It shouldn't be too hard to show, by direct substitution,  that f(s+|t|) and f(s-|t|) either have opposite signs or at least one of them is   zero. Then, by the intermediate value theorem there is at least one solution.  

Clearly  f'(x)=1-t cos(x)  and cos(x) is between 0 and 1. Since you assume that  0

Dear manuele santoprete

But maybe you wanted to know how to solve the equation.  There are three approaches:

1) use Newton's method to approximate the root.

2) Use a series expansion. I believe there are several different series. See, for example 

http://people.oregonstate.edu/~peterseb/mth480/docs/kepler-prob.pdf

where the use Bessel function.

3)  There are some definite integral expressions that solve the Kepler equation. See for example

 Ioakimids, N. I. and Papadakis, K. E. "A New Simple Method for the Analytical Solution of Kepler's Equation." Celest. Mech. 35, 305-316, 1985.

Ioakimids, N. I. and Papadakis, K. E. "A New Class of Quite Elementary Closed-Form Integrals Formulae for Roots of Nonlinear Systems." Appl. Math. Comput. 29, 185-196, 1989.

Other references:

 Wintner, The Analytical foundations of Celestial Mechanics (page 212).

Danby's book "Introdution to Celestial Mechanics ", discusses a series expansion due to Lagrange.

Maybe this link and the references in there can also be useful: http://mathworld.wolfram.com/KeplersEquation.html

Dear Manuele Santoprete

thanks a lot for your helpful answer

i solved my problem easily

Good luck

Bart Leplae found optically, that planets probably are driven by vortex.

Solar cycle induced by rotating medium.

Be next Kepler!