An iterative Algorithm for Hemicontractive Mappings in Banach Spaces

Abstract

We First introduce a three-step iterative algorithm for approximating the fixed points of the hemicontractive mappings in Banach spaces. Consequently, we prove the strong convergence of the proposed algorithm under some assumptions. Since three-step iterations include Ishikawa iterations as special cases, our result continue to hold for these problems. Our main results can be viewed as an important refinement of the previously known results.

1. Introduction

In recent years, several convergence results have been proved on iterative methods for approximating fixed points of pseudocontractive mappings (see, e.g., [1–10] and references therein). It is worth mentioning that such iterative type methods are known as Mann iterations and Ishikawa iterations. It is clear that a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iteration sequence failed to converge, but it does converge for the sequence obtained by the Ishikawa iterations, see [11]. In 2000, Noor [12] suggested and analyzed three-step iterative methods for finding the approximate solution of a continuous mapping in the Hilbert space using the technique of updating the solution. Three-step iterations are also known as Noor iteration. It is well known [13] that three-step iterative schemes include one-step (Mann) and two-step (Ishikawa) iterations as special cases. It raises an interesting question. Is there exist any Lipschitz pseudo-contractive mapping with a unique fixed point for which Ishikawa iteration sequence fail to convergence, but Does convergence for the sequence obtained from Noor iteration? This is an open and challenging problem. To the best of our knowledge, main result of Ishikawa [14], see Theorem IS has never been extended to more general Banach spaces. Motivated and inspired by the recent research activities in this filed, we suggest and analyze a three-step iterative scheme associated with hemi-contractive mappings in Banach spaces. We also prove the strong convergence of the sequence generated by the three-step iterations under mild conditions. Since three-step iterations include Ishikawa iterations as special cases, our results continue to hold for these problems. It is worth mentioning that our results may be considered as very significant, interesting and important extensions of the previously known results concerning pseudo-contractive mappings.

Let 𝐸 be a real Banach space and 𝐸^∗ be its dual space. The normalized duality mapping from 𝐸 to 2^𝐸∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝐸^∗∶𝑥,𝑥^∗=‖𝑥‖𝑥^∗,𝑥^∗=‖𝑥‖,∀𝑥∈𝐸, (1.1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing.

Let 𝐶 be a nonempty subset of 𝐸, a mapping 𝑇∶𝐶→𝐶 is called pseudo-contractive if there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦², (1.2) for all 𝑥,𝑦∈𝐶. Let 𝐹(𝑇)∶={𝑥∈𝐶∶𝑇𝑥=𝑥}. A mapping 𝑇∶𝐶→𝐶 is called hemicontractive if 𝐹(𝑇)≠∅ and 𝑇𝑥−𝑥^∗,𝑗𝑥−𝑥^∗≤𝑥−𝑥^∗2,∀𝑥∈𝐶,𝑥^∗∈𝐹(𝑇). (1.3) It is easy to see that the class of pseudo-contractive mappings with fixed points is a subclass of the class of hemicontractions. A mapping 𝑇∶𝐶→𝐶 is called Lipschitzian if there exists a constant 𝐿≥0 such that ‖𝑇𝑥−𝑇𝑦‖≤𝐿‖𝑥−𝑦‖ for each 𝑥,𝑦∈𝐶.

In 1974, Ishikawa [14] proved the following result for the pseudo-contractive mappings.

Theorem 1.1 (see [14]). If 𝐶 is a compact convex subset of a Hilbert space 𝐻, 𝑇∶𝐶→𝐶 is a Lipschitzian pseudo-contractive mapping. For 𝑥₀∈𝐶, define the sequence {𝑥_𝑛} iteratively by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑛≥0, (1.4) where {𝛼_𝑛}, {𝛽_𝑛} are sequences of positive numbers satisfying the conditions
(i)   0≤𝛼_𝑛≤𝛽_𝑛<1;
(ii)   lim_𝑛→∞𝛽_𝑛=0;
(iii)  ∑^∞_𝑛=1𝛼_𝑛𝛽_𝑛=∞.
Then the sequence {𝑥_𝑛} defined by (1.4) converges strongly to a fixed point of 𝑇.

