Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical banach spaces

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E=L_p$$\end{document}E=Lp, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document}1<p<∞, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A:E\rightarrow E^*$$\end{document}A:E→E∗ be a strongly monotone and Lipschitz mapping. A Krasnoselskii-type sequence is constructed and proved to converge strongly to the unique solution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Au=0$$\end{document}Au=0. Furthermore, our technique of proo f is of independent interest.

A is called strongly accretive if there exists k ∈ (0, 1) such that for each x, y ∈ E, there exists j(x − y) ∈ J (x − y) such that Several existence theorems have been established for the equation Au = 0 when A is of the monotone-type (see e.g., Deimling (1985;Pascali and Sburian 1978).
For approximating a solution of Au = 0, assuming existence, where A : E → E is of accretive-type, Browder (1967) defined an operator T : E → E by T := I − A, where I is the identity map on E. He called such an operator pseudo-contractive. A map T : E → E is then called pseudo-contractive if and is called strongly pseudo-contractive if there exists k ∈ (0, 1) such that It is trivial to observe that zeros of A corresspond to fixed points of T. For Lipschitz strongly pseudo-contractive maps, Chidume (1987) proved the following theorem.
Theorem C1 (Chidume 1987) Let E = L p , 2 ≤ p < ∞, and K ⊂ E be nonempty closed convex and bounded. Let T : K → K be a strongly pseudocontractive and Lipschitz map. For arbitrary x 0 ∈ K, let a sequence {x n } be defined iteratively by x n+1 = (1 − α n )x n + α n Tx n , n ≥ 0, where {α n } ⊂ (0, 1) satisfies the following conditions: Then, {x n } converges strongly to the unique fixed point of T.
The main tool used in the proof of Theorem C1 is an inequality of Bynum (1976). This theorem signalled the return to extensive research efforts on inequalities in Banach spaces and their applications to iterative methods for solutions of nonlinear equations. Consequently, this theorem of Chidume has been generalized and extended in various directions, leading to flourishing areas of research, for the past thirty years or so, by numerous authors (see e.g., Chidume 1986Chidume , 1990Chidume , 2002Chidume and Ali 2007;Chidume 2005, 2006;Chidume and Osilike 1999;Deng 1993a, b;Zhou 1997;Zhou andJia 1996, 1997;Liu 1995Liu , 1997Qihou 1990Qihou , 2002Weng 1991Weng , 1992Xiao 1998;Xu 1989Xu , 1991aXu , b, 1992Xu , 1998Roach 1991, 1992;Xu et al. 1995;Zhu 1994 and a host of other authors). Recent monographs emanating from these researches include those by Chidume (2009), Berinde (2007, Goebel and Reich (1984) and William and Shahzad (2014).
Unfortunately, the success achieved in using geometric properties developed from the mid 1980ies to early 1990ies in approximating zeros of accretive-type mappings has not carried over to approximating zeros of monotone-type operators in general Banach spaces. The first problem is that since A maps E to E * , for x n ∈ E, Ax n is in E * . Consequently, a recursion formula containing x n and Ax n may not be well defined. Another difficulty is that the normalized duality map which appears in most Banach space inequalities developed, and also appears in the definition of accretive-type mappings, does not appear in the definition of monotone-type mappings in general Banach spaces. This creats very serious technical difficulties.
Attemps have been made to overcome the first difficulty by introducing the inverse of the normalized duality mapping in the recursion formulas for approximating zeros of monotone-type mappings. But one major problem with such recursion formulas is that the exact form of the normalized duality map (or its inverse) is not known precisely in any space more general than L p spaces, 1 < p < ∞. Futhermore, the recursion formulas, apart from containing the normalized duality map and its inverse, generally involve computation of subsets and generalized projections, both of which are defined in a way that makes their computation almost impossible. We give some examples of some results obtained using these approximation schemes. Before we do this, however, we need the following definitions.
Let E be a real normed space and let a funtion φ(., .) : is called a generalized projection map. Now we present the following results.
In Hilbert space, suppose that a map A : where {α n } is a sequence in [0, 2γ ]. They proved that the sequence {x n } generated by (1.7) converges strongly to P VI(K ,A) (x 0 ), where P VI(K ,A) is the metric projection from K onto VI(K , A) (see e.g., Iiduka et al. 2004 for definition and explanation of the symbols).
Zegeye and Shahzad proved the following result.
Theorem 1.1 (Zegeye and Shahzad 2009) Let E be uniformly smooth and 2-uniformly convex real Banach space with dual E * . Let where c is the constants from the Lipschitz property of J −1 , then the sequence generated by We remark here that although the approximation methods used in the result of Iiduka et al. referred to above, and in Theorem 1.1 yield strong convergence to a solution of the problem under consideration, it is clear that they are not easy to implement. Furthermore, Theorem 1.1 excludes L p spaces, 2 < p < ∞, because these spaces are not 2-uniformly convex. The theorem, however, is applicable in L p spaces 1 < p < 2.
In this paper, we introduce an iterative scheme of Krasnoselskii-type to approximate the unique zero of a strongly monotone Lipschitz mapping in L p spaces, 1 < p < ∞. In these spaces, the formula for J is known precisely (see e.g., Cioranescu 1990;Chidume 2009). The Krasnoselskii sequence, whenever it converges, is known to converge as fast as a geometric progression. Furthermore, our iteration method which will not involve construction of subsets or the use of generalized projection is also of independent interest.

