• 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.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1319-1338
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• 2009

In this paper, using proximal-point mappings technique of Raccretive mappings and the property of the fixed point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of the solution set of the system of parametric generalized quasi-variational inclusions involving R-accretive mappings in real uniformly smooth Banach space. Further under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to parameters. The technique and results presented in this paper can be viewed as extension of the techniques and corresponding results given in [3, 23, 24, 32, 33, 34].

• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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• pp.153-169
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• 2000

Let E be a real q-uniformly smooth Banach space which admits a weakly sequentially continuous duality map, and K a nonempty closed convex subset of E. Let T : K -> K be a mapping such that $F(T)\;=\;{x\;{\in}\;K\;:\;Tx\;=\;x}\;{\neq}\;0$ and (I - T) satisfies the accretive-type condition: $\;{\geq}\;{\lambda}$\mid$$\mid$x-Tx$\mid$$\mid$^2$, for all $x\;{\in}\;K,\;x^*\;{\in}\;F(T)$ and for some ${\lambda}\;>\;0$. The weak and strong convergence of the Ishikawa iteration method to a fixed point of T are investigated. An application of our results to the approximation of a solution of a certain linear operator equation is also given. Our results extend several important known results from the Mann iteration method to the Ishikawa iteration method. In particular, our results resolve in the affirmative an open problem posed by Naimpally and Singh (J. Math. Anal. Appl. 96 (1983), 437-446).

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.161-185
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• 2016

We consider a mathematical model which describes the quasistatic frictional contact between a piezoelectric body and an electrically conductive obstacle, the so-called foundation. A nonlinear electro-viscoelastic constitutive law is used to model the piezoelectric material. Contact is described with Signorini's conditions and a version of Coulomb's law of dry friction in which the adhesion of contact surfaces is taken into account. The evolution of the bonding field is described by a first order differential equation. We derive a variational formulation for the model, in the form of a system for the displacements, the electric potential and the adhesion. Under a smallness assumption which involves only the electrical data of the problem, we prove the existence of a unique weak solution of the model. The proof is based on arguments of time-dependent quasi-variational inequalities, differential equations and Banach's fixed point theorem.

• Proceedings of the Korea Committee for Ocean Resources and Engineering Conference
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• 2004.11a
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• pp.179-182
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• 2004

This paper deals with a new mathematical formulation of nonlinear wave profile based on Banach fixed point theorem. As application of the formulation and its solution procedure, some numerical solutions was presented in this paper and nonlinear equation was derived. Also we introduce a new operator for iteration and getting solution. A numerical study was accomplished with Stokes' first-order solution and iteration scheme, and then we can know the nonlinear characteristic of Stokes' high-order solution. That is, using only Stokes' first-oder(linear) velocity potential and an initial guess of wave profile, it is possible to realize the corresponding high-oder Stokian wave profile with tile new numerical scheme which is the method of iteration. We proved the mathematical convergence of tile proposed scheme. The nonlinear strategy of iterations has very fast convergence rate, that is, only about 6-10 iterations arc required to obtain a numerically converged solution.

• East Asian mathematical journal
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• v.24 no.1
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• pp.35-43
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• 2008

Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let ${\{T_i\}}^N_{i=1}$ be N nonexpansive self-mappings of K with $F\;=\;{\cap}^N_{i=1}F(T_i)\;{\neq}\;{\theta}$ (here $F(T_i)$ denotes the set of fixed points of $T_i$). Suppose that one of the mappings in ${\{T_i\}}^N_{i=1}$ is semi-compact. Let $\{{\alpha}_n\}\;{\subset}\;[{\delta},\;1-{\delta}]$ for some ${\delta}\;{\in}\;(0,\;1)$ and $\{{\beta}_n\}\;{\subset}\;[\tau,\;1]$ for some ${\tau}\;{\in}\;(0,\;1]$. For arbitrary $x_0\;{\in}\;K$, let the sequence {$x_n$} be defined iteratively by $\{{x_n\;=\;{\alpha}_nx_{n-1}\;+\;(1-{\alpha}_n)T_ny_n,\;\;\;\;\;\;\;\;\; \atop {y_n\;=\;{\beta}nx_{n-1}\;+\;(1-{\beta}_n)T_nx_n},\;{\forall}_n{\geq}1,}$, where $T_n\;=\;T_{n(modN)}$. Then {$x_n$} convergence strongly to a common fixed point of the mappings family ${\{T_i\}}^N_{i=1}$. The result presented in this paper generalized and improve the corresponding results of Chidume and Shahzad [C. E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal. 62(2005), 1149-1156] even in the case of ${\beta}_n\;{\equiv}\;1$ or N=1 are also new.

• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.251-263
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• 1993

The purpose of this paper is to study the solvability of the equation (E). In Section 2, we give preliminary definitions. In Section 3, we prove related two results (Theorem 1 and Corollary 1) concerning the closedness property of accretive operators in the class of spaces whose nonempty bounded closed convex subsets have the fixed point property for nonexpansive self-mapping. Using therem 1, we derive a result (Theorem 2) on the range of accetive operators in (.pi.)$_{1}$ spaces with a view to establishing a new result, which improves a result of Kartsatos [8] and Webb [15]. Further, we give an interesting consequence (Corollary 3) of Theorem 2. In section 4, we apply Corollary 1 to obtain two results (Theorem 3 and 4) for the range of sums of two accretive operators, which generalize two results of Reich [12].

• Honam Mathematical Journal
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• v.30 no.2
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• pp.233-246
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• 2008

In this paper, we obtain the general solution of the following cubic type functional equation and establish the stability of this equation (0.1) $kf({{\sum}\limits^{n-1}_{j=1}}x_j+kx_n)+kf({{\sum}\limits^{n-1}_{j=1}}x_j-kx_n)+2{{\sum}\limits^{n-1}_{j=1}}f(kx_j)+(k^3-1)(n-1)[f(x_1)+f(-x_1)]=2kf({\sum\limits^{n-1}_{j=1}}x_j)=K^3{\sum\limits^{n-1}_{j=1}[f(x_j+x_n)+f(x_j-x_n)]$ for any integers k and n with k ${\geq}$ 2 and n ${\geq}$ 3.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.11 no.1
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• pp.25-32
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• 2011

In this paper, we study the existence and uniqueness of solutions for the impulsive semilinear fuzzy integrodifferential equations with nonlocal conditions and forcing term with memory in n-dimensional fuzzy vector space ($E^n_N$, $d_{\varepsilon}$) by using Banach fixed point theorem. That is an extension of the result of Kwun et al. [9] to impulsive system.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.539-553
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• 2011

We study the solvability of a perturbed quadratic functional-integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions defined, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions.