In the seventh episode of Durarara!!, we see the world through the eyes of Shizuo Heiwajima, a man who you simply do not want to mess with. Known for his bartender uniform and ability to throw traffic signs at people, he is feared and hated by many in Ikebukuro, despite an overwhelming disdain for his own violent tendencies.
Often, he simply falls victim to circumstances which either release a hidden fury inside him, or force him to pull off the kid gloves and kick ass, so to speak. Other times, however, the root cause of his violence is attributed to longtime rival Izaya Orihara, resident information broker and complete sociopath. Their dynamic is a rather strange one, even for a series like this.
In their first encounter, Izaya established himself as the only person Shizuo would not be able to defeat. The following night, Shizuo showed up on the door of Shinra and Celty's apartment, pulled said door off its hinges, and threatened to beat Izaya to death with it. Notwithstanding Celty's attempts to diffuse the situation, it still seems weird that Shizuo didn't exactly follow through on his threat.
But it goes deeper. During the second arc of the anime, Shizuo tends to actively pursue Izaya with the purpose of getting information from him. The fights which ensue between them have an almost joyous air to them, not one of hardened malice. It's almost like the two have some sort of mutual respect for each other...
Oddly enough, the term I would attribute to this phenomena is "mutual self-interest." Put simply, the act of doing good things for others increases the odds of good things happening to the self. Izaya's constant tormenting of Shizuo actually serves to fulfill a need.
Shizuo admits to being misunderstood by a lot of people, and thus being unable to connect with them. Izaya's interactions serve as a connection between the two men, and it's entirely possible that Shizuo is thankful for that connection. A similar perspective comes from Izaya being an individual of intrigue due to his unique ability to stand up to Shizuo in the first place. All in all, it can be said that Shizuo Heiwajima is empowered by Izaya Orihara, which is an odd trait to give to an apparent enemy.
But you don't have to take my word for it. In 1981, Apple Computer publicly delivered a message to IBM following the release of IBM's first computer. In this message, Apple was of course purporting their intent to compete against IBM for supremacy, but there's a deeper point to consider. Apple was also challenging IBM to be an ethical corporation.
The task of spreading the personal computer was done with the intent of expanding computer literacy, and thus human productivity. Apple challenged IBM to do the right things in order to expand the reach of the personal computer, to eliminate any and all barriers that could arise from competition in the marketplace. The advert made competition second to cooperation, and that in itself is quite powerful indeed.
The lesson we can take away from all of this is how important it can be to learn from our rivals and even our enemies. To forget this lesson is to miss out on a deeper level of introspection.