An Analysis upon Some Approximation of Fixed Points through Iterative Methods

Give C a chance to be a nonempty shut arched subset of a genuine Hilbert space H and let be an appropriate curved lower semi continuous function of H into Consider a convex minimization problem The number is called an optimal value, C is called an admissible set and is called an optimal set. Next, define a functionas follows : Then, is a proper lower semi continuous convex function of H into So, we consider the convex minimization problem where is a proper lower semi continuous convex function of H into For such a we can define a multivalued operator on H by for all . Such a is said to be the sub differential of Let C be a nonempty closed convex subset of a real Hibert space H. Then a mapping is called non-expansive on C if for all We denote by F) the set of fixed point of T. Let Then, we can define a multivalued operator B from H to H by for allInversely, if B is a multivalued operator from H to H. then we can define a set A in H x H by. So, it is natural to regard a set in H x H in the same light with a multivalued operator from H to H. Let Then, we define the domain of A and the range of A as follows: We also define a multivalued operator from H to H by for all From this definition, we have An operator is accretive if for

If A is accretive, we can define, for each positivethe resolvent

is a proper lower semi continuous convex function, then is a m-accretive administrator. For a m-accretive administrator A, we can think about the following beginning value problem:

(**)

where is an element of Then, it is well known that has a unique strong solution Putting we know that the family of mappings on satisfies the following conditions: (i) for every (ii) for every (iii) for each is continuous; (iv) for every and

parameter non-expansive semi group on see Brezis. We also know that where is the set of fixed points of Further, we have that for Accordingly, an arched minimization problem is proportionate to a fixed point problem for a non-expansive mapping or a group of non far reaching mappings. Further, we realize that one method for illuminating is the proximal point calculation initially presented by Martinet (1970). The proximal point calculation depends on the thought of resolvent i.e., introduced by Moreau. The proximal point algorithm is an iterative procedure, which starts at a point and generates recursively a sequence of points where is a sequence of positive valued continuous arched functions on a Hilbert space H. The problem is to discover a solution of the limited raised disparity framework, i.e., to discover such a point that Such a problem is known as the practicality problem. This problem is likewise connected with approximation of fixed points. In this article, we initially examine fixed point theorems for a non-expansive mapping or a group of non-expansive mappings. Specifically, we express a fixed point hypothesis which addressed positively a problem postured amid the Conference on Fixed Point Theory and Applications held at CIRM, Marseille-Luminy, 1989. At that point we examine nonlinear ergodic theorems of Baillon's write for nonlinear semi groups of non-expansive mappings. Specifically, we state nonlinear ergodic theorems which addressed certifiably the problem postured amid the Second World Congress on Nonlinear Analysts, Athens, Greece, 1996. Next, we manage weak and strong meeting theorems of Mann's compose and Halpern's write in a Banach space. At last, utilizing these results, we consider the achievability problem by arched blends of non-expansive withdrawals and the curved minimization problem of finding a minimizer of a raised function.

Let C be a nonempty closed convex subset of a Banach space E and let T be a mapping of C into C. Then we denote by R(T) the range of T. A mapping T of C into C is said to be asymptotically regular if for everyconverges to 0. Let D be a subset of C and let P be a mapping of C into D. Then P is said to be sunny if wheneverforandA mapping P of C into C is said to be a retraction if If a mapping P of C into C is a retraction, thenfor every A subset D of C is said to be a sunny non-expansive retract of C if there exists a sunny non-expansive retraction of C onto D. Let E be a Banach space. Then, for everywiththe modulus of convexity of E is defined by A Banach space E is said to be uniformly convex if

