖x−y‖. It is clear that each nonexpansive mapping satisfies the condition (C) but the converse is not true. For details and counterexamples see e.g. [10]. Definition 1.5 ([8]). Let K be a nonempty subset of Banach space X and λ ∈ (0, 1). We say that a mapping T : K → K satisfies (Cλ) -condition on if for all x, y ∈ K, we have λ‖x−T (x)‖≤‖x−y‖⇒‖T (x)−T (y)‖≤‖x−y‖. Note that if λ = 1 2 , then (Cλ)-condition implies (C)-condition. For more details and examples, see e.g. Falset et al. [8]. Throughout the paper, the pair (X,≤) will denote an ordered Banach space where X is a Banach space endowed with a partial order ” ≤ ”. Definition 1.6. A self-mapping T defined on an ordered Banach space (X,≤) is said to be monotone if for all x, y ∈ X, x ≤ y ⇒ T (x) ≤ T (y). Definition 1.7 ([1]). Let K be a nonempty subset of a Banach space X. A mapping T : K → K is said to be α-nonexpansive if for all x, y ∈ K and α < 1, ‖T (x) −T (y)‖2 ≤ α‖T (x) −y‖2 + α‖x− T (y)‖2 + (1−2α)‖x−y‖2 Definition 1.8 ([19]). Let K be a nonempty subset of an ordered Banach space (X,≤). A mapping T : K → K will be called a generalized α-nonexpansive mapping if there exists α ∈ (0, 1) such that { 1 2 ‖x−T (x)‖≤‖x−y‖ implies ‖T (x)−T (y)‖≤ α‖T (x)−y‖+ α‖T (y)− x‖+ (1−2α)‖x−y‖ for all x, y ∈ K with x ≤ y. Remark 1.9. When α = 0, a generalized-nonexpansive mapping is reduced to a mapping satisfying (C)-condition. The converse is false. For more details and counterexamples see e.g. [19] and [14, 13]. 2. Reich type (λ −α)-nonexpansive mappings Definition 2.1. Let K be a nonempty subset of an ordered Banach space (X,≤). A mapping T : K → K will be called Reich type (λ−α)-nonexpansive mappings if there exists λ ∈ (0, 1) and α ∈ [0, 1) such that (2.1) λ‖x−T (x)‖≤‖x−y‖⇒‖T (x)−T (y)‖≤ RαT (x, y), where RαT (x, y) := α(‖T (x) −y‖+‖T (y)−x‖) + (1 −2α)‖x−y‖ c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 293 R. Belbaki, E. Karapınar and A. Ould-Hammouda for all x, y ∈ K with x ≤ y. In addition, if the mapping T is monotone, we say that monotone Reich type (λ−α)-nonexpansive mapping. Remark 2.2. We point the following special cases: (1) When α = 0, a Reich type λ − α-nonexpansive mapping reduced to a mapping satisfying condition (Cλ), see e.g. [8]. (2) If λ = 1 2 , it becomes a generalized α-nonexpansive condition. Proposition 2.3. Let K be a nonempty subset of an ordered Banach space (X,≤) and T : K → K be a Reich type (λ − α)-nonexpansive mapping with a fixed point z ∈ K with x ≤ z. Then T is quasinonexpansive. Proof. Since z ∈ K fixed point, 0 = λ‖z −T (z)‖≤‖z −x‖ , we have ‖z −T (x)‖≤ α‖z −T (x)‖+ α‖T (z)−x‖+ (1−2α)‖z −x‖≤‖z −x‖. � Definition 2.4. Let T be a monotone self-mapping on a nonempty convex subset of an ordered Banach space (X,≤) . For a fix λ ∈ (0, 1) and for an initial point x1 ∈ K, the Krasnoselskii iteration sequence {xn}⊂ K is defined by (2.2) xn+1 = λT (xn) + (1−λ)xn , n ≥ 1. In the sequel we need the following lemmas. Lemma 2.5 ([17]). Let x, y, z ∈ X and λ ∈ (0, 1). Suppose p is the point of segment [x, y] which