International Journal of Analysis and Applications ISSN 2291-8639 Volume 4, Number 2 (2014), 192-200 http://www.etamaths.com CONVERGENCE OF A MODIFIED MULTI-STEP ITERATIVE SCHEME FOR P- NEARLY UNIFORMLY L-LIPSCHITZIAN ASYMPTOTICALLY PSEUDOCONTRATIVE MAPPINGS V.O. OLISAMA∗, A.A. MOGBADEMU, J.O. OLALERU Abstract. In this paper, it is shown that the modified multi-step iteration converges strongly to the common fixed point of a finite family of nearly uni- formly L- Lipschitzian asymptotically pseudocontractive mappings. The main result is an improvement and extension of well known results in the literature. 1. Preliminary. Let E be a real Banach space and let E∗ be its dual space. The normalized duality mappping J : E → 2E ∗ is defined by J(x) = {f ∈ X∗ : < x,f > = ‖x‖‖f‖}, where < .,. > denotes the generalized duality pairing and ‖x‖ = ‖f‖. We shall denote the single-valued normalized duality pairing by j. J satisfies the following properties: (1) J is an odd mapping, i.e J(−x) = −J(x). (2) J is positive homogeneous , i.e for any number λ > 0, J(λx) = λJ(x). (3) J is bounded, i.e. for any subset A of E J(A) is a bounded subset of E∗. (4) If E is smooth (or E∗ is strictly convex), Then J is singled-valued. Consistent with Goebel and Kirk [4] we give the following definitions, Let K be a nonempty closed convex subset of E and T : K → K be a map. Definition 1.1 A mapping T is said to be asymptotically nonexpansive if for each x,y ∈ K < Tnx−Tny >≤ kn‖x−y‖, 2010 Mathematics Subject Classification. 47H10. Key words and phrases. Modified multi-step iteration process, asymptotically pseudocontra- tive maps, uniformly L- Lipschitzian maps, nearly uniformly L-Lipschitzian maps, nearly asymp- totically pseudocontrative maps, Banach spaces. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 192 CONVERGENCE OF A MODIFIED MULTI-STEP ITERATIVE SCHEME 193 ∀n ≥ 0, where the sequence {kn}⊂ [1,∞) with limn→∞kn = 1. Definition 1.2 The mapping is said to be uniformly L- Lipschitzian if there exists a constant L ≥ 0 such that ‖Tn −Tny‖≤ L‖x−y‖, for any x,y ∈ K and ∀n ≥ 0. We give the definition of asymptotically pseudocontractive as in [14]. Definition 1.3[14] The mapping T is said to be asymptotically pseudocontractive if there exists a sequence {kn}⊂ [1,∞) with limn→∞kn = 1 and for any x,y ∈ K there exist j(x−y) ∈ J(x−y) such that < Tnx−Tny,j(x−y) >≤ kn‖x−y‖2, ∀n ≥ 0. Remark 1.4 Every asymptotically nonexpansive mapping is both asymptotically pseudocontractive and uniformly L- Lipschitzian. The converse is not true in gen- eral. Clearly, every operator which is asymptotically pseudocontractive in general may not admit a fixed point. The existence of fixed point result for asymptotically pseudocontractive maps depend on the space, nature of subset and further proper- ties of the operators (see, [1]). Infact, asymptotically