Constructive techniques for zeros of monotone mappings in certain Banach spaces

Let E be a 2-uniformly convex real Banach space with uniformly Gâteaux differentiable norm, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^*$$\end{document}E∗ its dual space. Let \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 bounded strongly monotone mapping such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{-1}(0)\ne \emptyset .$$\end{document}A-1(0)≠∅. For given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1\in E,$$\end{document}x1∈E, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x_n\}$$\end{document}{xn} be generated by the algorithm: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x_{n+1}= J^{-1}( Jx_n -\alpha _nAx_n),\,n\ge 1, \end{aligned}$$\end{document}xn+1=J-1(Jxn-αnAxn),n≥1,where J is the normalized duality mapping from E into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^*$$\end{document}E∗ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\alpha _n\}$$\end{document}{αn} is a real sequence in (0, 1) satisfying suitable conditions. Then it is proved that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x_n\}$$\end{document}{xn} converges strongly to the unique point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^*\in A^{-1}(0).$$\end{document}x∗∈A-1(0). Finally, our theorems are applied to the convex minimization problem.

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).
The extension of the monotonicity definition to operators from a Banach space into its dual has been the starting point for the development of nonlinear functional analysis. The monotone maps constitute the most manageable class because of the very simple structure of the monotonicity condition. The monotone mappings appear in a rather wide variety of contexts since they can be found in many functional equations. Many of them appear also in calculus of variations as subdifferential of convex functions. (Pascali and Sburian 1978, p. 101).
Let E be a real normed space, E * its topological dual space. The map J : E → 2 E * defined by is called the normalized duality map on E. where, , denotes the generalized duality pairing between E and E * .
A map A : E → E * is called monotone if for each x, y ∈ E, the following inequality holds: A is called strongly monotone if there exists k ∈ (0, 1) such that for each x, y ∈ E, the following inequality holds: A map A : E → E is called accretive if for each x, y ∈ E, there exists j(x − y) ∈ J (x − y) such that 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 In a Hilbert space, the normalized duality map is the identity map. Hence, in Hilbert spaces, monotonicity and accretivity coincide. For accretive-type operator A, solutions of the equation Au = 0, in many cases, represent equilibrium state of some dynamical system (see e.g., Chidume 2009, p. 116).
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. It is trivial to observe that zeros of A correspond to fixed points of T. For Lipschitz strongly pseudocontractive 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 pseudo-contractive and Lipschitz map. For arbitrary x 0 ∈ K , let a sequence {x n } be defined iteratively by x n+1 = (1 − n )x n + n Tx n , n ≥ 0, where { n } ⊂ (0, 1) satisfies the following conditions: Then, {x n } converges strongly to the unique fixed point of T.
By setting T := I − A in Theorem C1, the following theorem for approximating a solution of Au = 0 where A is a strongly accretive and bounded operator can be proved. Theorem C2 Let E = L p , 2 ≤ p < ∞. Let A : E → E be a strongly accretive and bounded map. Assume A −1 (0) � = ∅. For arbitrary x 0 ∈ K , let a sequence {x n } be defined iteratively by x n+1 = x n − n Ax n , n ≥ 0, where { n } ⊂ (0, 1) satisfies the following conditions: Then, {x n } converges strongly to the unique solution of Au = 0.
Unfortunately, the success achieved in using geometric properties developed from the mid 1980s to early 1990s in approximating zeros of accretive-type mappings has not carried over to approximating zeros of monotone-type operators in general Banach spaces. Part of the 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.
Attempts have been made to overcome this difficulty by introducing the inverse of the normalized duality mapping in the recursion formulas for approximating zeros of monotone-type mappings.
In this paper, we introduce an iterative scheme of Mann-type to approximate the unique zero of a strongly monotone bounded mapping in 2-uniformly convex real Banach with uniformly Gâteaux differentiable norm. Then we apply our results to the convex minimization problem. Finally, our method of proof is of independent interest.

