©2022 Ada Academica https://adac.eeEur. J. Math. Anal. 2 (2022) 1doi: 10.28924/ada/ma.2.1 Convergence and Stability of New Approximation Algorithms for Certain Contractive-Type Mappings Imo Kalu Agwu∗, Donatus Ikechi Igbokwe Department of Mathematics, Micheal Okpara University of Agriculture, Umudike, Umuahia Abia State, Nigeria agwu.imoh@mouau.edu.ng, igbokwedi@yahoo.com ∗Correspondence: agwu.imoh@mouau.edu.ng Abstract. We present new fixed points algorithms called multistep H-iterative scheme and multistepSH-iterative scheme. Under certain contractive-type condition, convergence and stability results wereestablished without any imposition of the ’sum conditions’, which to a large extent make some existingiterative schemes so far studied by other authors in this direction practically inefficient. Our resultscomplement and improve some recent results in literature. 1. Introduction There is an intimate connection existing between nonlinear problems and fixed point problemsof related contractive-type operators. As a result, researchers have focused more attention onfinding approximate fixed points of different contractive-type mappings in recent times; see, forexample, [5], [6], [7], [8], [9], [10], [11], [18], [25], [28], etc. and the reference contained in them. Let Xbe a normed linear space and Γ : X −→ X a given of X. We represent the set of fixed points of Γby F (Γ) = {q ∈ X : q = Γ(q)}.For the past forty years or so, some investigation of fixed points via iterative schemes have beena flourishing area of research for many mathematicians. Mann [21], Ishikawa [22] and Noor [19]iterative schemes, with their modifications, have been studied by different authors and differentinteresting results were obtained. However, to meet up with the demand of the modern fixedpoint theory, researchers have continually renewed their efforts toward constructing more efficientiterative schemes. In this direction, following Kirk’s introduction of his remarkable iterative schemein 1971, the results below have found thier place in the current literature.Let X and Γ be as earlier stated. Received: 1 Nov 2021. Key words and phrases. strong convergence; multistep H-iterative scheme; multistep sh-iterative scheme; stability;contractive operator; fixed point; normed linear space. 1 https://adac.ee https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 2 (a) For arbitrarily y0 ∈ X, let the sequence {yn}∞n=0 be defined iteratively as follows: yn+1 = ∑̀ j=0 αjΓ jyn, ∑̀ j=0 αj = 1,n ≥ 0. (1.1) The iteration method defined by (1.1) is due to Kirk [20]. (b) In [17], Olatinwo presented the algorithms below:(i) for an arbitrary point y0 ∈ X and for αn,t ≥ 0,αn,0 6= 0,αn,t ∈ [0, 1] and ` as a fixedinteger, define the sequence {yn}∞n=0 by yn+1 = ∑̀ t=0 αn,tΓ tyn, ∑̀ t=0 αn,t = 1,n ≥ 0 (1.2) (ii) for an arbitrary point y0 ∈ X and for ` ≥ m,αn,t βn,t ≥ 0,αn,0,βn,0 6= 0,αn,t,βn,t ∈ [0, 1] and `,m as fixed integers, define the sequence {yn}∞n=0 by yn+1 = αn,0yn + ∑̀ t=0 αn,tΓ jzn, ∑̀ t=0 αn,t = 1; zn = m∑ t=0 βn,tΓ tyn, ∑̀ t=0 βn,t = 1,n ≥ 0, (1.3) and called them Kirk-Mann and Kirk-Ishikawa algorithms, respectively. (c) Chugh and Kumar [25] presented the following iterative scheme: for an arbitrary point y0 ∈ X and for ` ≥ m ≥ p,αn,s,γn,r,βn,t ≥ 0,γn,0,αn,0,βn,0 6= 0,αn,s,γn,r,βn,t ∈ [0, 1] and `,m,p as fixed integers, define the sequence {yn}∞n=0 by yn+1 = γn,0yn + ∑̀ r=1 γn,rΓ rzn, ∑̀ r=0 γn,r = 1; zn = αn,0yn + m∑ s=1 αn,sΓ szn, m∑ s=0 αn,s = 1; zn = p∑ t=0 βn,tΓ tyn, p∑ t=0 βn,t = 1,n ≥ 0, (1.4) (d) Very recently, Akewe, Okeke and Olayiwola [26] presented the following general iterativescheme in the sense of Kirk [20]: (i) for an arbitrary point y0 ∈ X, for `1 ≥ `2 ≥ `3 ≥ ··· ≥ `u, for each i, αtn,s,γn,t ≥ 0,γn,0,αn,0, 6= 0, for each i, αin,s,γn,t ∈ [0, 1] and `1,`u as fixed integers for each u, https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 3 define the sequence {yn}∞n=0 by yn+1 = γn,0yn + `1∑ r=1 γn,rΓ rz1n , `1∑ k=0 αn,r = 1; ztn = α t n,0yn + `t+1∑ s=1 αtn,sΓ jzt+1n , `t+1∑ s=0 αtn,s = 1,t = 1, 2, · · · ,u − 2; zu−1n = `u∑ s=0 αu−1n,t Γ syn, `u∑ s=0 αu−1n,t = 1,u ≥ 2,n ≥ 0, (1.5) (ii) for an arbitrary point y0 ∈ X, retaining the conditions in (i), define the sequence {yn}∞n=0 by yn+1 = γn,0z 1 n + `1∑ r=1 γn,kΓ rz1n , `1∑ r=0 αn,r = 1; ztn = α t n,0z t+1 n + `t+1∑ s=1 αtn,sΓ szt+1n , `t+1∑ s=0 αtn,s = 1,t = 1, 2, · · · ,u − 2; zu−1n = `u∑ s=0 αu−1n,t Γ syn, `u∑ s=0 αu−1n,t = 1,u ≥ 2,n ≥ 0, (1.6) It is worthy to mention that in application, the stability of the iterative schemes studied aboveis quite invaluable. The first researcher to demonstrate this respecting the Banach contractionconditions is Ostrowski [13]. Afterwards, several authors have developed this subject basicallybecause of its indispensable position in the current trend of computer programing. Some recentworks in this direction could be seen in [1], [2], [3], [4], [12], [13], [14],[23], [24], [26] and the references therein. Remark 1.1. The stability and the convergence results in the papers studied were made possible due to the sum conditions imposed on the control parameters; see, for