A family of third-order iterative processes (that includes Chebyshev and Halley's methods) is studied in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates. This study allows us to compare the most famous third-order iterative processes.

Nonlinear equations in Banach spaces Newton's method Chebyshev's method Multipoint iteration Semilocal convergence R-order of convergence Region of accessibility Attraction basin a b s t r a c t From Chebyshev's method, new third-order multipoint iterations are constructed with their efficiency close to that of Newton's method and the same region of accessibility.

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A new smoothing method has been used to obtain the electron energy distribution function (EEDF) in plasmas by evaluating the second derivative of the I-V characteristic of a probe inmersed in the plasma. The smoothing method is based on the use of the instrument function. A comparison with other smoothing techniques has permitted us to show the advantages in using the new smoothing method. The experimental setup used to measure the I-V probe characteristic fast and accurately is also presented. The smoothing method was tested by measuring the EEDF in an argon dc discharge at different conditions of the gas pressure and discharge current. The plasma parameters (electron density and temperature) evaluated from the EEDF were compared with those evaluated by using other classic diagnostic methods obtaining a quite good consistency among them.

From a one-point iterative method of R-order at least three, we construct new two-point iterations to solve nonlinear equations in Banach spaces such that the computational cost is reduced, whereas the R-order of convergence is increased to at least four.Keywords Nonlinear equations in Banach spaces · Semilocal convergence · Recurrence relations · A priori error estimates · R-order of convergence Mathematics Subject Classification (2000) 45G10 · 47H17 · 65J15

A new theoretical model for the potential distribution in the surroundings of a cylindrical conductor placed inside a neutral plasma permitted us to analyse the effect of the positive ion thermal motion on the ion current collected by a cylindrical Langmuir probe. The new theoretical model includes the ABR (Allen - Boyd - Reynolds) theory as a limiting case, which is that of a negligible ion temperature compared to the electron temperature.