I= FIXED POINT THEOREMS IN GENERALIZED METRIC AND BANACH SPACES
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Title of Thesis
FIXED POINT THEOREMS IN GENERALIZED METRIC AND BANACH SPACES

Author(s)
Muhammad Arif Rafiq
Institute/University/Department Details
Bahauddin Zakariya University, Multan/ Center For Advanced Studies In Pure And Applied Mathematics
Session
2003
Subject
Applied Mathematics
Number of Pages
169
Keywords (Extracted from title, table of contents and abstract of thesis)
fixed point theorems, banach spaces, metric spaces, asymptotically nonexpansive mapping, nonlinear operator equations, demicontractive mappings

Abstract
Let X be an arbitrary Banach space and K be a nonempty closed convex subset of X. otherwise mentioned in the thesis.

The Mann iterative scheme was invented in 1953, see [61], and was used to obtain convergence to a fixed point for many functions for which the Banach principle fails. For example. B. E. Rhoades in [82] showed that. For any continuous selfmap of a closed and bounded interval, the Mann iteration converges to a fixed point of the function.

In 1974, Ishikawa [49] devised a new iteration scheme to establish convergence for a lapschitzin pseudocontractive map in a situation where the Mann iteration process failed to converge.

DEFINTTION 1. Let K be a nonempty convex subset of X and let T: K†’ K be a self map of K. (A-i) for any given xO „ K. the sequence {xn}ˆžn-o defined by xn+1 = (1-bn)xn + bn Tyn,(A1) yn = (1 €“ b1n)xn + b1nTxn. n ‰ 0. Is called the Ishikawa iterarive sequence. Where 0 ‰ bn ‰ bn ‰ 1.(A2) (A-ii) if b1n=0 for all n ‰ 0 in (A-i), then the sequence {Xn}ˆžn-o defined by x0 „ K.(A3) xn+1 = (1 €“ b1n)xn + b1nTxn. n ‰ 0. is called the Mann iterative sequence.

In specific situations. Additional conditions are placed on {b1n}ˆžn-o and {bn}ˆžn-o. In {83}. B. E. Rhoades modified the definition of Ishikawa by replacing (A2) by 0 ‰ bn ‰ b1n ‰ 1. n ‰ 0.(A4)

In recent years, the iteration processes described above have been studied extensively by various authors for – approximating fixed points of nonlinear mappings, – solutions of nonlinear operators equations, – variational inequalities, In Hilbert and Banach spaces; see [1, 8, 28, 39, 47-49, 55, 61, 65-67, 83-85, 89, 90, 92, 94, 100] and the references therein. Noor [65-66] introduced and analyzed three step it-erative methods to study the approximate solutions of variational inclusions (inequalities) in Hilbert spaces by using the techniques of updating the solution and the anxiliary principle. A similar idea goes back to the so called 0-schemes introduced by Glowinski and Le Tallec [39] to find a zero of the sum of two (or more) maximal monotone operators by using the La-grangian multiplier. Glowinski and Le Tallec {39} used three-step iterative schemes to find the approximate solutions of the elastoviscoplasticity problem, liquid crystal theory, and eigenvalue computation, and they showed that three-step approximations perform better numerically. Haubruge et al. [47] studied the convergence analysis of three-step schemes of Glowinski and Le Tallec [39] and applied variational inequalities, separable convex programming, and minimization of a sum of convex functions. They also proved that three-step iterations lead to highly parallelized algorithms under certain conditions. Noor [68] introduced and analyzed three step iterative methods for solving the non-linear equation Tx = x for asymptotically nonexpansive mappings in uniformly convex Banach spaces.