nonexpansive and asymptotically pseudocontractive were first introduced by Goebel and Kirk [4] and Schu [15] respectively. Since then many authors have studied several iterative process for asymptotically nonexpan- sive and asymptotically pseudocontractive in both Hilbert and Banach spaces (see, [7, 8, 12]). Schu[15] proved the convergence of Mann[6] iterative sequence to the fixed point of uniformly L-Lipschitzian and asymptotically pseudocontractive mappings in the setting of Hilbert space. In 2001, Chang kextended the work of Schu to the setting of real uniformly smooth Banach spaces. Also, Ofoedu [11] extended Theorem 1.2 of Chang[1] to the setting of arbitrary real Banach spaces and dropped the bound- edness assumption. He proved the following theorem: Theorem 1.5[11] Let E be a real Banach space, K be a nonempty closed convex subspace of E and T : K → K uniformly L-Lipschitzian and asymptotically pseu- docontractive mappings with a sequence {kn}n≥0 ⊂ [1,∞),kn → 1 and x∗ ∈ F(T). Let the sequence {αn} be a sequence in [0,1] satisfying the following conditions: (a-1) ∑∞ n=0 αn = ∞; (a-2) ∑∞ n=0 α 2 n < ∞; (a-3) ∑∞ n=0 αn(kn − 1) < ∞. For any x0 ∈ K, let {xn}∞n≥0 be an iterative sequence defined by xn+1 = (1 −αn)xn + αnTnxn, (1.1) 194 OLISAMA, A.A. MOGBADEMU AND J.O. OLALERU for all n ≥ 0. If there exists a strictly increasing function Φ : [0,∞) → [0,∞), Φ(0) = 0 such that < Tnx−x∗,j(x−x∗) > ≤ kn‖x−x∗‖2 − Φ(‖x−x∗‖) ∀ n ≥ 0. Then (1) {xn}∞n≥0 is bounded; (2) {xn}∞n≥0 converges strongly to x ∗ ∈ F(T). Further more, Chang Cho and Kim [2] improved on the theorem above by extending the parameters and modifying the iterative procedure. Infact, they proved the following theorem: Theorem 1.6[2] Let E be a real Banach space, K be a nonempty closed and convex subspace of E and T : K → K uniformly L-Lipschitzian and asymptotically pseudocontractive mappings with a sequence {kn}n≥0 ⊂ [1,∞), kn → 1 and x∗ ∈ F(T). Let {an},{bn} and{cn} be a real sequences in [0,1] satisfying the following conditions: (a-1) an + bn + cn = 1; (a-2) ∑∞ n=0(bn + cn) = ∞; (a-3) ∑∞ n=0(bn + cn) 2 < ∞; (a-4) ∑∞ n=0(bn + cn)(kn − 1) < ∞; (a-5) ∑∞ n=0 cn < ∞. For any x0 ∈ K, let {xn}∞n≥0 be a sequence in K iteratively defined by xn+1 = anxn + bnT nxn + cnun, for all n ≥ 0. where {un} is a bounded sequence in K. Suppose that there exists a strictly increasing continuous function Φ : [0,∞) → [0,∞), Φ(0) = 0 such that < Tnx−x∗,j(x−x∗) > ≤ kn‖x−x∗‖2 − Φ(‖x−x∗‖), ∀ n ≥ 0. Then (1) {xn}∞n≥0 is bounded; (2) {xn}∞n≥0 converges strongly to x ∗ ∈ F(T). Another extension of the fixed point theory is the iterative processes for approxi- mating fixed points of mappings. Several authors have studied and extended the Mann[6],Ishikawa[5], Noor[10] and multi-step[13] iterative process to evaluate the fixed point of uniformly L-Lipschitzian and asymptotically pseudocontractive map- pings in Hilbert and Banach spaces.