Defendants Toyota Motor Corporation, Toyota Motor North America, Inc., and Toyota Motor Sales, U.S.A., Inc. (collectively, “Toyota”) appeal a judgment by the United States District Court for the Eastern District of Texas that Toyota infringed claims 11 and 39 of U.S. Patent No. 5,343,970 (“the ’970 patent”) owned by Paice LLC (“Paice”) under the doctrine of equivalents. Paice LLC v. Toyota Motors Corp., No. 2:04-CV-211 (E.D. Tex. Aug. 16, 2006). Paice cross-appeals the district court’s judgment that Toyota did not literally infringe claims 11 and 39 of the ’970 patent, claim 15 of the U.S. Patent No. 6,209,672 (“the ’672 patent”), and claims 1 and 2 of U.S. Patent No. 6,554,088 (“the ’088 patent”).1 Paice also appeals the district court’s imposition of an ongoing royalty arrangement that allows Toyota to continue practicing the invention of the ’970 patent in exchange for a set royalty payment. For the reasons explained below, we affirm-in-part, vacate-in-part, and remand for further proceedings.