DEFINITION 2. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given x0 „ K, compute sequences {zn}, {yn}, {xn} by the iterative schemes xn+1 = bnTyn + (1 - bn)xn,(A5) yn = b1nTzn + (1 €“ b1n)xn, zn = bnnTxn + (1 €“ bnn)xn. n ‰ 0. Where {bn}, {b1n}, {bnn} are real numbers in [0, 1], is called the Noor-type iterative process. Note that the Mann and Ishikawa iterative sequences are all the special cases of the Noor iterative sequences. In 1995, L. S. Liu introduced the concepts of Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive mappings in [54] as follows. DEFINITION 3. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. (A-iii) for any given x0 „ K, the sequence {xn}ˆžn-o. defined by xn+1 = (1 - bn)xn + bnTyn + vn,(A6) yn = (1 €“ b1n)xn + b1nTyn + un, n ‰ 0. Is called the Ishikawa iterative sequence with errors. Here {un}nˆž=0 and {vn} nˆž=0 are two summable ˆž sequences in X, i.e., ˆ‘ n=0 (A-iv) in particular, if bin = o and un = o for all n ‰ 0 in (A-iii), then the sequence {xn} nˆž=0 defined by xo „ K,(A7) xn+1 = (1-bn)xn+bnTxn+un, n ‰ 0 is called the Mann iterative sequence with errors. Here {un}nˆž=0 is a summable sequence in X, i.e.,

Unfortunately, the definition of Liu, which depends on the convergence of the error terms, tends to contradict the randomness of errors. Hence in 1998, Y. Xu [102] devised a new iteration scheme to study the unique solution of the nonlinear strongly accretive operator equation Tx = f and the convergence problem of the revised iterative sequences for strongly pseudo-contractive mappings without the Lipschitz condition.

DEFINITION 4. Let K be nonempty convex subset of X : K †’K be a self map of K, (A-v) for any given xo „ K, the sequence {xn} nˆž=0 defined by xn+1 = anxn + bnTyn +cnvn,(A8) yn = a1nxn+b1nTxn+ c1nun, n ‰ 0 where {un}nˆž=0 and {vn} nˆž=0 are two arbitrary bounded sequences in K and {an}nˆž=0, {bn} nˆž=0, {cn}nˆž=0, {a1n} nˆž=0, {b1n}nˆž=0 and {c1n} nˆž=0 are real sequences in [0, 1] such that an + bn + cn + a1n + b1n + c1n =1 for all n ‰ 0 is called the Ishikawa iterative sequence with errors in the sense of Xu. (A-vi) if b1n = c1n=0 for all n ‰ 0 in (A-v), then the sequence {xn} nˆž=0 now defined by xo „ K,(A9) xn+1 = anxn+bnTxn+ cnvn, n ‰ 0 is called the Mann iterative sequence with errors in the sense of Xu. It is clear that the Mann and Ishikawa iterative sequences are all special cases of the Ishikawa iterative sequences with errors in the sense of Xu. Inspired and motivated by these facts, we suggest a new class of three-step iterative schems. Our schemes can be viewed as an extension for three-step and two step iterative schemes of Glowinski and Le Tallec [39], Mann [61], Ishikawa [49], Liu [54], Xu [102] and Noor [68]. Algorithms for these schemes are described as follows.