Since its publication in 1974, Theorem IS, as far as we know, has never been extended to more general Banach spaces.

In this paper, we suggest and analyze a three-step iteration below Algorithm 1.2 associated with hemi-contractive mappings having a strong convergence in the setting of Banach spaces under some appropriate conditions.

Algorithm 1.2. For arbitrary 𝑥₁∈𝐶, let the sequence {𝑥_𝑛} be generated by 𝑧_𝑛=1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑧_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑛≥1. (1.5)

It is clear that Algorithm 1.2 includes Mann (one-step) and Ishikawa (two-step) iterations as special cases. For some related works, please refere to [15–19].

2. Preliminaries

Let 𝐸 be a Banach space, the modulus of convexity of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜖)1=inf1−2𝑥+𝑦∶‖𝑥‖=1,𝑦=1,𝑥−𝑦≥𝜖. (2.1) A Banach space 𝐸 is called uniformly convex if and only if 𝛿_𝐸(𝜖)>0 for all 𝜖∈(0,2]. For 𝑝>1, the (generalized) duality mapping 𝐽_𝑝∶𝐸→2^𝐸∗ is defined as 𝐽_𝑝(𝑥)∶={𝑥^∗∈𝐸∶⟨𝑥,𝑥^∗⟩=‖𝑥‖^𝑝,‖𝑥^∗‖=‖𝑥‖^𝑝−1}. In particular, 𝐽=𝐽₂ is the normalized duality mapping on 𝐸. It is known that 𝐽_𝑝(𝑥)=‖𝑥‖^𝑝−2𝐽(𝑥), 𝑥≠0. A Banach space 𝐸 is called p-uniformly convex if there exists a constant 𝑐>0 such that 𝛿_𝐸(𝜖)≥𝑐𝜖^𝑝, 0<𝜖≤2. It is known (see e.g., [12]) that 𝐿_𝑝 is 2uniformlyconvex,if1<𝑝≤2,𝑝uniformlyconvex,if𝑝≥2. (2.2)

For proving our main results, we shall need the following lemmas.

Lemma 2.1 (see [20]). Let 𝑝>1 be a given real number. Then the following statements about a Banach space 𝐸 are equivalent:
(i)  𝐸 is p-uniformly convex;
(ii)  there is a constant 𝑐_𝑝>0 such that for every 𝑥,𝑦∈𝐸, 𝑗_𝑝(𝑥)∈𝐽_𝑝(𝑥), the following inequality holds: 𝑥+𝑦^𝑝≥‖𝑥‖^𝑝+𝑝𝑦,𝑗_𝑝(𝑥)+𝑐_𝑝𝑦^𝑝. (2.3)

Remark 2.2. Replacing 𝑥 by (𝑥+𝑦), 𝑦 by (−𝑦) in inequality (2.3) and using the Cauchy-Schwarz inequality, we can obtain 𝑥+𝑦^𝑝≤‖𝑥‖^𝑝+𝑝𝑦⋅𝑥+𝑦^𝑝−1. (2.4)

Lemma 2.3 (see [20]). Let 𝑝>1 be a given real number. Let 𝐸 be a 𝑝-uniformly convex Banach space. Then, there exists a constant 𝑑>0 such that 𝜆𝑥+(1−𝜆)𝑦^𝑝≤𝜆‖𝑥‖^𝑝+(1−𝜆)𝑦^𝑝−𝑊_𝑝(𝜆)𝑑𝑥−𝑦^𝑝, (2.5) for all 𝜆∈[0,1] and 𝑥,𝑦∈𝐸, where 𝑊_𝑝(𝜆)=𝜆^𝑝(1−𝜆)+𝜆(1−𝜆)^𝑝.