example, [20], [17], [25], [26], etc and the references therein. But in application, especially for N large enough, the iterative schemes defined by (1.1), (1.2), (1.3), (1.4) (1.5) and (1.6) become practically inefficient due to the difficulties involved in generating a family of such control parameters, the windy process involved for each sum and the computational cost. Base on the problems mentioned in Remark 1.1, it becomes necessary to ask the followingquestions: Question 1.1. Is it possible to construct an alternative iterative scheme that would address the problems generated by the sum conditions imposed on the control parameters while maintaining, in particular, the results in [26], which in a larger sense contains the results of the other papers studied? https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 4 Following the same argument as in [27] regarding the linear combination of the products ofcountably finite family of control parameters and the problems mentioned in Remark 1.1, in thispaper, we provide an affirmative answer to Question 1.1. 2. Preliminary Throughout the remaining sections, φ : R+ −→ R+,R+,N and H will denote monotone in-creasing subadditive function, the set of positive integers, the set of natural numbers and a realHilbert space, respectively. Also, the following definition, lemmas and propositions will be neededestablish our results. Definition 2.1. ( [13]) Suppose Y is a metric space and let Γ : Y −→ Y be a self-map of Y . Let {xn}∞n=0 ⊆ Y be a sequence generated by an iteration scheme xn+1 = g(Γ,xn), (2.1) where x0 ∈ Y is the initial approximation and g is some function. Suppeose {xn}∞n=0 converges to a fixed point q of Γ. Let {tn}∞n=0 ⊆ Y be an arbitrary sequence and set �n = d(tn,g(Γ,tn)),n = 1, 2, · · · Then, (2.1) is said to be Γ-stable if and only if limn→∞�n = 0 implies limn→∞yn = q. Note that in practice, the sequence {tn}∞n=0 could be obtained using the following approach: let x0 ∈ Y . Set xn+1 = g(Γ,xn) and let t0 = x0. Since, x1 = g(Γ,x0) following the rounding in thefunction Γ, the value t1 (which is estimated to be equal to x1) could be calculated to give t2, anapproximate value of g(Γ,t1). The procedure is continued to yield the sequence {tn}∞n=0, which isapproximately tha same as the sequence {xn}∞n=0. Lemma 2.1. (see, e.g., [26]) Let {τn}∞n=0 ∈R + : τn → 0 as n →∞. For 0 ≤ δ < 1, let {wn}∞n=0 be a sequence of positive numbers satisfying wn+1 ≤ δwn +τn,n = 0, 1, 2, · · · Then, wn → 0 as n →∞. Lemma 2.2. (see, e.g., [17]) Let (Y, ‖ .‖) be a normed space, the self-map Γ : Y −→ Y satisfies (1.13) and φ : R+ −→ R+ (retaining its usual meaning) be such that ψ(0) = 0,φ(Mt) = Mφ(t),M ≥ 0,t ∈R+. Then, ∀i ∈N and ∀s,t ∈ Y, we have ‖Γjs − Γjt‖≤ ρj‖s − t‖ + j∑ i=0 ( j i ) ρj−1φ(‖s − Γs‖). (2.2) Proposition 2.3. (see,e.g., [27]) Let {αi}∞i=1 ⊆ N, where k ∈ [0,R +] is fixed and N ∈ N is any integer with k + 1 ≤ N. Then, the following holds: αk + N∑ i=k+1 αi i−1∏ j=k (1 −αj) + N∏ j=k (1 −αj) = 1. (2.3) https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 5 Proposition 2.4. (see,e.g., [27]) Let t,u,v ∈ H. Let k ∈ [0,R+] be fixed and N ∈ N be such that k + 1 ≤ N. Let {vi}N−1i=1 ⊆ H and {αi} N i=1 ⊆ [0, 1]. Define y = αkt + N∑ i=k+1 αi i−1∏ j=k (1 −αj)vi−1 + N∏ j=k (1 −αj)v. Then, ‖y −u‖2 = αk‖t −u‖2 + N∑ i=k+1 αi i−1∏ j=k (1 −αj)‖vi−1 −u‖2 + N∏ j=k (1 −αj)‖v −u‖2 −αk [ N∑ i=k+1 αi i−1∏ j=k (1 −αj)‖t −vi−1‖2 + i−1∏ j=k (1 −αj)‖t −v‖2 ] −(1 −αk) [ N∑ i=k+1 αi i−1∏ j=k (1 −αj)‖vi−1 − (αi+1 + wi+1)‖2 +αN i−1∏ j=k (1 −αj)‖v −vN−1‖2 ] , where wk = ∑N i=k+1αi ∏i−1 j=k(1 −αj)vi−1 + ∏i−1 j=k(1 −αj)v,k = 1, 2, · · · ,N and wn = (1 −cn)v . 3. Main Results I Let H be a Hilbert space and let Γ : H −→ H be a self-map of X. For arbitrary x0 ∈ H definethe sequence {xn+1}∞n=0 iteratively, for s = 1, 2, · · · ,k − 2, as follows: xn+1 = δn,1xn + ∑`1 j=2δn,j ∏j−1 i=1(1 −δn,i)Γ j−1y1n + ∏`1 i=1(1 −δn,i)Γ `1y1n ; ysn = α s n,1xn + ∑`s+1 j=2 αsn,j ∏j−1 i=1(1 −α s n,i)Γ j−1ys+1n + ∏`s+1 i=1 (1 −αsn,i)Γ `1ys+1n ; yk−1n = ∑`k j=1 αk−1 n,j ∏j−1 i=1(1 −α k−1 n,i )Γj−1xn + ∏`k i=1 (1 −αk−1 n,i )Γ`kxn,k ≥ 2,n ≥ 1, (3.1) where `1 ≥ `2 ≥ `3 ≥ ··· ≥ `k , for each s, {{δn,i}∞n=0}`kj=1,{{αn,i}∞n=0}`kj=1 ∈ [0, 1] for each k and `1,`2, · · · ,`k are fixed integers (for each k). We shall call the iteration scheme defined by (3.1)the multistep IH-iteration scheme.Again, for any x0 ∈ X, we shall call the sequence {xn}∞n=0 defined recursively, for s = 1, 2, · · · ,k − 2, by xn+1 = δn,1y 1 n + ∑`1 j=2δn,j ∏j−1 i=1(1 −δn,i)Γ j−1y1n + ∏`1 i=1(1 −δn,i)Γ `1y1n ; ysn = α s n,1y s+1 n + ∑`s+1 j=2 αsn,j ∏j−1 i=1(1 −α s n,i)Γ j−1ys+1n + ∏`s+1 i=1 (1 −αsn,i)Γ `1ys+1n ; yk−1n = ∑`k j=1 αk−1 n,j ∏j−1 i=1(1 −α k−1 n,i )Γj−1xn + ∏`k i=1 (1 −αk−1 n,i )Γ`kxn,k ≥ 2,n ≥ 1, (3.2) where `1 ≥ `2 ≥ `3 ≥ ··· ≥ `k , for each s, {{δn,i}∞n=0}`kj=1,{{αn,i}∞n=0}`kj=1 ∈ [0, 1] for each k and `1,`2, · · · ,`k are fixed integers (for each k), the multistep DI-iteration scheme. https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 6 Theorem 3.1. Let H be a Hilbert space, Γ : H −→ H be a self-map of H satisfying the contractive condition ‖Γjx − Γjy‖≤ ρj‖x −y‖ + j∑ i=0 ( j