Working example: http://jsfiddle.net/Gajotres/6nG2n/
You are combining several different widgets, to enhance their markup you should run trigger('create').
Read more about it in my other answer: jQuery Mobile: Markup Enhancement of dynamically added content
$('[data-role="content"]').append('<div data-role="collapsible-set" id="collapsible" data-theme="a" data-content-theme="c"><div data-role="collapsible" data-collapsed="false"><h3 id="title_collapsible">Image</h3><center><div id=image_cesta><img src=data:imajpeg;base64,Image alt="image"></div><div data-role="fieldcontain"><fieldset data-role="controlgroup"><input type="range" class="slider" name="slider" value="0" min="0" max="100" data-highlight="false"></fieldset></div></center><div class="ui-grid-a"><div class="ui-block-a"><a id=SaveChangesCesta data-role="button" type="submit" >Guardar cambios</a></div><div class="ui-block-b"><a id="DeleteCesta" type="submit" >Eliminar de cesta</a></div></div></div></div><br><br>');
$('#index').trigger('create');