satisfies ‖x−p‖ = λ‖x−y‖ , then, (2.3) ‖z −p‖≤ λ‖z −y‖+ (1−λ)‖z −x‖ Lemma 2.6 ([15]). Let K be convex and T : K → K be monotone. Assume that x1 ∈ K, x1 ≤ T (x1). Then the sequence {xn} defined by (2.2) satisfies: xn ≤ xn+1 ≤ T (xn) ≤ T (xn+1), for n ≥ 1. Moreover, if {xn} has two subsequences which converge to y and z, then we must have y = z. It is easy to see that by the mimic of the idea used in Lemma 2.6, we get that T (xn+1) ≤ T (xn) ≤ xn+1 ≤ xn, by assuming the initial condition as T (x1) ≤ x1. Lemma 2.7. Let K be a nonempty convex subset of an ordered Banach space (X,≤) and {xn} is the iteration sequence defined by (2.2) in K. Let T : K → K be a monotone Reich type λ − α-nonexpansive mapping with λ ∈ (1 3 , 1) and α ∈ [0, 1). Suppose also that yn = T (xn), n ≥ 1. If, for x1 ∈ K with x1 ≤ y1 = T (x1) we have (2.4) ‖yn −xn+1‖≤ (3λ−1)‖yn −xn‖, for all n ∈ N, then the sequence {‖yn −xn‖} is decreasing. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 294 On Reich type λ − α-nonexpansive mapping Proof. On account of the definition of Krasnoselskii iteration we have (2.5) xn+1 = λyn + (1−λ)xn, with yn = T (xn). It means that xn+1 belongs to the segment ]xn, yn[, and hence we have (2.6) ‖xn −yn‖ = ‖xn −xn+1‖+‖xn+1 −yn‖. Furthermore, (2.7) yields that (2.7) ‖xn −xn+1‖ = ‖xn − [λT (xn) + (1−λ)xn]‖ = λ‖xn −T (xn)]‖. On account of the triangle inequality together with the fact that T is monotone Reich type (λ−α)-nonexpansive mapping, we derive that (2.8) ‖yn+1 −xn+1‖ ≤‖yn+1 −yn‖+‖yn −xn+1‖ = ‖T (xn+1) −T (xn)‖+‖yn −xn+1‖ ≤ α(‖T (xn) −xn+1‖+‖T (xn+1) −xn‖) +(1−2α)‖xn −xn+1‖+‖yn −xn+1‖ = (1 + α)‖yn −xn+1‖+ α‖yn+1 −xn‖+ (1 −2α)‖xn −xn+1‖ = (1 + α)‖yn −xn+1‖+ α‖yn+1 −xn‖ +(1 + α)‖xn −xn+1‖−3α‖xn −xn+1‖. On account of (2.6) the left hand side of the inequality of (2.8) turns into (2.9) = (1 + α)‖xn −yn‖+ α‖yn+1 −xn‖−3α‖xn −xn+1‖, Taking the inequality (2.7) into account, the expression (2.9) turns into (2.10) ≤ (1 + α)‖xn −yn‖+ α‖yn+1 −xn‖−3λα‖xn −yn‖ = (1 + α −3λα)‖xn −yn‖+ α‖yn+1 −xn‖ Employing the assumption (2.4) of the lemma, we estimate the expression (2.10) from above as (2.11) = (1 + α−3λα)‖xn −yn‖+ α (3λ−1)‖yn −xn‖ = ‖xn −yn‖. By combining (2.8)- (2.11), for each n, we deduce that ‖yn+1 −xn+1‖≤‖xn −yn‖, which complete the proof. � In the following proposition, we extend the Goebel-Kirk inequality [9] from the class of nonexpansive mappings into the class of monotone generalized(λ− α)-nonexpansive mapping. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 295 R. Belbaki, E. Karapınar and A. Ould-Hammouda Proposition 2.8. Let K be a nonempty convex subset of an ordered Banach space (X,≤) . Let T : K → K be a monotone Reich type (λ−α)-nonexpansive mapping with λ ∈ (1 3 , 1) and α ∈ (0, 1). For x1 ∈ K with x1 ≤ T (x1), we set yn = T (xn) where {xn} is the iteration sequence defined by (2.2) in K satisfies the assumption (2.4). Then, we have (2.12) ‖yi+n −xi‖≥ (1−λ) −n[‖yi+n −xi+n‖−‖yi−xi‖] + (1 + nλ)‖yi−xi‖, for all i, n ∈ N. Proof. Inspired the techniques used in [9], we shall use the