• Journal of Mechanical Science and Technology
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• v.20 no.11
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• pp.1950-1960
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• 2006

Based on Banach fixed point theorem, a method to calculate nonlinear superposition for three interacting Stokes' waves is proposed in this paper. A mathematical formulation for the nonlinear superposition in deep water and some numerical solutions were investigated. The authors carried out the numerical study with three progressive linear potentials of different wave numbers and succeeded in solving the nonlinear wave profiles of their three wave-interaction, that is, using only linear wave potentials, it was possible to realize the corresponding nonlinear interacting wave profiles through iteration of the method. The stability of the method for the three interacting Stokes' waves was analyzed. The calculation results, together with Fourier transform, revealed that the iteration made it possible to predict higher-order nonlinear frequencies for three Stokes' waves' interaction. The proposed method has a very fast convergence rate.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.13-23
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• 2009

Some strong convergence theorems of explicit iteration scheme for asymptotically nonexpansive semi-groups in Banach spaces are established. The results presented in this paper extend and improve some recent results in [T. Suzuki. On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131(2002)2133-2136; H. K. Xu. A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72(2005)371-379; N. Shioji and W. Takahashi. Strong convergence theorems for continuous semigroups in Banach spaces, Math. Japonica. 1(1999)57-66; T. Shimizu and W. Takahashi. Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211(1997)71-83; N. Shioji and W. Takahashi. Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. TMA, 34(1998)87-99; H. K. Xu. Approximations to fixed points of contraction semigroups in Hilbert space, Numer. Funct. Anal. Optim. 19(1998), 157-163.]

• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.125-139
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• 2015

In this paper we introduce the notion of the generalized Darbo fixed point theorem and prove some fixed and coupled fixed point theorems in Banach space via the measure of non-compactness, which generalize the result of Aghajani et al. [6]. Our results generalize, extend, and unify several well-known comparable results in the literature. One of the applications of our main result is to prove the existence of solutions for the system of integral equations.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.773-788
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• 2014

We introduce an iterative process which converges strongly to a common minimum-norm point of solutions of variational inequality problem for a monotone mapping and fixed points of a finite family of relatively nonexpansive mappings in Banach spaces. Our theorems improve most of the results that have been proved for this important class of nonlinear operators.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.1021-1034
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• 2021

This paper deals with a nonlinear Langevin equation involving two fractional orders with three-point boundary conditions. Our aim is to find the existence of solutions for the proposed Langevin equation by using the Banach contraction mapping principle and the Krasnoselskii's fixed point theorem. Three examples are also given to show the significance of our results.

• East Asian mathematical journal
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• v.25 no.4
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• pp.527-532
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• 2009

In this paper, we study the following extended generalized variational inequality problem, introduced by Noor (for short, EGVI) : Given a closed convex subset K in q-uniformly smooth Banach space B, three nonlinear mappings T : $K\;{\rightarrow}\;B^*$, g : $K\;{\rightarrow}\;K$, h : $K\;{\rightarrow}\;K$ and a vector ${\xi}\;{\in}\;B^*$, find $x\;{\in}\;K$, $h(x)\;{\in}\;K$ such that $\xi$, g(y)-h(x)> $\geq$ 0, for all $y\;{\in}\;K$, $g(y)\;{\in}\;K$. [see [2]: M. Aslam Noor, Extended general variational inequalities, Appl. Math. Lett. 22 (2009) 182-186.] By using sunny nonexpansive retraction $Q_K$ and the well-known Banach's fixed point principle, we prove existence results for solutions of (EGVI). Our results extend some recent results from the literature.

• Korean Journal of Mathematics
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• v.30 no.4
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• pp.665-678
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• 2022

The aim of this paper is to introduce a new iterative process and show that our iteration scheme is faster than other existing iteration schemes with the help of numerical examples. Next, we have established convergence and stability results for the approximation of fixed points of the contractive-like mapping in the framework of uniformly convex Banach space. In addition, we have established some convergence results for the approximation of the fixed points of a nonexpansive mapping.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.9 no.2
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• pp.137-140
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• 2009

In this paper, we introduce and formulate the definitions of Banach operator type k and Banach operator pair on intuitionistic fuzzy metric space. Thereafter we prove some properties and theorems on intuitionistic fuzzy metric space.

• 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.

• Korean Journal of Mathematics
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• v.23 no.2
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• pp.231-248
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• 2015

In this paper, we solve the following quadratic $\rho$-functional inequalities ${\parallel}f(\frac{x+y+z}{2})+f(\frac{x-y-z}{2})+f(\frac{y-x-z}{2})+f(\frac{z-x-y}{2})-f(x)-x(y)-f(z){\parallel}\;(0.1)\\{\leq}{\parallel}{\rho}(f(x+y+z+)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-f(z)){\parallel}$ where $\rho$ is a fixed complex number with ${\left|\rho\right|}<\frac{1}{8}$, and ${\parallel}f(x+y+z)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-4f(z){\parallel}\;(0.2)\\{\leq}{\parallel}{\rho}(f(\frac{x+y+z}{2})+f(\frac{x-y-z}{2})+f(\frac{y-x-z}{2})+f(\frac{z-x-y}{2})-f(x)-f(y)-f(z)){\parallel}$ where $\rho$ is a fixed complex number with ${\left|\rho\right|}$ < 4. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces.