• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.177-189
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• 2015

In this paper, we prove ${\Delta}$-convergence theorems for Ishikawa iteration of asymptotically and multivalued nonexpansive mapping in CAT(0) spaces. This results we obtain are analogs of Banach spaces results of Sokhuma [13].

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.665-678
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• 2019

We construct an iteration scheme involving a hybrid pair of mappings, one a single-valued asymptotically nonexpansive mapping t and the other a multivalued nonexpansive mapping T, in a complete CAT(0) space. In the process, we remove a restricted condition (called the end-point condition) from results of Akkasriworn and Sokhuma [1] and and use this to prove some convergence theorems. The results concur with analogues for Banach spaces from Uddin et al. [16].

• Korean Journal of Mathematics
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• v.16 no.2
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• pp.215-231
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• 2008

Let E be a uniformly convex Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm, C a nonempty closed convex subset of E, and $T:C{\rightarrow}{\mathcal{K}}(E)$ a multivalued nonself-mapping such that $P_T$ is nonexpansive, where $P_T(x)=\{u_x{\in}Tx:{\parallel}x-u_x{\parallel}=d(x,Tx)\}$. For $f:C{\rightarrow}C$ a contraction and $t{\in}(0,1)$, let $x_t$ be a fixed point of a contraction $S_t:C{\rightarrow}{\mathcal{K}}(E)$, defined by $S_tx:=tP_T(x)+(1-t)f(x)$, $x{\in}C$. It is proved that if C is a nonexpansive retract of E and $\{x_t\}$ is bounded, then the strong ${\lim}_{t{\rightarrow}1}x_t$ exists and belongs to the fixed point set of T. Moreover, we study the strong convergence of $\{x_t\}$ with the weak inwardness condition on T in a reflexive Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm. Our results provide a partial answer to Jung's question.

• Journal of the Chungcheong Mathematical Society
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• v.6 no.1
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• pp.53-57
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• 1993

Let E and F be Banach spaces with $X{\subset}E$ and $Y{\subset}F$. Suppose that X is weakly compact, convex and has the fixed point property for a nonexpansive mapping, and Y has the fixed point property for a multivalued nonexpansive mapping. Then $(X{\oplus}Y)_p$, $1{\leq}$ P < ${\infty}$ has the fixed point property for a multi valued nonexpansive mapping. Furthermore, if X has the generic fixed point property for a nonexpansive mapping, then $(X{\oplus}Y)_{\infty}$ has the fixed point property for a multi valued nonexpansive mapping.

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.1-11
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• 2021

In this paper we study the weak and strong convergence to minimizers of convex function of proximal point algorithm SP-iteration of three multivalued nonexpansive mappings in a Hilbert space.

• Journal of the Chungcheong Mathematical Society
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• v.3 no.1
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• pp.103-110
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• 1990

Let K be a nonempty weakly compact convex subset of a Banach space X and T : K ${\rightarrow}$ C(X) a nonexpansive mapping satisfying $P_T(x){\cap}clI_K(x){\neq}{\emptyset}$. We first show that if I - T is semiconvex type then T has a fixed point. Also we obtain the same result without the condition that I - T is semiconvex type in a Banach space satisfying Opial's condition. Lastly we extend the result of [5] to the case, that T is an 1-set contraction mapping.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.35-48
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• 2023

Let 𝔼 be a CAT(0) space and K be a nonempty closed convex subset of 𝔼. Let T : K → 𝓟(K) be a multimap such that F(T) ≠ ∅ and ℙ_T(x) = {y ∈ Tx : d(x, y) = d(x, Tx)}. Define sequence {x_n} by x_n+1 = (1 - α)𝜈_n⊕αw_n, y_n = (1 - β)u_n⊕βw_n, z_n = (1-γ)x_n⊕γu_n where α, β, γ ∈ [0; 1]; u_n ∈ ℙ_T (x_n); 𝜈_n ∈ ℙ_T (y_n) and w_n ∈ ℙ_T (z_n). (1) If ℙ_T is quasi-nonexpansive, then it is proved that {x_n} converges strongly to a fixed point of T. (2) If a multimap T satisfies Condition(I) and ℙ_T is quasi-nonexpansive, then {x_n} converges strongly to a fixed point of T. (3) Finally, we establish a weak convergence result. Our results extend and unify some of the related results in the literature.