TY - JOUR

T1 - Mann iteration with power means

AU - Bischi, G. I.

AU - Cavalli, Fausto

AU - Naimzada, A.

PY - 2015

Y1 - 2015

N2 - We analyse the recurrence (Formula presented.) , where (Formula presented.) is a weighted power mean of (Formula presented.) , which has been proposed to model a class of non-linear forward-looking economic models with bounded rationality. Under suitable hypotheses on weights, we prove the convergence of the sequence (Formula presented.) Then, to simulate a fading memory, we consider exponentially decreasing weights. Since, in this case, the resulting recurrence does not fulfil the hypotheses of the previous convergence theorem, it is studied by reducing it to an equivalent two-dimensional autonomous map, which shares the asymptotic behaviours with a particular one-dimensional map. This allows us to prove that a long memory with sufficiently large weights has a stabilizing effect. Finally, we numerically investigate what happens when the memory ratio is not sufficiently large to provide stability, showing that, depending on the power mean and the memory ratio, either a delayed or early cascade of flip bifurcations occurs.

AB - We analyse the recurrence (Formula presented.) , where (Formula presented.) is a weighted power mean of (Formula presented.) , which has been proposed to model a class of non-linear forward-looking economic models with bounded rationality. Under suitable hypotheses on weights, we prove the convergence of the sequence (Formula presented.) Then, to simulate a fading memory, we consider exponentially decreasing weights. Since, in this case, the resulting recurrence does not fulfil the hypotheses of the previous convergence theorem, it is studied by reducing it to an equivalent two-dimensional autonomous map, which shares the asymptotic behaviours with a particular one-dimensional map. This allows us to prove that a long memory with sufficiently large weights has a stabilizing effect. Finally, we numerically investigate what happens when the memory ratio is not sufficiently large to provide stability, showing that, depending on the power mean and the memory ratio, either a delayed or early cascade of flip bifurcations occurs.

KW - Forward-looking models

KW - Learning

KW - Mann iterations

KW - Non-autonomous difference equations

KW - Forward-looking models

KW - Learning

KW - Mann iterations

KW - Non-autonomous difference equations

UR - http://hdl.handle.net/10807/85361

U2 - 10.1080/10236198.2015.1080252

DO - 10.1080/10236198.2015.1080252

M3 - Article

SN - 1023-6198

VL - 21

SP - 1212

EP - 1233

JO - Journal of Difference Equations and Applications

JF - Journal of Difference Equations and Applications

ER -