i ) ρj−iφ(‖x − Γx‖), (3.3) where x,y ∈ H, 0 ≤ ρj < 1, and let φ retain its usual meaning with φ(0) = 0 and φ(Mt) = Mφ(t),M ≥ 0,t ∈ R+. For arbitrary x0 ∈ H, let {ωn}∞n=0 be the multistep H-iteration scheme defined by (3.1). Then, (i) Γ defined by (3.3) has a fixed point q; (ii) the multistep IH-iteration scheme converges strongly to q ∈ Γ. Proof. Firstly, we show that Γ satisfying condition of (3.3) has a fixed point. Assume there existstwo points q1,q2 ∈ F (Γ) with 0 < ‖q1 −q2‖. Then, we have 0 < ‖q1 −q2‖ = ‖Γjq1 − Γjq2‖ ≤ ρj‖q1 −q2‖ + j∑ i=0 ( j i ) ρj−iφ(‖q √ 1 − Γq1‖) = ρj‖q1 −q2‖ + j∑ i=0 ( j i ) ρj−iφ(0) ⇒ (1 −ρj)ρj‖q1 −q2‖≤ 0. Using the fact that ρj ∈ [[0, 1), we get 0 < 1 −ρj and ‖q1 −q2‖≤ 0.Since the norm is a nonnegative function, we get ‖q1 −q2‖ = 0; q1 = q2 = q(say). Therefore, Γconverges uniquely to a point of F (Γ).Now, we show that the sequence defined by (3.1) converges strongly to q ∈ F (Γ). Using (3.3)and Proposition 2.4 with xn+1 = y,u = q,xn = t, j = i,k = 1, Γj−1y1n = vj−1 and Γ`1y1n = v , wehave ‖xn+1 −q‖2 ≤ δn,1‖xn −q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖Γj−1y1n − Γ j−1q‖2 + `1∏ i=1 (1 −δn,i)‖Γ`1y1n − Γ `1q‖2 (3.4) But from (3.3), with y = y1n , we have ‖Γj−1y1n − Γ j−1q‖ ≤ ρj‖y1n −q‖ + j∑ i=0 ( j i ) ρj−1φ(‖q − Γq‖) = ρj‖y1n −q‖ (3.5) https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 7 Proposition 2.3, (3.4) and (3.5) imply ‖xn+1 −q‖2 ≤ δn,1‖xn −q‖2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i)‖y1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖y1n −q‖ 2 = δn,1‖xn −q‖2 + ( 1 −δ1n,1 − `1∏ i=1 (1 −δn,i)(ρj)2 ) ‖y1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖y1n −q‖ 2 = δn,1‖xn −q‖2 + ( 1 −δ1n,1 ) ‖y1n −q‖ 2 (3.6) Since `1,`k are fixed integers and αsn,i ∈ [0, 1] for each s, we have, using Proposition 2.3, thefollowing estimates for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖y1n −q‖ 2 ≤ αn,1‖xn −q‖2 + `2∑ j=2 αn,j j−1∏ i=1 (1 −αn,i)‖Γj−1y2n − Γ j−1q‖2 + `2∏ i=1 (1 −αn,i)‖Γ`2y2n − Γ `2q‖2 ≤ α1n,1‖xn −q‖ 2 + `2∑ j=2 αn,j(ρ j)2 j−1∏ i=1 (1 −αn,i)‖y2n −q‖ 2 + `2∏ i=1 (1 −αn,i)(ρj)2‖y2n −q‖ 2 ≤ α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) [ α2n,1‖xn −q‖ 2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)‖y 3 n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖y3n −q‖ 2 ] + `2∏ i=1 (1 −α1n,i)(ρ j)2 [ α2n,1‖xn −q‖ 2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)‖y 3 n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖y3n −q‖ 2 ] https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 8 = α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ‖y3n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖y3n −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ‖y3n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ‖y3n −q‖ 2 ≤ α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) [α3n,1‖xn −q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖y4n −q‖ 2 ] + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖xn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖y4n −q‖ 2 ] +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )[ α3n,1‖xn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖y4n −q‖ 2 ] + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )[ α3n,1‖xn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖y4n −q‖ 2 ] https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 9 = α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖y4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖y4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖y4n −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖y4n −q‖2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 10 = α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖xn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2 ) ‖y4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2) ) ‖y4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2) ) ‖y4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2 ) ‖y4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y4n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 11 = α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖xn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  (1 −α3n,1)(ρj)2‖y4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2)‖y4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2)‖y4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2‖y4n −q‖ 2 ≤ α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖xn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 12 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1(1 −α3n,1)(ρj)2‖xn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2)‖xn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2)‖xn −q‖2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2‖xn −q‖2 + · · · +  `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)3 j−1∏ i=1 (1 −α2n,i) ×···× `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i )  ×  `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) αsn,1‖xn −q‖2 + (ρj)2 ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ×(ρj)2 ( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) (ρj)2 ( `4∏ i=1 (1 −α3n,i)(ρ j)2 ) ×···× (ρj)2 `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2  ×(ρj)2 ( `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ‖xn −q‖2 < α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) α3n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i) ) ‖xn −q‖2 + ( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1‖xn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 13 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i)) )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + · · · + ( 1 −α1n,1 − `2∏ i=1 (1 −α2n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) × ( 1 −α3n,1 − `4∏ i=1 (1 −α3n,i) ) ×···× 1 −α`s−2n,1 − `s−1∏ i=1 (1 −α`s−2 n,i )  × ( 1 −α`s−1n,1 − `s∏ i=1 (1 −α`s−1 n,i ) ) αsn,1‖xn −q‖ 2 + ( `2∏ i=1 (1 −α1n,i) ) × ( `3∏ i=1 (1 −α2n,i) )( `4∏ i=1 (1 −α3n,i) ) ×···× `s−1∏ i=1 (1 −α`s−2 n,i )  × ( `s∏ i=1 (1 −α`s−1 n,i ) ) ‖xn −q‖2 (3.7) < α1n,1‖xn −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) ) α3n,1 ( 1 −α2n,1 ) ‖xn −q‖2 +α3n,1 ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) ‖xn −q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + · · · + + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) αsn,1‖xn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 14 = α1n,1‖xn −q‖ 2 + α2n,1 ( 1 −α1n,1 − `2∏ i=1 (1 −α1n,i) ) ‖xn −q‖2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖xn −q‖ 2 + ( (1 −α1n,1 − `2∏ i=1 (1 −α1n,i) ) α3n,1 ( 1 −α2n,1 ) ‖xn −q‖2 +α3n,1 ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) ‖xn −q‖2 + ( (1 −α1n,1 − `2∏ i=1 (1 −α1n,i) )( 1 −α2n,1 ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖xn −q‖ 2 + · · · + + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) αsn,1‖xn −q‖ 2 < [α1n,1 + α 2 n,1 ( 1 −α1n,1 ) + (1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) + (1 −α1n,1) ( 1 −α2n,1 ) α3n,1(1 −α 3 n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) ]‖xn −q‖2 (3.8) (3.6) and (3.8) imply that ‖xn+1 −q‖2 ≤ {δn,1 + (1 −δn,1) [α1n,1 + α 2 n,1 ( 1 −α1n,1 ) + (1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) +(1 −α1n,1) ( 1 −α2n,1 ) (1 −α3n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) ]}‖xn −q‖2 (3.9) Using Lemma 2.3, we obtain (from (3.9)) that the sequence {xn}∞n=0 converges strongly to q ∈ F (Γ);and this completes the proof. � Theorem 3.2. Let H be a Hilbert space, Γ : H −→ H be a self-map of H satisfying the contractive condition ‖Γjx − Γjy‖≤ ρj‖x −y‖ + j∑ i=0 ( j i ) ρj−1φ(‖x − Γx‖), (3.10) where x,y ∈ H, 0 ≤ ρj < 1, and let φ retain its usual meaning with φ(0) = 0 and φ(Mt) = Mφ(t),M ≥ 0,t ∈ R+. For arbitrary x0 ∈ H, let {ωn}∞n=0 be the multistep DI-iteration scheme defined by (3.2). Then, (i) Γ defined by (3.10) has a fixed point q; https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 15 (ii) the multistep SH-iteration scheme converges strongly tp q ∈ Γ. Proof. We first show that Γ satisfying condition of (3.10) has a fixed point. Assume there existstwo points q1,q2 ∈ F (Γ) with 0 < ‖q1 −q2‖. Then, we have 0 < ‖q1 −q2‖ = ‖Γjq1 − Γjq2‖ ≤ ρj‖q1 −q2‖ + j∑ i=0 ( j i ) ρj−1φ(‖q √ 1 − Γq1‖) = ρj‖q1 −q2‖ + j∑ i=0 ( j i ) ρj−1φ(0) ⇒ (1 −ρj)ρj‖q1 −q2‖≤ 0. Using the fact that ρj ∈ [[0, 1), we get 0 < 1 −ρj and ‖q1 −q2‖≤ 0.Since the norm is a nonnegative function, we get ‖q1 −q2‖ = 0; q1 = q2 = q(say). Therefore, Γconverges uniquely to a point of F (Γ).Now, we show that the sequence defined by (3.1) converges strongly to q ∈ F (Γ). Using (3.2)and Proposition 2.4 with xn+1 = y,u = q,y1n = t, j = i,k = 1, Γj−1y1n = vj−1 and Γ`1y1n = v , weget ‖xn+1 −q‖2 = δn,1‖y1n −q‖ 2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖Γj−1y1n − Γ j−1q‖2 + `1∏ i=1 (1 −δn,i)‖Γ`1y1n − Γ `1q‖2 (3.11) But from (3.10), with y = y1n , we have ‖Γj−1y1n − Γ j−1q‖ ≤ ρj‖y1n −q‖ + j∑ i=0 ( j i ) ρj−1φ(‖q − Γq‖) = ρj‖y1n −q‖ (3.12) Proposition 2.3, (3.11) and (3.12) imply ‖xn+1 −q‖2 ≤ δ1n,1‖y 1 n −q‖ 2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i)‖y1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖y1n −q‖ 2 = δ1n,1‖y 1 n −q‖ 2 + ( 1 −δ1n,1 − `1∏ i=1 (1 −δn,i)(ρj)2 ) ‖y1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖y1n −q‖ 2 = ‖y1n −q‖ 2 (3.13) https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 16 Since `1,`k are fixed integers and αsn,i ∈ [0, 1] for each s, we have (using Proposition 2.3, (3.2)and (3.12)) the following estimates for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖y1n −q‖ 2 ≤ α1n,1‖y 2 n −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)‖Γ j−1y2n − Γ j−1q‖2 + `2∏ i=1 (1 −α1n,i)‖Γ `2y2n − Γ `2q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 ‖y2n −q‖2 ≤ α1n,1 + `2∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 [α2n,1‖y3n −q‖2 + `3∑ j=2 α2n,j j−1∏ i=1 (1 −α2n,i)‖Γ j−1y3n − Γ j−1q‖2 + `3∏ i=1 (1 −α2n,i)‖Γ `3y3n − Γ `3q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 [α2n,1‖y3n −q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i)‖y2n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖y3n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖y3n −q‖ 2 (3.14) ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖y 4 n −q‖ 2 + `4∑ j=2 α3n,j j−1∏ i=1 (1 −α3n,i)‖Γ j−1y4n − Γ j−1q‖2 + `4∏ i=1 (1 −α3n,i)‖Γ `4y4n − Γ `4q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 17 × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖y 4 n −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖y 4 n −q‖ 2 + `4∏ i=1 (1 −α3n,i)(ρ j)2‖y4n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) ‖y4n −q‖ 2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 )[ α4n,1‖y 5 n −q‖ 2 + `5∑ j=2 α4n,j j−1∏ i=1 (1 −α4n,i)‖Γ j−1y5n − Γ j−1q‖2 + `5∏ i=1 (1 −α4n,i)‖Γ `5y5n − Γ `5q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 )[ α4n,1‖y 5 n −q‖ 2 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i)‖y 5 n −q‖ 2 + `5∏ i=1 (1 −α4n,i)(ρ j)2‖y5n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 18 × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖y5n −q‖ 2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖xn −q‖2 (3.15) (3.13) and (3.15) imply that ‖xn+1 −q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖xn −q‖2 (3.16) https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 19 Since ρj ∈ [0, 1], we obtain using Proposition 2.3, for j = 1, 2, 3, · · · ,s − 1, that Q ≤ P = 1, (3.17) where Q = ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) and P = ( α3n,1 + `4∑ j=2 α3n,j j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i) ) × ( α4n,1 + `5∑ j=2 α4n,j j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i) ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i ) ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i ) ) Applying (3.17) in (3.16), we obtain, using Lemma 2.3 that the sequence {xn}∞n=0 defined by (3.2)converges strongly to the fixed point q in F (Γ). Thus, the proof is completed. � Example 3.1. Let the operator Γ : [0, 1] −→ [0, 1] be defined as Γz = z 3 ,∀z ∈ [0, 1]. Clearly, Γ is quasi-contractive satisfying (2.2) with a unique fixed point 0; see, for example, [26] for details. Set α1n,1 = δ 1 n,1 = 1 √ n + 1 ,n = 1, 2, · · · ,n0, f or n0 ∈N; δn,i = 1 −δ1n,1, f or i = 1, 2, · · · ,`1 and αsn,i = 1 − 2α 1 n,1, f or i = 1, 2, · · · ,`s+1,s = 1, 2, · · · ,n0. It is not hard to see that all the conditions of Theorem 3.1 and Theorem 3.2 has been satisfied by Example 3.1. https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 20 4. Main Results II Here, we consider stablity results for the multistep IH-iteration scheme and the multistep DI-iteration scheme defined by (3.1) and (3.2) for operators satisfying (2.2), respectively. Theorem 4.1. Let H be a Hilbert space, Γ : H −→ H be a self-map of H satisfying the contractive condition ‖Γjx − Γjy‖≤ ρj‖x −y‖ + j∑ i=0 ( j i ) ρj−1φ(‖x − Γx‖), (4.1) where x,y ∈ H, 0 ≤ ρj < 1, and let φ retains its usual meaning with φ(0) = 0 and φ(Mt) = Mφ(t),M ≥ 0,t ∈ R+. For arbitrary x0 ∈ H, let {xn}∞n=0 be the multistep DI-iteration scheme defined by (3.2). Assume F (Γ) 6= ∅,q ∈ F (Γ). Then, the multisetp DI-iterative scheme is Γ-stable. Proof. Let {vn}∞n=0, be a real sequences in H. Suppose {tn}∞n=0 ⊂ X is an arbitrary sequence, set �n = ‖tn+1 −δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n − `1∏ i=1 (1 −δn,i)Γ`1v1n‖ 2 (4.2) where, for s = 1, 2, · · · ,k − 2, vsn = α s n,1v s+1 n + `s+1∑ j=2 αsn,j j−1∏ i=1 (1 −αsn,i)Γ j−1vs+1n + `s+1∏ i=1 (1 −αsn,i)Γ `1vs+1n (4.3) and, for k ≥ 2, vk−1n = `k∑ j=1 αk−1 n,j j−1∏ i=1 (1 −αk−1 n,i )Γj−1tn + `k∏ i=1 (1 −αk−1 n,i )Γ`ktn,n ≥ 1, (4.4) Now, suppose �n → 0 as n →∞. Then, we show that tn → q as n →∞ using contractive mappingdefined by (4.1).Indeed, using Proposition 2.4 with u = q,v1n = t, j = i,k = 1, Γj−1v1n = vj−1 and Γ`1v1n = v„we obtain ‖tn+1 −q‖2 = ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q −[δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n − tn+1]‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 21 ≤ ‖− [tn+1 −δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n − `1∏ i=1 (1 −δn,i)Γ`1v1n ]‖ 2 +‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q‖ 2 = ‖tn+1 −δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n − `1∏ i=1 (1 −δn,i)Γ`1v1n‖ 2 +‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q‖ 2 = �n + ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q‖ 2 ≤ �n + δn,1‖v1n,1 −q‖ 2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖Γj−1v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)‖Γ`1v1n −q‖ 2 ≤ �n + δn,1‖v1n,1 −q‖ 2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i)‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 ≤ �n + ( δn,1 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i) + `1∏ i=1 (1 −δn,i)(ρj)2 ) ×‖v1n −q‖ 2 (4.5) Since `1,`k are fixed integers and αsn,i ∈ [0, 1] for each s, using (3.2) and (3.12), the estimationsbelow are obtained, for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 :, ‖v1n −q‖ 2 ≤ α1n,1‖v 2 n −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)‖Γ j−1v2n − Γ j−1q‖2 + `2∏ i=1 (1 −α1n,i)‖Γ `2v2n − Γ `2q‖2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 ‖v2n −q‖2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 22 ≤ α1n,1 + `2∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 [α2n,1‖v3n −q‖2 + `3∑ j=2 α2n,j j−1∏ i=1 (1 −α2n,i)‖Γ j−1v3n − Γ j−1q‖2 + `3∏ i=1 (1 −α2n,i)‖Γ `3v3n − Γ `3q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2 [α2n,1‖v3n −q‖2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i)‖v3n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖v3n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖v3n −q‖ 2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖v 4 n −q‖ 2 + `4∑ j=2 α3n,j j−1∏ i=1 (1 −α3n,i)‖Γ j−1v4n − Γ j−1q‖2 + `4∏ i=1 (1 −α3n,i)‖Γ `4v4n − Γ `4q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖v 4 n −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α3n,i)(ρ j)2‖v4n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) ‖v4n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 23 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 )[ α4n,1‖v 5 n −q‖ 2 + `5∑ j=2 α4n,j j−1∏ i=1 (1 −α4n,i)‖Γ j−1v5n − Γ j−1q‖2 + `5∏ i=1 (1 −α4n,i)‖Γ `5v5n − Γ `5q‖2 ] ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 )[ α4n,1‖v 5 n −q‖ 2 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i)‖v 5 n −q‖ 2 + `5∏ i=1 (1 −α4n,i)(ρ j)2‖v5n −q‖ 2 ] = α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v5n −q‖ 2 ≤ α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 24 ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖tn −q‖2 (4.6) (4.5) and (4.6)imply that ‖tn+1 −q‖2 ≤ ( δn,1 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i) + `1∏ i=1 (1 −δn,i)(ρj)2 ) × α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖tn −q‖2 + �n (4.7) Note that (4.7) is valid since Γq = q and φ(0) = 0.Now, since ρj ∈ [0, 1], we obtain using Proposition 2.3, for j = 1, 2, 3, · · · ,s − 1, that τn < ηn = 1, (4.8) where τn = ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 25 × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) and ηn = ( α3n,1 + `4∑ j=2 α3n,j j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i) ) × ( α4n,1 + `5∑ j=2 α4n,j j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i) ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i ) ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i ) ) Putting (4.8) in (4.7), we obtain, using Lemma 2.3 that the sequence {tn}∞n=0 converges strongly tothe point q in F (Γ).On the other hand, suppose tn → q as n → ∞. Then, we show that � → 0 as n → ∞. Indeed,from (3.5) with v1n = y1n , (4.2) and Proposition 2.4 with u = q,v1n = t, j = i,k = 1, Γj−1v1n = vj−1and Γ`1v1n = v„ we have �n = ‖tn+1 −δn,1v1n,1 − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n − `1∏ i=1 (1 −δn,i)Γ`1v1n‖ 2 = ‖tn+1 −q − δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q ‖2 ≤ ‖tn+1 −q‖2 + ‖δn,1v1n,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q‖ 2 ≤ ‖tn+1 −q‖2 + δn,1‖v1n,1 −q‖ 2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖Γj−1v1n − Γ j−1q‖2 + `1∏ i=1 (1 −δn,i)‖Γ`1v1n − Γ `1q‖2 ≤ ‖tn+1 −q‖2 + δn,1‖v1n,1 −q‖ 2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 26 = ‖tn+1 −q‖2 + δn,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)(ρj)2 + `1∏ i=1 (1 −δn,i)(ρj)2  ×‖v1n −q‖ 2 (4.9) Putting (4.6) into (4.9), and using (4.8), we get �n ≤ ‖tn+1 −q‖2 + δn,1 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)(ρj)2 + `1∏ i=1 (1 −δn,i)(ρj)2  × α1n,1 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) + `2∏ i=1 (1 −α1n,i)(ρ j)2  × ( α2n,1 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −αn,i) + `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( α3n,1 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) + `4∏ i=1 (1 −α3n,i)(ρ j)2 ) × ( α4n,1 + `5∑ j=2 α4n,j(ρ j)2 j−1∏ i=1 (1 −α4n,i) + `5∏ i=1 (1 −α4n,i)(ρ j)2 ) ×···× ( α `s−2 n,1 + `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i ) + `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2 ) × ( α `s−1 n,1 + `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) + `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ×‖tn −q‖2 ≤ ‖tn+1 −q‖2 + τn‖tn −q‖2 (4.10) Thus, from our assumption, we obtain from (4.10) that �n → 0 as n → ∞. Hence, the multistep DI-iteration scheme (3.2) is Γ-stable. Thus, tje proof is completed. � Theorem 4.2. Let H be a Hilbert space, Γ : H −→ H be a self-map of H satisfying the contractive condition ‖Γjx − Γjy‖≤ ρj‖x −y‖ + j∑ i=0 ( j i ) ρj−1φ(‖x − Γx‖), (4.11) where x,y ∈ H, 0 ≤ ρj < 1, and let φ retains its usual meaning with φ(0) = 0 and φ(Mt) = Mφ(t),M ≥ 0,t ∈ R+. For arbitrary x0 ∈ H, let {ωn}∞n=0 be the multistep IH-iteration scheme defined by (3.1). Assume F (Γ) 6= ∅,q ∈ F (Γ). Then, the multisetp IH-iteration scheme is Γ-stable. https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 27 Proof. Let {tn}∞n=0 and {vn}∞n=0, for i = 1, 2, · · · ,s − 1, be two real sequences in H. Set �n = ‖tn+1 −δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n − `1∏ i=1 (1 −δn,i)Γ`1v1n‖ 2 (4.12) where, for s = 1, 2, · · · ,k − 2, vsn = α s n,1tn + `s+1∑ j=2 αsn,j j−1∏ i=1 (1 −αsn,i)Γ j−1vs+1n + `s+1∏ i=1 (1 −αsn,i)Γ `1vs+1n (4.13) and, for k ≥ 2, vk−1n = `k∑ j=1 αk−1 n,j j−1∏ i=1 (1 −αk−1 n,i )Γj−1tn + `k∏ i=1 (1 −αk−1 n,i )Γ`ktn,n ≥ 1, (4.14) Now, suppose �n → 0 as n →∞. Then, we show that tn → q as n →∞ using contractive mappingdefined by (4.1).Indeed, Using Proposition 2.4 with u = q,tn = t, j = i,k = 1, Γj−1v1n = vj−1 and Γ`1v1n = v„we obtain ‖tn+1 −q‖2 = ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q −[δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n − tn+1]‖ 2 ≤ ‖− [tn+1 −δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n − `1∏ i=1 (1 −δn,i)Γ`1v1n ]‖ 2 +‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q‖ 2 = ‖tn+1 −δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n − `1∏ i=1 (1 −δn,i)Γ`1v1n‖ 2 +‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q‖ 2 = �n + ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q‖ 2 ≤ �n + δn,1‖tn −q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖Γj−1v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)‖Γ`1v1n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 28 ≤ �n + δn,1‖tn −q‖2 + `1∑ j=2 δn,j(ρ j)2 j−1∏ i=1 (1 −δn,i)‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 = �n + δn,1‖tn −q‖2 + ( 1 −δn,1 − `1∏ i=1 (1 −δn,i) ) (ρj)2‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 < �n + δn,1‖tn −q‖2 + (1 −δn,1)‖v1n −q‖ 2 (4.15) Since `1,`k are fixed integers and αsn,i ∈ [0, 1] for each s, the estimations below are obtained for n = 1, 2, · · · and 1 ≤ s ≤ k − 1 : ‖v1n −q‖ 2 ≤ αn,1‖tn −q‖2 + `2∑ j=2 αn,j j−1∏ i=1 (1 −αn,i)‖Γj−1v2n − Γ j−1q‖2 + `2∏ i=1 (1 −αn,i)‖Γ`2v2n − Γ `2q‖2 ≤ α1n,1‖tn −q‖ 2 + `2∑ j=2 αn,j(ρ j)2 j−1∏ i=1 (1 −αn,i)‖v2n −q‖ 2 + `2∏ i=1 (1 −αn,i)(ρj)2‖v2n −q‖ 2 ≤ α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) [ α2n,1‖tn −q‖ 2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)‖v 3 n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖v3n −q‖ 2 ] + `2∏ i=1 (1 −α1n,i)(ρ j)2 [ α2n,1‖tn −q‖ 2 + `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)‖v 3 n −q‖ 2 + `3∏ i=1 (1 −α2n,i)(ρ j)2‖v3n −q‖ 2 ] = α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ‖v3n −q‖2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 29 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖v3n −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ‖v3n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ‖v3n −q‖ 2 ≤ α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) [α3n,1‖tn −q‖2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖v4n −q‖ 2 ] + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )[ α3n,1‖tn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖v4n −q‖ 2 ] +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )[ α3n,1‖tn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖v4n −q‖ 2 ] + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )[ α3n,1‖tn −q‖ 2 + `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i)‖v 4 n −q‖ 2 + `4∏ i=1 (1 −α4n,i)(ρ j)2‖v4n −q‖ 2 ] = α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 30 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖v4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖v4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖v4n −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) `4∑ j=2 α3n,j(ρ j)2 j−1∏ i=1 (1 −α3n,i) ‖v4n −q‖2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 = α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 31 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖tn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2 ) ‖v4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) ×  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2) ) ‖v4n −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) × ( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2) ) ‖v4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( (1 −α3n,1 − `4∏ i=1 (1 −α3n,i))(ρ j)2 ) ‖v4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 )( `4∏ i=1 (1 −α4n,i)(ρ j)2 ) ‖v4n −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 32 = α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖tn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  (1 −α3n,1)(ρj)2‖v4n −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2)‖v4n −q‖ 2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2)‖v4n −q‖ 2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) (1 −α3n,1)(ρ j)2‖v4n −q‖ 2 ≤ α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)(ρ j)2α2n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) ‖tn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 33 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) ) `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) α3n,1(1 −α3n,1)(ρj)2‖tn −q‖2 + ( `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2)‖tn −q‖2 +  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i) ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2)‖tn −q‖2 + ( `3∏ i=1 (1 −α2n,i)(ρ j)2 )( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) α3n,1(1 −α 3 n,1)(ρ j)2‖tn −q‖2 + · · · +  `2∑ j=2 α1n,j(ρ j)2 j−1∏ i=1 (1 −α1n,i)  `3∑ j=2 α2n,j(ρ j)2 j−1∏ i=1 (1 −α2n,i)  ×  `4∑ j=2 α3n,j(ρ j)3 j−1∏ i=1 (1 −α2n,i) ×···× `s−1∑ j=2 α `s−2 n,j (ρj)2 j−1∏ i=1 (1 −α`s−2 n,i )  ×  `s∑ j=2 α `s−1 n,j (ρj)2 j−1∏ i=1 (1 −α`s−1 n,i ) αsn,1‖tn −q‖2 + (ρj)2 ( `2∏ i=1 (1 −α1n,i)(ρ j)2 ) ×(ρj)2 ( `3∏ i=1 (1 −α2n,i)(ρ j)2 ) (ρj)2 ( `4∏ i=1 (1 −α3n,i)(ρ j)2 ) ×···× (ρj)2 `s−1∏ i=1 (1 −α`s−2 n,i )(ρj)2  ×(ρj)2 ( `s∏ i=1 (1 −α`s−1 n,i )(ρj)2 ) ‖tn −q‖2 < α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i) ) ‖tn −q‖2 + ( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 34 + ( `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( `3∏ i=1 (1 −α2n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + ( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i)) )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + ( `3∏ i=1 (1 −α2n,i) )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + · · · + ( 1 −α1n,1 − `2∏ i=1 (1 −α2n,i) )( 1 −α2n,1 − `3∏ i=1 (1 −α2n,i) ) × ( 1 −α3n,1 − `4∏ i=1 (1 −α3n,i) ) ×···× 1 −α`s−2n,1 − `s−1∏ i=1 (1 −α`s−2 n,i )  × ( 1 −α`s−1n,1 − `s∏ i=1 (1 −α`s−1 n,i ) ) αsn,1‖tn −q‖ 2 + ( `2∏ i=1 (1 −α1n,i) ) × ( `3∏ i=1 (1 −α2n,i) )( `4∏ i=1 (1 −α3n,i) ) ×···× `s−1∏ i=1 (1 −α`s−2 n,i )  × ( `s∏ i=1 (1 −α`s−1 n,i ) ) ‖tn −q‖2 (4.16) < α1n,1‖tn −q‖ 2 + `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) ) α3n,1 ( 1 −α2n,1 ) ‖tn −q‖2 +α3n,1 ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) ‖tn −q‖2 + ( `2∑ j=2 α1n,j j−1∏ i=1 (1 −α1n,i) )( 1 −α2n,1 ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 35 + ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + · · · + + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) αsn,1‖tn −q‖ 2 = α1n,1‖tn −q‖ 2 + α2n,1 ( 1 −α1n,1 − `2∏ i=1 (1 −α1n,i) ) ‖tn −q‖2 + `2∏ i=1 (1 −α1n,i)α 2 n,1‖tn −q‖ 2 + ( (1 −α1n,1 − `2∏ i=1 (1 −α1n,i) ) α3n,1 ( 1 −α2n,1 ) ‖tn −q‖2 +α3n,1 ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) ‖tn −q‖2 + ( (1 −α1n,1 − `2∏ i=1 (1 −α1n,i) )( 1 −α2n,1 ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + ( 1 −α2n,1 )( `2∏ i=1 (1 −α1n,i) ) α3n,1(1 −α 3 n,1)‖tn −q‖ 2 + · · · + + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) αsn,1‖tn −q‖ 2 < [ α1n,1 + α 2 n,1 ( 1 −α1n,1 ) + (1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) + (1 −α1n,1) ( 1 −α2n,1 ) α3n,1(1 −α 3 n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 )] ‖tn −q‖2 (4.17) (4.15) and (4.17) imply that ‖tn+1 −q‖2 ≤ {δn,1 + (1 −δn,1) [α1n,1 + α 2 n,1 ( 1 −α1n,1 ) + (1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) +(1 −α1n,1) ( 1 −α2n,1 ) (1 −α3n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 ) ]}‖tn −q‖2 (4.18) Using Lemma 2.3, we obtain (from (4.18)) that the sequence {xn}∞n=0 converges strongly to q ∈ F (Γ).Conversely, suppose tn → q as n → ∞. Then, we show that � → 0 as n → ∞. Indeed, from(3.5) with v1n = y1n , (4.12) and Proposition 2.4 with u = q,tn = t, j = i,k = 1, Γj−1v1n = vj−1 and https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 36 Γ`1v1n = v„ we have �n = ‖tn+1 −δn,1tn − `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n − `1∏ i=1 (1 −δn,i)Γ`1v1n‖ 2 = ‖tn+1 −q − δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q ‖2 ≤ ‖tn+1 −q‖2 + ‖δn,1tn + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)Γj−1v1n + `1∏ i=1 (1 −δn,i)Γ`1v1n −q‖ 2 ≤ ‖tn+1 −q‖2 + δn,1‖tn −q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)‖Γj−1v1n − Γ j−1q‖2 + `1∏ i=1 (1 −δn,i)‖Γ`1v1n − Γ `1q‖2 ≤ ‖tn+1 −q‖2 + δn,1‖tn −q‖2 + `1∑ j=2 δn,j j−1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 = ‖tn+1 −q‖2 + δn,1‖tn −q‖2 + ( 1 −δn,1 − `1∏ i=1 (1 −δn,i) ) (ρj)2‖v1n −q‖ 2 + `1∏ i=1 (1 −δn,i)(ρj)2‖v1n −q‖ 2 = ‖tn+1 −q‖2 + δn,1‖tn −q‖2 + (1 −δn,1)‖v1n −q‖ 2 (4.19) (4.17) and (4.19) imply �n ≤ ‖tn+1 −q‖2 + { δn,1 + (1 −δn,1) [ α1n,1 + α 2 n,1 ( 1 −α1n,1 ) +(1 −α1n,1)α 3 n,1 ( 1 −α2n,1 ) + (1 −α1n,1) ( 1 −α2n,1 ) α3n,1(1 −α 3 n,1) + · · · + ( 1 −α1n,1 )( 1 −α2n,1 )( 1 −α3n,1 ) ×···× ( 1 −α`s−2n,1 ) × ( 1 −α`s−1n,1 )]} ‖tn −q‖2 (4.20) Again, from our assumption, we obtain from (4.20) that �n → 0 as n → ∞. Hence, the multistep IH-iteration scheme (3.1) is Γ-stable, and this completes the proof. � Remark 4.1. The following areas are still open: https://doi.org/10.28924/ada/ma.2.1 Eur. J. Math. Anal. 10.28924/ada/ma.2.1 37 (i) to reconstruct, approximate the fixed points and the stability results of some existing iter- ative schemes in the current literature, other than the ones under study, for finite family of certain class of contractive-type map;(ii) to compare convergent rates of the iterative schemes defined by (3.1) and (3.2) with those of (1.5) and (1.6). Competing interestThe authors declare that there is no conflict of interest. References [1] B. E. Rhoade, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl.Math. 24(11) (1993) 691-03.[2] B. E. Rhoade, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl.Math. 21 (1990) 1-9.[3] M. O. Osilike, A. 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Okeke, New iteration scheme for approximating a common fixed point of a finitefamily of mappings, J. Math. 2020 (2020) 3287968. https://doi.org/10.1155/2020/3287968.[28] I. K. Agwu, D. I. Igbokwe, New iteration algorithm for equilibrium problems and fixed point problems of two finitefamilies of asymptotically demicontractive multivalued mappings, (in press). https://doi.org/10.28924/ada/ma.2.1 https://doi.org/10.2307/2032162 https://doi.org/10.2307/2032162 https://doi.org/10.1090/S0002-9939-1974-0336469-5 https://doi.org/10.1090/S0002-9939-1974-0336469-5 https://doi.org/10.1186/1687-1812-2014-45 https://doi.org/10.1186/1687-1812-2014-45 https://doi.org/10.1155/2020/3287968 1. Introduction 2. Preliminary 3. Main Results I 4. Main Results II References