The RMT said the strike was “rock solid”.
Southern CEO Charles Horton has offered to meet the RMT on Wednesday for talks.
The RMT has held a series of walkouts since April in the dispute over the role of conductors on Southern services.
In a letter to RMT general secretary Mick Cash, Mr Horton has now said: “I’m prepared to free my diary from tomorrow morning onwards to meet and to show your serious intent, I would like the RMT to call off the rest of the strike action planned for this week.”
Mr Cash, who has not yet responded to Mr Cash, had earlier called for fresh talks with Mr Horton, in a bid to break the deadlock over whether there would be a second member staff, as well as the driver, on every train.
The RMT say having new on-board supervisors on the trains instead of conductors will lower safety standards.
The two sides don’t seem to be coming any closer together.
The row is getting increasingly bitter and it doesn’t look as if there is any quick resolution to this.
Earlier train services between Brighton and Gatwick Airport via Balcombe were disrupted because of over-running engineering work by Network Rail as a result of a broken down engineers’ train.
Russell Woollen said: “@SouthernRailUK why risk doing engineering works before a day like this?? You can’t make it up. Appalling.”
Southern, which runs trains linking parts of Kent, Surrey, Sussex and Hampshire with London, is introducing driver-only operation and wants conductors to accept new on-board supervisor positions.
The RMT revealed it had received a legal challenge to the strike hours before the start of the walkout, but said it would go ahead while it examined the details.
The union previously advised conductors to accept new contracts to “protect their position” – but said it would seek to overturn them.
Strike dates announced by the RMT:
- 00:01 Tuesday 11 October to 23:59 Thursday 13 October
- 00:01 Tuesday 18 October to 23:59 Thursday 20 October
- 00:01 Thursday 3 November to 23:59 Saturday 5 November
- 00:01 Tuesday 22 November to 23:59 Wednesday 23 November
- 00:01 Tuesday 6 December to 23:59 Thursday 8 December
Mayor of Seaford Lindsay Freeman said the south coast town’s summer tourist trade had dropped and some commuters had given up jobs in London because of the strikes.
Ms Freeman said she was very concerned about the impact of a strike planned for 5 November, when up to 40,000 people travel to bonfire celebrations in the East Sussex town of Lewes.