Abstract
It is proved that any first-order globally periodic linear inhomogeneous autonomous difference equation defined by a linear operator with closed range in a Banach space has an equilibrium. This result is extended for higher order linear inhomogeneous system in a real or complex Euclidean space. The work was highly motivated by the early works of Smith (1934, 1941) and the papers of Kister (1961) and Bas (2011).
Highlights
Let X be a set and let p be a positive integer
Where idX is the identical function on X and p is the least positive integer with this property
The following question was posed by Smith: does any continuous periodic transformation of a Euclidean nspace always admit a fixed point? Smith knew that the answer is true if the period p of the transformation is a prime number or a power of a prime number
Summary
Let X be a set and let p be a positive integer. It is said that the transformation T : X → X is p-periodic if. (b) We say that (2) is globally periodic if there is a positive integer p ≥ s for which the equation is globally pperiodic; that is, every solution of it is p-periodic. Even if there is a metric on X and h is continuous, it is still an open problem to determine whether (2) has or not an equilibrium point, or equivalently, the transformation (4) has a fixed point, if (2) is globally periodic. One can see that the opposite statement is true if the inhomogeneous equation has a steady state solution which is obviously p-periodic for any p ≥ 1.
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