Preliminaries
Let E be a normed linear space. E is said to be smooth if exist for each x, y ∈ S E (Here S E := {x ∈ E : ||x|| = 1} is the unit sphere of E). E is said to be uniformly smooth if it is smooth and the limit is attained uniformly for each x, y ∈ S E , and E is Fréchet differentiable if it is smooth and the limit is attained uniformly for y ∈ S E .
A normed linear space E is said to be strictly convex if: The modulus of convexity of E is the function δ E : Observe that every p-uniformly convex space is uniformly convex. It is well known that E is smooth if and only if J is single valued. Moreover, if E is a reflexive smooth and strictly convex Banach space, then J −1 is single valued, one-to-one, surjective and it is the duality mapping from E * into E. Finally, if E has uniform Gâteaux differentiable norm, then J is norm-to-weak * uniformly continuous on bounded sets.
In the sequel, we shall need the following results and definitions.
Theorem 2.1 ) Let p > 1 be a given real number. Then the following are equivalent in a Banach space: There is a constant c 1 > 0 such that for every x, y ∈ E and j x ∈ J p (x), The following inequality holds: (iii) There is a constant c 2 > 0 such that for every x, y ∈ E and j x ∈ J p (x), j y ∈ J p (y), the following inequality holds: 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 E * , introduced by Alber has been studied by Alber (1996), Alber and Guerre-Delabiere (2001), Kamimura and Takahashi (2002), Reich (1979) and a host of other authors. This functional φ will play a central role in what follows. If E = H , a real Hilbert space, then Eq. (2.3) reduce to φ(x, y) = �x − y� 2 for x, y ∈ H . It is obvious from the definition of the function φ that Define a functional V : E × E * → R by Then, it is easy to see that Lemma 2.3 (Alber 1996) Let E be a reflexive strictly convex and smooth real Banach space with E * as its dual. Then, for all x ∈ E and x * , y * ∈ E * .

Lemma 2.4 (Kamimura and Takahashi 2002) Let E be a smooth uniformly convex real
Banach space, and let {x n } and {y n } be two sequences of E. If either {x n } or {y n } is bounded and φ(x n , y n ) → 0 as n → ∞, then �x n − y n � → 0 as n → ∞.
Lemma 2.5 (Tan and Xu 1993) Let {a n } be a sequence of non-negative real numbers satisfying the following relation: Such that ∞ n=0 σ n < ∞. Then lim n→∞ a n exists. If addition, the sequence {a n } has a subsequence that converges to 0. Then {a n } converges to 0.
The following results will be useful.

Lemma 2.6 (Alber and Ryazantseva 2006) For p > 1, let X be a p-uniformly convex and smooth real Banach space and S a bounded subset of X. Then there exists a positive constant α such that
Lemma 2.7 Let E be a 2-uniformly convex smooth real Banach space. Then the following inequality holds: a n+1 ≤ a n + σ n n ≥ 0.
where 0 ≤ c 1 ≤ 1 has the same meaning as in Theorem 2.1.
Proof Using (ii) of Theorem 2.1, we have Interchanging x and y, we obtain

Main results
We now prove the following result Theorem 3.1 Let E be a 2-uniformly convex real Banach space with uniformly Gâteaux differentiable norm and E * its dual space. Let A : E → E * be a bounded and k-strongly monotone mapping such that A −1 (0) � = ∅. For arbitrary x 1 ∈ E, let {x n } be the sequence defined iteratively by: where J is the normalized duality mapping from E into E * and {α n } ⊂ (0, 1) is a real sequence satisfying the following conditions: (i) Proof The proof is in two steps: Step 1: We prove that {x n } is bounded. Since A −1 (0) � = ∅, let x * ∈ A −1 (0).There exists r > 0 such that: We show that φ(x n , x * ) ≤ r for all n ≥ 1. The proof is by induction. We have φ(x 1 , x * ) ≤ r. Assume that φ(x n , x * ) ≤ r for some n ≥ 1. We show that φ(x n+1 , x * ) ≤ r. From the induction assumption and Lemma 2.6, there exists α * > 0 such that �x n − x * � 2 ≤ rα * . Since A is bounded, we have: where L is a Lipschitz constant of J −1 . Define Using the definition of x n+1 , we compute as follows: Using Lemma 2.3, with y * = α n Ax n , we have: Using the strong monotonocity of A, Schwartz inequality and the Lipzchitz property of J −1 , we obtain Using Lemma 2.7, it follows that Finally, using inequality (3.2), the definition of γ 0 (3.4), and the induction assumption, we have Therefore, φ(x * , x n+1 ) ≤ r. Thus, by induction, φ(x * , x n ) ≤ r for all n ≥ 1. So, by inequality (2.4), {x n } is bounded.
Step 2: We now prove that {x n } converges strongly to the unique point x * of A −1 (0). Following the same arguments as in Step 1, using the fact the sequence {x n } is bounded and A is bounded, there exists a positive constant M such that Therefore, Using the hypothesis ∞ n=0 α 2 n < ∞ and Lemma 2.5, it follows that lim n→∞ φ(x * , x n ) exists. From (3.6), we have Using the fact that ∞ n=0 α n = ∞, it follows that lim inf �x * − x n � 2 = 0. Therefore, there exists a subsequence {x n k } of {x n } such that x n k → x * as k → ∞. We have Since {x n } is bounded and J is norm-to weak * uniformly continuous on bounded subsets of E, it follows that {φ(x * , x n )} has a subsequence that converges to 0. Thus, by Lemma (), {φ(x * , x n )} converges strongly to 0. Applying Lemma(), we obtain that �x n − x * � → 0 as n → ∞. This completes the proof.
Corollary 3.2 Let E = L p , 1 < p ≤ 2 and A : E → E * be a bounded and strongly monotone mapping. For arbitrary x 1 ∈ E, let {x n } be the sequence defined iteratively by: where J is the normalized duality mapping from E into E * and {α n } ⊂ (0, 1) is a real sequence satisfying the following conditions: (i) ∞ n=1 α n = ∞; (ii) ∞ n=0 α 2 n < ∞. Then, there exists γ 0 > 0 such that if α n < γ 0 , ∀ n ≥ 1 the sequence {x n } converges strongly to the unique solution of the equation Au = 0.
Proof Since L p spaces, 1 < p ≤ 2 are 2-uniformly convex Banach space with uniformly Gâteaux differentiable norm, then the proof follows from Theorem 3.1.

