Despite a record-breaking Christmas week, Sainsbury's third-quarter like-for-like sales declined 1.7%, though this beat City expectations of a 3.2% fall, and the grocer announced a new investment in lowering 700 prices.For the 14 weeks to 3 January, the FTSE 100 supermarket group, the first of the Big Four to report after the festive period, said total retail sales fell 0.4% excluding petrol, or 2.5% including fuel, while like-for-like sales were down 1.7% without and 3.9% with fuel.With food price deflation likely to continue, chief executive Mike Coupe said the outlook for the rest of the financial year was "set to remain challenging" and he warned that fourth quarter sales were expected to be slip further back to the level seen in the first half, which saw like-for-likes fall 2.1%.He explained: "Our performance in the third quarter showed an improving trend quarter-on-quarter. However, given the uncertainty in the trading environment, food price deflation and the price reductions we announced this week, we currently expect our fourth quarter like-for-like to be similar to that of our first half."Speaking to CNBC on the morning of the results, chief financial officer John Rogers said management "do not foresee today the need to raise equity for our business, you can never say never, but we don't foresee that need."To try and swim against the tide, Coupe revealed that this week Sainsbury's was lowering prices on over 700 items - although this pales in comparison with rival Asda's £300m investment announced the day before and a predicted major campaign to be announced by Tesco on Thursday.As the British public moves increasingly from weekly shops at larger megastores to more frequent and local shopping, Sainsbury's convenience stores enjoyed growth of over 16% in the quarter. In the week before Christmas it took more than 6m convenience customer transactions and saw its largest ever day for convenience sales on 24th December.The online business was said to have had its biggest Christmas, delivering more than 110,000 orders three days to 23 December, but no total sales figure was given for the period, which analysts suggested might mask less-than-impressive results. Independent analyst Nick Bubb noted that Sainsbury underperformed Waitrose's 2.8% LFL growth in the same five weeks."Sainsbury have warned that Q4 LFL sales will be worse so the stockmarket is unlikely to be cheered that much this morning, particularly given the shadow of a big new price campaign announcement from Tesco tomorrow morning," he said.Clive Black at broker Shore Capital issued a 'sell' recommendation on the shares and remained concerned about falling earnings and dividends, as well as the company's position in the squeezed-middle of the supermarket price war."We believe that Morrison's and Tesco UK may be in a superior competitive position for the prevailing trading environment, one where it looks like Sainsbury could be vulnerable player in the middle, undercut by more price and cost aggressive retailers beneath and losing share to the growing premium retailers in the form of Marks & Spencer and Waitrose above."After Coupe "reset" longer-term expectations in November, Black said he expects the next four years to see Sainsbury's annual earnings per share decline by 25% compared to the 2014 level, with dividends falling on an ongoing basis too.