  We study the frustrated and dimerized ferromagnetic-antiferromagnetic
$J_1$-$J_1'$-$J_2$ chain using the density-matrix renormalization group (DMRG)
method. Based on numerical calculations of the second derivative of energy,
spin gap, spin-spin correlations, string order parameter (SOP), and
entanglement spectrum (ES), we obtain the ground-state phase diagram for a wide
range of $J_1'/J_1$ and $J_2/|J_1|$ values. This phase diagram reveals a
ferromagnetic phase and two distinct valence-bond-solid (VBS) phases. The first
VBS phase, referred to as $\mathcal{D}_3$-VBS, is typified by the formation of
valence bonds between third-neighbor spin-1/2's, persisting as a continuation
from the $J_1'/J_1=1$ limit. Alternatively, the second VBS phase, referred to
as mixed-VBS, exhibits a coexistence of both second- and third-neighbor valence
bonds, interpreted as a continuation from the $J_1'/J_1=0$ case. Remarkably,
both VBS states are identified as being of Haldane-type, marked by a finite SOP
and 2-fold ES degeneracy. Unexpectedly, our analysis uncovers a significant
enhancement of the valence bond stability at the boundary of the two VBS
phases. This study provides the first empirical demonstration of a nontrivial
quantum phase transition between different topological VBS states in spin-1/2
chains. Moreover, we find that the ground state of the relevant
quasi-one-dimensional material LiCuSbO$_4$ is classified as the
$\mathcal{D}_3$-VBS state. Collectively, these results mark a substantial
stride forward in our comprehension of quantum phase transitions and
topological states.
