How to automatically scroll a PageView with some delay that is made without using a builder?

You need to add a PageController to your PageView. Then on initState() you can start a Timer.periodic() where you just jump from page to page. Like this:

Note: You have to cancel the timer when dispose the page or other events.

int _currentPage = 0;

'''
Timer? _timer;
'''

Timer _timer;
PageController _pageController = PageController(
  initialPage: 0,
);

@override
void initState() {
  super.initState();
  _timer = Timer.periodic(Duration(seconds: 5), (Timer timer) {
    if (_currentPage < 2) {
      _currentPage++;
    } else {
      _currentPage = 0;
    }

    _pageController.animateToPage(
      _currentPage,
      duration: Duration(milliseconds: 350),
      curve: Curves.easeIn,
    );
  });
}

  @override
  void dispose() {
    super.dispose();
    _timer?.cancel();
  }

@override
Widget build(BuildContext context) {
  return PageView(
    controller: _pageController,
    children: [
      Container(
        decoration: BoxDecoration(
          image: DecorationImage(
            image: new AssetImage(images[0]), 
            fit: BoxFit.cover,
          ),
        ),
      ),
      Container(
        decoration: BoxDecoration(
          image: DecorationImage(
            image: new AssetImage(images[1]), 
            fit: BoxFit.cover,
          ),
        ),
      ),
      Container(
        decoration: BoxDecoration(
          image: DecorationImage(
            image: new AssetImage(images[2]), 
            fit: BoxFit.cover,
          ),
        ),
      ),
    ],
  );
}

By the way you need to import 'dart:async' for using Timer.

The most effective way for me was to implement animation controllers. AnimationController and Animation. Inside a StatefulWidget with a page controller in the page view.

import 'package:flutter/material.dart';

class ChangePageViewAuto extends StatefulWidget {
  @override
  _ChangePageViewAutoState createState() => _ChangePageViewAutoState();
}

class _ChangePageViewAutoState extends State<ChangePageViewAuto>
    with SingleTickerProviderStateMixin {

  //declare the variables
  AnimationController _animationController;
  Animation<double> _nextPage;
  int _currentPage = 0;
  PageController _pageController = PageController(initialPage: 0);


  @override
  void initState() {
    super.initState();
    //Start at the controller and set the time to switch pages
    _animationController =
        new AnimationController(vsync: this, duration: Duration(seconds: 10));
    _nextPage = Tween(begin: 0.0, end: 1.0).animate(_animationController);

    //Add listener to AnimationController for know when end the count and change to the next page
    _animationController.addListener(() {
      if (_animationController.status == AnimationStatus.completed) {
        _animationController.reset(); //Reset the controller
        final int page = 4; //Number of pages in your PageView
        if (_currentPage < page) {
          _currentPage++;
          _pageController.animateToPage(_currentPage,
              duration: Duration(milliseconds: 300), curve: Curves.easeInSine);
        } else {
          _currentPage = 0;
        }
      }
    });
  }

  @override
  void dispose() {
    _animationController.dispose();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    _animationController.forward(); //Start controller with widget
          print(_nextPage.value);
    return PageView.builder(
        itemCount: 4,
        scrollDirection: Axis.horizontal,
        controller: _pageController,
        onPageChanged: (value) {
          //When page change, start the controller 
          _animationController.forward();
        },
        itemBuilder: (BuildContext context, int index) {
          
          return Container();
        });
  }
}

the @GaboBrandX's answer is the correct one, so I added some code to reverse the animation, for example, it will animate from 0,1,2 pages and when reaching the end, it will reverse to 2,1,0.

 final PageController _pageController = PageController(initialPage: 0);
 int _currentPage = 0;

bool end = false;
@override
void initState() {
super.initState();
Timer.periodic(Duration(seconds: 2), (Timer timer) {
  if (_currentPage == 2) {
    end = true;
  } else if (_currentPage == 0) {
    end = false;
  }

  if (end == false) {
    _currentPage++;
  } else {
    _currentPage--;
  }

  _pageController.animateToPage(
    _currentPage,
    duration: Duration(milliseconds: 1000),
    curve: Curves.easeIn,
  );
});}