method of the induction to prove our assertion. It is evident that (2.12) is trivially true for all i if n = 0. We assume that the inequality (2.12) holds for a given n and for all i. By replacing i by i + 1 in (2.12), we get ‖yi+n+1 −xi+1‖≥ (1−λ) −n[‖yi+n+1 −xi+n+1‖ −‖yi+1 −xi+1‖] + (1 + nλ)‖yi+1 −xi+1‖. (2.13) On the other hand, due to Krasnoselskii iteration, we have xn+1 = λyn + (1− λ)xn with yn = T (xn) and also (2.14) ‖xi+1 −xi‖ = ‖λyi + (1−λ)xi −xi‖ = λ‖yi −xi‖. The observation in (2.14) provide to apply Lemma 2.5 that yields ‖yi+n+1 −xi+1‖≤ λ‖yi+n+1 −yi‖+ (1 −λ)‖yi+n+1 −xi‖. Regarding that T is a monotone Reich type (λ−α)-nonexpansive mapping, we have ‖yi+n+1 −xi+1‖ ≤ (1−λ)‖yi+n+1 −xi‖+ λ ∑n k=0 ‖yi+k+1 −yi+k‖ ≤ (1−λ)‖yi+n+1 −xi‖ +λ ∑n k=0(α‖xi+k+1 −yi+k‖+ α‖yi+k+1 −xi+k‖ +(1−2α)‖xi+k+1 −xi+k‖) So, we derive that ‖yi+n+1 −xi‖≥ (1−λ) −1‖yi+n+1 −xi+1‖− (1 −λ) −1λαBin − (1−λ)−1λ(1 −2α)Ain, (2.15) where Ain = n ∑ k=0 ‖xi+k+1 −xi+k‖ and Bin = n ∑ k=0 [‖xi+k+1 −yi+k‖+‖yi+k+1 −xi+k‖]. Taking the assumption (2.4) and (2.14) into account, we derive that (2.16) ‖yi+k+1 −xi+k‖+‖yi+k −xi+k‖ ≤ (3λ− 1)‖yi+k −xi+k‖+‖yi+k −xi+k‖ = 3λ‖yi+k −xi+k‖ = 3‖xi+k −xi+k+1‖, c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 296 On Reich type λ − α-nonexpansive mapping for all k ∈ 0, 1, ..., n. Regarding the definition of Krasnoselskii iteration we have xi+k+1 = λyi+k + (1 − λ)xi+k, with yi+k = T (xi+k). In other words, xi+k ≤ xi+k+1 ≤ yi+k and we have (2.17) ‖xi+k −yi+k‖ = ‖xi+k+1 −xn+1‖+‖xi+k+1 −yi+k‖. Now, by revisiting the inequality (2.16) by keeping the equality (2.17) in mind, we find (2.18) ‖yi+k+1 −xi+k‖+‖xi+k+1 −yi+k‖ = ‖yi+k+1 −xi+k‖+‖yi+k −xi+k‖ −‖xi+k −xi+k+1‖ ≤ 2‖xi+k −xi+k+1‖ which implies Bin ≤ 2Ain. Accordingly, the inequality (2.15) becomes (2.19) ‖yi+n+1 −xi‖≥ (1 −λ) −1‖yi+n+1 −xi+1‖−λ(1 −λ) −1 Ain. Employing the inequality (2.13) in (2.19), we find that ‖yi+n+1 −xi‖≥(1−λ) −(n+1)[‖yi+n+1 −xi+n+1‖−‖yi+1 −xi+1‖] + (1 −λ)−1(1 + nλ)‖yi+1 −xi+1‖−λ(1 −λ) −1Ain. On account of (2.14), the estimation above turns into ‖yi+n+1 −xi‖≥(1−λ) −(n+1)[‖yi+n+1 −xi+n+1‖−‖yi+1 −xi+1‖] + (1 −λ)−1(1 + nλ)‖yi+1 −xi+1‖−λ 2(1−λ)−1Cin, (2.20) where Cin := n ∑ k=0 ‖yi+k−xi+k‖. By bearing, Lemma 2.7, in mind, we find that Cin := n ∑ k=0 ‖yi+k −xi+k‖≤ (n + 1)‖yi −xi‖. Consequently, (2.20) can be estimated above as ‖yi+n+1 −xi‖≥(1− λ) −(n+1)[‖yi+n+1 −xi+n+1‖−‖yi+1 −xi+1‖] + (1−λ)−1(1 + nλ)‖yi+1 −xi+1‖−λ 2(1−λ)−1(n + 1)‖yi −xi‖ = (1−λ)−(n+1)[‖yi+n+1 −xi+n+1‖−‖yi −xi‖] + [(1−λ)−1(1 + nλ) − (1−λ)−(n+1)]‖yi+1 −xi+1‖ + [(1−λ)−(n+1) −λ2(1−λ)−1(n + 1)]‖yi −xi‖, (2.21) by adding and substraction the same term (1−λ)−(n+1)‖yi+1 −xi+1‖. Notice that (1 − λ)−1(1 + nλ) − (1 − λ)−(n+1) ≤ 0. Thus, regarding this observation c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 297 R. Belbaki, E. Karapınar and A. Ould-Hammouda together with Lemma 2.7, the inequality (2.21) changed into ‖yi+n+1 −xi‖≥ (1 −λ) −(n+1)[‖yi+n+1 −xi+n+1‖−‖yi −xi‖] + [(1−λ)−1(1 + nλ) − (1−λ)−(n+1)]‖yi −xi‖ + [(1−λ)−(n+1) −λ2(1−λ)−1(n + 1)]‖yi −xi‖ = (1 −λ)−(n+1)[‖yi+n+1 −xi+n+1‖−‖yi −xi‖] + (1 + (n + 1)λ)‖yi −xi‖ which completes the proof of Proposition 2.8. � Theorem 2.9. Let K be a nonempty, convex and compact subset of an ordered Banach space (X,≤) . Let T : K → K be a monotone Reich type λ − α- nonexpansive mapping with λ ∈ (1 3 , 1). For x1 ∈ K with x1 ≤ T (x1), we set yn = T (xn) where {xn} is the iteration sequence defined by (2.2) in K satisfies the assumption (2.4). Then {xn} converges to some x ∈ K with xn ≤ x and, (2.22) lim n ‖xn −T (xn)‖ = 0 Proof. We shall divide the proof in two cases: α = 0 and α ∈ (0, 1). Suppose, first, that α = 0 . Due to the definition (2.2) of the sequence {xn}, we have λ‖xn −yn‖ = ‖xn −xn+1‖, for all n ≥ 1. On account of Lemma 2.6, we have xn ≤ xn+1, for all n ≥ 1. Therefore condi- tion (2.1) implies that, ‖T (xn) −T (xn+1)‖ = ‖yn −yn+1‖≤ R α T (xn, xn+1) = ‖xn −xn+1‖, since α = 0. Employing Lemma 1.3, the inequality above yields that lim n ‖xn −T (xn)‖ = 0. In the following, we shall consider the second case α ∈ (0, 1). The proof of this case mainly adopted from the proof of Theorem 3.1 in [15]. Since K is compact, there exists a subsequence of {xn} which converges to x ∈ K. On account of Lemma 2.6, the sequence {xn} converges to x and xn ≤ x, for n ≥ 1. To show our assertion (2.22), suppose, on the contrary, that lim n ‖xn −T (xn)‖ = R > 0. As x1 ≤ xn ≤ x, we then have (2.23) ‖xn −x1‖≤‖x−x1‖ for all n ≥ 1. Due to triangle inequality we have ‖yi+n −xi‖ = ‖T (xi+n) −xi‖≤‖T (xi+n) −xi+n‖+‖xi+n −x1‖+‖x1 −xi‖ ≤‖T (x1) −x1‖+ 2‖x−x1‖ (2.24) c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 298 On Reich type λ − α-nonexpansive mapping for any i, n ≥ 1, due to (2.23) and Lemma 2.7. Since all conditions are satisfied in Proposition 2.8, we have (2.12). Letting i →∞ in the inequality (2.12), we derive that (2.25) lim i→∞ ‖yi+n − xi‖≥ (1 + nλ)R, where we used that lim i→∞ (‖T (xi) −xi‖−‖T (xi+n) −xi+n‖) = R −R = 0, for any n ≥ 1. Combining (2.24) and (2.25), we find (1 + nλ)R ≤ lim i→∞ ‖yi+n −xi‖≤‖T (x1)−x1‖+ 2‖x−x1‖ Thus, the inequality can be fulfilled only if R = 0 which yields the inequality (2.22). � Lemma 2.10. Let K be a nonempty subset of an ordered Banach space (X,≤) and T : K → K be a monotone Reich type (λ−α)-nonexpansive mapping with λ ∈]0, 1 2 ]. Then for each x, y ∈ K with x ≤ y : (i) ‖T (x)−T 2(x)‖≤‖x−T (x)‖ (ii) either λ‖x−T (x)‖≤‖x−y‖ or λ‖T (x)−T 2(x)‖≤‖T (x)−y‖ (iii) either ‖T (x)−T (y)‖≤ α‖T (x)−y‖+ α‖x−T (y)‖+ (1−2α)‖x−y‖ or ‖T 2(x) −T (y)‖≤ α‖T (x)−T (y)‖+ α‖T 2(x) −y‖+ (1−2α)‖T (x)−y‖ Proof. (i) Since we have λ‖x − T (x)‖ ≤ ‖x − T (x)‖ for all λ ∈]0, 1 2 ], by the definition of Reich type (λ−α)-nonexpansive mapping we get the desired results. Indeed, ‖T (x)−T 2(x)‖≤ α‖x−T 2(x)‖+ (1−2α)‖x−T (x)‖. Thus (i) hold for α = 0. (ii) Suppose, on the contrary, that λ‖x − T (x)‖ > ‖x − y‖ and ‖T (x) − T 2(x)‖ > ‖T (x) − y‖. Then, by triangle inequality together with the assumption (i), we find that ‖x−T (x)‖≤‖x−y‖+‖T (x)−y‖ < λ‖x−T (x)‖+ λ‖T (x) −T 2(x)‖ ≤ 2λ‖x−T (x)‖. Since λ ≤ 1 2 we obtain ‖x−T (x)‖ < ‖x−T (x)‖ which is a contradiction. Thus, we obtain the desired result. (iii) The proof of (iii) follows from (ii). We skip the details. � c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 299 R. Belbaki, E. Karapınar and A. Ould-Hammouda Lemma 2.11. Let K be a nonempty subset of an ordered Banach space (X,≤) and T : K → K be a monotone Reich type (λ−α)-nonexpansive mapping with λ ∈ (0, 1 2 ]. Then for each x, y ∈ K with x ≤ y, ‖x−T (y)‖≤ ( 3 + α 1−α )‖x−T (x)‖+‖x−y‖. Proof. It is the mimic of the proof of Lemma 3.8 of [19]. So, we skip the details. � Using the above two lemmas, we can prove the following. Theorem 2.12. Let K be a nonempty convex and a compact subset of an ordered Banach space (X,≤) and be T : K → K a monotone Reich type (λ−α)- nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. Select x1 ∈ K such that x1 ≤ T (x1), and for n ≥ 1, denote yn = T (xn) where {xn} is the iteration sequence defined by (2.2) in K satisfying, for all n ∈ N, the assumption (2.4) . Then {xn} converges strongly to a fixed point of T. Proof. By Theorem 2.9, we have lim n ‖xn −T (xn)‖ = 0. Since K is compact, there exist a subsequence {xnk} of {xn} and z ∈ K such that {xnk} converges to z. Employing Lemma 2.11, we have, ‖xnk −T (z)‖≤ ( 3 + α 1−α )‖xnk −T (xnk)‖+‖xnk −z‖ for all k ∈ N. Thus, the sequence {xnk} converges to T (z) and hence T (z) = z. Since z is a fixed point of T , by Proposition 2.3, we find that ‖xn+1 −z‖≤ λ‖T (xn) −z‖+ (1−λ)‖xn −z‖≤‖xn −z‖ for all n ∈ N. Therefore {xn} converges to z . � We say that a Banach space X has the Opial property [18] if for every weakly convergent sequence {xn} in X with a limit z, fulfils lim inf n→∞ ‖xn −z‖ < lim inf n→∞ ‖xn −y‖, for all y ∈ X with y 6= z. It is a very rich class, for examples, all Hilbert spaces, sequence spaces ℓp, (1 < p < ∞), and finite dimensional Banach spaces have the Opial property. Unexpectedly, Lp[0, 2π], (p 6= 2) do not have the Opial property [9],[10]. Proposition 2.13. Let K be a nonempty subset of an ordered Banach space (X,≤) with the Opial property and T : K → K be a monotone Reich type (λ − α)-nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. If {xn} converges weakly to z and lim n ‖xn −T (xn)‖ = 0, then T (z) = z. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 300 On Reich type λ − α-nonexpansive mapping Proof. By Lemma 2.11, we have, ‖xn −T (z)‖≤ ( 3 + α 1−α )‖xn −T (xn)‖+‖xn −z‖ for n ∈ N and hence, lim inf n ‖xn −T (z)‖≤ lim inf n ‖xn −z‖ We claim that T (z) = z. Indeed, if T (z) 6= z, the Opial property implies, lim inf n ‖xn −z‖ < lim inf n ‖xn −T (z)‖ which is a contradiction with inequality (2.22) . � Theorem 2.14. Let K be a nonempty convex and weakly compact subset of an ordered Banach space (X,≤) with the Opial property and T : K → K be a monotone Reich type (λ − α)-nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. . Select x1 ∈ K such that x1 ≤ T (x1), and for n ≥ 1 , denote yn = T (xn) where {xn} is the iteration sequence defined by (2.2) in K satisfying, for all n ∈ N, the