Application to convex minimization problems
In this section, we study the problem of finding a minimizer of a convex function f defined from a real Banach space E to R.
The following basic results are well known.
Lemma 4.1 Let f : E → R be a real-valued differentiable convex function and a ∈ E. df : E → E * denotes the differential map associated to f. Then the following hold. Proof Let x 0 ∈ E and r > 0. Set B := B(x 0 , r). We show that df(B) is bounded. From lemma 4.1, there exists γ > 0 such that φ(x * , x n k ) = �x� 2 − 2�x * , Jx n k � + �x n k � 2 .
(3.7) x n+1 = J −1 (Jx n − α n Ax n ), n ≥ 1, Let z * ∈ df (B) and x * ∈ B such that z * = df (x * ). Since B is open, for all u ∈ E, there exists t > 0 such that x * + tu ∈ B. Using the fact that z * = df (x * ) the convexity of f and inequality (4.1), it follows that so that Therefore �z * � ≤ γ . Hence df(B) is bounded. Definition 4.3 A function f : E → R is said to be strongly convex if there exists α > 0 such that for every x, y ∈ E with x � = y and ∈ (0, 1), the following inequality holds: Lemma 4.4 Let E be normed linear space and f : E → R a real-valued differentiable convex function. Assume that f is strongly convex. Then the differential map df : E → E * is strongly monotone, i.e., there exists a positive constant k such that We now prove the following theorem.
Theorem 4.5 Let E be a 2-uniformly convex real Banach space with uniformly Gâteaux differentiable norm and let f : E → R be a differentiable, bounded, strongly convex realvalued function which satisfies the growth condition: f (x) → +∞ as �x� → +∞. For arbitrary x 1 ∈ E, let {x n } be the sequence defined iteratively by: where J is the normalized duality mapping from E into E * and {α n } ⊂ (0, 1) is a real sequence satisfying the following conditions: (i) ∞ n=1 α n = ∞; (ii) ∞ n=0 α 2 n < ∞. Then, f has a unique minimizer a * ∈ E and there exists γ 0 > 0 such that if α n < γ 0 , the sequence {x n } converges strongly to a * .
Proof Since E is reflexive, then from the growth condition, the continuity and the strict convexity of f, f has a unique minimizer a * characterized by df (a * ) = 0 (Lemma 4.1). Finally, from Lemmas 4.2 and 4.4, the differential map df : E → E * is bounded and strongly monotone. Therefore, the proof follows from Theorem 3.1.

Conclusion
In this work, we proposed a new iteration scheme for the approximation of zeros of monotone mappings defined in certain Banach spaces. Our results are used to approximate minimizers of convex functions. The results obtained in this paper are important improvements of recent important results in this field.