( see[7,8,16,17]). We remarked that in all these theorems above, for certain application the con- tinuity assumption becomes a rather strong condition. In this direction, a natural question arises that whether there is any class of(not necessarily continuous) map- ping more general than the class of asymptotically nonexpansive and asymptotically pseudocontractive (which has asymptotically nonexpansiveness)? Motivated by this inspired question, Sahu[14] introduced the classes of nearly contraction and nearly asymptotically nonexpansive mappings. He gave the following definition: Definition 1.7 [14] Let K be a nonempty subset of a Banach space E andj sequence CONVERGENCE OF A MODIFIED MULTI-STEP ITERATIVE SCHEME 195 {an} in [0,∞) with an → 0. A mapping T : K → K is called nearly Lipschitzian with respect to {an} if for each n ∈ N, there exists a constant kn ≥ 0 such that : < Tnx−Tny > ≤ kn(‖x−y‖ + an) for all x,y ∈ K. Remark 1.8 It is important to note that Lipschitzian mappings are always con- tinuous but nearly Lipschitzian mappings need not be continuous. The class of nearly asymptotically nonexpansive mappings contains the class of asymptotically nonexpasive mappings and is contained in the class of mappings of asymptotically nonexpansive type. Hence, according to Sahu[14] nearly asymptotically pseudocon- tractive mapping is a generalisation of asymptotically pseudocontractive mapping. Example 1.9[16] Let E = R and T : K → K be defined by: T(x) =   x 2 x ∈ [0, 1); 0 x = 1. Then T is a discontinuous mapping which is not Lipschitzian, but nearly 1 2 Lips- chitzian with sequence { 1 2n }. Sahu[14] proved some theorems in an attempt to develop asymptotically fixed point theory for a more general class of demicontinuous nearly Lipschirzian mappings in Banach spaces. And later extended this theorem to uniformly convex Banach s- paces. He proved the following theorem: Theorem 1.10[14, Theorem 3.8] Let C be a nonempty closed convex subset of a uniformly convex Banach space X and T : C → a demicontinuos nearly Lips- chitzian mapping with sequences {(an,η(Tn))} such that limn→∞η(Tn) ≤ 1. Then the following statements are equivalent: (a) T has a fixed point; (b) there exists a bounded sequence {Tnx0} in C; (c) there exists a bounded sequence {yn} in C such that limm→∞(limn→∞‖yn −Tmyn‖) = 0. Very recently, Thakur[16] stated and proved the following theorem: Theorem 1.11[16] Let E be a real Banach space, K be a nonempty closed con- vex subset of E and Ti : K → K,i = 1, 2 be two asymptotically generalised Φ- hemicontractive nearly uniformly Li Lipschitzian mappings with sequence {an} and F(T1) ∩F(T2) 6= φ where F(Ti) is the set of fixed point of Ti in K. Let {αn} and {βn} be two sequences in [0, 1] satisfying the following conditions: (i) ∑∞ n=1 αn = ∞; (ii) ∑∞ n=1 α 2 n < ∞; (iii) ∑∞ n=1 βn < ∞; (iv) ∑∞ n=0 αn(kn − 1) < ∞. Let {xn} be a sequence in K generated from arbitrary x1 ∈ K by Xn+1 = (1 −αn)xn + αT 1nyn, yn = (1 −βn)xn + βT 2nxn, n ∈ N. (1.2) Then, {xn} converges strongly to x∗ ∈ F(T1) ∩F(T2). 