assumption (2.4) . Then {xn} converges weakly to a fixed point of T . Proof. By Theorem 2.9, we have lim n ‖xn −T (xn)‖ = 0. Since K is weakly compact, there exist a subsequence {xnk} of {xn} and z ∈ K such that {xnk} converges weakly to z . By Proposition 2.13, we deduce that z is a fixed point of T . As in the proof of Theorem 2.12, we can prove that {‖xn − z‖} is a nonincreasing sequence. We prove our assertion by reductio de absurdum. Suppose, on the contrary, that {xn} does not converge to z. Then there exist a subsequence {xnj} of {xn} which converges weakly to ω and ω 6= z. We note that T (ω) = ω . From the Opial property, lim n ‖xn −z‖ = lim k ‖xnk −z‖ < lim k ‖xnk −ω‖ = lim n ‖xn −ω‖ = lim j ‖xnj −ω)‖ < lim j ‖xnj −z‖ = lim n ‖xn −z‖, a contradiction that complete the proof. � The following theorem directly follows from Theorems 2.12 and 2.14. So, to avoid the repetition, we skip the details. Theorem 2.15. Let K be a convex subset of an ordered Banach space (X,≤), and T : K → K be a monotone Reich type (λ−α)-nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. Assume that either of the following holds: (i) K is compact; (ii) K is weakly compact and X has the Opial property. Then T has a fixed point. Finally, we will give a generalization of a fixed point theorem due to Browder [5], Göhde [11] and Suzuki [21]. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 301 R. Belbaki, E. Karapınar and A. Ould-Hammouda Theorem 2.16. Let K be a convex and weakly compact subset of a UCED ordered Banach space (X,≤). Let T : K → K be a monotone Reich type (λ−α)-nonexpansive mapping with λ ∈ (1 3 , 1 2 ]. Then T has a fixed point. Proof. We construct an iterative sequence {xn} in K by starting x1 ∈ K as xn+1 = 1 2 (T (xn) + xn) with ‖T (xn+1) −xn‖≤‖T (xn) −xn‖, for all n ∈ N. Then by Theorem 2.9, we have lim n ‖xn −T (xn)‖ = 0 holds. Define a continuous convex function f from K to [0, +∞) by f(x) = lim sup n ‖xn −x‖ for all x ∈ K . Since K is weakly compact and f is weakly lower semicontinuous, there exists z ∈ K, such that f(z) = min{f(x) : x ∈ K} Since, by Lemma 2.11: ‖xn −T (z)‖≤ ( 3 + α 1−α )‖xn −T (xn)‖+‖xn −z‖, we then have, f(T (z)) ≤ f(z) . Since f(z) is the minimum, f(T (z)) = f(z) holds. If T (z) 6= z, then since f is strictly quasiconvex (Lemma 1.2) we have, f(z) ≤ f( z + f(z) 2 ) < max{f(z), f(T (z))} = f(z). which is a contradiction. Hence T (z) = z. � 3. Application to L1([0, 1]) As an application, we consider L1([0, 1]) the Banach space of real valued functions defined on [0, 1] with absolute value Lebesgue integrable, i.e., ∫ 1 0 |f(x)|dx < ∞. We recall some definitions which can be found in e.g. [3]. As usual, f = 0 if and only if the set {x ∈ [0, 1] : f(x) = 0} has Lebesgue measure 0, then, we say f = 0 almost everywhere. An element of L1([0, 1]) is therefore seen as a class of functions. The norm of any f ∈ L1([0, 1]) is given by ‖f‖ = ∫ 1 0 |f(x)|dx From now on, we will write L1 instead of L1([0, 1]) . Recall that f ≤ g if and only if f(x) ≤ g(x) almost everywhere, for