• The Pure and Applied Mathematics
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• v.10 no.3
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• pp.145-161
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• 2003

Weak commutativity of a pair of maps was introduced by Sessa [On a weak commutativity condition of mappings in fixed point considerations. Publ. Inst. Math. (Beograd) (N.S.) 32(40) (1982),149-153] in fixed point considerations. Thereafter a number of generalizations of this notion has been obtained. The purpose of this paper is to present a brief development of weaker forms of commuting maps, and to obtain two fixed point theorems for noncommuting and noncontinuous maps on noncomplete metric spaces.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.83-104
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• 2011

In this note, compound-commuting additive maps on matrix spaces are studied. We show that compound-commuting additive maps send rank one matrices to matrices of rank less than or equal to one. By using the structural results of rank-one nonincreasing additive maps, we characterize compound-commuting additive maps on four types of matrices: triangular matrices, square matrices, symmetric matrices and Hermitian matrices.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.157-165
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• 2014

In this paper, we obtain some common fixed point theorems of single and multivalued maps under hybrid contractive conditions satisfying weakly commuting in IFMS. Our results extend previous ones in IFMS.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.629-643
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• 2019

In common fixed point problems in metric spaces several versions of weak commutativity have been considered. Mappings which are not compatible have also been discussed in common fixed point problems. Here we consider common fixed point problems of non-compatible and R-weakly commuting mappings in probabilistic semimetric spaces with the help of a control function. This work is in line with research in probabilistic fixed point theory using control functions. Further we support our results by examples.

• The Pure and Applied Mathematics
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• v.20 no.3
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• pp.181-198
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• 2013

C. Vetro [4] gave the concept of weak non-Archimedean in fuzzy metric space. Using the same concept for Menger PM spaces, Mishra et al. [22] proved the common fixed point theorem for six maps, Also they introduced semi-compatibility. In this paper, we generalized the theorem [22] for family of maps and proved the common fixed point theorems using the pair of semi-compatible and reciprocally continuous maps for one pair and R-weakly commuting maps for another pair in Menger WNAPM-spaces. Our results extends and generalizes several known results in metric spaces, probabilistic metric spaces and the similar spaces.

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.77-86
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• 2006

We consider the set of commuting pairs of matrices and their preservers over binary Boolean algebra, chain semiring and general Boolean algebra. We characterize those linear operators that preserve the set of commuting pairs of matrices over a general Boolean algebra and a chain semiring.

• Honam Mathematical Journal
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• v.35 no.2
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• pp.109-117
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• 2013

In the present paper, we utilize the notion of converse commuting mappings due to L$\ddot{u}$ [On common fixed points for converse commuting self-maps on a metric spaces, Acta. Anal. Funct. Appl. 4(3) (2002), 226-228] and prove a common fixed point theorem in Menger space using an implicit relation. We also give an illustrative example to support our main result.

• Communications of the Korean Mathematical Society
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• v.20 no.1
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• pp.23-34
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• 2005

Let R be a ring with left identity. Let G : $R{\times}R{\to}R$ be a symmetric biadditive mapping and g the trace of G. Let ${\alpha}\;:\;R{\to}R$ be an endomorphism and ${\beta}\;:\;R{\to}R$ an epimorphism. In this paper we show the following: (i) Let R be 2-torsion-free. If g is (${\alpha},{\beta}$)-skew-commuting on R, then we have G = 0. (ii) If g is (${\beta},{\beta}$)-skew-centralizing on R, then g is (${\beta},{\beta}$)-commuting on R. (iii) Let $n{\ge}2$. Let R be (n+1)!-torsion-free. If g is n-(${\alpha},{\beta}$)-skew-commuting on R, then we have G = 0. (iv) Let R be 6-torsion-free. If g is 2-(${\alpha},{\beta}$)-commuting on R, then g is (${\alpha},{\beta}$)-commuting on R.

• Honam Mathematical Journal
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• v.16 no.1
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• pp.41-45
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• 1994

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1205-1211
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• 2010

In the present article, we investigate the possibility of a real-valued map on the space of tuples of commuting trace-class self-adjoint operators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with continuous group homomorphisms from the Milnor's K-group of the real numbers into the additive group of real numbers. Using this connection, it is shown that any such trace map must be trivial, but it is proposed that the target group of a nontrivial trace should be a linearized version of Milnor's K-theory as with the case of universal determinant for commuting tuples of matrices rather than just the field of constants.