• Management Science and Financial Engineering
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• v.20 no.1
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• pp.17-19
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• 2014

In this paper, we consider an interesting variant of the inverse minimum traveling salesman problem. Given an instance (G, w) of the minimum traveling salesman problem defined on a metric space, we fix a specified Hamiltonian cycle $HC_0$. The task is then to adjust the edge cost vector w to w' so that the new cost vector w' satisfies the triangle inequality condition and $HC_0$ can be returned by the minimum spanning tree algorithm in the TSP-instance defined with w'. The objective is to minimize the total deviation between the original and the new cost vectors with respect to the $L_1$-norm. We call this problem the inverse metric traveling salesman problem against the minimum spanning tree algorithm and show that it is closely related to the inverse metric spanning tree problem.

• The Pure and Applied Mathematics
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• v.22 no.2
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• pp.185-197
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• 2015

The author considers a Noor-type iterative scheme to approximate com- mon fixed points of an infinite family of uniformly quasi-sup(f_n)-Lipschitzian map- pings and an infinite family of g_n-expansive mappings in convex cone metric spaces. His results generalize, improve and unify some corresponding results in convex met- ric spaces [1, 3, 9, 16, 18, 19] and convex cone metric spaces [8].

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.117-135
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• 2021

Our purpose in this paper is to introduce the concept of complex valued convex metric spaces and introduce an analogue of the Picard-Ishikawa hybrid iterative scheme, recently proposed by Okeke [24] in this new setting. We approximate (common) fixed points of certain contractive conditions through these two new concepts and obtain several corollaries. We prove that the Picard-Ishikawa hybrid iterative scheme [24] converges faster than all of Mann, Ishikawa and Noor [23] iterative schemes in complex valued convex metric spaces. Also, we give some numerical examples to validate our results.

• The Mathematical Education
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• v.21 no.3
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• pp.7-11
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• 1983

A measurable function f defined on a measurable subset A of the real line R is called pth power summable on A if │f│$^{p}$ is integrable on A and the set of all pth power summable functions on A is denoted by L$^{p}$ (A). For each member f in L$^{p}$ (A), we define ∥f∥$_{p}$ =(equation omitted) For real numbers p and q where (equation omitted) and (equation omitted), we discuss the Holder's inequality ∥fg∥$_1$<∥f∥$_{p}$ ∥g∥$_{q}$ , f$\in$L$^{p}$ (A), g$\in$L$^{q}$ (A) and the Minkowski inequality ∥＋g∥$_{p}$ <∥f∥$_{p}$ ＋∥g∥$_{p}$ , f,g$\in$L$^{p}$ (A). In this paper also discuss that L$_{p}$ (A) becomes a metric space with the metric $\rho$ : L$^{p}$ (A) $\times$L$^{p}$ (A) longrightarrow R where $\rho$(f,g)=∥f-g∥$_{p}$ , f,g$\in$L$^{p}$ (A). Then, in this paper prove the Riesz-Fischer theorem, i.e., the space L$^{p}$ (A) is complete and that the space L$^{p}$ (A) is separable.

• The Mathematical Education
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• v.26 no.1
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• pp.41-45
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• 1987

In this paper, we give several fixed point theorems in a complete metric space for two multi-valued mappings commuting with two single-valued mappings. In fact, our main theorems show the existence of solutions of functional equations f($\chi$)=g($\chi$)$\in$S$\chi$∩T$\chi$ and $\chi$=f($\chi$)=g($\chi$)$\in$S$\chi$∩T$\chi$ under certain conditions. We also answer an open question proposed by Rhoades-Singh-Kulsherestha. Throughout this paper, let (X, d) be a complete metric space. We shall follow the following notations : CL(X) = {A; A is a nonempty closed subset of X}, CB(X)={A; A is a nonempty closed and founded subset of X}, C(X)={A; A is a nonempty compact subset of X}, For each A, B$\in$CL(X) and $\varepsilon$>0, N($\varepsilon$, A) = {$\chi$$\in$X; d($\chi$, ${\alpha}$) < $\varepsilon$ for some ${\alpha}$$\in$A}, E$\sub$A, B/={$\varepsilon$ > 0; A⊂N($\varepsilon$ B) and B⊂N($\varepsilon$, A)}, and (equation omitted). Then H is called the generalized Hausdorff distance function fot CL(X) induced by a metric d and H defined CB(X) is said to be the Hausdorff metric induced by d. D($\chi$, A) will denote the ordinary distance between $\chi$$\in$X and a nonempty subset A of X. Let R$\^$+/ and II$\^$+/ denote the sets of nonnegative real numbers and positive integers, respectively, and G the family of functions ${\Phi}$ from (R$\^$+/)$\^$s/ into R$\^$+/ satisfying the following conditions: (1) ${\Phi}$ is nondecreasing and upper semicontinuous in each coordinate variable, and (2) for each t>0, $\psi$(t)=max{$\psi$(t, 0, 0, t, t), ${\Phi}$(t, t, t, 2t, 0), ${\Phi}$(0, t, 0, 0, t)} $\psi$: R$\^$+/ \longrightarrow R$\^$+/ is a nondecreasing upper semicontinuous function from the right. Before sating and proving our main theorems, we give the following lemmas:

• The Pure and Applied Mathematics
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• v.23 no.2
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• pp.201-201
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• 2016

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.327-340
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• 2003

We are devoted to dealing with the conformal theory of a Rizza manifold with a generalized Finsler metric $G_{ij}$ (x,y) and a new conformal invariant non-linear connection $M^{i}$ $_{j}$ (x,y) constructed from the generalized Cern's non-linear connection $N^{i}$ $_{j}$ (x,y) and almost complex structure $f^{i}$ $_{j}$ (x). First, we find a conformal invariant connection ( $M_{j}$ $^{i}$ $_{k}$ , $M^{i}$ $_{j}$ , $C_{j}$ $^{i}$ $_{k}$ ) and conformal invariant tensors. Next, the nearly Kaehlerian (G, M)-structures under conformal change in a Rizza manifold are investigate.

• The Pure and Applied Mathematics
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• v.24 no.2
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• pp.79-98
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• 2017

We establish a coupled coincidence point theorem for generalized compatible pair of mappings $F,G:X{\times}X{\rightarrow}X$ under generalized symmetric Meir-Keeler contraction on a partially ordered metric space. We also deduce certain coupled fixed point results without mixed monotone property of $F:X{\times}X{\rightarrow}X$. An example supporting to our result has also been cited. As an application the solution of integral equations are obtain here to illustrate the usability of the obtained results. We improve, extend and generalize several known results.

• East Asian mathematical journal
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• v.16 no.2
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• pp.339-354
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• 2000

• Honam Mathematical Journal
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• v.32 no.4
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• pp.651-661
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• 2010

In this paper, we obtain a necessary and sufficient condition for a left invariant connection in the tangent bundle over a closed Lie group with a left invariant metric to be a Yang-Mills connection. Moreover, we have a necessary and sufficient condition for a left invariant connection with a torsion-free Weyl structure in the tangent bundle over SU(2) with a left invariant Riemannian metric g to be a Yang-Mills connection.