• Title/Summary/Keyword: Lipschitz spaces

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SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV2α(I)

  • Aziz, Wadie;Guerrero, Jose Atilio;Merentes, Nelson
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.649-659
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    • 2015
  • The space $BV^2_{\alpha}(I)$ of all the real functions defined on interval $I=[a,b]{\subset}\mathbb{R}$, which are of bounded second ${\alpha}$-variation (in the sense De la Vall$\acute{e}$ Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function $h:I{\times}\mathbb{R}{\rightarrow}\mathbb{R}$, and prove, in particular, that if H maps $BV^2_{\alpha}(I)$ into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.

PROXIMITY MAPS FOR CERTAIN SPACES

  • Lee, Mun-Bae;Park, Sung-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.259-271
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    • 1997
  • Let K be a nonempty subset of a normed linear space X and let x $\in$ X. An element k$_0$ in K satisfying $\$\mid$$x - k$_0$$\$\mid$$ = d(x, K) := (equation omitted) $\$\mid$$x - k$\$\mid$$ is called a best approximation to x from K. For any x $\in$ X, the set of all best approximations to x from K is denoted by P$_K$(x) = {k $\in$ K : $\$\mid$$ x - k $\$\mid$$ = d(x, K)}. (omitted)

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ON THE "TERRA INCOGNITA" FOR THE NEWTON-KANTROVICH METHOD WITH APPLICATIONS

  • Argyros, Ioannis Konstantinos;Cho, Yeol Je;George, Santhosh
    • Journal of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.251-266
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    • 2014
  • In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fr$\acute{e}$chet-derivative of the operator involved is p-H$\ddot{o}$lder continuous (p${\in}$(0, 1]). Numerical examples involving two boundary value problems are also provided.

ON A QUADRATICALLY CONVERGENT ITERATIVE METHOD USING DIVIDED DIFFERENCES OF ORDER ONE

  • Argyros, Ioannis K.
    • The Pure and Applied Mathematics
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    • v.14 no.3
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    • pp.203-221
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    • 2007
  • We introduce a new two-point iterative method to approximate solutions of nonlinear operator equations. The method uses only divided differences of order one, and two previous iterates. However in contrast to the Secant method which is of order 1.618..., our method is of order two. A local and a semilocal convergence analysis is provided based on the majorizing principle. Finally the monotone convergence of the method is explored on partially ordered topological spaces. Numerical examples are also provided where our results compare favorably to earlier ones [1], [4], [5], [19].

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Weak and Strong Convergence of Hybrid Subgradient Method for Pseudomonotone Equilibrium Problems and Nonspreading-Type Mappings in Hilbert Spaces

  • Sriprad, Wanna;Srisawat, Somnuk
    • Kyungpook Mathematical Journal
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    • v.59 no.1
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    • pp.83-99
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    • 2019
  • In this paper, we introduce a hybrid subgradient method for finding an element common to both the solution set of a class of pseudomonotone equilibrium problems, and the set of fixed points of a finite family of ${\kappa}$-strictly presudononspreading mappings in a real Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative method under some suitable conditions. These convergence theorems are investigated without the Lipschitz condition for bifunctions. Our results complement many known recent results in the literature.

A VISCOSITY TYPE PROJECTION METHOD FOR SOLVING PSEUDOMONOTONE VARIATIONAL INEQUALITIES

  • Muangchoo, Kanikar
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.2
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    • pp.347-371
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    • 2021
  • A plethora of applications from mathematical programmings, such as minimax, mathematical programming, penalization and fixed point problems can be framed as variational inequality problems. Most of the methods that used to solve such problems involve iterative methods, that is why, in this paper, we introduce a new extragradient-like method to solve pseudomonotone variational inequalities in a real Hilbert space. The proposed method has the advantage of a variable step size rule that is updated for each iteration based on previous iterations. The main advantage of this method is that it operates without the previous knowledge of the Lipschitz constants of an operator. A strong convergence theorem for the proposed method is proved by letting the mild conditions on an operator 𝒢. Numerical experiments have been studied in order to validate the numerical performance of the proposed method and to compare it with existing methods.

CONVERGENCE AND STABILITY OF ITERATIVE ALGORITHM OF SYSTEM OF GENERALIZED IMPLICIT VARIATIONAL-LIKE INCLUSION PROBLEMS USING (𝜃, 𝜑, 𝛾)-RELAXED COCOERCIVITY

  • Kim, Jong Kyu;Bhat, Mohd Iqbal;Shaf, Sumeera
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.4
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    • pp.749-780
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    • 2021
  • In this paper, we give the notion of M(., .)-𝜂-proximal mapping for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. As an application, we introduce and investigate a new system of variational-like inclusions in Banach spaces. By means of M(., .)-𝜂-proximal mapping method, we give the existence of solution for the system of variational inclusions. Further, propose an iterative algorithm for finding the approximate solution of this class of variational inclusions. Furthermore, we discuss the convergence and stability analysis of the iterative algorithm. The results presented in this paper may be further expolited to solve some more important classes of problems in this direction.

A NEW EXPLICIT EXTRAGRADIENT METHOD FOR SOLVING EQUILIBRIUM PROBLEMS WITH CONVEX CONSTRAINTS

  • Muangchoo, Kanikar
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.1
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    • pp.1-22
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    • 2022
  • The purpose of this research is to formulate a new proximal-type algorithm to solve the equilibrium problem in a real Hilbert space. A new algorithm is analogous to the famous two-step extragradient algorithm that was used to solve variational inequalities in the Hilbert spaces previously. The proposed iterative scheme uses a new step size rule based on local bifunction details instead of Lipschitz constants or any line search scheme. The strong convergence theorem for the proposed algorithm is well-proven by letting mild assumptions about the bifunction. Applications of these results are presented to solve the fixed point problems and the variational inequality problems. Finally, we discuss two test problems and computational performance is explicating to show the efficiency and effectiveness of the proposed algorithm.

ADMISSIBLE INERTIAL MANIFOLDS FOR INFINITE DELAY EVOLUTION EQUATIONS

  • Minh, Le Anh
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.669-688
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    • 2021
  • The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$\{{\frac{du}{dt}}+Au=F(t,\;u_t),\;t{\geq}s,\\\;u_s({\theta})={\phi}({\theta}),\;{\forall}{\theta}{\in}(-{{\infty}},\;0],\;s{\in}{\mathbb{R}},$$ where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

A NEW RELAXED TSENG METHOD FOR FINDING A COMMON SOLUTION OF FIXED POINT AND SPLIT MONOTONE INCLUSION PROBLEMS

  • Lusanda Mzimela;Akindele Adebayo Mebawondu;Adhir Maharaj;Chinedu Izuchukwu;Ojen Kumar Narain
    • Nonlinear Functional Analysis and Applications
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    • v.29 no.1
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    • pp.225-258
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    • 2024
  • In this paper, we study the problem of finding a common solution to a fixed point problem involving a finite family of ρ-demimetric operators and a split monotone inclusion problem with monotone and Lipschitz continuous operator in real Hilbert spaces. Motivated by the inertial technique and the Tseng method, a new and efficient iterative method for solving the aforementioned problem is introduced and studied. Also, we establish a strong convergence result of the proposed method under standard and mild conditions.