• Korean Journal of Mathematics
• /
• 제24권3호
• /
• pp.397-407
• /
• 2016

In this paper, the concepts of k-smoothness, k-very smoothness and k-strongly smoothness of Banach spaces are dealt with together briefly by introducing three types k-denting point regarding different topology of conjugate spaces of Banach spaces. In addition, the characterization of first type ${\omega}^*-k$ denting point is described by using the slice of closed unit ball of conjugate spaces.

• 대한수학회지
• /
• 제40권1호
• /
• pp.29-40
• /
• 2003

Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and T : KlongrightarrowK be a strictly hemi-contractive operator. Under some conditions we obtain that the Mann iteration method with errors both converges strongly to a unique fixed point of T and is almost T-stable on K. The results presented in this paper generalize the corresponding results in [l]-[7], [20] and others.

• 대한수학회지
• /
• 제31권3호
• /
• pp.333-337
• /
• 1994

Many authors[3][4][5] constructed and examined some processes for the fixed point of strictly pseudocontractive mapping in various Banach spaces. In fact the fixed point of strictly pseudocontractive mapping is the zero of strongly accretive operators. So the same processes are used for the both circumstances. Reich[3] proved that Mann-iteration precess can be applied to approximate the zero of strongly accretive operator in uniformly smooth Banach spaces. In the above paper he asked whether the fact can be extended to other Banach spaces the duals of which are not necessarily uniformly convex. Recently Schu[4] proved it for uniformly continuous strictly pseudocontractive mappings in smooth Banach spaces. In this paper we proved that Mann-iteration process can be applied to approximate the fixed point of strictly pseudocontractive mapping in certain Banach spaces.

• 대한수학회보
• /
• 제30권2호
• /
• pp.201-212
• /
• 1993

It is our purpose in this paper to first establish an inequality in real uniformly smooth Banach spaces with modulus of smoothness of power type q > 1 that generalizes a well known Hilbert space inequality. Using our inequality, we shall then extend the above result of Qihou [15] on the Ishikawa iteration process from Hilbert spaces to these much more general Banach spaces. Furthermore, we shall prove that the Mann iteration process converges strongly to the unique fixed point of a quasi-contractive map in this general setting. No compactness assumption on K is required in our theorems.

• Journal of applied mathematics & informatics
• /
• 제17권1_2_3호
• /
• pp.329-341
• /
• 2005

In this paper, we introduce new random iterative sequences with errors approximating a unique random common fixed point for three random set-valued strongly pseudo-contractive mappings and show the convergence of the random iterative sequences with errors by using an approximation method in real uniformly smooth separable Banach spaces. As applications, we study the existence of random solutions for some kind of random nonlinear operator equations group in separable Hilbert spaces.

• Kyungpook Mathematical Journal
• /
• 제54권3호
• /
• pp.375-385
• /
• 2014

We investigate strong convergence of Halpern's iteration for a countable family of strongly relatively nonexpansive mappings in the framework of uniformly convex and uniformly smooth Banach spaces. Our results extend those announced by many authors.

• Journal of applied mathematics & informatics
• /
• 제10권1_2호
• /
• pp.261-275
• /
• 2002

Let T be a local strongly accretive operator from a real uniformly smooth Banach space X into itself. It is proved that Ishikawa iterative schemes with errors converge strongly to a unique solution of the equations T$\_$x/ = f and x + T$\_$x/ = f, respectively, and are almost T$\_$b/-stable. The related results deal with the strong convergence and almost T$\_$b/-stability of Ishikawa iterative schemes with errors for local strongly pseudocontractive operators.

• Journal of applied mathematics & informatics
• /
• 제33권1_2호
• /
• pp.13-31
• /
• 2015

In this paper, we introduce a general iterative algorithm for finding a common element of the common fixed point set of an infinite family of ${\lambda}_i$-strict pseudo-contractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in q-uniformly smooth Banach spaces. Then, we analyze the strong convergence of the iterative sequence generated by the proposed iterative algorithm under mild conditions.

• Journal of applied mathematics & informatics
• /
• 제8권3호
• /
• pp.1027-1033
• /
• 2001

In this paper we obtain the necessary and sufficient conditions for convergence concerning the Mann iterative sequences with errors for the class of continuous accretive operators in smooth Banach spaces.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제11권4호
• /
• pp.293-308
• /
• 2004

Let K be a nonempty convex subset of an arbitrary Banach space X and $T\;:\;K\;{\rightarrow}\;K$ be a uniformly continuous strictly hemi-contractive operator with bounded range. We prove that certain Ishikawa iteration scheme with errors both converges strongly to a unique fixed point of T and is almost T-stable on K. We also establish similar convergence and almost stability results for strictly hemi-contractive operator $T\;:\;K\;{\rightarrow}\;K$, where K is a nonempty convex subset of arbitrary uniformly smooth Banach space X. The convergence results presented in this paper extend, improve and unify the corresponding results in Chang [1], Chang, Cho, Lee & Kang [2], Chidume [3, 4, 5, 6, 7, 8], Chidume & Osilike [9, 10, 11, 12], Liu [19], Schu [25], Tan & Xu [26], Xu [28], Zhou [29], Zhou & Jia [30] and others.