Cross sections for the reactions e^{+}e^{-}→K^{+}K^{-}π^{+}π^{-}, K^{+}K^{-}π^{0}π^{0}, and K^{+}K^{-}K^{+}K^{-} measured using initial-state radiation events

We study the processes e+e-\u2192K+K-\u3c0+\u3c0-\u3b3, K+K-\u3c00\u3c00\u3b3, and K+K-K+K-\u3b3, where the photon is radiated from the initial state. About 84\u2009000, 8000, and 4200 fully reconstructed events, respectively, are selected from 454\u2009\u2009fb-1 of BABAR data. The invariant mass of the hadronic final state defines the e+e- center-of-mass energy, so that the K+K-\u3c0+\u3c0-\u3b3 data can be compared with direct measurements of the e+e-\u2192K+K-\u3c0+\u3c0- reaction. No direct measurements exist for the e+e-\u2192K+K-\u3c00\u3c00 or e+e-\u2192K+K-K+K- reactions, and we present an update of our previous result based on a data sample that is twice as large. Studying the structure of these events, we find contributions from a number of intermediate states and extract their cross sections. In particular, we perform a more detailed study of the e+e-\u2192\u3d5(1020)\u3c0\u3c0\u3b3 reaction and confirm the presence of the Y(2175) resonance in the \u3d5(1020)f0(980) and K+K-f0(980) modes. In the charmonium region, we observe the J/\u3c8 in all three final states and in several intermediate states, as well as the \u3c8(2S) in some modes, and measure the corresponding products of branching fraction and electron width