var testObject = {
    oThis : this,
    testVariable : "somestring",
    init : function(){

       console.log(this.testVariable); // outputs testVariable as expected

       this.testObject.submit(function(){

            var anotherThis = this;
            console.log(this.testVariable) // undefined
            console.log(oThis.testVariable) // undefined
            console.log(testObject.testVariable) // outputs testVariable
            console.log(anotherThis.testVariable) // undefined

    }

}

init : function(){

    this.testObject.submit = this.submitForm;

 },
 submitForm : function(){
     // do validation here
     console.log(this.testVariable) // outputs testvariable

     .
     .
     .

     return valid;
 }

init: function() {
   var _this = this;
   this.testObject.submit(function() {
        console.log(_this.testVariable); // outputs testVariable
   });
}

init: function() {
  // whatever
  var instance = this;
  instance.submitForm = function() {
    console.log(instance.testVariable);
    // ...
  };
}

var testInstance = new testObject();

testInstance.oThis;

init: function(){

var self = this; // this allows you to access the parent object from different contexts

this.testObject.submit(function(){

    console.log(self.testVariable);

}