Int. J. Anal. Appl. (2022), 20:65 Generalization of Fixed Point Approximation of Contraction and Suzuki Generalized Non-Expansive Mappings in Banach Domain N. Muhammad, A. Asghar, M. Aslam, S. Irum, M. Iftikhar, M. M. Abbas, A. Qayyum∗ Department of Mathematics, Institute of Southern Punjab Multan, Pakistan ∗Corresponding author: atherqayyum@isp.edu.pk Abstract. By principal motivation from the results of the new iterative scheme that produces faster results than K-iteration. In this article, we study generalized results by a new iteration scheme to approximate fixed points of generalized contraction and Suzuki non-expansive mappings. We establish strong convergence results of generalized contraction mappings of closed convex Banach space and also deduce data dependent results. Furthermore, we prove some weak and strong convergence theorems in the sense of generalized Suzuki non-expansive mapping by applying condition (C). 1. Introduction Mappings play a vital role in the field of inequalities (see of example [17–20]). The mappings which have Lipschitzís constant equal to 1 are called as non-expansive mappings. Let Z be a non empty bounded closed convex subset of k. A Banach space Z has the fixed point property (FPP) for non expansive mapping if for every non-empty bounded closed convex subset of Z contains a fixed point. Meanwhile, in 1965 major struggle has been proposed to study the theory of fixed point of non-expansive mappings in the setting of reflexive and non-reflexive Banach domain. Since then, a number of generalizations and extensions of non-expansive mappings and their results have been obtained by many authors. We can say that FPP provides basis of physical appearance of the Banach space. When K is a weakly compact convex subset of Z, a non-expansive self-mapping of K requires not have fixed point. However, if the norm of Z has suitable ordered properties (i.e., uniform convex and some others) each non-expansive self-mapping of every weakly closed convex subset of Z has a fixed point. In this case, K is called a weak fixed point property. All over in this article we assume that Received: Oct. 2, 2022. 2010 Mathematics Subject Classification. 47H09, 47H10, 47J25. Key words and phrases. Fixed point, contraction mapping, suzuki generalized non expansive mapping, condition (C). https://doi.org/10.28924/2291-8639-20-2022-65 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-65 2 Int. J. Anal. Appl. (2022), 20:65 K is a non-empty subset of a Banach space Z and Q(T ), the set of all fixed points of the mapping T over K. A mapping T : K → K is called to be a non-expansive if ‖Txo −Tyo‖ ≤ ‖xo −yo‖, of all xo, yo ∈ K. This is also called quasi non-expansive if Q(T ) 6= φ and ‖Txo −p‖ ≤ ‖xo −p‖, of all xo ∈ K and of all p ∈ Q(T ). It is known as Q(T ) is non-empty while Z is uniformly convex, K be a bounded closed convex subset of X and T be a non-expansive mapping [2]. In 2008, Japanese mathematician Suzuki [3] presented idea of generalized non-expansive mappings which is also called condition (C) and defined as A self-mapping T on K is said to be condition (C) that, 1 2 ‖xo −Txo‖≤‖xo −yo‖ =⇒‖Txo −Tyo‖≤‖xo −yo‖ ,∀xo,yo ∈ K of such mappings, Suzuki also obtained the existence of fixed point and convergence results. In [4], he proved that condition(C) have faster results as compared to non-expansive mappings. For a self- mapping T be defined over [0, 3] T (xo) = [0, if xo 6= 0 1, if xo = 0]. By this, we claim that T satisfy the condition (C), but T is not a non-expansive mapping. In extension, Picard’s iterative scheme is approximate the fixed point of contraction mappings in the Banach contraction principle. Over time, many mathematicians [6–10] played a fundamental role