Southwest Airlines Co. and United Airlines Holdings Inc. reported modest improvements in passenger demand in the second quarter as the U.S. and other countries start to emerge from coronavirus lockdowns.

Southwest said it recorded month-to-date net positive bookings through May 18, meaning new passenger bookings outpaced trip cancellations, a reversal of the booking trends in March and April.

With the improving outlook, the company revised its projected year-over-year decline in operating revenues for May to a range of 85% to 90% from a range of 90% to 95% previously. Revenues plunged by 90% to 95% in April.

"The revenue environment remains uncertain and may require additional capacity reductions depending on passenger demand," Southwest said in a regulatory filing. The company expects its capacity to drop by 45% to 55% in June, compared with an estimated decline of 60% to 70% in May.

Southwest estimated its average daily core cash burn in the second quarter at $25 million.

Meanwhile, United Airlines said customer cancellation rates, which hit record highs in April, have declined as of May 18. In addition, passenger demand in the U.S. and certain international destinations for the rest of the second quarter has moderately improved.

United now expects its scheduled capacity to fall by 75% year over year in July, lower than the estimated capacity decline of 90% in May and June.

The company projected less than $4.5 billion and close to $2 billion in adjusted capex for 2020 and 2021, respectively.