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The existence and stability of stable bright solitons in one-dimensional (1D)media with a spatially periodical modulated Kerr nonlinearity are demonstratedby means of the linear-stability analysis and in direct numerical simulations.The nonlinear potential landscape can balance the fractional-order diffractionand thus stabilizes the solitons, making the model unique and governed by therecently introduced fractional Schr\"{o}dinger equation with a self-focusingcubic nonlinear lattice. Both 1D fundamental and multihump solitons (in formsof dipole and tripole ones) are found, which occupy one or three cells of thenonlinear lattice respectively, depending on the soliton's power (intensity).We find that the profiles of the predicted soliton families are impactedintensely by the L\'{e}vy index $\alpha$ which denotes the level of fractionalLaplacian, so does to their stability. The stabilization of soliton families ispossible if $\alpha$ exceeds a threshold value, below which the balance betweenfractional-order diffraction and the spatially modulated focusing nonlinearitywill be broken.

One-dimensional solitons in fractional Schr\"{o}dinger equation with a spatially modulated nonlinearity: nonlinear lattice