196 OLISAMA, A.A. MOGBADEMU AND J.O. OLALERU The purpose of this sequel is to improve Theorem 1.10 and Theorem 1.11 by ex- tending to a finite family of nearly p- uniformly L- Lipschitzian asymptotically pseudocontractive mappings. 2. Main Results The following concepts and Lemmas will be used. Definition 2.1 [13]. Let T1,T2, ...,Ti : K → K be finite family of maps. For any given x1 ∈ K, the multi-step iteration {xn}∞(n=1) ⊂ K is defined by xn+1 = (1 − bn)xn + bnT 1nyn1, n ≥ 1 yn i = (1 − bni)xn + bniTinyni+1, i = 1, 2, ...,p− 2, yn p−1 = (1 − bnp−1)xn + bnp−1Tpnxn, p ≥ 2. (2.1) Lemma 2.2 [9]. Let J : E → 2E∗ be the normalized duality mapping. Then for any x,y ∈ E, we have ‖x + y‖2 ≤‖x‖2 + 2 < y,j(x + y) >,∀j(x + y) ∈ J(x,y) Lemma 2.3 [3]. Let {dn},{en} and {hn} be three positive real sequences and Φ : [0,∞) → [0,∞) be a strictly increasing function with Φ(x) = 0 ⇔ x = 0 satisfying the following inequality: d2n+1 ≤ dn 2 −enΦ(dn+1) + hn,∀n ≥ 0 where en ∈ [0, 1], with ∑∞ n=0 en = +∞ and hn = o(en). Then limn→∞dn = 0. Theorem 2.4 Let K be a nonempty closed convex subset of a real Banach space E, and Ti : K → K, (i = 1, 2, .....,p, p ≥ 2) be a finite family of nearly Li - uniformly Lipschitzian mappings with sequence {kn}n≥1 ⊂ [1,∞), kn → 1 and ∑ n≥1(kn − 1) < ∞ such that ∩ p≥2 i=1 F(Ti) 6= φ. Let {bn}n≥1,{b i n}n≥1 and {bi+1n }n≥1 be the real sequences in [0,1] satisfying: (i)bn,b i n,b i+1 n → 0, as n →∞, (i = 1, 2, ...,p− 2). (ii) ∑ n≥1 bn = ∞. For any x1 ∈ K, define {xn}n≥0 by the iterative process (2.1). Suppose there exists a strictly increasing function Φ : [0,∞) → [0,∞), Φ(0) = 0 such that < Ti nxn −ρ,j(xn −ρ) > ≤ kn‖xn −ρ‖2 − Φ(‖xn −ρ‖) (2.2) ∀x ∈ K, (i = 1, 2, ....,p, p ≥ 2). Then {xn}n≥0 converges strongly to ρ ∈∩ p≥2 i=1 F(Ti). Proof: First we show that for any n ≥ 1,{xn} is a bounded sequence. Since T1,T2, ....,Tp, p ≥ 2 are nearly uniformly Li - Lipschizian mappings, we have that, ∀x,y ∈ K ‖Tni x−T n i y‖≤ Li(‖x−y‖ + an), (i = 1, 2, ....,p, p ≥ 2). CONVERGENCE OF A MODIFIED MULTI-STEP ITERATIVE SCHEME 197 Setting max{kn : n ≥ 1} = k and L = max{L1,L2, ....,Lp}. There exists x1 ∈ K with x1 6= Tx1 such that a0 = (k + L)‖x1 − ρ‖2 ∈ D(Φ). If Φ(a) → +∞ as a → +∞, then a0 ∈ D(Φ). If sup{Φ(a) : a ∈ [0,∞)} = a1 < +∞ with a1 < a0, then there exists a sequence {τn} ⊂ K such that τn → ρ as n → ∞ with τn 6= ρ, thus there exist a positive integer n0 such that (k + L)‖x1 −ρ‖2 < a12 for all n ≥ n0. Redefining x1 = τn0, (k + L)‖x1 −ρ‖2 ∈ D(Φ) and setting D(Φ) = Φ−1(a0), we obtain ‖x1 −ρ‖ ≤ D. Also let A1 = {x ∈ K : ‖x1 −ρ‖ ≤ D},A2 = {x ∈ K : ‖x1 −ρ‖ ≤ 2D}. Now, we