in the development of the current literature. Sahu V. K [5] and many other tried their best as compared to the previous one and added outstanding work. Inspired by the above, now we generalize some results by a new iteration scheme to approximate fixed point of generalized contraction and Suzuki’s non-expansive mappings. Also we discuss strong convergence theorems of generalized contraction mappings with closed convex Banach space and some data dependence results are also deduce. Moreover, we prove some weak and strong convergence results in type of generalized Suzuki non-expansive mappings by using condition (C). 2. Preliminaries (2.1) [12] Opial property if for each sequence {$n} in X, (where X be a Banach space) converging weakly to xo ∈ X take lim n→∞ sup‖$n −xo‖ < lim n→∞ sup‖$n −yo‖∀ yo ∈ X such that yo 6= xo. (2.2) Let K be a non-empty bounded sequence convex subset of a Banach space Z and consider {$n} be in Z of x0 ∈ Z, that d(z,{$n} = lim n→∞ sup‖$n −xo‖ . The asymptotic radius of {$n} comparative to K is given that d (K,{$n}) = inf {d(xo,{$n}) : xo ∈ K} . Int. J. Anal. Appl. (2022), 20:65 3 The asymptotic center of {$n} relative to K is the set B(K,$n}) = {xo ∈ K : d(xo,{$n}) = d(K,{$n})}. (2.3) A uniformly convex Banach space, X and {$n} be a real sequence such that 0 < s ≤ $n ≤ t < 1 ,∀ n ≥ 1. Consider {$n} and {ωn} be two sequences of K given that limn→∞ sup‖$n‖≤ d, limn→∞ sup‖ωn‖ ≤ d and limn→∞ sup‖$n + (1 −$n)ωn‖ = d holds of some d ≥ 0. Then, limn→∞‖$n −ωn‖ = 0. It is called a uniformly convex Banach space, B(K,{$n}) contains exactly one point. (2.4) A mapping T : K → K is called demi-closed with respect to yo ∈ K if of each sequence {$n} in K and K be a closed, convex and non-empty subset of a Banach space K of each xo ∈ K,{$n} converges weakly at xo and {T$n} converges strongly at yo =⇒ Txo = yo. (2.5) [16] Let{un}∞n=0 and {vn} ∞ n=0 are two fixed points iteration sequences that converges to the same fixed point q. If ‖un −q‖ ≤ an and ‖vn −q‖ ≤ bn, of all n ≥ 0, wherever {an}∞n=0 and {bn} ∞ n=0 be two real convergent sequences. Then we say that {un}∞n=0 converge faster than {vn} ∞ n=0 to q if {an}∞n=0 converges faster as compare to {bn} ∞ n=0. (2.6) Let K be a non-empty subset of a Banach space Z. Consider that a mapping T : K → K is said to be generalized contraction when ∃ 0 ≤ h ≤ 1 such that ‖Ts −Tt‖ ≤ h max[‖s − t‖ ,‖s −Ts‖ ,‖t −Tt‖ ,‖s −Tt‖ + ‖t −Ts‖] ∀ s,t ∈ K. (2.7) A Banach space K is known as uniformly convex if of each � belongs to (0, 2] there is a δ > 0 such that of s,t ∈ K ‖s‖ ≤ 1, ‖t‖ ≤ 1, ‖s − t‖ > �, Implies that ‖s + t‖ 2 ≤ δ. [1] Let X be a non-empty set and ϕ is collection of X, then (1) X belongs to τ. (2) Absolute union of number of τ belongs to τ. (3) Limited intersection of τ belongs to τ. Than τ be a topology over X so, (X,τ) is called topological space. Topology also helpful in different properties like convergence, existence, convex and many other. 4 Int. J. Anal. Appl. (2022), 20:65 3. Some Basic Results Proposition (3.1) ( [3]) Let Z be a non-empty subset K of a Banach space K and T be a self mappings. (a1) If T be non-expansive mapping then T satisfies condition (C). (a2) Every mapping satisfying condition (C) with a fixed point is quasi non-expansive. (a3) If T satisfies condition (C) ‖xo −Tyo‖≤ 3‖Txo −yo‖ =⇒‖Txo −Tyo‖ + ‖xo −yo‖∀xo,yo ∈ C. Lemma (3.2) Let {λm}∞m=0 and {µm} ∞ m=0 be a non negative real sequences satisfying the given inequality λm+1 ≤ (1 − ξm)λm + µm, also ξm ∈ (0, 1) ∀ m ∈ N, Σ∞m=0 ξm = ∞ and µm ξm → 0 as m →∞, then limm→∞λm = 0. Lemma (3.3) ( [15]) Let{λn}∞n=0 be a non-negative real sequence for which assume that ∃ n0 ∈ N such that ∀ n ≥ n0, the given inequalities satisfies λn+1 ≤ (1 − νn)λn + νnµn, also νn ∈ (0, 