Lemma 2.4 (see [4]). Let {𝜌_𝑛}∶{𝜎_𝑛} be two nonnegative sequences and for all integers 𝑛≥𝑁₀ (for some fixed 𝑁₀), 𝜌_𝑛+1≤𝜌_𝑛+𝜎_𝑛.
(i)  if ∑^∞_𝑛=1𝜎_𝑛<∞, then lim_𝑛→∞𝜌_𝑛 exists;
(ii)  if ∑^∞_𝑛=1𝜎_𝑛<∞ and {𝜌_𝑛} has a sequence converging to zero, then lim_𝑛→∞𝜌_𝑛=0.

3. Main Results

In the sequel, 𝑐_𝑝 and 𝑑 will denote the constants appearing in inequalities (2.3) and (2.5), respectively. For the rest of this paper, we shall assume that 𝐸 be a real p-uniformly convex Banach space such that 2^{−(𝑝−2)}𝑑𝑝>(𝑝−1)𝑐_𝑝⁻¹ and 𝑝≤1+𝑐_𝑝. For 𝐿_𝑝 spaces with 1<𝑝≤2, the following inequalities hold (see [12, pages 1131-1132]): 𝑥+𝑦²≥‖𝑥‖²+2𝑦,𝐽(𝑥)+𝑐_𝑝𝑦²,𝜆𝑥+(1−𝜆)𝑦²≤𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝑊₂(𝜆)𝑝−1𝑥−𝑦², (3.1) for 𝜆∈[0,1] and for all 𝑥,𝑦∈𝐸, where 𝑐_𝑝=[1+𝑡_𝑝^(𝑝−1)][(1+𝑡_𝑝)^{−(𝑝−1)}], and for 0<𝑡_𝑝<1, 𝑡_𝑝 is the unique solution of the equation 𝑔(𝑡)=(𝑝−2)𝑡^(𝑝−1)+(𝑝−1)𝑡^(𝑝−2)−1=0.

Remark 3.1. We observe that the function ℎ∶[0,1]→[0,∞) defined by ℎ(𝑥)=(1+𝑥^𝑝−1)/(1+𝑥)^𝑝−1 is increasing on [0,1] (ℎ^′(𝑥)=(1+𝑥)^𝑝−2(𝑝−1)(𝑥^𝑝−2−1)/(1+𝑥)^2𝑝−2≥0), hence for 𝐿_𝑝(1<𝑝≤2), we have 𝑐_𝑝≥1 and 𝑑=𝑝−1. Therefore, the conditions 2^{−(𝑝−2)}𝑑𝑝>(𝑝−1)𝑐_𝑝⁻¹ and 𝑝≤1+𝑐_𝑝 are satisfied.

Lemma 3.2. Let 𝐸 be a real 𝑝-uniformly convex Banach space, ∅≠𝐶⊂𝐸 nonempty closed convex and bounded, and 𝑇∶𝐶→𝐶 a hemi-contractive mapping with 𝐹(𝑇)≠∅. Then, for each 𝑥∈𝐶 and for each integer 𝑛≥1, the following inequality holds: 𝑐_𝑝𝑇𝑥−𝑥^∗𝑝≤𝑝−1𝑥−𝑥^∗𝑝+‖𝑥−𝑇𝑥‖^𝑝,∀𝑥^∗∈𝐹(𝑇). (3.2)

Proof. Replacing 𝑥 by (1/2)(𝑥−𝑥^∗) and 𝑦 by (−1/2)(𝑇𝑥−𝑥^∗) in inequality (2.3), we can get ‖𝑥−𝑇𝑥‖^𝑝≥𝑥−𝑥^∗𝑝−𝑝2^𝑝−1𝑇𝑥−𝑥^∗,𝑗_𝑝12𝑥−𝑥^∗+𝑐_𝑝𝑇𝑥−𝑥^∗𝑝≥𝑥−𝑥^∗𝑝−𝑝𝑥−𝑥^∗𝑝+𝑐_𝑝𝑇𝑥−𝑥^∗𝑝. (3.3) Since 𝑗_𝑝((1/2)(𝑥−𝑥^∗))∈𝐽_𝑝((1/2)(𝑥−𝑥^∗))=2^{−(𝑝−1)}‖𝑥−𝑥^∗‖^(𝑝−2)𝐽(𝑥−𝑥^∗) so that 𝑐_𝑝𝑇𝑥−𝑥^∗𝑝≤𝑝−1𝑥−𝑥^∗𝑝+‖𝑥−𝑇𝑥‖^𝑝. (3.4) This completes the proof.