Preliminaries
In the sequel, we shall need the following results and definitions.
Lemma 2.1 (see e.g., Chidume 2009, p. 55) Let E = L p , 1 < p < 2. Then, there exists a constant c p > 0 such that for all x, y in L p the following inequalities hold: Let E be a smooth real Banach space with dual E * . The function φ : E × E → R, defined by, where J is the normalized duality mapping from E into 2 E * , introduced by Alber has been studied by Alber (1996), Alber and Guerre-Delabriere (2001), Kamimura and Takahashi (2002), Reich (1996) and a host of other authors. If E = H, a real Hilbert space, then Eq (2.3) reduces to φ(x, y) = �x − y� 2 for x, y ∈ H . It is obvious from the definition of the function φ that Define V : X × X * → R by Then, it is easy to see that Corollary 2.2 Let E = L p , 1 < p ≤ 2. Then J −1 is Lipschitz, i.e., there exists L 1 > 0 such that for all u, v ∈ E * , the following inequality holds: Proof This follows from inequality (2.2).
Lemma 2.4 (Alber 1996) Let X be a reflexive striclty convex and smooth Banach space with X * as its dual. Then, for all x ∈ X and x * , y * ∈ X * . Definition 2.5 An operator T : X → X * is called ψ-strongly monotone if there exists a continuous, strictly increasing function ψ : R → R with ψ(0) = 0 such that Let X and Y be Banach spaces with X * and Y * as their respective duals.

Definition 2.6 An operator
Clearly, every continuous map is hemicontinuous.

Main results
Convergence in L p spaces, 1 < p ≤ 2.
Theorem 4.1 Let E = L p , 1 < p ≤ 2. Let A : E → E * be a strongly monotone and Lipschitz map. For x 0 ∈ E arbitrary, let the sequence {x n } be defined by: where ∈ 0, δ . Then, the sequence {x n } converges strongly to x * ∈ A −1 (0) and x * is unique.
Proof Let ψ(t) = kt in inequality (2.11). By Lemma 2.7, A −1 (0) � = ∅. Let x * ∈ A −1 (0). Using the definition of x n+1 we compute as follows: Applying Lemma 2.4, we have Using the strong monotonocity of A, Lipschitz property of j −1 and the Lipschitz property of A, we have that: Thus, φ(x * , x n ) converges, since it is monotone decreasing and bounded below by zero. Consequently, This yields x n → x * as n → ∞. Suppose there exists y * ∈ A −1 (0), y * � = x * . Then, substituting x * by y * in the above argument, we obtain that x n → y * as n → ∞. By uniqueness of limit x * = y * . So, x * is unique. completing the proof.
Remark 1 We remark that for E = L p , 2 ≤ p < ∞, if A : E → E * satisfies the following conditions: there exists k ∈ (0, 1) such that (4.1) and A −1 (0) � = ∅, then the Krasnoselskii-type sequence (4.1) converges strongly to the unique solution of Au = 0. In fact, we prove the following theorem.
In the following theorem, δ p := k 2m p L p p−1 p−1 . Theorem 5.1 Let X = L p , 2 ≤ p < ∞. Let A : X → X * be a Lipschitz map. Assume that there exists a constant k ∈ (0, 1) such that A satisfies the following condition: and that A −1 (0) � = ∅. For arbitrary x 0 ∈ X, define the sequence {x n } iteratively by: where ∈ (0, δ p ). Then, the sequence {x n } converges strongly to the unique solution of the equation Ax = 0.
Proof We first prove that {x n } is bounded. This proof is by induction.