1) ∀ n ∈ N, Σ∞n=0νn = ∞ and µn ≥ 0 ∀, n ∈ N, so 0 ≤ limn→∞ sup λn ≤ limn→∞ sup µn. Lemma (3.4) [13] Let K be a uniformly convex Banach space and T be a self-mapping over a weakly compact convex subset K. Consider that T fulfil condition (C), then T has a fixed point. Lemma (3.5) Suppose that X be a subset K of a Banach space with the Opial’s property [12] and T be a self mapping over X. Suppose T satisfies the criteria of condition (C). If {$n} converges weakly to τ and limn→∞‖$n−T$n = 0‖ , then Tτ = τ. It is I −T demi-closed at 0. Here we define our new iterative process, it has better approximations and have faster rate of con- vergence then previous all (for further details see [11]). Now we generalized our results by this faster iterative scheme u0 ∈ K zn = T [(1 −δn)un + δnTun] yn = T [(1 −αn)Tun + αnTzn] un+1 = Tyn. (1) 4. Main Results In this section we generalize results via new faster iterative scheme to approximate fixed point of generalized contraction and Suzuki’s non-expansive mappings. We generalize strong convergence results in closed convex Banach space and some data dependence results are also deduce. Theorem 4.1. Suppose K be a non-empty closed convex subset of a Banach space Z and T : K → K be generalized contraction mapping. Assume {hn}∞n=0 be an iterative sequence which is generated by (1), with the real sequence {ηn} ∞ n=0 and {κn} ∞ n=0 in [0, 1] satisfying Σ ∞ n=0ηnγn = 0 then, {hn} ∞ n=0 converges strongly to a unique fixed point of T. Int. J. Anal. Appl. (2022), 20:65 5 Proof. The well-known Banach principle has guarantees of existence and uniqueness of fixed point g. We prove that {hn} converges to a fixed point g, by using (1) we get ‖zn −g‖ = ‖T [(1 −γn) hn + γnThn] −g‖ ≤ h max[‖(1 −γn)h + γnThn −g‖ , ‖((1 −γn)hn + γnThn) −T{(1 −γn)hn + hnγn}‖ , ‖g −Tg‖ ,‖(1 −γn)hn + γnThn −Tg‖ + ‖g −T ((1 −γn)hn + γnhn)‖] ≤ h max[‖(1 −γn) hn + γnThn −Tg‖ ,‖((1 −γn)hn + γnThn) −zn‖ , ‖(1 −γn)hn + γnThn −g‖ + ‖g −zn‖]. (2) Case#1 Let ‖zn −g‖≤ h[‖(1 −γn) hn + γnThn −g‖]. (3) Case#2 Let ‖zn −g‖ ≤ h[‖(1 −γn) hn + γnThn −zn‖] = h[‖(1 −γn) hn + γnThn −g + g −zn‖] ≤ [‖(1 −γn) hn + γnThn −g‖ + ‖zn −g‖] ‖zn −g‖ ≤ h 1 −h [‖(1 −γn) hn + γnhn −g‖]. (4) Case#3 Let ‖zn −g‖ ≤ h[‖(1 −γn) hn + γnThn −g‖ + ‖zn −g‖] ‖zn −g‖ ≤ h 1 −h [‖(1 −γn) hn + γnThn −g‖]. Let η = max[h, h 1−h ] ∈ (0, 1) ‖zn −g‖ ≤ η‖(1 −γn) hn + γnhn −g‖ ≤ η‖(1 −γn) hn − (1 −γn)g + γnThn −γng‖ ≤ η [(1 −γn)‖hn −g‖ + γn‖Thn −g‖ . Now ‖Thn −g‖ ≤ h max[‖hn −g‖ ,‖Th−g‖ ,‖Tg −g‖ + ‖g −Thn‖] = h max[‖hn −g‖ ,‖Thn −g‖ ,‖Thn −g‖ + ‖hn −g‖] = h max[‖hn −g‖ ,‖Thn −g‖ + ‖hn −g‖] ≤ η‖hn −g‖ . 6 Int. J. Anal. Appl. (2022), 20:65 ‖zn −g‖ ≤ η [(1 −γn)‖hn −g‖ + γnη‖hn −g‖ ≤ η [(1 −γn + γnη)]‖hn −g‖ ≤ η [(1 −γn)(1 −α)]‖hn −g‖ . (5) Similarly ∥∥h′n −g∥∥ = ‖T (1 −ηn)Thn + ηnTzn −Tg‖ ≤ h max[‖(1 −ηn)Thn + ηnTzn −g‖ , ‖(1 −ηn)Thn + ηnTzn −T ((1 −ηn)Thn + ηnTzn)‖ , ‖g −Tg‖ ,‖((1 −ηn)Thn + ηnTzn) −Tg‖ + ‖g −T ((1 −ηn)Thn + ηnTzn)‖ . Case#1 ∥∥h′n −g∥∥ ≤ h[‖(1 −ηn)Thn + ηnTzn −g‖]. (6) Case#2 ∥∥h′n −g∥∥ ≤ h[∥∥(1 −ηn)Thn + ηnTzn −h′n∥∥] = h[ ∥∥(1 −ηn)Thn + ηnTzn −g + g −h′n∥∥]∥∥h′n −g∥∥ ≤ h1 −h[‖(1 −ηn)Thn + ηnTzn −g‖]. (7) Case#3 ∥∥h′n −g∥∥ ≤ h[‖(1 −ηn)Tqn + ηnTzn −q‖ + ∥∥q −h′n∥∥]∥∥h′n −g∥∥ ≤ h1 −h[‖(1 −ηn)Thn + ηnTzn −q‖]. Let η = max{h, h 1−h}∈ (0, 1)∥∥h′n −g∥∥ ≤ η [‖(1 −ηn)Thn + ηnTzn −g‖] = η [(1 −ηn)‖Thn −g‖ + ηn‖Tzn −g‖] ≤ η [(1 −ηn)‖Thn −g‖ + ηηn‖zn −g‖] ≤ η [η (1 −ηn)‖hn −g‖ + η 2ηn(1 −γn(1 −η))‖hn −g‖] ≤ η2[(1 −ηn)‖hn −g‖ + ηηn(1 −γn(1 −η))‖hn −g‖] ≤ η2[(1 −ηn)‖hn −g‖ + ηηn(1 −γn(1 −η))‖hn −g‖] ≤ η2[(1 −ηn + ηηn −ηηnγn(1 −η)]‖hn −g‖ ≤ η2[(1 −ηn(1 −η) −ηηnγn(1 −ηn))]‖hn −g‖ ∥∥h′n −g∥∥ ≤ η2[(1 − (1 −η) ηn(1 + ηγn)]‖hn −g‖ (8) Int. J. Anal. Appl. (2022), 20:65 7 ‖hn+1 −g‖ = ∥∥Th′n −g∥∥ = ‖T [T (1 −ηn) Thn + ηnTzn] −g‖ ≤ max[ ∥∥h′n −g∥∥ ,∥∥h′n −Th′n∥∥ ,‖g −Tg‖ ,∥∥h′n −Tg∥∥ + ∥∥g −Th′n∥∥ ≤ h max‖T (1 −ηn) Thn + ηnTzn −g‖ , ‖T ((1 −ηn)Thn + ηnTzn) −T (T (1 −ηn)Thn + ηnTzn))‖ , ‖Tg −g‖ ,‖T (T (1 −ηn)Thn −ηnTzn)) −g‖ +‖T ((1 −ηn)Thn + ηnTzn) −g‖] = h max[‖hn −g‖ , ∥∥h′n −Th′n∥∥ , 0,∥∥Th′n −g∥∥ + ∥∥h′n −g∥∥ = h max[ ∥∥h′n −g∥∥ ,∥∥hn+1 −h′n∥∥ ,‖hn+1 −g‖ + ∥∥h′n −g∥∥ . Case#1 ‖hn+1 −g‖≤ h ∥∥h′n −g∥∥ . Case#2 ‖hn+1 −g‖≤ h 1 −h ∥∥h′n −g∥∥ . Case#3 ‖hn+1 −g‖≤ h 1 −h ∥∥h′n −g∥∥ η = max{h, h 1−h}∈ (0, 1) ‖hn+1 −g‖ ≤ η ∥∥h′n −g∥∥ ≤ η[η2(1 −ηn(1 + ηγn) (1 −η))‖hn −g‖] ≤ η3(1 −ηn(1 + ηγn) (1 −η))‖hn −g‖]. (9) Repetition of above scheme gives the following inequalities ‖hn+1 −g‖ ≤ η3(1 −ηn(1 + ηγn) (1 −η))‖hn −g‖ ‖hn −g‖ ≤ η3(1 −ηn−1(1 + ηγn−1) (1 −η))‖hn−1 −g‖ ‖hn−1 −g‖ ≤ η3(1 −ηn−2(1 + ηγn−2) (1 −η))‖hn−2 −g‖ ... ‖h1 −g‖ ≤ η3(1 −η0(1 + ηγ0) (1 −η))‖h0 −g‖ . (10) From (10) we can easily derive ‖hn+1 −g‖≤‖h0 −g‖η3(n+1)Πnk=01 −ηk(1 + ηγk) (1 −η) (11) 8 Int. J. Anal. Appl. (2022), 20:65 Where 1 −ηk(1 + ηγk) (1 −η) < 1 because η ∈ (0, 1) and ηnγn ∈ (0, 1) ∀ n ∈ N. We know that 1 −h ≤ %−h ∀ x ∈ (0, 1) then by (11), we have ‖hn+1 −g‖≤ ‖h0 −q‖η3(n+1) %(1−η)Σn k=0 ηk(1 + ηγk) (12) Taking limit of both sides in (12), we get limn→∞‖hn −g‖ i.e hn → g of n →∞ as required. � Theorem 4.2. Suppose that K be a non-empty closed convex subset of a Banach space Z and T : K → K be a generalized contraction mappings. Consider {xn}∞n=0 be an iterative sequence that is generated from (1) with real sequences {αn}∞n=0 and {βn} ∞ n=o in [0, 1] satisfying the criteria of Σ∞n=0αnβn = ∞. Then, iteration scheme (1) be T- stable. Proof. Let {un}∞n=0 ⊂ Z be arbitrary sequence in K. Suppose that the sequence generated from (1) be xn+1 = f (T,xn) converging to a unique fixed point q (by theorem 4.1) and �n = ||un+1−f (T,un)|| we prove that limn→∞ �n = 0 ⇐⇒ limn→∞un = q. Assume limn→∞ �n = 0 we take ‖un+1 −q‖ ≤ ‖un+1 − f (T,un)‖ + ‖f (T,un) −q‖ = �n + ‖T (T ((1 −βn)Tun + βnT ((1 −αn)un + αnTun))) −q‖ ≤ α3(1 − (1 −α))αn(1 + βnα)‖un −q‖ + �n. Since α ∈ (0, 1) and αn,βn ∈ [0, 1] ∀n ∈ N and limn→∞�n = 0. So, by above inequality and lemma 3.2 which leads limn→∞‖un −q‖ = 0. Hence limn→∞un = q. Conversely Consider that limn→∞un = q we get �n = ‖un+1 − f (T,un)‖ ≤ ‖un+1 −q‖ + ‖f (T,un) −q‖ ≤ ‖un+1 −q‖ + α3(1 − (1 −α)αn(1 + αβn)‖un −q‖ ⇐⇒ limn→∞�n = 0. Hence, (1) is stable w.r.t T . � Theorem 4.3. Suppose that K be a non-empty closed convex subset of a Banach space Z and T : K → K be a generalized contraction mapping of a fixed point p . It is given that u0 = x0 ∈ C, let {un}∞n=0 and {xn} ∞ n=0 be iterative sequences generated by (1) respectively, with real sequences {αn}∞n=0 and {βn} ∞ n=0 in [0, 1] satisfying (S1) α ≤ αn < 1 and β ≤ βn < 1, for some results like α,β > 0 and ∀ n ∈ N. Then, {xn}∞n=0 converges to p faster as than {un}∞n=0 does. Int. J. Anal. Appl. (2022), 20:65 9 Proof. By (11) we get ‖xn+1 −p‖≤‖x0 −p‖α3(n+1)Πnk=0(1 − (1 −α))αk(1 + αβk). (13) The following inequality is due to (9) and Lemma (3.2) which is obtained from (1), also converges to a unique fixed point p. ‖un+1 −p‖≤‖u0 −p‖α2(n+1)Πnk=0(1 − (1 −α))αk(1 + αβk) (14) Together with supposition (S1) and (13) implies that ‖xn+1 −p‖ ≤ ‖x0 −p‖α3(n+1)Πnk=0[(1 − (1 −α))α(1 + αβ)] = ‖x0 −p‖α3(n+1)[(1 − (1 −α))α(1 + αβ)]n+1. (15) Similarly (15) and supposition (S1) ‖un+1 −p‖ = ‖u0 −p‖α2(n+1)Πnk=0(1 − (1 −α))α(1 + βα) = ‖u0 −p‖α2(n+1)[(1 − (1 −α))α(1 + βα)]n+1 (16) Define an = ‖x0 −p‖α3(n+1)[(1 − (1 −α))α(1 + αβ)]n+1 bn = ‖u0 −p‖α2(n+1)[(1 − (1 −α))α(1 + αβ)]n+1 Then Ψn = an bn = ‖x0 −p‖α3(n+1)[(1 − (1 −α))α(1 + αβ)]n+1 ‖u0 −p‖α2(n+1)[(1 − (1 −α))α(1 + αβ)]n+1 = ∥∥αn+1∥∥ (17) Since lim n→∞ Ψn+1 Ψn = lim n→∞ αn+2 αn+1 = α < 1 By applying the ratio test we get Σ∞n=0Ψn < ∞ Hence from (17), we have lim n→∞ an bn = lim n→∞ Ψn = 0 Implies that {xn}∞n=0 is faster than {un} ∞ n=0. Now we prove following data dependence results. � 10 Int. J. Anal. Appl. (2022), 20:65 Theorem 4.4. Suppose that T̃ be an approximate operator of a generalized contraction mapping T. Consider that {hn}∞n=0 be an iterative sequence generated from equation (1) for T and we define an iterative sequence {h̃n}∞n=0 which is given as h̃0 ∈ K z̃n = T [(1 −γn)h̃n + γnT̃ h̃n] h̃′n = T̃ [(1 −αn)T̃ h̃n + αnT̃ z̃n] h̃n+1 = T̃ h̃ ′ n (18) With real sequences {αn}∞n=0 and {γn} ∞ n=0 in [0, 1] which satisfying the (I) 1 2 ≤ αnγn ∀ n ∈ N (II) Σ∞n=0αnγn = ∞ if Tq = q and T̃ q̃ =q̃ such that limn→∞ h̃n = q̃, then we get ‖q − q̃‖≤ 7� 1 −α Where � ≥ 0 is a fixed number. Proof. It follows from (1) and (18) ‖zn − z̃n‖ = ∥∥∥T (1 −γn)hn + γnThn) − T̃ ((1 −γn)h̃n −γnT̃ h̃n)∥∥∥ ≤ ∥∥∥T (1 −γn)hn + γnThn) −T (1 −γn)h̃n + γnT̃ h̃n)∥∥∥ + ∥∥∥T ((1 −γn)h̃n + γnT̃ h̃n) − T̃ ((1 −γn)h̃n + γnT̃ h̃n)∥∥∥ ≤ α ∥∥∥(1 −γn)hn + γnThn − (1 −γn)h̃n −γnT̃ h̃n)∥∥∥ + � ≤ α[(1 −γn) ∥∥∥hn + h̃n∥∥∥ + γn ∥∥∥Thn − T̃ h̃n∥∥∥ + � ≤ α[(1 −γn) ∥∥∥hn − h̃n∥∥∥ + γn{∥∥∥Thn − T̃ h̃n∥∥∥ + ∥∥∥Th̃n − T̃ h̃n∥∥∥} + � ≤ α[(1 −γn) ∥∥∥hn − h̃n∥∥∥ + γnα∥∥∥hn − h̃n∥∥∥ + γn�] + � ≤ α[1 −γn (1 −α) ∥∥∥hn − h̃n∥∥∥ + γn�] + �. (19) Using (19), we have∥∥∥h′n − h̃′n∥∥∥ = ∥∥∥T ((1 −αn)Thn + αnTzn) − T̃ ((1 −αn)T̃ h̃n + T̃ z̃n)∥∥∥ ≤ T ((1 −αn)Thn + αnTzn) −T ((1 −αn)T̃ h̃n + αnT̃ z̃n) +T ((1 −αn)T̃ h̃n + αnT̃ z̃n) − T̃ ((1 −αn)T̃ h̃n + αnT̃ z̃n) ≤ α ∥∥∥(1 −αn)Thn + αnTzn − (1 −αn)T̃ h̃n −αnT̃ z̃n)∥∥∥ + � ≤ α[(1 −αn) ∥∥∥Thn − T̃ h̃n∥∥∥ + αn ∥∥∥Tzn − T̃ z̃n∥∥∥ + � Int. J. Anal. Appl. (2022), 20:65 11 ≤ α[(1 −αn) ∥∥∥Thn −Th̃n∥∥∥ + ∥∥∥Th̃n − T̃ h̃n∥∥∥ +αn[‖Tzn −Tz̃n‖ + ∥∥∥Tz̃n − T̃ z̃n∥∥∥] + � ≤ α[(1 −αn)α ∥∥∥xn − h̃n∥∥∥ +αnα[α (1 −γn) (1 −α) ||hn − h̃n|| + γn� + �] + � ≤ α2[(1 −αn) ∥∥∥xn − h̃n∥∥∥ + α3αn[1 −γn(1 −α) ∥∥∥hn − h̃n∥∥∥ +α3αnγn + α 2αn]� ≤ α2[1 −αn + αnα + α (1 −α) αnγn)] ∥∥∥hn − h̃n∥∥∥ +α�(1 + ααnγn) + � ≤ α2[1 − (1 −α)αn −α (1 −α) αnγn)] ∥∥∥hn − h̃n∥∥∥ +α�(1 + ααnγn) + � ≤ α2[1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +α�(1 + ααnγn) + �. (20) By using (20), we have∥∥∥h′n+1 − h̃′n−1∥∥∥ = ∥∥∥Th′n − T̃ h̃′n∥∥∥ ≤ ∥∥∥h′n − h̃′n∥∥∥ + � ≤ α3[1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +α2�(1 + ααnγn)] + �α + � ≤ [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +�(1 + ααnγn)] + � + � ≤ [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +αnγn� + 3� ≤ [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +αnγn� + 3 (1 −αnγn + αnγn) � (21) By supposition (I) we have 1 −αnγn ≤ αnγn∥∥∥hn+1 − h̃n+1∥∥∥ ≤ [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +7αnγn� = [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +αnγn (1 −α) 7� 1 −α . (22) 12 Int. J. Anal. Appl. (2022), 20:65 Let Ψn = ∥∥∥hn − h̃n∥∥∥ , φn = (1 −α) αn(1 + αγn), φn = 7�1−α, then from lemma 3.2 together with (22) we get 0 ≤ lim n→∞ sup ∥∥∥hn − h̃n∥∥∥ ≤ lim n→∞ sup 7� 1 −α (23) Since by theorem 4.1 we have limn→∞hn = q and from supposition I and II we get limn→∞h̃n = q̃ by using these together with (23) and we get ‖q − q̃‖≤ 7� 1−α as required. � 5. Convergence Results of Suzuki Generalized Non-Expansive Mappings of Condition (C) In this section, we prove some weak and strong convergence theorems of a sequence generated from new iterative scheme of Suzuki generalized non-expansive mappings with condition (C) by uniformly convex Banach spaces. Lemma 5.1. Suppose that K be a non-empty uniformly closed convex subset of a Banach space Z. Let T : K → K be a mapping satisfying condition (C) with Q(T ) 6= 0. For arbitrary chosen h0 ∈ K, Consider that a sequence {hn} is generated from (1), then limn→∞‖hn −q‖ exists for any g ∈ Q(T ). Proof. Consider that g ∈ Q(T ) and z ∈ K. So T satisfies condition (C) ≤ 0 1 2 ‖g −Tg‖ = 0 ≤‖g −z‖⇒‖Tg −Tz‖≤‖g −z‖ so by proposition (a2) we get ‖zn −g‖ = ‖T [(1 −γn)hn + γnδTn] −g‖ = ‖T [(1 −γn)hn + γnTn] −Tg‖ ≤ ||(1 −γn)hn + γnTn −g|| ≤ (1 −γn)‖hn −g‖ + γn‖Txn −g‖ ≤ (1 −γn)‖hn −g‖ + γn‖hn −g‖ ≤ ‖hn −g‖ . (24) By using (24) we have ∥∥h′n −g∥∥ = ‖T ((1 −δn)Thn + δnTzn) −g‖ ≤ ‖(1 −δn)Thn + δnTzn −g‖ ≤ (1 −δn)‖Txn −g‖ + δn‖Tzn −g‖ ≤ (1 −δn)‖hn −g‖ + δn‖zn −g‖ ≤ (1 −δn)‖hn −g‖ + δn‖hn −g‖ = ‖hn −g‖ . (25) Int. J. Anal. Appl. (2022), 20:65 13 Similarly by using (25) we have ‖hn+1 −g‖ = ∥∥Th′n −g∥∥ ≤ ∥∥h′n −g∥∥ ≤ ‖hn −g‖ (26) ⇒‖hn −g‖ be a bounded and decreasing ∀ g ∈ Q (T ) So, limn→∞‖hn −g‖ exist as required. � Theorem 5.1. Suppose that K is a non-empty and uniformly closed convex of subset of a Banach space Z. and let T : K → K be a mapping satisfying condition (C). For arbitrary chosen h0 ∈ K, consider that the sequence {hn} be generated from (1) ∀ n ≥ 1, where {αn} and {βn} are two sequences of real numbers in [u,v] for some u,v with 0 < u ≤ v < 1. So, Q(T ) 6= θ ⇐⇒ {hn} is bounded sequence and limn→∞‖Thn −hn‖ = 0. Proof. Suppose that Q(T ) 6= φ and let g ∈ Q(T ). Then, from Lemma 5.1, limn→∞‖hn −g‖ exists and {hn} is bounded. lim n→∞ ‖hn −g‖ = r (27) From (24) and (27), we have lim n→∞ sup‖zn −g‖≤ lim n→∞ sup‖hn −g‖ = r (28) So by proposition 3.1 (a2) lim n→∞ sup‖Thn −g‖≤ lim n→∞ sup‖hn −g‖ = r (29) Also ‖hn+1 −g‖ = ∥∥Th′n −g∥∥ ≤ ∥∥h′n −g∥∥ = ‖T ((1 −αn)Thn + αnTzn) −g‖ ≤ ‖(1 −αn)Thn + αnTzn −g‖ ≤ (1 −αn)‖Thn −g‖ + αn‖Tzn −g‖ ≤ (1 −αn)‖hn −g‖ + αn‖zn −g‖ ≤ ‖hn −g‖−αn‖hn −g‖ + αn‖zn −g‖ (30) This implies ‖hn+1 −g‖−‖hn −g‖ αn ≤ ‖zn −g‖−‖yn −g‖ 14 Int. J. Anal. Appl. (2022), 20:65 ‖hn+1 −g‖−‖hn −g‖ ≤ ‖hn+1 −g‖−‖hn −g‖ αn ≤ ‖zn −g‖−‖hn −g‖ =⇒ ‖xn+1 −p‖≤‖zn −p‖ Therefore r ≤ limn→∞ inf ‖zn −g‖ from (28) and (30), we have r = ||zn −g|| = lim n→∞ ‖T ((1 −γn)hn + γnTn)g‖ ≤ lim n→∞ ‖γn(Thn −g) + (1 −γn)hn −g‖ (31) From (27) , (29) and (31) together with Lemma 3.3 we get limn→∞‖Thn −hn‖ = 0. Conversely Suppose that {hn} is bounded and lim n→∞ ‖Thn −hn‖ = 0 Consider that g ∈ (c,{hn}) by proposition 3.1 we get r (Tg,{hn}) = lim n→∞ sup‖hn −Tg‖ ≤ lim n→∞ sup 3‖hn −Tg‖ + ‖hn −g‖ ≤ lim n→∞ sup‖hn −g‖ = r (g,{hn}) . This implies that Tg ∈ B(K,{hn}). Since Z is uniformly convex, B(K,{hn}) singleton and we get Tg = g. Hence Q(T ) 6= φ. We are able to prove weak convergence theorem. � Theorem 5.2. Let K be a non-empty closed convex subset of a uniformly convex Banach space Z, with Opial property, and consider that T : K → K be a mapping satisfying condition (C). For arbitrary chosen x0 ∈ C, let the sequence {xn} is generated from (1) of all n ≥ 1, where {αn} and {βn} are sequences of a real numbers in [l,m] for some l,m with 0 < l ≤ m < 1 such that Q(T ) 6= φ. Then {xn} converges weakly to a fixed point of T. Proof. From Theorem 5.1 we get {xm} be bounded and limn→∞‖Txm −xm‖ = 0. Since, X be a uniformly convex and reflexive thereof, from Eberlin’s theorem ∃ a subsequence {xmu} of {xm} which converges weakly to some points q1 ∈ Z. Since, K is closed and convex from Mazur’s theorem q1 ∈ K. From lemma 3.4, q1 ∈ Q(T ). Now, we prove that {xm} converges weakly to q1. In fact if this is false than there must exist a subsequence {xmv} of {xm} such that {xmv} converges weakly to Int. J. Anal. Appl. (2022), 20:65 15 q2 ∈ K and q2 6= q1. From lemma 3.5 q2 ∈ Q(T ). Since, limn→∞‖xm −p‖ exists ∃, p ∈ Q(T ). By Theorem 5.1 and by Opial’s property, we get lim m→∞ inf ‖xm −q1‖ = lim u→∞ inf ‖xmu −q1‖ < lim u→∞ inf ‖xmu −q2‖ = lim m→∞ inf ‖xm −q2‖ = lim v→∞ inf ‖xmv −q2‖ < lim v→∞ inf ‖xmv −q1‖ = lim m→∞ inf ‖xm −q1‖ which is contradiction so q1 = q2. This implies {xm} converges weakly to a fixed point of T . Now we establish strong convergence results. � Theorem 5.3. Suppose