public void update()
//Touched left side of screen
if (Gdx.input.getX() > Gdx.graphics.getWidth() / 2)
//give actor upforce and and a bit of force to the left
someActor.setVelocity(new Vector2(-25, 100));
//touched right side of screen
//give actor upforce and a bit to the right
someActor.setVelocity(new Vector2(25, 100));
The problem is this won't work on it's own. You would need a class for your sprite and give it appropriate down force as well. The amounts you use depend on the units you give in your game.
You could add some drag to the sideway movement by multiplying the X velocity by something like .95. For the downforce/gravity you could use need to subtract something each update so it will eventually fall down. You could use realistic numbers here or make something up and tweak it until it's right.
To move your sprite you use the velocity the actor has.
Your best bet is to get familiar with the basic vector math. Start coding and if you run into unsolvable problems come back and ask something more specific.