any f, g ∈ L1. We adopt the convention f ≤ g if and only if g ≤ f . We remark that order intervals are closed for convergence almost everywhere and convex. Recall that an order interval is a subset of the form [f,→) = {g ∈ L1 : f ≤ g} or (←, f] = {g ∈ L1 : g ≤ f}, c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 302 On Reich type λ − α-nonexpansive mapping for any f ∈ L1. As a direct consequence of this, the subset [f, g] = {h ∈ L1 : f ≤ h ≤ g} = [f,→) ∩ (←, g] is closed and convex, for any f, g ∈ L1. Let K be a nonempty subset of L1 which is equipped with a vector order relation ≤ . A map T : K → K is called monotone if for all f ≤ g we have T (f) ≤ T (g). Remark 3.1. Since L1([0, 1]) fails to be uniformly convex, Theorem 2.12 can’t not be used to get a fixed point result for monotone generalized λ − α non- expansive mappings in L1([0, 1]). As an alternative, we will use an interest- ing property for the convergence almost everywhere contained in the following lemma. Lemma 3.2 ([4]). If (fn) is a sequence of uniformly L p -bounded functions on a measure space, and if fn → f almost everywhere, then lim inf n ‖fn‖ p p = lim inf n ‖fn −f‖ p p +‖f‖ p p for all 0 < p < ∞. In particular, this result holds when p = 1. On account of Lemma 2.11 and Lemma 3.2, we shall prove the following. Theorem 3.3. Let K ⊂ L1 be nonempty, convex and compact for the conver- gence almost everywhere. Let T : K → K be a monotone Reich type (λ − α)- nonexpansive mapping with α ∈]1 3 , 1 2 ]. Select f1 ∈ K such that f1 ≤ T (f1),, and for n ≥ 1, denote gn = T (fn) where (fn) is the iteration sequence defined by (2.2) in K satisfying, for all n ∈ N, the assumption (2.4) . Then the sequence (fn) converges almost everywhere to some f ∈ K which is a fixed point of T , i.e., T (f) = f. Moreover, f1 ≤ f. Proof. Theorem 2.9 implies that (fn) converges almost everywhere to some f ∈ K where fn → f, for any n ≥ 1. Since (fn) is uniformly bounded, lemma 3.2 [4] implies lim inf n ‖fn −T (f)‖ = lim inf n ‖fn −f‖+‖f −T (f)‖ Theorem 2.9 implies lim inf n ‖fn −T (fn)‖ = 0. Therefore we get lim inf n ‖fn −T (f)‖ = lim inf n ‖fn −f‖+‖f −T (f)‖ On the other hand, we know that each fn ≤ f for each n ≥ 1, so, by assumption (2.1), we have, lim inf n ‖fn−f‖+‖f−T (f)‖≤ lim inf n (α‖fn−T (f)‖+α‖T (fn)−f‖+(1−2α)‖fn−f‖) c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 303 R. Belbaki, E. Karapınar and A. Ould-Hammouda And, by Lemma 2.11, we have, lim inf n ‖fn−f‖+‖f−T (f)‖≤ lim inf n (α 3 + α 1 −α ‖fn−T (f)‖+α‖T (fn)−f‖+(1−2α)‖fn−f‖) Again, by application of the Theorem 2.9, we obtain, lim inf n ‖fn −f‖+‖f −T (f)‖≤ lim inf n (1−α)‖fn −T (f)‖+ α‖T (fn)−f‖) And like, lim inf n ‖fn −f‖ = ‖T (fn)−f‖ we then have, lim inf n ‖fn −f‖+‖f −T (f)‖≤ lim inf n ‖fn −T (f)‖, that implies ‖f −T (f)‖ = 0 or T (f) = f. � Acknowledgements. The authors thanks to anonymous referees for their remarkable comments, suggestion and ideas that helps to improve this paper. References [1] K. Aoyama and F. 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