We investigate the periodic Anderson-Holstein model by using the dynamical mean-field theory combined with the exact diagonalization method. For the strong electron-phonon coupling g greater than or similar to g(c), the system shows an anomalous heavy-fermion behaviour which is accompanied by a large lattice fluctuation and an extreme phonon softening. We also calculate an effective potential for the ions and find that a simple harmonic potential for g g, changes into a double-well potential for g greater than or similar to g(c). The effective pairing interaction between the conduction electrons shows a maximum at g approximate to g(c) where the superconducting transition temperature is expected to be maximum. (c) 2005 Elsevier B.V. All rights reserved.