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To be specific, i have a two class Request and Utils,

class Request
{
public function __construct()
{
//constructor method
}
public function request()
{
  $utils=new Utils;
  $consolidated_errors=$utils->array_remove_empty($all_params_error);
}   
public function process()
{
  $utils=new Utils;
  $consolidated_errors=$utils->another_method($all_params_error);
 }                                                                   
}

And class Utils,

class Utils
{
public function __construct()
{
 //constructor method
}
public function array_remove_empty()
{
  //returns a variable.
  }        
public function another_method()
{
  //returns a variable.
 }     

}

you can see that i am initializing the class two times in request class , and my question is that any way initializing the class globally and using through out the class?

You are looking for Singleton pattern

Following demonstrate very basic Singleton example for your class

public class Utils {
  private static Utils uniqInstance;

  private Utils() {
  }

  public static synchronized Utils getInstance() {
    if (uniqInstance == null) {
      uniqInstance = new Utils();
    }
    return uniqInstance;
  }
  // other useful methods here
} 

get the instance using static-factory pattern

The above code does not look like Java to me, but anyway,

You could create the class at a class level private Utils myUtuils = new Utils ();

or have the class as a static class and then just use it directly in your method

public function process()
{
  consolidated_errors= Utils.another_method($all_params_error);
 }                                                                   
}