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INERTIAL PICARD NORMAL S-ITERATION PROCESS
Samir Dashputre,Padmavati,Kavita Sakure 경남대학교 수학교육과 2021 Nonlinear Functional Analysis and Applications Vol.26 No.5
Many iterative algorithms like that Picard, Mann, Ishikawa and S-iteration are very useful to elucidate the fixed point problems of a nonlinear operators in various topological spaces. The recent trend for elucidate the fixed point via inertial iterative algorithm, in which next iterative depends on more than one previous terms. The purpose of the paper is to establish convergence theorems of new inertial Picard normal S-iteration algorithm for nonexpansive mapping in Hilbert spaces. The comparison of convergence of InerNSP and InerPNSP is done with InerSP (introduced by Phon-on et al. [25]) and MSP (introduced by Suparatulatorn et al. [27]) via numerical example.
Approximation of Common Fixed Points of Non-self Asymptotically Nonexpansive Mappings
김종규,Samir Dashputre,S. D. Diwan 영남수학회 2009 East Asian mathematical journal Vol.25 No.2
Let E be a uniformly convex Banach space and K a nonempty closed convex subset which is also a nonexpansive retract of E. For i=1,2,3, let Ti: K → E be an asymptotically nonexpansive mappings with sequence [수식] such that [수식], as [수식] (the set of all common fixed points of Ti, i = 1, 2, 3). Let {an},{bn} and {cn} are three real sequences in [0,1] such that [수식] for n ∈ N and some ϵ ≥ 0. Starting with arbitrary x₁∈K, define sequence {xn} by setting [수식] Assume that one of the following conditions holds: (1) E satisfies the Opial property, (2) E has Frechet differentiable norm, (3) E* has Kedec -Klee property, where E* is dual of E. Then sequence {xn} converges weakly to some p ∈ F(T).
APPROXIMATION OF COMMON FIXED POINTS OF NON-SELF ASYMPTOTICALLY NONEXPANSIVE MAPPINGS
Kim, Jong-Kyu,Dashputre, Samir,Diwan, S.D. The Youngnam Mathematical Society 2009 East Asian mathematical journal Vol.25 No.2
Let E be a uniformly convex Banach space and K a nonempty closed convex subset which is also a nonexpansive retract of E. For i = 1, 2, 3, let $T_i:K{\rightarrow}E$ be an asymptotically nonexpansive mappings with sequence ${\{k_n^{(i)}\}\subset[1,{\infty})$ such that $\sum_{n-1}^{\infty}(k_n^{(i)}-1)$ < ${\infty},\;k_{n}^{(i)}{\rightarrow}1$, as $n{\rightarrow}\infty$ and F(T)=$\bigcap_{i=3}^3F(T_i){\neq}{\phi}$ (the set of all common xed points of $T_i$, i = 1, 2, 3). Let {$a_n$},{$b_n$} and {$c_n$} are three real sequences in [0, 1] such that $\in{\leq}\;a_n,\;b_n,\;c_n\;{\leq}\;1-\in$ for $n{\in}N$ and some ${\in}{\geq}0$. Starting with arbitrary $x_1{\in}K$, define sequence {$x_n$} by setting {$$x_{n+1}=P((1-a_n)x_n+a_nT_1(PT_1)^{n-1}y_n)$$ $$y_n=P((1-b_n)x_n+a_nT_2(PT_2)^{n-1}z_n)$$ $$z_n=P((1-c_n)x_n+c_nT_3(PT_3)^{n-1}x_n)$$. Assume that one of the following conditions holds: (1) E satises the Opial property, (2) E has Frechet dierentiable norm, (3) $E^*$ has Kedec -Klee property, where $E^*$ is dual of E. Then sequence {$x_n$} converges weakly to some p${\in}$F(T).
GENERALIZED α-NONEXPANSIVE MAPPINGS IN HYPERBOLIC SPACES
Jong Kyu Kim,Samir Dashputre,Padmavati,Kavita Sakure 경남대학교 수학교육과 2022 Nonlinear Functional Analysis and Applications Vol.27 No.3
This paper deals with the new iterative algorithm for approximating the fixed point of generalized α-nonexpansive mappings in a hyperbolic space. We show that the proposed iterative algorithm is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur and Piri iteration processes for contractive mappings in a Banach space. We also establish some weak and strong convergence theorems for generalized α-nonexpansive mappings in hyperbolic space. The examples and numerical results are provided in this paper for supporting our main results.
CONVERGENCE THEOREMS FOR GENERALIZED α-NONEXPANSIVE MAPPINGS IN UNIFORMLY HYPERBOLIC SPACES
J. K. Kim,Samir Dashputre,Padmavati,Rashmi Verma 경남대학교 수학교육과 2024 Nonlinear Functional Analysis and Applications Vol.29 No.1
In this paper, we establish strong and $\Delta$-convergence theorems for new iteration process namely S-R iteration process for a generalized $\alpha$-nonexpansive mappings in a uniformly convex hyperbolic space and also we show that our iteration process is faster than other iteration processes appear in the current literature's. Our results extend the corresponding results of Ullah et al. [5], Imdad et al. [16] in the setting of uniformly convex hyperbolic spaces and many more in this direction.
Jong Kyu Kim,Ramesh Prasad Pathak,Samir Dashputre,Shailesh Dhar Diwan,Rajlaxmi Gupta 경남대학교 수학교육과 2018 Nonlinear Functional Analysis and Applications Vol.23 No.1
In this paper, we study the existence of fixed points, demiclosedness principle and the structure of fixed point sets for the class of nearly asymptotically nonexpansive nonselfmappings in CAT(0) spaces, and also we discuss the strong and triangle-convergence theorems for an iterative scheme introduced by Khan. Our results are improvements of the variouswell-known results of fixed point theory which is established in uniformly convex Banach spaces as well as CAT(0) spaces.