Min and Max for each image of NDVI to calculate Em and LST

If I understand your process okay, the below script will work for the "calculating Emissivity " section. It is calculating and adding the "FV" band to each image of each Landsat collection based on image specific min and max values.

Code Editor script

// I want to calculate the mean annual land surface temp(LST)  for my case study for (1984-2019). 

// define the geometry 
var Scotty = ee.Geometry.Polygon(
  [[[-121.29890539422013, 61.306061839957344],
    [-121.30568601860978, 61.30371278494993],
    [-121.31246664299942, 61.30029191652167],
    [-121.31538488640763, 61.29781876724274],
    [-121.30208112969376, 61.29579888402434],
    [-121.29178144707657, 61.29847829278191],
    [-121.2836275316713, 61.304042487495025],
    [-121.29006483330704, 61.306886028086545]]]);

Map.centerObject(Scotty, 13);
//cloud mask landsat7,landsat5, and landsat8 based on the pixel_qa band of Landsat SR data.

// function for cloud masking on three types of lan dsat
var cloudmasklandsat7and5and8= function(image){
    var Qlandsat5and7= image.select('pixel_qa');
    var cloudShadowBitMask = (1 << 3);
    var cloudsBitMask = (1 << 5);
    var mask5and7=Qlandsat5and7.clip(Scotty).bitwiseAnd(cloudShadowBitMask).eq(0)
                                   .and(Qlandsat5and7.bitwiseAnd(cloudsBitMask).eq(0));
    return image.updateMask(mask5and7);
};

//collecting Images (1984,2013) by masking the cloud (landsat7and5)
var landsat5= ee.ImageCollection('LANDSAT/LT05/C01/T1_SR')
                  .filterDate('1984-01-01', '2012-12-31')
                  .filterBounds(Scotty)
                  .filter(ee.Filter.lt('CLOUD_COVER', 25))
                  .map(cloudmasklandsat7and5and8);
var landsat7= ee.ImageCollection('LANDSAT/LE07/C01/T1_SR')
                  .filterDate('1984-01-01', '2012-12-31')
                  .filterBounds(Scotty)
                  .filter(ee.Filter.lt('CLOUD_COVER', 25))
                  .map(cloudmasklandsat7and5and8);

//merge landsat5 and landsat7 
var landsat7and5=landsat7.merge(landsat5);

//collecting Images (2013,2019) by masking the cloud (landsat8)
var landsat8 = ee.ImageCollection('LANDSAT/LC08/C01/T1_SR')
                  .filterDate('2013-01-01', '2019-12-31')
                  .filterBounds(Scotty)
                  .filter(ee.Filter.lt('CLOUD_COVER', 25))
                  .map(cloudmasklandsat7and5and8);

var visParams8= {
  bands: ['B4', 'B3', 'B2'],
  min: 0,
  max: 3000,
  gamma: 1.4,
};

var visParams7and5= {
  bands: ['B3', 'B2', 'B1'],
  min: 0,
  max: 3000,
  gamma: 1.4,
};

// Visualization of all the images collected from three types of landsat
Map.addLayer(landsat7, visParams7and5,'Landsat7');
Map.addLayer(landsat8, visParams8,'Landsat8');
Map.addLayer(landsat5, visParams7and5,'Landsat5');
Map.addLayer(landsat7and5, visParams7and5,'Landsat7and5');

// NDVI   calculation for each image seperately
function setNdviMinMax(img) {
  var minMax = img
    .select('NDVI')
    .reduceRegion({
      reducer: ee.Reducer.minMax(),
      scale: 30,
      maxPixels: 1e13
    });
  return img.set({
    'NDVI_min': minMax.get('NDVI_min'),
    'NDVI_max': minMax.get('NDVI_max'),
  });
}

// calculate NDCI for each collected image from landsat8
// Function landsat8 NDVI
var NDVI8=function(image){
  var ndvi = image.addBands(image.normalizedDifference(['B5', 'B4']).rename('NDVI'));
  return setNdviMinMax(ndvi);
};

// Function landsat7and5 NDVI
var NDVI7and5=function(image){
  var ndvi = image.addBands(image.normalizedDifference(['B4', 'B3']).rename('NDVI'));
  return setNdviMinMax(ndvi);
};

// Mapping the NDVI functions on all the collected images seperately 
var landsat8Ndvi = landsat8.map(NDVI8)
  .filter(ee.Filter.notNull(['NDVI_min', 'NDVI_max']));
var landsat5Ndvi = landsat5.map(NDVI7and5)
  .filter(ee.Filter.notNull(['NDVI_min', 'NDVI_max']));
var landsat7Ndvi = landsat7.map(NDVI7and5)
  .filter(ee.Filter.notNull(['NDVI_min', 'NDVI_max']));
var landsat7and5Ndvi = landsat7and5.map(NDVI7and5)
  .filter(ee.Filter.notNull(['NDVI_min', 'NDVI_max']));

//NDVI visualization
var ndviParams = {
  min: -1,
  max: 1.0,
  palette: [
    'FFFFFF', 'CE7E45', 'DF923D', 'F1B555', 'FCD163', '99B718', '74A901',
    '66A000', '529400', '3E8601', '207401', '056201', '004C00', '023B01',
    '012E01', '011D01', '011301'
  ],
};

Map.addLayer(landsat8Ndvi.select('NDVI'), ndviParams, 'landsat8Ndvi');
Map.addLayer(landsat5Ndvi.select('NDVI'), ndviParams, 'landsat5Ndvi');
Map.addLayer(landsat7Ndvi.select('NDVI'), ndviParams, 'landsat7Ndvi');
Map.addLayer(landsat7and5Ndvi.select('NDVI'), ndviParams, 'landsat7and5Ndvi');


// calculating Emissivity 
  //1: estimate min and max of each image in  NDVI layer
function addMinMaxBands(img) {
  var imgMin = ee.Image(img.getNumber('NDVI_min')).toFloat();
  var imgMax = ee.Image(img.getNumber('NDVI_max')).toFloat();
  return img.addBands(
    ee.Image.cat(imgMin, imgMax).rename(['NDVI_min', 'NDVI_max'])
  );
}

landsat8Ndvi = landsat8Ndvi.map(addMinMaxBands);
landsat5Ndvi = landsat5Ndvi.map(addMinMaxBands);
landsat7Ndvi = landsat7Ndvi.map(addMinMaxBands);
landsat7and5Ndvi = landsat7and5Ndvi.map(addMinMaxBands);


  //2; fractional vegetation for each image 
function addFVband(img) {
  var ndvi = img.select('NDVI');
  var ndviMin = img.select('NDVI_min').toDouble();
  var ndviMax = img.select('NDVI_max').toDouble();

  var fvBand = ndvi
    .subtract(ndviMin)
    .divide(ndviMax.subtract(ndviMin))
    .rename('FV');

  return img.addBands(fvBand);
}

landsat8Ndvi = landsat8Ndvi.map(addFVband);
landsat5Ndvi = landsat5Ndvi.map(addFVband);
landsat7Ndvi = landsat7Ndvi.map(addFVband);
landsat7and5Ndvi = landsat7and5Ndvi.map(addFVband);

print(landsat8Ndvi);
Map.addLayer(landsat8Ndvi.select('FV'), null, 'FV');