Remark 3.3. We note that the function 𝑓∶[0,∞)→(−∞,+∞) defined by 𝑓(𝑥)=𝐿^𝑝𝑥^𝑝−𝑑𝑝(1−𝑥)2^{−(𝑝−2)}+(𝑝−1)𝑐_𝑝⁻¹ is strictly increasing on (0,∞). Hence, it has at most one zero on (0,∞), provided that 𝑓(0)=(𝑝−1)𝑐_𝑝⁻¹−𝑑𝑝2^{−(𝑝−2)}<0. In this case, since 𝑓(1)=𝐿^𝑝+(𝑝−1)𝑐_𝑝⁻¹>0, it follows that the zero 𝑡_𝑝∈(0,1).

Lemma 3.4. Let 𝐸 be a real 𝑝-uniformly convex Banach space such that 2^{−(𝑝−2)}𝑑𝑝>(𝑝−1)𝑐_𝑝⁻¹ and 𝑝≤1+𝑐_𝑝. Let 𝐶 be a nonempty closed convex and bounded subset of 𝐸, let 𝑇∶𝐶→𝐶 be a Lipschitz hemi-contractive mapping with Lipschitz constant 𝐿≥0 and 𝐹(𝑇)≠∅. Let {𝛼_𝑛}, {𝛽_𝑛} and {𝛾_𝑛} be three real sequences in [0,1] satisfying the following conditions: 𝜖≤1−𝑑𝑐_𝑝1−𝛼_𝑛2^{−(𝑝−2)}≤𝛽_𝑛≤𝑏,^∞_𝑛=1𝛾_𝑛<∞. (3.5) for all integers 𝑛≥1, some 𝜖>0 and 𝑏∈(0,𝑡_𝑝), where 𝑡_𝑝 is the unique solution of the equation: 𝐿^𝑝𝑥^𝑝−𝑑𝑝(1−𝑥)2^{−(𝑝−2)}+𝑝−1𝑐_𝑝⁻¹=0, (3.6) on (0,∞).
For arbitrary 𝑥₁∈𝐶, let the sequence {𝑥_𝑛} be generated by 𝑧_𝑛=1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑧_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑛≥1. (3.7) Then, lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.