ALGORITHM 1.1. Let K be a nonempty convex subset of X and let T: K †’ K be a self map of K. for a given xo „ K, compute sequences {zn}, {yn}, {xn}, by the iterative schemes xn+1 = anxn + bnTyn +cnvn,(A10) yn = a1nxn+b1nTzn+ c1nun, zn = a11nxn+b11nTzn+ c11nwn, n ‰ 0 where {vn}nˆž=0, {un} nˆž=0, {wn}nˆž=0 are three arbitrary bounded sequences in K and {an}, {bn}, {cn}, {a1n}, {b1n}, {c1n}, {a11n}, {b11n}, {c11n} are real numbers in [0, 1] and an + bn + cn =1=a1n + b1n + c1n=+ a11n + b11n + c11n. We will call it the three-steps iterative process in the sense of Xu. ALGORITHM 1.2 Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given xo „ K, compute sequences {zn}, {yn}, {xn}, by the iterative schemes xn+1 = (1 - bn)xn+ bnTyn +vn,(A11) yn = (1 - b1n)xn+b1nTzn+un, zn = (1 - b11n)xn+b11nTxn+ c11nwn, n ‰ 0 where {vn}nˆž=0, {un} nˆž=0, {wn}nˆž=0 are three summable sequences in K and {bn}, {b1n}, {b11n} are real numbers in [0, 1]. We will call it the three-step iterative process in the sense of Liu. If c11n = 0 = c1n, = cn, then Algorithm 1.1 reduces to ALGORITHM 1.3. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given xo „ K,compute sequences {zn}, {yn}, {xn}, by the iterative schemes xn+1 = bnTyn +(1 - bn)xn, yn = b1nTyn +(1 - b1n)xn, zn = b11nTyn +(1 - b11n)xn, n ‰ 0 where {bn}, {b1n}, {b11n} are real numbers in [0, 1]. Algorithm 1.3 is called a Noor-type iterative process. If c11n = 0 = c1n, = cn, then Algorithm 1.1 reduces to ALGORITHM 1.4. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given xo „ K,compute sequences {yn}, {xn} by the iterative schemes xn+1 = anxn +bnTyn+cn vn, yn = a1nxn +b1nTyn+c1n vn, n ‰ 0 where {an}, {bn}, {cn}, {a1n}, {b1n}, {c1n} are real numbers in [0, 1]. Algorithm 1.4 is called a Ishikawa-type iterative process in the sense of Xu. If c11n = 0 = b11n, = b11n,= c1n, then Algorithm 1.1 reduces to ALGORITHM 1.5. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given xo „ K,compute sequences {xn} by the iterative schemes xn+1 = anxn +bnTxn+cnvn, n ‰ 0 where {an}, {bn}, {cn}, are real numbers in [0, 1]. Algorithm 1.5 is called a Mann-type iterative process in the sense of Xu. If b11n = 0 and wn,= 0 then Algorithm 1.2 reduces to ALGORITHM 1.6. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given xo „ K,compute sequences {yn}, {xn} by the iterative schemes. xn+1 = (1 - bn)xn+bnTyn+vn, yn = (1 - b1n)xn+b1nTyn+un, n ‰ 0 where {bn} and {b1n}, are real numbers in [0, 1]. Algorithm 1.6 is called a Ishikawa-type iterative process in the sense of Liu. If b11n = 0 and wn,= 0 = un then Algorithm 1.2 reduces to ALGORITHM 1.7. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given xo „ K,compute sequences {xn} by the iterative schemes. xn+1 = (1 - bn)xn+bnTyn+ un, n ‰ 0 where {bn} are real numbers in [0, 1]. Algorithm 1.7 is called a Mann-type iterative process in the sense of Liu. If b11n = 0 = c11n = c1n, = cn, then Algorithm 1.1 reduces to ALGORITHM 1.8. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given xo „ K,compute sequences {yn}, {xn} by the iterative schemes. xn+1 = bnTyn + (1 - bn)xn yn = b1nTyn + (1 - b1n)xn n ‰ 0 where {bn}, {b1n} are real numbers in [0, 1]. Algorithm 1.8 is called a Ishikawa-type iterative process in the sense of Liu. If b11n = 0 = c11n = b1n = c1n, = cn, then Algorithm 1.1 reduces to ALGORITHM 1.9. Let K be a nonempty convex subset of X and let T : K †’ K be a self map of K. for a given xo „ K,compute sequences {xn} by the iterative schemes. xn+1 = bnTyn + (1 - bn)xn n ‰ 0 where {bn} are real numbers in [0, 1]. Algorithm 1.9 is called a Mann-type iterative process.

In applications some particular of the features of the corresponding Banach space play an important role. In what follows we consider several classes of Banach spaces which have some importance in fixed point theory. In this respect it is worth mentioning that many fixed point theorems have a more general form only on such spaces. This research is oriented towards the themes of establishing the convergence of.

The iterative scheme due to Xu [102] (ALGORITHM 1.4), A new class of three-step iterative shceme (ALGORITHM 1.1, To approximate fixed points of certain classes of nonlinear mappings and solutions of nonlinear operator equations is Banach spaces. It is shown that the previously known iterative schemes due to Mann [61], Ishikwaa [49], Liu [54], Xu [102] and Noor [68] are all the special cases of this new class of three-step iterative schemes.

