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Lipschitz type mappings in metric spaces with fixed points and maximum fixed points of magic squares. (English) Zbl 1259.54022

Lipschitz type compatible pairs of mappings are considered by employing generalized Meir-Keeler type conditions to obtain fixed points. As an application, a method for construction of magic squares of various orders like \(4n \times 4n\), \((4n + 2) \times (4n + 2)\) with a view of obtaining maximum fixed points is proposed. It is claimed that the maximum number of possible fixed points of a \(4n \times 4n\) magic square is half of the total number of elements of the magic square. Prior to some simple remarks, the work ends with a plan to attempt a proof in some later paper of a hypothesis that the maximum number of possible fixed points of \((8n - 2) \times (8n - 2)\) magic square is equal to \((8n - 2)(8n - 3)/2\) whenever \((8n - 2)/2\) is a prime number and that the maximum number of possible fixed points of \((8n + 2) \times (8n + 2)\) magic square is equal to \((8n + 2)(8n + 1)/2 - 2\), if \((8n + 2)/2\) is a prime number.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
05B15 Orthogonal arrays, Latin squares, Room squares
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