that K be a non-empty compact convex subset of a uniformly convex Banach space Z. Let T : K → K be a mapping satisfying condition (C). For arbitrary chosen l0 ∈ K, consider that a sequence {ln} is generated from (1) ∀, n ≥ 1, where {αn} and {βn} are two sequences of real numbers in [u,v] for some u,v with 0 < u ≤ v < 1. Then {xn} converges strongly to a fixed point for T. Proof. By lemma 3.4, Q(T ) 6= φ and by theorem 5.1 we have limn→∞‖Tln − ln‖ = 0. Since K is compact and ∃ a subsequence {lnk} of {ln} such as {lnk} converges strongly to p for some p ∈ K. From proposition a3 we get ‖lnk −Tp‖≤ 3‖Tlnk − lnk‖ + ‖lnk −p‖ For all n ≥ 1. Suppose that k → ∞, than we get Tp = p, i.e., p ∈ Q(T ). Since, from lemma 5.1, limn→∞‖ln −p‖ exists for all p ∈ Q(T ), since xn converges strongly to p. Senter and Dotson [14] both mathematicians discovered notion of mapping which satisfying condition (I) as. A mapping T : K → K is called to satisfy condition (I), if ∃ an increasing function f : [0,∞) → [0,∞) in f (0) = 0 and f (r ′) > 0 for all r ′ > 0 such as ‖l −Tl‖ ≥ f (d(l,Q(T ))) for all l ∈ K, and d(l, Q(T )) = infp ∈ Q(T )‖l −p‖. � Theorem 5.4. Suppose that K be a non-empty uniformly closed convex subset of a Banach space Z, and consider that T : K → K be a mapping which satisfying condition (C). For arbitrary chosen y0 ∈ K, let the sequence {yn} be generated from (1) for all n ≥ 1. Since {αn} and {βn} are sequences of real numbers in [u,v] for some u,v with 0 < u ≤ v < 1 such that Q(T ) 6= φ. If T fulfil condition (I), so {yn} converges strongly to a fixed point of T . 16 Int. J. Anal. Appl. (2022), 20:65 Proof. By Lemma 5.1, we get limn→∞‖yn −p‖ exist ∀ p ∈ Q(T ) and limn→∞d(yn,Q(T )) exists. Consider that limn→∞‖yn −Q‖ = s′ for some s′ ≥ 0. If s′ = 0 then results follows. Suppose that s′ > 0, from proposition (3.1) and condition (I), f (d(yn,Q(T ))) ≤‖Tyn −yn‖ (32) Since, Q(T ) 6= φ, from theorem 5.2 with (32) ⇐⇒ limn→∞‖Tyn −yn‖ = 0 lim n→∞ f (d(yn,Q(T ))) = 0 (33) Since f is an increasing function and by (33), we have limn→∞d(yn,Q(T )) = 0. Thus we get a subsequence {ynk} of {yn} and a sequence {y ′k}⊂ Q(T ) such that∥∥ynk −y ′k∥∥ < 12k ∀, k ∈ N than by applying (26), we have∥∥ynk+1 −y ′k∥∥ ≤ ∥∥ynk −y ′k∥∥ < 12k ≤ 1 2k+1 + 1 2k 1 2k−1 → 0 as k →∞ This shows that {y ′k} is a Cauchy sequence with Q(T ) and so it converges to a point p. Since Q(T ) is closed, therefore p ∈ Q(T ) and then {ynk} converges strongly to p. Since limn→∞‖yn −p‖ exists, we get yn → p ∈ Q(T ). � 6. Conclusion In this article we discussed generalized results by using new iterative scheme to approximate fixed point of generalized contraction and Suzuki non-expansive mappings. Here we developed new strongly convergence results of generalized contraction mappings of closed convex Banach space and also pro- duced some new data dependence results. In addition, we proved some weak and strong convergence results in sense of generalized Suzuki non expansive mapping by applying condition (C). Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] A. Asghar, A. Qayyum, N. Muhammad, Different Types of Topological Structures by Graphs, Eur. J. Math. Anal. 3 (2022), 3. https://doi.org/10.28924/ada/ma.3.3. [2] F.e. Browder, Nonexpansive Nonlinear Operators in a Banach Space, Proc. Natl. Acad. Sci. U.S.A. 