Iterative Algorithms for Single-valued and Multi-valued Nonexpansive-type Mappings in Real Lebesgue Spaces

Iterative Algorithms for Single-valued and Multi-valued Nonexpansive-type Mappings in Real Lebesgue Spaces

Chapter One

PREAMBLE TO THE STUDY

Fixed Point Theory is concerned with solutions of the equation

x = Tx (1.0.1)

where T is a (possibly) nonlinear operator defined on a metric space. Any x that solves (1.0.1) is called a fixed point of T and the collection of all such elements is denoted by F (T ).   For a multi-valued mapping T  :  X      2^X, a fixed point of T is any x in X such that x Tx.

Fixed Point Theory is inarguably the most powerful and effective tools used in modern nonlinear analysis today. It is still an area of current intensive research as it has vast applicability in establishing existence and uniqueness of solutions of diverse mathematical models like solutions to optimization prob- lems, variational analysis, and ordinary differential equations. These models represent various phenomena arising in different fields, such as steady state temperature distribution, neutron transport theory, economic theories, chem- ical equations, optimal control of systems, models for population, epidemics and flow of fluids.

CHAPTER TWO

Theoretical Framework

In this chapter,  we aim to highlight some definitions on which the problems are formulated and introduce some concepts and ideas used in the rest of the chapters. This will include an overview of the geometry of some Banach spaces and some well known iterative methods for single valued and multivalued pseu- docontractive mappings.

Notions and Definitions

Unless otherwise specified, X represents a Banach space with norm . . The dual space X^∗ of X is the Banach space of all bounded linear functionals on X. It is endowed with the norm

x^∗  _X∗  :=  sup   x, x^∗  ,

ǁxǁ=1

where ⟨., .⟩  represent the pairing between the elements of X  and X^∗.  Given any sequence {x_n} in X, we take x_n → x^∗ to mean {x_n} converges strongly to x^∗ and x_n ~ x^∗ to mean that {x_n} converges weakly to x^∗. The set of real numbers including +∞ is represented by R¯

 Some Well known Definitions

Definition 2.1.1 A mapping T : X X is called L Lipschitzian if there exists L > 0 such that

ǁTx − Tyǁ ≤ Lǁx − yǁ,   ∀x, y, ∈ X. (2.1.1)

Remark 2.1.1 If L = 1 in the inequality (2.1.1), the mapping is called non- expansive and if L < 1, it is called a strict contraction. It is well known that F (T ) is closed and convex whenever T is nonexpansive.

Definition 2.1.2 A mapping T : X X is pseudocontractive in the termi- nology of Browder and Petryshyn [23] if

ǁx − yǁ ≤ ǁ(x − y) + r[(x − Tx) − (y − Ty)]ǁ,  ∀x, y ∈ X,  r > 0. (2.1.2)

Remark: By the result of Kato [62], stated in Lemma (2.1.1) this is equivalent to

⟨(I − T )x − (I − T )y, j(x − y)⟩ ≥ 0.

Thus, a mapping T is pseudocontractive if and only if the the complementary operator A := I T is accretive.

A well known proper subclass of the class of pseudocontractive mappings is the class of strictly pseudocontractive mapping.

Definition 2.1.3 Given a real Hilbert space H and a closed convex subset K

of H, let T : K → K be a mapping. Then T is said to be

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CHAPTER THREE

Contributions on Iterative Algorithms for Some Single-valued Pseudocontractive-type Mappings

Most important iteration procedures for single valued mappings currently in the literature [16], can be summarised as follows:

x_n+1= Tx_n,  n ≥ 0 1890 Picard

⇑ λ = 1

• x_n+1= 2 (x_n + Tx_n), n ≥ 0 ≥ 0 1955 Krasnoselski

1

⇑ λ = 2

x_n+1= (1 − λ)x_n + λTx_n, n ≥ 0, 0 ≤ λ ≤ 1, 1957 (Krasnoselski-)Shaeffer

⇑ a_n = λ(const.)

