  The fractional versions of graph theoretic-invariants multiply the range of
applications in scheduling, assignment and operational research problems. In
this paper, we introduce the fractional version of local metric dimension of
graphs. The local resolving neighborhood $L(xy)$ of an edge $xy$ of a graph $G$
is the set of those vertices in $G$ which resolve the vertices $x$ and $y$. A
function $f:V(G)\rightarrow[0, 1]$ is a local resolving function of $G$ if
$f(L(xy))\geq1$ for all edges $xy$ in $G$. The minimum value of $f(V(G))$ among
all local resolving functions $f$ of $G$ is the fractional local metric
dimension of $G$. We study the properties and bounds of fractional local metric
dimension of graphs and give some characterization results. We determine the
fractional local metric dimension of strong and cartesian product of graphs.
