With homotopy methods, this paper discusses the existence and the number of zeroes of nonlinear mappings on bounded regions and extends some classical theorems.
This paper discusses the number of zeroes of the com- plex function F:CC defined by F(Z)(a_ke^(KZ)+be^(-kz)+ao +a_1Re(Z)+…+a_m(Re(Z))~m, where Re(Z) is the real part of Z,|a_n|+|b_n|0.Let and .We prove that if n_1n_20 and O is a regular value of F,then F has at least n_1+n_2 zeroes in domain Rx(0, 2x],and n_1+n_2 of them can be located with homotopy method simultaneously. Furthermore,if a_1=…a_m=0 and n_1n_20,then F has exactly n_1+n_2 zeroes in domain R×(0,2x].
