Arbitrary X1 Research Articles

A Remark on the Theorem of Ishikawa

Given a Lipschitz pseudocontractive mapping T from a closed convex and bounded subset K of a real Hilbert space H onto itself, and an arbitrary x1 ∈ K, a Krasnolselskii-type sequence defined by xn+1 = (1− λ)xn + λTyn, yn = (1− λ)xn + λTxn is proved to be an approximate fixed point sequence of T , for a suitable λ ∈ (0, 1). Under some suitable compactness assumptions on K or on T , the sequence converges strongly to a fixed point of T . The algorithm is simple and natural, and the theorems presented here improve the theorem of Ishikawa [1] and other similar results in the literature.

(n + 1)-Ary derivations of simple n-ary algebras

By a δ-derivation we mean a linear map φ such that φ(xy) = δ(φ(x)y + xφ(y)) for some fixed element δ of the ground field. For more details about δ-derivations, we ask the reader to consult [1-6]. A natural generalization of δ-derivations to n-ary algebras is a generalized derivation of an n-ary algebra. By an (n+1)-ary derivation of an n-ary algebra A we mean a tuple (f0, f1, . . . , fn) ∈ End(A)n+1 such that for arbitrary x1, . . . , xn ∈ A, it is true that

Convergence theorems of finite-step iteration with errors for non-self asymptotically nonexpansive in the intermediate sense mappings

Let K be a nonempty closed convex nonexpansive retract of a uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K ? E be non-self asymptotically nonexpansive in the intermediate sense mapping with F(T) = ?. Let {?ni}, {?ni} and {?ni} are sequences in [0, 1] with ?n(i) + ?n(i) + ?n(i) = 1 for all i = 1, 2, . . . , N. From arbitrary x1 ? K , define the sequence {xn } iteratively by (8), where {u(i) } for all i = 1, 2, . . . , N are bounded sequences in K with P? u(i) < ?. (i) If the dual E

Approximating common fixed points of finite family of asymptotically nonexpansive non-self mappings

Let K be a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T1 , T2 , ... , TN : K ? E be N asymptotically nonexpansive nonself mappings with sequences {rin} such that ??(n=1) rin < ?, for all 1 ? i ? N and n n=1 n F = ?N(i-1) F (Ti) ? ?. Let {?in}, {?in} and {?in} are sequences in [0, 1] with i=1 ?in + ?in + ?in = 1 for all i = 1, 2, ... , N . From arbitrary x1 ? K , define the sequence {xn} iteratively by (6), where {uin} are bounded sequences in K with ??(n=1) uin < ?. (i) If the dual E*of E has the Kadec-Klee property, then {xn} converges weakly to a common fixed point x*? F ; (ii) if {T1 , T2 , ... , TN} satisfies condition (B), then {xn} converges strongly to a common fixed point x*? F. .

STRONG CONVERGENCE THEOREMS OF ISHIKAWA ITERATION PROCESS WITH ERRORS FOR FIXED POINTS OF LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES

Let q > 1 and E be a real q-uniformly smooth Banach space, K be a nonempty closed convex subset of E and T : K → K be a Lipschitz continuous mapping. Let {un} and {vn} be bounded sequences in K and {αn} and {βn} be real sequences in [0, 1] satisfying some restrictions. Let {xn} be the sequence generated from an arbitrary x1 ∈ K by the Ishikawa iteration process with errors: yn = (1 − βn)xn + βnTxn + vn, xn+1 = (1−αn)xn +αnTyn +un, n ≥ 1. Sufficient and necessary conditions for the strong convergence {xn} to a fixed point of T is established.

Approximating fixed points of non‐self asymptotically nonexpansivemappings in Banach spaces

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be an asymptotically nonexpansive mapping with {kn}⊂[1, ∞) such that and F(T) is nonempty, where F(T) denotes the fixed points set of T. Let {αn}, {αn′}, and {αn′′} be real sequences in (0,1) and ε ≤ αn, αn′, αn′′≤1 − ε for all n ∈ ℕ and some ε > 0. Starting from arbitrary x1 ∈ K, define the sequence {xn} by x1 ∈ K, zn = P(αn′′T(PT) n−1xn + (1 − αn′′)xn), yn = P(αn′T(PT) n−1zn + (1 − αn′)xn), xn+1 = P(αnT(PT) n−1yn + (1 − αn)xn). (i) If the dual E* of E has the Kadec-Klee property, then { xn} converges weakly to a fixed point p ∈ F(T); (ii) if T satisfies condition (A), then {xn} converges strongly to a fixed point p ∈ F(T).

