Where values for, , are computed by ( 3.9) with. If we multiply the first equation in system ( 3.2) by, the second by and the third by, and then using in such obtained system the change of variables įollowing the idea in, we use a transformation which reduces nonlinear systems ( 1.1) and ( 3.2) to third-order systems of nonhomogeneous linear difference equations. Recall that we may assume that, for every and for each. Here we study well-defined solutions of system ( 1.1) when neither of the sequences, , or initial conditions, ,, , is equal to zero. If or for some then similar results are obtained analogously. If for some then from ( 3.2) it follows that, for each such that. įirst we consider the case when some of initial values of solutions of system ( 3.2) is equal to zero. Using the same notation for coefficients as in ( 1.1) except for the coefficients, assuming that for all and. Hence we consider, without loss of generality, the system
Where, ,, we see that we may assume that, for every and for each. Noticing that in this case, system ( 1.1) can be written in the form Here we consider system ( 1.1) in the case when for all and. Explicit Formulae for the Case for All and So that from the first and the second equation we get, from which it follows that 3. So that from the first and the third equations we get, from which it follows thatįinally, if, , then the third equation in ( 1.1) becomes If, , then the second equation in ( 1.1) becomes So that from the second and the third equations and since the solution is well defined we get, , from which it follows that If, then the first equation in ( 1.1) becomes
Here we give explicit formulae for solutions of system ( 1.1) and present some consequences on asymptotic behavior of the solutions for the case when coefficients are periodic with period three. Some related results on systems of difference equations can be found in (see also references therein). For some related scalar difference equations see, for example, and the related references therein. For a different approach in dealing with the scalar difference equation see. Some related transformations are used also in. This idea appeared for the first time in for the case of the scalar equation with constant coefficients corresponding to system ( 1.1) and was also used later in. We show that in the main case, system ( 1.1) is transformed to a third-order system of nonhomogeneous linear first-order difference equations, which can be explicitly solved. We also adopt the customary notation and. Instead of that we assume, throughout the paper, that solutions of ( 1.1) are well defined. In fact, for “majority” initial values of system ( 1.1), solutions are well defined, but we will not discuss the problem here. So that there are a lot of well defined solutions of the system. Note that the solutions of ( 1.1) such that all sequences, ,, ,, and initial values in system ( 1.1) are positive, are also positive, that is, For some old results see, for example, classic book. Some recent results on solving difference equations can be found, for example, in. In we have shown that system ( 1.1) for the case when the sequences, ,, ,, are constant can be explicitly solved (if solutions are well defined).
The cases when both and are equal to zero for some fixed and an, are not interesting so they are excluded. Where all elements of the sequences, ,, ,, and initial values, ,, are real numbers. This paper studies the system of three difference equations Studying nonlinear difference equations and systems is an area of a great interest nowadays (see, e.g., and the references therein). Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced. We show that the system of three difference equations, , and, , where all elements of the sequences, ,, ,, and initial values, ,, , are real numbers, can be solved.