How to optimize jQuery to run on asp.net web form updatepanel?
Some notes in no particular order:

1

//get all elemnt with class .resource-download
$.each($(".resource-download"), function () {
    var fileExtension = getFileExtension(this.href);
    if (fileExtension == "" || fileExtension != "pdf") {
        $(this).addClass("hideLinke");
    }
});

//load script agine 
function init() {
    $.each($(".resource-download"), function () {
        var fileExtension = getFileExtension(this.href);
        if (fileExtension == "" || fileExtension != "pdf") {
            $(this).addClass("hideLinke");
        }
    });
}


better to define the function and call it rather than copy paste the code. 

function init() {
    $.each($(".resource-download"), function () {
        var fileExtension = getFileExtension(this.href);
        // see point 2
        if (fileExtension == "" || fileExtension != "pdf") {
            $(this).addClass("hideLinke");
        }
    });
}

init();


2

if (fileExtension == "" || fileExtension != "pdf") {


Is pointless, "" !== "pdf" so you should just have the second condition.

if (fileExtension !== "pdf") {


3
It's better to be in the habit of using === and !== to avoid subtle bugs caused by type coercion.

4

$.each($(".resource-download"), function ()  {}); is better written as $(".resource-download").each(function() {});