This thesis consists of five chapters. The first chapter contains basic definitions and facts which are directly related to our work. We have mentioned in this chapter the results without proof in order to avoid making the thesis unnecessarily voluminous. We have also avoided giving the definitions (which are available in text books) by presuming that the reader is familiar with these.

In chapter two, we consider the convergence properties of the Ishikawa iterative scheme with errors in the sense of Xu [102] for solving the nonlinear equation Tx = x for asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this chapter generalize the corresponding results in [55,85, 89, 90].

In chapter three, (A-vii) we establish the strong convergence of the Ishikawa iterative schemes by Xu [102] to the unique fixed point of a ø- hemi contractive mapping defined on a nonempty convex subset of a normed linear space,

(A-viii) under some conditions we also obrain the the Mann iteration method by [102] converges strongly to a unique fixed point of a ø- hemi contractive mapping and is T-stable on K in an arbitrary smooth Banach space,

(A-ix) it is our purpose to establish the convergence, almost stability and stability of the Ishikawa iteration scheme by Xu [102] for Lipschitz strictly hemicontractive operators in arbitrary Banach spaces.

Our results extend, improve and unify the corresponding results in References [6,10, 12-13, 15, 17, 19, 21, 29, 56, 78, 88, 93]. In chapter four, (A-x) it is proved that the new three-step iteration methods in the sense of Xu converge strongly to the solution of the equation Tx = f for a Lipschitz ø- strongly accretive operator in an arbitrary real Banach space. Related results deal with the iterative approximation of fixed points of Lipschitz ø- pseudocontractions with the new iteration methods in arbitrary real Banach spaces,

(A-xi) we will show that Mann and Ishikawa iteration schemes introduced by Xu [102] are equivalent for various classes of functions.

Our results extend, generalize and unify several recent results (see for example [0, 12, 14, 17, 19, 25, 27]). Lastly in chapter five, (A-xii) iterative methods for the approximation of fixed points of asymptotically demicontractive mappings are constructed using the more general modified new three-step iteration methods with errors interoduced by us in a real Banach space. Our results show that a recent result of Osilike [76] (which is itself a generalization of a theorem of Qihou [81] can be extended from real q-uniformly smooth Banach spaces, 1 < q < ˆž , to arbitrary real Banach spaces, and to the more general modified three-step iteration methods with errors in the sense of Xu. Furthermore, the boundedness assumption imposed on the subset K in ([76, 89]) are removed in our present more general result, moreover, our iteration parameters are independent of any geometric properties of the underlying Banach space.

(A-xiii) it is proved that the new three-step iteration process with errors in the sense of Xu converges weakly to a fixed point of a demicontractive mapping in a real q-uniformly smooth Banach space. If K is compact and convex, then the convergence is strong. Related results deal with other sufficient conditions for strong or weak convergence of the iteration processes used.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
330.33 KB
2 1 Preliminaries And Some Fundamental Results 12
246.08 KB
3 2 Fixed Point Iterations For Asymptotically Nonexpansive Mappings In Banach Spaces 25
249.67 KB
  2.1 Introduction 25
  2.2 Main Results 27
4 3 Iteration Schemes For Strictly Hemicontractive Operators In Arbitrary Banach Spaces 40
790.81 KB
  3.1 Introduction 40
  3.2 Preliminaries 42
  3.3 Main Results 44
5 4 Solutions Of Certain Nonlinear Operator Equations 75
650.36 KB
  4.1 Introduction 75
  4.2 Strong Convergence Of Iterative Methods To Solutions Of Certain Nonlinear Operator Equations 77
  4.3 On The Equivalence Of Mann And Ishikawa Iteration Methods With Errors 88
6 5 Fixed Points Of Demicontractive Mappings 105
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  5.1 Introduction 105
  5.2 Approximation Of Fixed Points Of Asymptotically Demicontractive Mappings In Arbitrary Banach Spaces 108
  5.3 Iterative Approximations Of Fixed Points Of Demicontractive Maps 129
  5.4 References 159