  The main aim of this paper is to solve an inverse source problem for a
general nonlinear hyperbolic equation. Combining the quasi-reversibility method
and a suitable Carleman weight function, we define a map of which fixed point
is the solution to the inverse problem. To find this fixed point, we define a
recursive sequence with an arbitrary initial term by the same manner as in the
classical proof of the contraction principle. Applying a Carleman estimate, we
show that the sequence above converges to the desired solution with the
exponential rate. Therefore, our new method can be considered as an analog of
the contraction principle. We rigorously study the stability of our method with
respect to noise. Numerical examples are presented.