convex if for with

convex. Let E be a Banach space and let be its dual, that is, the space of all continuous linear functionals on E. The value of at will be denoted by With each we associate the set Using the Hahn-Banach theorem, it is immediately clear that for any Then the multivalued operator is called the duality mapping of E. Let be the unit sphere of E. Then a Banach space E is said to be smooth provided exists for each At the point when this is the case, the standard of E is said to be Gateaux differentiable. It is said to be Frechet differentiable if for every x in {/, this cutoff is achieved consistently for y in U. The space E is said to have a consistently Gateaux differentiable standard if for each the limit is attained uniformly for It is notable that if E is smooth, at that point the duality mapping J is single valued. It is additionally realized that if E has a Frechet differentiable standard, at that point J is standard to standard continuous. A shut curved subset C of a Banach space E is said to have typical structure if for each shut bounded arched subset K of C, which contains no less than two points, there exists a component of K which isn't a diametral point of K. Baillon and Schoneberg (1981) likewise presented the following weakening of the idea of ordinary structure: A shut curved subset C of a Banach space is said to have asymptotic typical structure if for each shut bounded raised subset K of C, which contains no less than two points and each sequence in K satisfying as, there is a point such that where is the measurement of K. It is notable that a shut curved subset of a consistently arched Banach space has ordinary structure and a conservative raised subset of a Banach space has typical structure. A Banach space E is said to fulfill OpiaVs condition if and simply where denotes the weak convergence to x. Let S be a semitopological semigroup, i.e., a semigroup with Hausdorff topology such that for eachthe mappings and of S into itself are continuous. Let B(S) be the Banach space of all bounded real valued functions on S with supremum norm and let X be a subspace of B(S) containing constants. Then, an element of is called a mean

for every A real valued function on X is called a sub mean on X if the following properties are satisfied: (i) for every (ii) for every and (iii) For implies (iv) for every constant function c. Obviously every mean on I is a submean. The idea of submean was first presented by Mizoguchi and Takahashi (1990). For a submean on X and sometimes we use instead ofFor eachand, we define elements and of B (S) given by and for all Let X be a subspace of B(S)containing constants which is invariant under(resp.). Then a mean on X is said to be left invariant (resp. right invariant)

invariant iffor all and Let S be a semi topological semi group. Then S is called left (resp. right) reversible if any two closed right (resp. left) ideals of S have non-void intersection. If S is left reversible, is a directed system when the binary relation on S is defined by if and only if Similarly, we can define the binary relation on a right reversible semi topological semi group S.

In this area, we talk about fixed point theorems for a non-expansive mapping or a group of non-expansive mappings. The main fixed point theorem for non-expansive mappings was built up in 1965 by Browder (1965). He demonstrated that if C is a bounded shut raised subset of a Hilbert space H and T is a non-expansive mapping of C into itself, at that point T has a fixed point in C. Very quickly, both Browder and Gohde demonstrated that the same is valid if E is a consistently arched Banach space. Kirk likewise demonstrated the following theorem: Theorem 4.1 Let E be a reflexive Banach space and let C be a nonempty bounded shut raised subset of E which has ordinary structure. Give T a chance to be a

After kirk's theorem, numerous fixed point theorems concerning non-expansive mappings have been demonstrated in a Hilbert space or a Banach space. Specifically, Baillon and Schoneberg presented the idea of asymptotic ordinary structure and generalized Kirk's fixed point theorem as follows: Theorem 4.2 Let E be a reflexive Banach space and let C be a nonempty bounded shut raised subset of E which has asymptotic typical structure. Give T a chance to be a non-expansive mapping of C into itself. At that point F(T) is nonempty. Then again, DeMarr demonstrated the following fixed point theorem for a commu¬tative group of non-expansive mappings.

Browder demonstrated the following fixed point theorem without conservativeness: Theorem 4.4 Let C be a bounded shut raised subset of a consistently arched Banach space E and let S be a commutative group of non-expansive mappings of C into itself. At that point S has a typical fixed point in C Further, we endeavor to stretch out these theorems to a noncommutative semi group of non-expansive mappings. Give S a chance to be a semi topological semi group and given C a chance to be a nonempty subset of a Banach space E. At that point a family of mappings of C into itself is called a non-expansive semi group on C if it satisfies the following: (i) for all and (ii) For each the mapping is continuous; (iii) For each is a non-expansive mapping of C into itself. For a non-expansive semi group 011 C, we denote by F(S) the set of common fixed points of Let S be a semi topological semi group, let C(S) be the Banach space of all bounded continuous functions on S and let be the space of all bounded right uniformly continuous functions on S, i.e., all such that the mapping is continuous. Then is a closed sub algebra of containing constants and invariant under and . DeMarr's fixed point theorem, that is, he demonstrated that any discrete left manageable semi group has a typical fixed point. Mitchell generalized Takahashi's result by demonstrating that any discrete left reversible semi group has a typical fixed point. Lau demonstrated the following theorem:

Banach space E. Thenhas a left invariant mean if and only if has a common fixed point in c. Lim generalized Kirk‘s result, Browder‘s result and Mitchell‘s result by showing the following theorem:

Takahashi and Jeong also generalized Browder‘s result by using the concept of sub mean.

convex Banach space E. Suppose that has a left, sub invariant sub mean. Then has a common fixed point in C. To prove Theorem 4.7, we need the following lemma:

We may comment on the relationship between has an invariant mean‖ and “S is left reversible‖. As well known, they do not imply each other in general. But if has sufficiently many functions to separate closed sets, then has an invariant mean‖ would imply is left and right reversible‖. Recently, Lau and Takahashi generalized Lim‘s result and Takahashi and Jeong‘s result.

To prove Theorem 4.9, we need two lemmas.

Theorem 4.8 answers certifiably a problem postured amid the Conference on Fixed Point Theory and Applications held at CIRM, Marseille-Luminy, 1989, regardless of whether Lim's result and Takahashi and Jeong's result can be completely reached out to such Banach spaces for amiable semi groups. We don't know whether "ordinary structure "in Theorem 4.8 would be supplanted by "asymptotic typical structure".

The main nonlinear ergodic theorem for non-expansive mappings was set up in 1975 by Baillon in the system of a Hilbert space.

This theorem was stretched out to a consistently raised Banach space whose standard is Frechet differentiable by Bruck. Theorem 4.10 Let C be a shut curved subset of a consistently raised Banach space E with a Frechet

In their theorems, puttingfor each we have that P is a non-expansive retraction of C onto F(T) such that for all and for each where is the closure of the convex hull of A. We talk about nonlinear ergodic theorems for a nonlinear semigroup of non-expansive mappings in a Hilbert space or a Banach space. Before talking about them, we give a definition. Let be a net of means on Then is said to be asymptotically invariant if for each and and Let us give an example of asymptotically invariant nets. Let and let N be the set of positive integers. Then for and the real valued function defined by is a mean. Further since for and as and is commutative, is an asymptotically invariant net of means. If C is a nonempty subset of a Hilbert space H and is a non-expansive semi group on C such that is bounded for some then we know that for each and the functions and are in

functionis in we can define the value of at this function. By linearity of and of the inner product, this is linear in moreover, since it is continuous in y. So, by the Riesz theorem, there exists an such that for every We write such an by . Presently we can express a nonlinear ergodic theorem for noncommutative semi groups of non-expansive mappings in a Hilbert space.

Utilizing Theorem 4.11, we have Theorem 4.9. By a similar method, we can demonstrate the following nonlinear ergodic theorems: Theorem 4.12 Let C be a shut curved subset of a Hilbert space H and let T be a one-parameter non-

Next, we express a nonlinear ergodic theorem for non-expansive semigroups in a Banach space. Before expressing it, we give a definition. A net of continuous linear functional on is called strongly regular if it satisfies the following conditions: (ii) (iii) for every

We have not known whether Theorem 4.14 would hold in the case when is noncommutative. As of late, Lau, Shioji and Takahashi tackled the problem as follows: Theorem 4.15 Let C be a shut raised subset of a consistently curved Banach space E, let S be a semi

This is a generalization of Takahashi's result for an amiable semigroup of nonexpan¬sive mappings on a Hilbert space. Facilitate they stretched out Rode's result to an agreeable semigroup of non-expansive mappings on a consistently arched Banach space whose standard is Frechet differentiable. Theorem 4.16 Let E be a consistently raised Banach

4.15 and the following lemma which has been demonstrated in Lau, Nishiura and Takahashi.

The following theorem has been demonstrated in Takahashi and Lau, Nishiura and Takahashi when E is a Hilbert space.

