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app/code/local/Companyname/Deliverydate/Block/Adminhtml/Mappingdate.php

<?php

class Companyname_Deliverydate_Block_Adminhtml_Mappingdate 
 extends Mage_Adminhtml_Block_System_Config_Form_Field_Array_Abstract


 public function __construct()
 
 $helper = Mage::helper('deliverydate');
 $dateFormat = Mage::app()->getLocale()- 
 >getDateFormat(Mage_Core_Model_Locale::FORMAT_TYPE_SHORT);
 $this->addColumn( 'date', array(
 'name' => 'date',
 'label' => $helper->__('Date'),
 'image' => $this->getSkinUrl('images/grid-cal.gif'),
 'input_format' => Varien_Date::DATE_INTERNAL_FORMAT,
 'value' => date('d/m/Y'),
 'format' => $dateFormat
 )); 

 parent::__construct();
 

system.xml

<?xml version="1.0"?>
<config>
 <tabs>
 <bootsgrid translate="label" module="deliverydate">
 <label>Bootsgrid</label>
 <sort_order>400</sort_order>
 </bootsgrid>
 </tabs>
 <sections>
 <deliverydate translate="label" module="deliverydate">
 <class>separator-top</class>
 <label>Delivery Date</label>
 <tab>bootsgrid</tab>
 <sort_order>130</sort_order>
 <show_in_default>1</show_in_default>
 <show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
 <groups>
 <deliverydate_general translate="label">
 <label>General</label>
 <frontend_type>text</frontend_type>
 <sort_order>10</sort_order>
 <show_in_default>1</show_in_default>
 <show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
 <fields>
 <enabled translate="label">
 <label>Enabled</label>
 <frontend_type>select</frontend_type>
 <source_model>adminhtml/system_config_source_yesno</source_model>
 <sort_order>1</sort_order>
 <show_in_default>1</show_in_default>
 <show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
 <comment>Select Yes to enable this feature.</comment>
 </enabled>
 <!-- <on_which_page translate="label">
 <label>Show on?</label>
 <frontend_type>select</frontend_type>
 <source_model>Bootsgrid_Deliverydate_Model_Config_Option</source_model>
 <sort_order>2</sort_order>
 <show_in_default>1</show_in_default>
 <show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
 <comment>Choose at which step delivery date should be captured.</comment>
 </on_which_page> -->
<!-- <deliverydate_html translate="label">
 <label>Design HTML</label>
 <frontend_type>textarea</frontend_type>
 <sort_order>3</sort_order>
 <show_in_default>1</show_in_default>
 <show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
 <comment></comment>
 </deliverydate_html>
 --> <deliverydate_format>
 <label>Date Format</label>
 <frontend_type>select</frontend_type>
 <source_model>Bootsgrid_Deliverydate_Model_Config_Dateformat</source_model>
 <sort_order>4</sort_order>
 <show_in_default>1</show_in_default>
 <show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
 <comment>Choose date format in which date will be displayed in frontend and admin panel.</comment>
 </deliverydate_format>
 <!-- <deliverytime_format>
 <label>Time Format</label>
 <frontend_type>select</frontend_type>
 <source_model>Bootsgrid_Deliverydate_Model_Config_Timeformat</source_model>
 <sort_order>4</sort_order>
 <show_in_default>1</show_in_default>
 <show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
 <comment>Choose time format in which time will be displayed in frontend and admin panel.</comment>
 </deliverytime_format> -->
 <deliverydate_dayoff>
 <label>Day Off</label>
 <frontend_type>multiselect</frontend_type>
 <source_model>Bootsgrid_Deliverydate_Model_Config_Dayoff</source_model>
 <sort_order>5</sort_order>
 <show_in_default>1</show_in_default>
 <show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
 <comment>Choose days on which shipping will not be available.</comment>
 </deliverydate_dayoff>
 <mapping_date>
 <label>Date Off</label>
 <frontend_model>deliverydate/adminhtml_mappingdate</frontend_model>
 <backend_model>adminhtml/system_config_backend_serialized_array</backend_model>
 <sort_order>10</sort_order>
 <show_in_default>1</show_in_default> <show_in_website>1</show_in_website>
 <show_in_store>0</show_in_store>
 </mapping_date>
 </fields>
 </deliverydate_general>
 </groups>
 </deliverydate>
 </sections>
</config>

Refered this

To add date & time picker, first of all you need to create date.php file at following location in your custom extension

[Vendor][ModuleName] BlockAdminhtmlSystemConfig

<?php
class [Vendor][ModuleName] BlockAdminhtmlSystemConfig date extends Mage_Adminhtml_Block_System_Config_Form_Field

 protected function _getElementHtml(Varien_Data_Form_Element_Abstract $element)
 
 $date = new Varien_Data_Form_Element_Date;
 $format = Mage::app()->getLocale()->getDateFormat(Mage_Core_Model_Locale::FORMAT_TYPE_MEDIUM);

 $data = array(
 'name' => $element->getName(),
 'html_id' => $element->getId(),
 'image' => $this->getSkinUrl('images/grid-cal.gif'),
 );
 $date->setData($data);
 $date->setValue($element->getValue(), $format);
 $date->setFormat(Mage::app()->getLocale()->getDateFormat(Mage_Core_Model_Locale::FORMAT_TYPE_MEDIUM));
 $date->setClass($element->getFieldConfig()->validate->asArray());
 $date->setForm($element->getForm());

 return $date->getElementHtml();
 

?>

Once you have created date.php file, you have to add below code in System.xml file.

<startdate translate="label">
<label>Offer Start Date</label>
<frontend_model>notificationbar/adminhtml_system_config_date</frontend_model>
<backend_model>adminhtml/system_config_backend_serialized_array</backend_model>
<sort_order>150</sort_order>
<show_in_default>1</show_in_default>
<show_in_website>1</show_in_website>
<show_in_store>1</show_in_store>
</startdate>
<starttime translate="label comment">
<label>Offer Start Time</label>
 <frontend_type>time</frontend_type>
<sort_order>152</sort_order>
<show_in_default>1</show_in_default>
<show_in_website>1</show_in_website>
 <show_in_store>1</show_in_store>
</starttime>