Proof. We shall use 𝑀 to denote the possible different constants appearing in the following reasoning.
Let 𝑥^∗∈𝐹(𝑇). Using inequality (2.5), we have 𝑥_𝑛+1−𝑥^∗𝑝=1−𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑇𝑦_𝑛−𝑥^∗𝑝≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗𝑝+𝛼_𝑛𝑇𝑦_𝑛−𝑥^∗𝑝−𝑊_𝑝𝛼_𝑛𝑑𝑥_𝑛−𝑇𝑦_𝑛^𝑝. (3.8) From (3.2), we have 𝑐_𝑝𝑇𝑥_𝑛−𝑥^∗𝑝≤𝑝−1𝑥_𝑛−𝑥^∗𝑝+𝑥_𝑛−𝑇𝑥_𝑛^𝑝,𝑐_𝑝𝑇𝑦_𝑛−𝑥^∗𝑝≤𝑝−1𝑦_𝑛−𝑥^∗𝑝+𝑦_𝑛−𝑇𝑦_𝑛^𝑝. (3.9)(3.10) Moreover, we also have 𝑦_𝑛−𝑥^∗𝑝=1−𝛽_𝑛𝑥_𝑛−𝑥^∗+𝛽_𝑛𝑇𝑧_𝑛−𝑥^∗𝑝≤1−𝛽_𝑛𝑥_𝑛−𝑥^∗𝑝+𝛽_𝑛𝑇𝑧_𝑛−𝑥^∗𝑝−𝑊_𝑝𝛽_𝑛𝑑𝑥_𝑛−𝑇𝑧_𝑛^𝑝,𝑦_𝑛−𝑇𝑦_𝑛^𝑝=1−𝛽_𝑛𝑥_𝑛−𝑇𝑦_𝑛+𝛽_𝑛𝑇𝑧_𝑛−𝑇𝑦_𝑛^𝑝≤1−𝛽_𝑛𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝛽_𝑛𝑇𝑧_𝑛−𝑇𝑦_𝑛^𝑝−𝑊_𝑝𝛽_𝑛𝑑𝑥_𝑛−𝑇𝑧_𝑛^𝑝. (3.11)(3.12) At the same time, applying (2.4), we can obtain the following estimates: 𝑇𝑧_𝑛−𝑥^∗𝑝=𝑇𝑥_𝑛−𝑥^∗+𝑇𝑧_𝑛−𝑇𝑥_𝑛^𝑝≤𝑇𝑥_𝑛−𝑥^∗𝑝+𝑝𝑇𝑧_𝑛−𝑇𝑥_𝑛𝑇𝑧_𝑛−𝑥^∗𝑝−1≤𝑇𝑥_𝑛−𝑥^∗𝑝+𝑝𝐿𝑧_𝑛−𝑥_𝑛𝑇𝑧_𝑛−𝑥^∗𝑝−1=𝑇𝑥_𝑛−𝑥^∗𝑝+𝑝𝐿𝛾_𝑛𝑇𝑥_𝑛−𝑥_𝑛𝑇𝑧_𝑛−𝑥^∗𝑝−1≤𝑇𝑥_𝑛−𝑥^∗𝑝+𝑀𝛾_𝑛,𝑥_𝑛−𝑇𝑥_𝑛^𝑝=𝑇𝑧_𝑛−𝑇𝑥_𝑛+𝑥_𝑛−𝑇𝑧_𝑛^𝑝≤𝑥_𝑛−𝑇𝑧_𝑛^𝑝+𝑝𝑇𝑧_𝑛−𝑇𝑥_𝑛𝑥_𝑛−𝑇𝑥_𝑛^𝑝−1≤𝑥_𝑛−𝑇𝑧_𝑛^𝑝+𝑀𝛾_𝑛,𝑇𝑧_𝑛−𝑇𝑦_𝑛^𝑝=𝑇𝑥_𝑛−𝑇𝑦_𝑛+𝑇𝑧_𝑛−𝑇𝑥_𝑛^𝑝≤𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝑝𝑇𝑧_𝑛−𝑇𝑥_𝑛𝑇𝑧_𝑛−𝑇𝑦_𝑛^𝑝−1≤𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝑀𝛾_𝑛. (3.13)(3.14)(3.15) Substitute (3.13) and (3.14) into (3.11) to get 𝑦_𝑛−𝑥^∗𝑝≤1−𝛽_𝑛𝑥_𝑛−𝑥^∗𝑝+𝛽_𝑛𝑇𝑥_𝑛−𝑥^∗𝑝−𝑊_𝑝𝛽_𝑛𝑑𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛, (3.16) this together with (3.9) implies that 𝑦_𝑛−𝑥^∗𝑝≤1−𝛽_𝑛𝑥_𝑛−𝑥^∗𝑝+𝛽_𝑛𝑐_𝑝⁻¹𝑝−1𝑥_𝑛−𝑥^∗𝑝+𝑥_𝑛−𝑇𝑥_𝑛^𝑝−𝑊_𝑝𝛽_𝑛𝑑𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛=1+𝛽_𝑛𝑐_𝑝⁻¹𝑝−1−𝑐_𝑝𝑥_𝑛−𝑥^∗𝑝+𝛽_𝑛𝑐_𝑝⁻¹−𝑊_𝑝𝛽_𝑛𝑑𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛. (3.17) Set 𝑡_𝑛=𝛽_𝑛𝑐_𝑝⁻¹(𝑝−1−𝑐_𝑝), 𝑟_𝑛=𝛽_𝑛𝑐_𝑝⁻¹−𝑊_𝑝(𝛽_𝑛)𝑑. Then, 𝑦_𝑛−𝑥^∗𝑝≤1+𝑡_𝑛𝑥_𝑛−𝑥^∗𝑝+𝑟_𝑛𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛. (3.18) From (3.12), (3.14), and (3.15), we have 𝑦_𝑛−𝑇𝑦_𝑛^𝑝≤1−𝛽_𝑛𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝−𝑊_𝑝𝛽_𝑛𝑑𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛. (3.19)
Substitution of (3.18) and (3.19) into (3.10) yields 𝑐_𝑝𝑇𝑦_𝑛−𝑥^∗𝑝≤𝑝−11+𝑡_𝑛𝑥_𝑛−𝑥^∗𝑝+𝑝−1𝑟_𝑛𝑥_𝑛−𝑇𝑥_𝑛^𝑝+1−𝛽_𝑛𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝−𝑊_𝑝𝛽_𝑛𝑑𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛=𝑝−11+𝑡_𝑛𝑥_𝑛−𝑥^∗𝑝+𝑝−1𝑟_𝑛−𝑊_𝑝𝛽_𝑛𝑑×𝑥_𝑛−𝑇𝑥_𝑛^𝑝+1−𝛽_𝑛𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝑀𝛾_𝑛. (3.20) Substitution of this inequality into (3.8) now gives 𝑥_𝑛+1−𝑥^∗𝑝≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗𝑝+𝛼_𝑛𝑐_𝑝⁻¹𝑝−11+𝑡_𝑛𝑥_𝑛−𝑥^∗𝑝+𝑝−1𝑟_𝑛−𝑊_𝑝𝛽_𝑛𝑑×𝑥_𝑛−𝑇𝑥_𝑛^𝑝+1−𝛽_𝑛𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝−𝑊_𝑝𝛼_𝑛𝑑𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝑀𝛾_𝑛=1−𝛼_𝑛+𝛼_𝑛𝑐_𝑝⁻¹𝑝−11+𝑡_𝑛𝑥_𝑛−𝑥^∗𝑝+𝛼_𝑛𝑐_𝑝⁻¹1−𝛽_𝑛−𝑊_𝑝𝛼_𝑛𝑑𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝛼_𝑛𝑐_𝑝⁻¹𝑝−1𝑟_𝑛−𝑊_𝑝𝛽_𝑛𝑑𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝛼_𝑛𝛽_𝑛𝑐_𝑝⁻¹𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝑀𝛾_𝑛, (3.21) that is, 𝑥_𝑛+1−𝑥^∗𝑝≤1+𝛼_𝑛𝑝−1𝑐_𝑝⁻¹1+𝑡_𝑛−1𝑥_𝑛−𝑥^∗𝑝−𝑊_𝑝𝛼_𝑛𝑑−𝑐_𝑝⁻¹𝛼_𝑛1−𝛽_𝑛𝑥_𝑛−𝑇𝑦_𝑛^𝑝−𝑐_𝑝⁻¹𝛼_𝑛𝑊_𝑝𝛽_𝑛𝑑−𝑝−1𝑟_𝑛𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝛼_𝑛𝛽_𝑛𝑐_𝑝⁻¹𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝑀𝛾_𝑛. (3.22) Observe that 𝑐_𝑝⁻¹(𝑝−1)(1+𝑡_𝑛)−1=𝑐_𝑝⁻²(𝑝−1−𝑐_𝑝)[(𝑝−1)𝛽_𝑛+𝑐_𝑝] and that by condition (3.5), since 𝑊_𝑝(𝛼_𝑛)≥𝛼_𝑛(1−𝛼_𝑛)2^{−(𝑝−2)}, we get 𝑊_𝑝(𝛼_𝑛)𝑑−𝑐_𝑝⁻¹𝛼_𝑛(1−𝛽_𝑛)≥0. so that 𝑥_𝑛+1−𝑥^∗𝑝≤1+𝛼_𝑛𝑐_𝑝⁻²𝑝−1−𝑐_𝑝𝑝−1𝛽_𝑛+𝑐_𝑝𝑥_𝑛−𝑥^∗𝑝−𝛼_𝑛𝑐_𝑝⁻¹𝑊_𝑝𝛽_𝑛𝑑−𝑝−1𝑟_𝑛𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝛼_𝑛𝛽_𝑛𝑐_𝑝⁻¹𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝+𝑀𝛾_𝑛. (3.23) Since 𝑇 is Lipschitzian, we have 𝑇𝑥_𝑛−𝑇𝑦_𝑛^𝑝≤𝐿^𝑝𝑥_𝑛−𝑦_𝑛^𝑝=𝐿^𝑝𝛽_𝑛𝑥_𝑛−𝑇𝑧_𝑛^𝑝≤𝐿^𝑝𝛽^𝑝_𝑛𝑥_𝑛−𝑇𝑧_𝑛^𝑝=𝐿^𝑝𝛽^𝑝_𝑛𝑥_𝑛−𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑇𝑧_𝑛^𝑝≤𝐿^𝑝𝛽^𝑝_𝑛𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑝𝑇𝑥_𝑛−𝑇𝑧_𝑛𝑥_𝑛−𝑇𝑧_𝑛^𝑝−1≤𝐿^𝑝𝛽^𝑝_𝑛𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛. (3.24) By the assumption 𝑝≤1+𝑐_𝑝, hence 𝑥_𝑛+1−𝑥^∗𝑝≤𝑥_𝑛−𝑥^∗𝑝−𝛼_𝑛𝛽_𝑛𝑐_𝑝⁻¹𝑑𝑝1−𝛽_𝑛2^{−(𝑝−2)}−𝑝−1𝑐_𝑝⁻¹−𝛽^𝑝_𝑛𝐿^𝑝×𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛. (3.25) Since 𝑏∈(0,𝑡_𝑝), it follows that 𝛿=𝑑𝑝(1−𝑏)2^{−(𝑝−2)}−(𝑝−1)𝑐_𝑝⁻¹−𝑏^𝑝𝐿^𝑝>0. We can choose some 𝜖 such that 𝜖^′=1−(1−𝜖)2^{−(𝑝−2)}𝑐_𝑝𝑑>0. Then, condition (3.5) implies 𝛼_𝑛≥𝜖^′>0. Furthermore, inequality (3.25) now yields the following estimates: 𝑥_𝑛+1−𝑥^∗𝑝≤𝑥_𝑛−𝑥^∗𝑝−𝜖𝜖^′𝑐_𝑝⁻¹𝛿𝑥_𝑛−𝑇𝑥_𝑛^𝑝+𝑀𝛾_𝑛≤𝑥_𝑛−𝑥^∗𝑝+𝑀𝛾_𝑛. (3.26) Since ∑^∞_𝑛=0𝛾_𝑛<∞, it follows from Lemma 2.3 that lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖^𝑝 exists. Let lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖^𝑝=𝑟. Inequality (3.26) also yields 0<𝜖𝜖^′𝑐_𝑝⁻¹𝛿𝑥_𝑛−𝑇𝑥_𝑛^𝑝≤𝑥_𝑛−𝑥^∗𝑝−𝑥_𝑛+1−𝑥^∗𝑝+𝑀𝛾_𝑛→0. (3.27) Hence, lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. This completes the proof.

