A kinetic theory in the form of an integral equation is provided to study the electrostatic oscillations in a collisionless plasma immersed in a uniform magnetic field and a nonuniform transverse electric field. In the low temperature limit (ky&rgr;i≪1, wherekyis the wave vector in theydirection and &rgr;iis the ion gyroradius) the dispersion differential equation is recovered for the transverse Kelvin–Helmholtz modes for arbitrary values ofk∥, wherek∥is the component of the wave vector in the direction of the external magnetic field assumed in thezdirection. For higher temperatures (ky&rgr;i>1) the ion‐cyclotron‐like modes described earlier in the literature by Ganguli, Lee, and Palmadesso [Phys. Fluids28, 761 (1985)] are recovered. In this article the integral equation is reduced to a second‐order differential equation and a study is made of the kinetic Kelvin–Helmholtz and the ion‐cyclotron‐like modes that constitute the two branches of oscillation in a magnetized plasma including a transverse inhomogeneous dc electric field.