Using a selected invertible Lip-a operator M as the scale operator called the spectral scale, we introduce the M-resolvent set, M-spectrum, M-spectral radius, resolvent set, spectrum and spectral radius for a nonlinear Lipschitz-α operator between two Banach spaces.

First we prove that 0 is an eigenvalue of the operator with geometric multiplicity one,next we prove that all points on the imaginary axis except for zero belong to the resolvent set of the operator,last we prove that 0 is an eigenvalue of the adjoint operator of the operator.

First we consider the spectral properties of the operator corresponding to this system and obtain that all points on the imaginary axis except for zero belong to resolvent set of the operator, zero is an eigenvalue of the operator and its adjoint operator with geometric multiplicity one.

First we prove that all points on the imaginary axis except for zero belong to the resolvent set of the operator corresponding to the model, second prove that 0 is an eigenvalue of the operator and its adjoint operator with geometric multiplicity and algebraic multiplicity one,last by using theabove results we obtain that the time-dependent solution of the model str.

In this paper, we analyse the form rule of M set , through adjusting the value of color and made the elaborate structure of the escape area based on the escape time algorithm visible.

To effectively divide the parameter space of the mapping of upper halfplane limit,we construct relevant generalized M sets.

With the purpose of reducing the calculating time and improve the efficiency for constructing M sets,the research applies the simplex method of the unbound optimization theory for the set of initial iterating points, which make the determinant of Jacobin matrix of the mapping equal to zero.

A Method of Rendering M-set Based on the Time-escaped Algorithm;