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This app is white with a multicolored musical note on it. If you're prompted to click Download iTunes, do so before continuing. You'll need to restart your computer during this process. Connect your iPhone's charging cable to the iPhone and the computer's USB port. Some Mac computers don't have USB ports, though you can buy a USB adapter for these computers. It's an iPhone-shaped icon in the top-left side of the iTunes window. The "Summary" page will open after clicking it. This button is on the right side of the "Backups" section near the top of the page.  If prompted, turn off "Find My iPhone" before you proceed. You may want to create a new backup first so that you can later return to your phone's present state. To create a new backup, click Back Up Now. You'll see a drop-down menu appear. This must be a date at which your deleted text messages were still on the iPhone. It's on the right side of the "Backups" area of the page. Doing so will prompt your iPhone to begin backing up.  If your selected backup is password-protected, first enter the password. You may need to update your iPhone's software before iTunes will restore your iPhone. Once it does, you should be able to view your deleted text messages in the Messages app, which is the green app with a white speech bubble icon.
Open iTunes on a computer. Attach your iPhone to the computer. Click the "Device" icon. Click Restore Backup. Click the box next to "iPhone Name" when prompted. Click a backup date. Click Restore. Wait for your restore to finish.