On the other hand, Mann introduced an iteration procedure for approximating fixed points of a mapping T in a Hilbert space as follows: and where is a sequence in. Afterward, Reich talked about this iteration technique in a consistently raised Banach space whose standard is Frechet differentiable and gotten the following theorem:

This theorem has been known for those consistently arched Banach spaces that fulfill Opial's condition. Tan and Xu demonstrated the following intriguing result which generalizes the result of Reich. Theorem 4.19 Let C be a shut arched subset of a consistently curved Banach space E which fulfills Opial's condition or whose standard is Frechet

Takahashi and Kim additionally demonstrated the following theorem:

point of T. To prove Theorem 4.22, Suzuki and Takahashi used the following two lemmas. Let I be an infinite subset of positive integers N. If is a sequence of nonnegative numbers, then we denote by the subsequence of

Compare Theorem 4.22 with Theorem 4.19 of Tan and Xu. This indicates that the assumptionin Theorem 4.12 is superfluous. We do not know whether the assumptionsandin Theorem 4.16 are replaced by and We also know the following strong convergence theorem which is connected with Rhoades, Tan and Xu, and Takahashi and Kim.

Let C be a closed convex subset of a Banach space E, and let T,be selfmaps on C. Then Das and Debata considered the following iteration scheme: and for where and are real sequences inThey demonstrated a strong converence theorem concerning Roades' result. Takahashi and Tamura got the following weak convergence theorem. Theorem 4.25 Let E be a consistently arched Banach space E which fulfills Opial's condition or whose standard is Frechet differentiable, given C a chance to

Further, Takahashi and Tamura obtained the following theorem:

To apply convergence theorems of Mann's compose to the attainability problem, we have to stretch out Theorem 4.25 to a group of limited mappings. Give C a chance to be a nonempty curved subset of a Banach space E. Let be finite mappings of C into itself and let be real numbers such that for every Then, we define a mapping W of C into itself as follows: Such a W is called the W-mapping generated by and

We will at last demonstrate a weak convergence theorem of Mann's compose for a non-expansive semigroup in a Banach space. Theorem 4.28 Let E be a consistently curved Banach space E with a Frechet differentiable standard. Give C a chance to be a nonempty shut curved subset of E

Utilizing Theorem 4.28, we can demonstrate a weak convergence theorem of Mann's compose for a one-parameter non-expansive semigroup.

fixed point

In this section, we examine strong convergence theorems for non-expansive mappings. Give C a chance to be a nonempty shut curved subset of a genuine Hilbert space H. In 1967, Browder got the following strong convergence theorem: For a given and each define a contraction by for all where T is a non-expansive mapping of C into itself. Then, there exists a unique fixed point of in C such that then converges strongly as to a fixed point of T. After Browder's result, such a problem has been researched by a few creators. Specifically, Reich and Takahashi and Ueda additionally stretched out Browder's result to strong convergence theorems for resolvents of accretive administrators in a Banach space. Before expressing them, we give two definitions. A shut raised subset C of a Banach space E is said to have the fixed point property for non-expansive mappings if each non-expansive mapping of C into itself has a fixed point in each nonempty bounded shut arched subset of C with the end goal that T leaves invariant. Let A bean accretive administrator in a Banach space E. At that point An is said to fulfill the range condition if for every Presently we can demonstrate the principal strong convergence theorem for resolvents of accretive administrators. Theorem 4.30 Let E be a reflexive Banach space with a consistently Gateaux differ¬entiable standard and

As direct consequences of Theorem 4.30, we obtain the following two results.

We additionally know the following theorem: Theorem 4.33 Let C be a shut arched subset of a Banach space E and let T be a non-expansive

Theorem 4.32 generalizes Browder‘s strong convergence theorem. In fact, from

we have (* * *) Putting we have from Thorem 4.33 that A is accretive and satisfies the range condition. Since from , we have, by Theorem 4.32, Recently, Wittmann dealt with the following iterative process in a Hilbert space: and. for where is a sequence. The following theorem was proved by Wittmann.

Shioji and Takahashi stretched out Wittmann's theorem to a Banach space by utilizing Theo¬rem 4.30 as follows: Theorem 4.35 Let E be a consistently raised Banach space with a consistently Gateaux differentiable standard. Give C a chance to be a nonempty shut curved subset of E. Give T a chance to be a non-expansive mapping of C into itself with the end goal

Atsushiba and Takahashi demonstrated a strong convergence theorem for limited non-expansive mappings which is associated with the achievability problem. Theorem 4.37 Let E be a consistently raised Banach space with a consistently Gateaux differ¬entiable standard. Give C a chance to be a nonempty shut

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