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## On periodic p-harmonic functions on Cayley tree. (803)

Citations: | 2 - 2 self |

### Citations

101 |
On a system of functional equations.
- Choczewski, Kuczma
- 1985
(Show Context)
Citation Context ...ry of functional equations is not very developed even for case in which the unknown function is defined on R, only the solutions of functional equations of some special forms are known (e.g. see [3], =-=[4]-=-). Thus, it is natural to find periodic (simple) solutions of ∆pu = 0 first. The main goal of this paper is to describe the set of p-harmonic functions which are periodic with respect to subgroups of ... |

77 |
Potential theory on infinite networks
- Soardi
- 1994
(Show Context)
Citation Context ...∆pu for a function u on V (= Gk) by ∇u(x, y) = r(x, y) −1 (u(y) − u(x)), ∆pu(x) = ∑ y∈S(x) |∇u(x, y)| p−2 ∇u(x, y), where 1 < p < ∞. Let D ⊂ V . If ∆pu = 0 in D, then we say that u is p-harmonic in D =-=[9]-=--[11]. Let {u1, ..., un} be an m-tuple of p-harmonic functions in D. If p = 2, then the linear combination of p-harmonic functions u1, ..., um need not be p-harmonic. The m-tuple of p-harmonic functi... |

6 |
Description of periodic Gibbs measures of the Ising model on the Cayley tree
- Rozikov
(Show Context)
Citation Context ...k does not have normal subgroups of odd index (= 1). It has a normal subgroup of arbitrary even index. 2Also there are normal subgroups of infinite index. Some of them can be described as following =-=[7]-=-. Fix M ⊆ Nk such that |M| > 1. |•| is the cordinality of •. Let the mapping πM : {a1, ..., ak+1} −→ {ai, i ∈ M} ∪ {e} be defined by ⎧ ⎨ ai, if i ∈ M πM(ai) = ⎩ e, if i /∈ M. Denote by GM the free pro... |

3 |
Random walks in random media on a Cayley tree
- Rozikov
- 2001
(Show Context)
Citation Context ...t edges. The Cayley tree can be represent as the group Gk which is the free product of k +1 second order cyclic groups [1],[2],[6]. The group representation of the Cayley tree was used in [1],[2],[5]-=-=[8]-=- to study models of statistical mechanics and describe the sets of periodic Gibbs measures, also to study random walk trajectories in a random medium on the Cayley tree. These problems are related to ... |

3 |
Nonlinear Poisson equations on an infinite network, Mem
- Yamasaki
(Show Context)
Citation Context ...for a function u on V (= Gk) by ∇u(x, y) = r(x, y) −1 (u(y) − u(x)), ∆pu(x) = ∑ y∈S(x) |∇u(x, y)| p−2 ∇u(x, y), where 1 < p < ∞. Let D ⊂ V . If ∆pu = 0 in D, then we say that u is p-harmonic in D [9]-=-=[11]-=-. Let {u1, ..., un} be an m-tuple of p-harmonic functions in D. If p = 2, then the linear combination of p-harmonic functions u1, ..., um need not be p-harmonic. The m-tuple of p-harmonic functions i... |

2 |
Description of non periodic extreme Gibbs measures of some models on Cayley tree
- Ganikhodjaev, Rozikov
- 1997
(Show Context)
Citation Context ...y tree, i.e., an infinite tree in which each vertex has exactly k + 1 incident edges. The Cayley tree can be represent as the group Gk which is the free product of k +1 second order cyclic groups [1],=-=[2]-=-,[6]. The group representation of the Cayley tree was used in [1],[2],[5]-[8] to study models of statistical mechanics and describe the sets of periodic Gibbs measures, also to study random walk traje... |

1 |
The group representation and authomorphisms of the Cayley tree
- Ganikhadjaev
- 1994
(Show Context)
Citation Context ...ayley tree, i.e., an infinite tree in which each vertex has exactly k + 1 incident edges. The Cayley tree can be represent as the group Gk which is the free product of k +1 second order cyclic groups =-=[1]-=-,[2],[6]. The group representation of the Cayley tree was used in [1],[2],[5]-[8] to study models of statistical mechanics and describe the sets of periodic Gibbs measures, also to study random walk t... |

1 |
Functional equations and their applications
- Kuczma
- 1966
(Show Context)
Citation Context ... theory of functional equations is not very developed even for case in which the unknown function is defined on R, only the solutions of functional equations of some special forms are known (e.g. see =-=[3]-=-, [4]). Thus, it is natural to find periodic (simple) solutions of ∆pu = 0 first. The main goal of this paper is to describe the set of p-harmonic functions which are periodic with respect to subgroup... |

1 |
A description of harmonic functions via properties of the group representation of the Cayley tree
- Normatov, Rozikov
- 1997
(Show Context)
Citation Context ...ident edges. The Cayley tree can be represent as the group Gk which is the free product of k +1 second order cyclic groups [1],[2],[6]. The group representation of the Cayley tree was used in [1],[2],=-=[5]-=--[8] to study models of statistical mechanics and describe the sets of periodic Gibbs measures, also to study random walk trajectories in a random medium on the Cayley tree. These problems are related... |

1 |
Parabolic and hyperbolic infinite networks,Hiroshima
- Yamasaki
- 1977
(Show Context)
Citation Context ... un, n ∈ Z. (8) 5From (8) we obtain un = C 1 p−1 n−1 ∑ s=−∞ 2) un+1 < un then −C > 0. In this case we get rs,s+1, n ∈ Z. (9) Thus we have proved the following un = (−C) 1 +∞ ∑ p−1 s=n rs,s+1, n ∈ Z. =-=(10)-=- THEOREM 3. Let R be Hij-periodic, and ∑ ∞ s=−∞ rs,s+1 < +∞. Then there are two family U1, U2 of Hij-periodic p-harmonic functions on the Cayley tree of k ≥ 1 such that U1 = {u : u(x) = un = C 1 p−1 n... |