Remark 3.5. The interest and importance of Lemma 3.2 lie in the fact that strong convergence of the sequence {𝑥_𝑛} is achieved under certain mild compactness assumptions either on 𝑇 or on its domain.

Now, we give a strong convergence theorem as follows.

Theorem 3.6. Let 𝐸 be a real p-uniformly convex Banach space such that 2^{−(𝑝−2)}𝑑𝑝>(𝑝−1)𝑐_𝑝⁻¹ and 𝑝≤1+𝑐_𝑝. Let 𝐶 be a nonempty closed convex and bounded subset of 𝐸, 𝑇∶𝐶→𝐶 be a completely continuous Lipschitz hemi-contractive mapping with Lipschitz constant 𝐿≥0 and 𝐹(𝑇)≠∅. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three real sequences in [0,1] satisfying the following conditions: 𝜖≤1−𝑑𝑐_𝑝1−𝛼_𝑛2^{−(𝑝−2)}≤𝛽_𝑛≤𝑏,^∞_𝑛=1𝛾_𝑛<∞, (3.28) for all integers 𝑛≥1, some 𝜖>0 and 𝑏∈(0,𝑡_𝑝), where 𝑡_𝑝 is the unique solution of the equation: 𝐿^𝑝𝑥^𝑝−𝑑𝑝(1−𝑥)2^{−(𝑝−2)}+𝑝−1𝑐_𝑝⁻¹=0, (3.29) on (0,∞).
For arbitrary 𝑥₁∈𝐶, let the sequence {𝑥_𝑛} be defined by (3.7). Then, {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. By Lemma 3.2  lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. Since 𝑇 is completely continuous, there exists a subsequence {𝑇𝑥_𝑛𝑖} of {𝑇𝑥_𝑛} such that 𝑇𝑥_𝑛𝑖→𝑦^∗. This implies, by Lemma 3.2 that 𝑥_𝑛𝑖⟶𝑦^∗. (3.30) By the continuity of 𝑇 and Lemma 3.2, we obtain 𝑇𝑦^∗=𝑦^∗, that is, 𝑦^∗ is a fixed point of 𝑇. Replacing the 𝑥^∗ by 𝑦^∗ in inequality (3.26), we obtain that 𝑥_𝑛+1−𝑦^∗𝑝≤𝑥_𝑛−𝑦^∗𝑝−𝜖𝜖^′𝑐_𝑝⁻¹𝛿𝑥_𝑛−𝑇𝑥_𝑛^𝑝≤𝑥_𝑛−𝑦^∗𝑝+𝑀𝛾_𝑛. (3.31)
From (3.30), we know that {‖𝑥_𝑛−𝑦^∗‖} has a sequence converging to zero. We note that the condition ∑^∞_𝑛=0𝛾_𝑛<∞. Hence from inequality (3.31) and Lemma 2.3, we can conclude that ‖𝑥_𝑛−𝑦^∗‖→∞ as 𝑛→∞, that is, {𝑥_𝑛} converges to a fixed point of 𝑇. This completes the proof.

From Theorem 3.6, we can obtain the following result.

Corollary 3.7. Let 𝐸 be a real p-uniformly convex Banach space such that 2^{−(𝑝−2)}𝑑𝑝>(𝑝−1)𝑐_𝑝⁻¹ and 𝑝≤1+𝑐_𝑝. Let 𝐶 be a nonempty closed convex and bounded subset of 𝐸, 𝑇∶𝐶→𝐶 be a completely continuous Lipschitz hemi-contractive mapping with Lipschitz constant 𝐿≥0 and 𝐹(𝑇)≠∅. Let {𝛼_𝑛} and {𝛽_𝑛} be two real sequences in [0,1] satisfying the following condition: 𝜖≤1−𝑑𝑐_𝑝1−𝛼_𝑛2^{−(𝑝−2)}≤𝛽_𝑛≤𝑏, (3.32) for all integers 𝑛≥1, some 𝜖>0 and 𝑏∈(0,𝑡_𝑝), where 𝑡_𝑝 is the unique solution of the equation: 𝐿^𝑝𝑥^𝑝−𝑑𝑝(1−𝑥)2^{−(𝑝−2)}+𝑝−1𝑐_𝑝⁻¹=0 (3.33) on (0,∞).
For arbitrary 𝑥₁∈𝐶, let the sequence {𝑥_𝑛} be defined by (1.4). Then, {𝑥_𝑛} converges strongly to a fixed points of 𝑇.

Acknowledgment

This research was partially supported by the Youth Foundation of Taizhou University (2011QN11).