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041. [3] T. Suzuki, Fixed Point Theorems and Convergence Theorems for Some Generalized Nonexpansive Mappings, J. Math. Anal. Appl. 340 (2008), 1088–1095. https://doi.org/10.1016/j.jmaa.2007.09.023. https://doi.org/10.28924/ada/ma.3.3 https://doi.org/10.1073/pnas.54.4.1041 https://doi.org/10.1016/j.jmaa.2007.09.023 Int. J. Anal. Appl. (2022), 20:65 17 [4] J. Garcia-Falset, E. Llorens-Fuster, T. Suzuki, Fixed Point Theory for a Class of Generalized Nonexpansive Mappings, J. Math. Anal. Appl. 375 (2011), 185–195. https://doi.org/10.1016/j.jmaa.2010.08.069. [5] V.K. Sahu, H.K. Pathak, R. Tiwari, Convergence Theorems for New Iteration Scheme and Comparison Results, Aligarh Bull. Math. 35 (2016), 19-42. [6] W. Kassab, T. Turcanu, Numerical Reckoning Fixed Points of (%E)-Type Mappings in Modular Vector Spaces, Mathematics, 7 (2019), 390. https://doi.org/10.3390/math7050390. [7] S. Dhompongsa, W. Inthakon, A. Kaewkhao, Edelstein’s Method and Fixed Point Theorems for Some Generalized Nonexpansive Mappings, J. Math. Anal. Appl. 350 (2009), 12–17. https://doi.org/10.1016/j.jmaa.2008.08. 045. [8] J. Garcia-Falset, E. Llorens-Fuster, T. Suzuki, Fixed Point Theory for a Class of Generalized Nonexpansive Mappings, J. Math. Anal. Appl. 375 (2011), 185–195. https://doi.org/10.1016/j.jmaa.2010.08.069. [9] I. Uddin, M. Imdad, J. Ali, Convergence Theorems for a Hybrid Pair of Generalized Nonexpansive Mappings in Banach Spaces, Bull. Malays. Math. Sci. Soc. 38 (2014), 695–705. https://doi.org/10.1007/s40840-014-0044-6. [10] M. De la Sen, M. Abbas, On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces, J. Math. 2019 (2019), 1356918. https://doi.org/10.1155/2019/1356918. [11] N. Muhammad, A. Asghar, S. Irum, A. Akgül, E.M. Khalil, M. Inc, Approximation of Fixed Point of Generalized Non-Expansive Mapping via New Faster Iterative Scheme in Metric Domain, AIMS Math. 8 (2023), 2856–2870. https://doi.org/10.3934/math.2023149. [12] Z. Opial, Weak Convergence of the Sequence of Successive Approximations of Non-Expansive Mappings, Bull. Amer. Math. Soc. 73 (1967), 595-597. [13] J. Schu, Weak and Strong Convergence to Fixed Points of Asymptotically Nonexpansive Mappings, Bull. Austral. Math. Soc. 43 (1991), 153–159. https://doi.org/10.1017/s0004972700028884. [14] H.F. Senter, W.G. Dotson, Approximating Fixed Points of Nonexpansive Mappings, Proc. Amer. Math. Soc. 44 (1974), 375–380. https://doi.org/10.1090/s0002-9939-1974-0346608-8. [15] S.M. Soltuz, Data Dependence of Mann Iteration, Octogon Math. Mag. 9 (2001), 825-828. https://dl.acm. org/doi/10.5555/605858.605878. [16] V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, (2007). [17] A. Qayyum, A weighted Ostrowski-Grüss Type Inequality of Twice Differentiable Mappings and Applications, Int. J. Math. Comput. 1 (2008), 63-71. [18] M.M. Saleem, Z. Ullah, T. Abbas, M.B. Raza, A. Qayyum, A New Ostrowski’s Type Inequality for Quadratic Kernel, Int. J. Anal. Appl. 20 (2022), 28. https://doi.org/10.28924/2291-8639-20-2022-28. [19] T. Hussain, M.A. Mustafa, A. Qayyum, A New Version of Integral Inequalities of a Linear Function of Bounded Variation, Turk. J. Inequal. 6 (2022), 7-16. [20] J. Amjad, A. Qayyum, S. Fahad, M. Arslan, Some New Generalized Ostrowski Type Inequalities With New Error Bounds, Innov. J. Math. 1 (2022), 30–43. https://doi.org/10.55059/ijm.2022.1.2/23. https://doi.org/10.1016/j.jmaa.2010.08.069 https://doi.org/10.3390/math7050390 https://doi.org/10.1016/j.jmaa.2008.08.045 https://doi.org/10.1016/j.jmaa.2008.08.045 https://doi.org/10.1016/j.jmaa.2010.08.069 https://doi.org/10.1007/s40840-014-0044-6 https://doi.org/10.1155/2019/1356918 https://doi.org/10.3934/math.2023149 https://doi.org/10.1017/s0004972700028884 https://doi.org/10.1090/s0002-9939-1974-0346608-8 https://dl.acm.org/doi/10.5555/605858.605878 https://dl.acm.org/doi/10.5555/605858.605878 https://doi.org/10.28924/2291-8639-20-2022-28 https://doi.org/10.55059/ijm.2022.1.2/23 1. Introduction 2. Preliminaries 3. Some Basic Results 4. Main Results 5. Convergence Results of Suzuki Generalized Non-Expansive Mappings of Condition ( C) 6. Conclusion References