(4) x_n+1 = (1 − a_n)x_n + a_nTx_n, n ≥ 0, a_n ∈ [0, 1],

lim a_n = 0, a_n = 1953 Mann

n→∞

⇑ b_n = 0

(5)  x_n+1 = (1 − a_n)x_n + a_nT [(1 − b_n)x_n + b_nTx_n], n ≥ 0, 0 ≤ a_n ≤ b_n ≤  1,

lim b_n = 0,

n→

a_nb_n = ∞ 1974 Ishikawa

n=0

There is a need for an iterative procedure that fills the gap between (4) and

(5) above in the sense that here a_n = b_n = λ simply for some λ (0, 1). In this chapter, we state a theorem in this regard and demonstrate how suc

CHAPTER FOUR

Contributions on Iterative Algorithms for a General Class of Multivalued Strictly Pseudocontractive mappings In this chapter we will survey some techniques for approximating fixed points of a more general class of multivalued pseudocontractive mappings which we will define shortly.

First, we recall the single valued definition of strictly pseudocontractive map- ping due to Browder and Petryshin [23] as follows:

CHAPTER FIVE

Contribution on Countable Family of Multi-valued Strictly

Pseudocontractive Mappings

In this Chapter, we discuss the extension of the main theorem of the last chapter to finite family and then countable family of generalized k strictly pseudocontractive multivalued mappings in Hilbert spaces.

The extension of the main theorem of the last chapter to a finite family is quite straight forward. It makes use of the following identity valid in Hilbert spaces.

Lemma 5.0.5 ([36]) Let H be a real Hilbert space and let {x_i, i = 1, 2, …, m} ⊆

CHAPTER SIX

Contribution on Iterative Method for Multivalued Tempered Lipschitz Pseudocontractive mappings

Introduction

In this section, we will improve on the algorithm of Chidume and Okpala and develop an iterative algorithm for a much larger class of a lipschitzpseudocontractive mapping. We will show that our iterative sequence is an ap- proximating fixed point sequence for the mapping. Furthermore, under some mild assumption like hemicompact (or, in particular, compact), we will prove strong convergence of the

We will demonstrate with examples that our theorems have some edge over other results like those of Chidume et al. [35], Chidume and Ezeora [36], Pa- nyanak [88], Song and Wang [100], among others. It also complement several known results in the literature.

Few iterative algorithms have been developed for single valued Lipschitz pseudocntractive-type mappings in real Hilbert spaces. However, till now, there is no known algorithm that have been developed for the Multivalued analogue. It is natural, therefore, for us to try to develop a theory for the multi-valued analogues of these mappings. This is the purpose of this Chap- ter. More precisely, we propose a theory for the class of tempered Lipschitz pseudocontractive mappings as a multi valued analogue for the class of Lips- chitchz pseudocontractive mappings.

Bibliography

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• Alber, “Metric and generalized projection operators in Banach spaces: properties and applications”. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (A. G. Kartsatos, Ed.), Marcel Dekker, New York (1996), pp. 15-50.
• I. Alber and S. Reich, “An iterative method for solving a class ofnonlinear operator equations in Banach spaces”, Panamer. Math. J., 4(2) (1994), 39-54.
• Alber and S, Guerre-Delabriere, “On the projection methods for fixedpoint problems”, Analysis (Munich), vol. 21 (2001), no. 1, pp. 17-39.
• Alber and I. Ryazantseva,“ Nonlinear Ill Posed Problems of Monotone Type”, Springer, London, UK, (2006).
• Alber Y. and Butnariu D. “Convergence of Bregman projection meth-ods for solving consistent convex feasibility problems in reflexive Banach spaces” J. Optim. Theory Appl. 92 (1997) 33-61.
• Alber and S, Guerre-Delabriere, “Principles of weakly contractive maps in Hilbert spaces” in New results in Operator Theory and Its Applications,
• Gohbergand  Lyubich, Eds, vol 98 of Operator Theory: Advances and Applications, pp. 7-22, Birkhauser, Basel, Switzerland(1997).
• A. Alghamdi, N. Shahzad, and H. Zegeye, “Strong Convergence The-orems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces”, Journal of Applied Mathematics Vol 2014,(2014) Article ID 58068
• Aoyama, F. Kohsaka, and W. Takahashi, “Proximal point methods formonotone operators in Banach spaces”, Taiwanese Journal of Mathemat- ics, vol. 15, no. 1,(2011) pp. 259-281.
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