THREE-STEP ITERATIVE CONVERGENCE THEOREMS WITH ERRORS IN BANACH SPACES

Let q > 1 and E be a real q-uniformly smooth Banach space, K be a nonempty closed convex subset of E and T : K → K be a single-valued mapping. Let {un}∞ n=1, {vn}∞ n=1, {wn}∞ n=1 be three sequences in K and {αn}∞ n=1, {βn}∞ n=1 and {γn}∞ n=1 be real sequences in [0, 1] satisfying some restrictions. Let {xn} be the sequence generated from an arbitrary x1 ∈ K by the three-step iteration process with errors: xn+1 = (1−αn)xn+αnTyn+un, yn = (1 − βn)xn + βnTzn + vn, zn = (1 − γn)xn + γnTxn + wn, n ≥ 1. Sufficient and necessary conditions for the strong convergence {xn} to a fixed point of T is established. We also derive the corresponding new results on the strong convergence of the three-step iterative process.

Fixed point theorems for asymptotically pseudocontactive maps in certain Banach Spaces

Let E be a real q-uniformly smooth Banach space (for example, Lp or lp spaces ,1‹p‹∞) and let T: E→E be asymptotically pseudocontractive mapping with nonempty fixed point set F(T) and a real sequence {kn} n=1∞ . Let {xn}n≥1 be the sequence generated from an arbitrary x1 in E by xn+1=(1-αn) xn + αnT nxn, n≥1. If the range of T is bounded and there exists a strictly increasing function Φ : [ o , ∞) →[ o , ∞) with Φ(0)=0 such that (Tnxn-p, j(xn-p)) ≤ kn||xn-p||2 _ Φ(||xn-p||), p∈F(T), then {xn}∞n=1 converges strongly to p, provided that {kn} and {αn} satisfy certain properties. This result compliments the results of Chang, Park and Cho (2000), by dropping the Lipschitz condition on T. 2000 Mathematics Subject Classification : 47H09 47J05. KEY WORDS: Fixed points, asymptotically pseudocontractive maps, modified Mann iterative methods, q-uniformly smooth spaces. Global Journal of Mathematical Sciences Vol.3(2) 2004: 137-142

Fixed point iteration for pseudocontractive maps

Let K K be a compact convex subset of a real Hilbert space, H H ; T : K → K T:K\rightarrow K a continuous pseudocontractive map. Let { a n } , { b n } , { c n } , { a n ′ } , { b n ′ } \{a_{n}\}, \{b_{n}\}, \{c_{n}\}, \{a_{n}^{’}\}, \{b_{n}^{’}\} and { c n ′ } \{c_{n}^{’}\} be real sequences in [0,1] satisfying appropriate conditions. For arbitrary x 1 ∈ K , x_{1}\in K, define the sequence { x n } n = 1 ∞ \{x_{n}\}_{n=1}^{\infty } iteratively by x n + 1 = a n x n + b n T y n + c n u n ; y n = a n ′ x n + b n ′ T x n + c n ′ v n , n ≥ 1 , x_{n+1} = a_{n}x_{n} + b_{n}Ty_{n} + c_{n}u_{n}; y_{n} = a_{n}^{’}x_{n} + b_{n}^{’}Tx_{n} + c_{n}^{’}v_{n}, n\geq 1, where { u n } , { v n } \{u_{n}\}, \{v_{n}\} are arbitrary sequences in K K . Then, { x n } n = 1 ∞ \{x_{n}\}_{n=1}^{\infty } converges strongly to a fixed point of T T . A related result deals with the convergence of { x n } n = 1 ∞ \{x_{n}\}_{n=1}^{\infty } to a fixed point of T T when T T is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

The Hilbert Riemannian structure of equivalent Gaussian measures associated with the Fisher information

Rao has firstly introduced the Riemannian structure associated with the Fisher information matrix over a finite dimensional parametrized statistical model. He proposed the Riemannian distance as a measure of dissimilality between two probability measures, (cf. [2], for example.) In [1], Amari introduced a pair of dual affine connections with respect to the metric and discussed of the differential geometry of the space of a finite dimensional parametrized statistical model. It provides a differential geometrical meaning to statistical inference. In the present paper, we realize the above idea for a family of equivalent (i.e.,mutually absolute continuous) Gaussian measures on a Banach space. Our main result is as follows. Let 5 b e a real separable Banach space and P be a centered gaussian measure on the topological dual of B (cf. [6]). The covariance of P naturally determines the Hubert space H. i.e., for arbitrary x1 and x2eB, let

Hyperbolic Pairs in the Method of Conjugate Gradients

(starting from an arbitrary x1 and r1 = b Ax1, Pt = r1) converges to the solution of(1) in n steps or less. Theoretically, the process (2) can be carried out in such a manner that only one multiplication of A times a vector is required at each step, although in practice two such multiplications are often employed. In many applications we are led to equations of the form (1) where A is nonsingular and symmetric but not positive definite. For example the problem of minimizing the quadratic form