Search for the Rare Decays Dₛ⁺→ h⁺(h⁰)e⁺e⁻

Using 7.33~fb⁻¹ of e⁺e⁻ collision data collected by the BESIII detector at center-of-mass energies in the range of √s=4.128 - 4.226~GeV, we search for the rare decays Dₛ⁺→ h⁺(h⁰)e⁺e⁻, where h represents a kaon or pion. By requiring the e⁺e⁻ invariant mass to be consistent with a φ(1020), 0.98<M(e⁺e⁻)<1.04 ~GeV/c², the decay Dₛ⁺→π⁺φ,φ→ e⁺e⁻ is observed with a statistical significance of 7.8σ, and evidence for the decay Dₛ⁺→ρ⁺φ,φ→ e⁺e⁻ is found for the first time with a statistical significance of 4.4σ. The decay branching fractions are measured to be B(Dₛ⁺→π⁺φ, φ→ e⁺e⁻ )=(1.17+0.23-0.21±0.03)× 10⁻⁵, and B(Dₛ⁺→ρ⁺φ, φ→ e⁺e⁻ )=(2.44+0.67-0.62± 0.16)× 10⁻⁵, where the first uncertainties are statistical and the second systematic. No significant signal for the three four-body decays of Dₛ⁺→ π⁺π⁰e⁺e⁻,\ Dₛ⁺→ K⁺π⁰e⁺e⁻, and Dₛ⁺→ KS⁰π⁺e⁺e⁻ is observed. For Dₛ⁺→ π⁺π⁰e⁺e⁻, the φ mass region is vetoed to minimize the long-distance effects. The 90$%$ confidence level upper limits set on the branching fractions of these decays are in the range of (7.0-8.1)× 10⁻⁵.