Set at the present day Earth, the daylight on average is ten times as bright as today, approximately 111,000 lux at noon for example and subject to geographic and the air quality etc. I am looking for natural or man-made phenomenon that can produce super bright daylight across the equatorial region within a time frame of a couple of weeks, however the changes do not apply to luminous intensity only luminosity ideally 1,110,000 lux. Night is not affected only during the day.
Using a few narrow wavelengths in the visual spectrum (like LED bulbs) will add light, but not nearly as much heat as full-spectrum (blackbody) radiation like the sun's. Making light ten times as bright as the sun would however still add considerable heat.
According to Sewell Direct, an LED produces ca. 85 lumen/watt, while an old-fashioned , full-spectrum incandescent light bulb produces ca. 14 lumen/watt, or six times the energy per lumen. According to Wikipedia, direct sunlight has a luminous efficacy of about 93 lumens per watt of radiant flux, which is even more than an LED bulb. Hence, even with the best light technology we have today, adding artificial light would add more heat than sunlight. It is difficult to imagine a light source that adds only one-tenth the heat of sunlight or one-twelfth the heat of LED bulbs per lumen.
To make it work, you would need to give the ground a far higher albedo (fraction of sunlight that is reflected back into space) than today. You would still feel the heat of the light far more, but the air and ground would not be any hotter. The albedo of seawater, wet soil or forest is roughly 5-15%, meaning that ca. 90% of the light is retained as heat. To reduce this to one-tenth, the albedo would have to be 91%, which is higher than fresh snow or white clouds. Hence, a lot of the ground and sea would have to be mirror bright.
Addendum: Another source states that recent white-light LED lamps are capable of 150 lumens per watt. That reduces the albedo requirements enough to make a snow-white ground reflect enough light and heat.
Everybody's got dilated pupils, maybe due to some agent in the atmosphere or water supplies. Therefore they perceive the world more brightly, but no extra light is actually being sent to Earth.
About why only by day... The ogre spider (don't google it if you have arachnophobia) grows an extra layer of cells in its retina every night, and reabsorbs it at dawn. Just reverse the effect for brighter days but unnafected nights.