show that xn ∈ A1 for any n ≥ 1. If n = 1, then x1 ∈ A1. Assume that for some n, xn ∈ A1, we show that xn+1 ∈ A1. Suppose xn+1 is not in A1, then ‖xn −ρ‖ > D. Now we denote ν0 = min{ D D(1 + 2L) , Φ(D) 10D2 , Φ(D) 10(2DL + 3DL2) } (2.3) Since bn,b i n,b i+1 n ,kn−1 → 0, as n →∞, (i = 1, 2, ...,p−2), without loss of generali- ty, let 0 ≤ bn,bin,bi+1n ,kn−1 ≤ ν0 for any n ≥ 1. Thus we have ‖ynp−1 −ρ‖ = ‖(1 − bnp−1)xn + bnp−1Tpnxn −ρ‖ = ‖(xn −ρ) − bp−1n (xn −Tnp xn)‖ ≤ ‖xn −ρ‖ + bp−1n (‖xn −ρ‖ + ‖Tnp xn −ρ‖) ≤ ‖xn −ρ‖ + ν0(‖xn −ρ‖ + L(‖xn −ρ‖ + an)) ≤ D + ν0(D + LD + Lan) < 2D. ‖ynp−2 −ρ‖ = ‖(1 − bnp−2)xn + bnp−2Tp−1nyp−1n −ρ‖ ≤ ‖xn −ρ‖ + bp−2n ‖Tnp−1yp−1n −xn‖ ≤ ‖xn −ρ‖ + ν0[L(‖yp−1n −ρ‖ + an) + ‖xn −ρ‖)] ≤ D + ν0(L(2D + an) + D) ≤ 2D. Recursively, we have ‖yi−1n −ρ‖≤ 2D, Thus, ‖yin −ρ‖≤ 2D for i = 1, 2, ...,p− 2. Also, ‖xn+1 −yni‖ ≤ ‖xn+1 −xn‖ + ‖yn −xn‖ ≤ ‖xn+1 −xn‖ + bnp−2‖Tnyp−1n −xn‖ ≤ D(1 + 2L) + D(1 + L) ≤ D(2 + 3L). But, ‖xn −Tni xn‖ ≤ ‖xn −ρ‖ + ‖T n i xn −ρ‖ ≤ D + L(D + an) ≤ D(1 + L) and 198 OLISAMA, A.A. MOGBADEMU AND J.O. OLALERU ‖xn −Tni y i n)‖ ≤ ‖xn −ρ‖ + ‖Tni y i n −ρ‖ ≤ D + L(‖yin −ρ‖ + an) ≤ D + L(2D + an) ≤ D(1 + 2L). Then, ‖Tnxn+1 −Tnyin‖ ≤ L(‖xn+1 −yin‖ + an) ≤ L[(‖yin −xn‖ + ‖xn+1 −xn‖ + an)] ≤ binL((‖xn −Tni xn)‖ + bn‖xn −T n i y i n‖ + an) ≤ binL((D(1 + L) + bn(D(1 + 2L)) + an ≤ ν0(L(D(1 + L) + D(1 + 2L) + an)) = ν0(2DL + 3DL 2) ≤ Φ(D) 10D . Applying Lemma(2.2) and the estimates above we obtain, ‖xn+1 −ρ‖2 = ‖(1 − bn)xn + bnT 1ny1n −ρ‖2 = ‖xn −ρ− bn(Tn1 y1n −xn)‖2 ≤ ‖xn −ρ‖2 − 2bn < Tn1 y1nxn,j(xn+1 −ρ) > = ‖xn −ρ‖2 − 2bn < Tn1 xn+1 −ρ,j(xn+1 −ρ) > −2bn < xn+1,j(xn+1 −ρ) > +‖2bn < Tn1 y1n −Tn1 xn+1,j(xn+1 −ρ) > +2bn < xn+1 −xn,j(xn+1 −ρ) > ≤ ‖xn −ρ‖2 + 2bn(kn‖xn −ρ‖2 − Φ(‖xn+1 −ρ‖)) −2bn‖xn+1 −ρ‖2 + 2bn‖Tn1 y1n −Tn1 xn+1‖‖xn+1 −ρ‖ +2bn‖xn+1 −xn‖‖xn+1 −ρ‖ = ‖xn −ρ‖2 + 2bn(kn − 1)‖xn+1 −ρ‖2 − 2bnΦ‖xn+1 −ρ‖) +2bnL(‖y1n −xn+1‖ + an)‖xn+1 −ρ‖ + 2bn‖xn+1 −xn‖‖xn+1 −ρ‖ ≤ ‖xn −ρ‖2 + 2bn(kn − 1)‖xn −ρ‖2 − 2bnΦ(D) +2bn( Φ(D) 10D ) × 2D + 2bn(D(1 + 2L))) × 2D ≤ D2 + 8bn(kn − 1)D2 − 2bnΦ(D) +D2 + 8bn(kn − 1)D2 − 85bnΦ(D) + 4bnD(D + 2L) (2.4) Since bn → 0 and (kn − 1) → 1 as n →∞,then (2.4) becomes ‖xn −ρ‖2 ≤ D2 which is a contradiction. Hence, xn+1 ∈ A1. Therefore, {xn} is bounded. Next we prove that ‖xn−ρ‖→ 0 as n →∞. We have shown above that {‖xn−ρ‖} is a bounded sequence and so is {‖yin −ρ‖}. Let R0 = supn≥{‖xn −ρ‖} + sup{‖yn −ρ‖}. But ‖xn+1 −yin‖ ≤ (‖yin −xn‖ + ‖xn+1 −xn‖) ≤ bin(‖xn −Tinxn‖) + bn‖xn −Tni y i n‖ ≤ bin(‖xn −ρ‖ + ‖Tinxn −ρ‖) + bn(‖xn −ρ + ‖Tni y i n‖) ≤ (bn + bin)M0 + (bn + bin)(1 + L)M0 (2.5) Using Lemma(2.2),equations (2.4) and (2.5) we have CONVERGENCE OF A MODIFIED MULTI-STEP ITERATIVE SCHEME 199 ‖xn+1 −ρ‖2 = ‖(1 − bn)xn + bnT 1ny1n −ρ‖2 = ‖xn −ρ− bn(Tn1 y1n −xn)‖2 ≤ ‖xn −ρ‖2 − 2bn < Tn1 y1n −ρ,j(xn+1 −ρ) > = ‖xn −ρ‖2 − 2bn < Tn1 xn+1 −ρ,j(xn+1 −ρ) > −2bn < xn+1,j(xn+1 −ρ) > +‖2bn < Tn1 y1n −Tn1 xn+1,j(xn+1 −ρ) > +2bn < xn+1 −xn,j(xn+1 −ρ) > ≤ +2bn(kn)‖xn −ρ‖2 − Φ(‖xn+1 −ρ‖)) −2bn‖xn+1 −ρ‖2 + 2bn‖Tn1 y1n −Tn1 xn+1‖‖xn+1 −ρ‖ +2bn‖xn+1 −xn‖‖xn+1 −ρ‖ = ‖xn −ρ‖2 + 2bn(kn − 1)‖xn+1 −ρ‖2 − 2bnΦ(xn+1 −ρ‖) +2bnL(‖y1n −xn+1‖ + an)‖xn+1 −ρ‖ + 2bn‖xn+1 −xn‖‖xn+1 −ρ‖ ≤ ‖xn −ρ‖2 + 2bn(kn − 1)R20 − 2bnΦ(‖xn+1 −ρ‖) +2bnL(bn + b i n)R0 + (bn + b i n)(1 + L)R0) + an)R0 + 4bnR 2 (2.6) ≤ ‖xn −ρ‖2 − bnΦ(‖xn+1 −ρ‖) + Bn Where Bn = 2bn(kn − 1)R20 + 2bnL(bn + bin)R0 +(bn + b i n)(1 + L)R0 + an)R0 + 4bnR 2. Taking dn = ‖xn −ρ‖2,en = bn, and hn = Bn. Then (2.6) becomes d2n+1 ≤ d 2 n −enΦ(dn+1) + hn,∀n ≥ N0. Therefore, by Lemma 2.3, we obtain lim →∞dn = 0. Hence xn → ρ as n →∞.This completes the proof. Also, using Lemma 2.3 and the conditions of the parameters, we obtain: dn → 0 as n →∞. This ends the proof. We make the following remarks: (1) Clearly,Is it possible to drop the continu- ity condition in Theorem 1.5 and extend to a finite family of nearly p- uniformly L- Lipschitzian asymptotically pseudocontractive mappings? (2) we have dropped the continuity condition in Theorem 1.6 and show that the modified multi-step converges to the common fixed point of T? Corollary 2.5 The result in Theorem 2.4 is also true for two and three nearly Li - uniformly Lipschitzian asymptotically pseudocontractive mappings which extends the work of Sahu[14]. Corollary 2.6 Let K be a nonempty closed convex subset of a real Banach s- pace E, Ti : K → K, (i = 1, 2, .....,p, p ≥ 2) be p uniformly Li Lipschitzian mappings with sequence {kn}n≥0 ⊂ [1,∞),kn → 1 and ∑ n≥0(kn − 1) < ∞ such that ρ ∈∩p≥2i=1 F(Ti) 6= φ. Let {bn}n≥0,{b i n}n≥0 and {bi+1n }n≥0 be the real sequences in [0,1] satisfying: 200 OLISAMA, A.A. MOGBADEMU AND J.O. OLALERU (i)bn,b i n,b i+1 n → 0, as n →∞, (i = 1, 2, ...,p− 2). (ii) ∑ n≥0 bn = ∞. For any x0 ∈ K, define {xn}n≥0 by the iterative process (2.1). Suppose there exists a strictly increasing function Φ : [0,∞) → [0,∞), Φ(0) = 0 such that < Ti nxn −ρ,j(xn −ρ) > ≤ kn‖xn −ρ‖2 − Φ(‖xn −ρ‖) (2.7) ∀x ∈ K, (i = 1, 2, ....,p, p ≥ 2). Then {xn}n≥0 converges strongly to ρ ∈∩ p≥2 i=1 F(Ti). References [1] S. S. Chang, Some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings , Proc. Amer. Math. Soc., 129(2001), 845-853. [2] S. S. Chang, Y. J. Cho and J. K. 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Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, Jour.of Math. and Appl., 158(1999), 407-413. [16] B. S. Tharkur, Strong convergence for asymptotically generalised Φ -Hemicontractive map- pings, ROMAI J. 8(2012), 165-171. [17] Zhique Xue and Guiwen Lv, Strong convergence theorems for uniformly L- Lipschitzian asymptotically pseudocontractive mappings in Banach spaces Xue and Lv Jour. of Ineq. and Appl.. (2013) 1/79. Department of Mathematics, University of Lagos. Nigeria ∗Corresponding author