The Equilibria Problem Of Relative Flows In Transportation Networks

After Pigou first studied traffic network equilibrium problem in 1912, many peopleconsidered this problem and obtained perfect results. On the base of general trafficnetwork theory, this paper deals with a special network where are two relative -ows inthe network. We can see this situation in our everyday life and it can not be ignored.This paper mainly studies the following three contents:(1)After stating generalWardrop's network equilibrium principle, we describe a lemma which is the base ofthe definition of the equilibrium. By using the above lemma, the traffic network equi-librium problem is equivalent to a variational inequality problem. After discussing thesituation that there are two relative equilibrium -ows in a same network, immediately,we state the definition of relative equilibrium -ows. (2)The equivalent condition isalso posed through taking advantage of the relations between a equilibrium problem anda variation inequality as well as a projection operator. That is ,the relative equilibrium-ows are fixed points of a set of projection operators. So the problem of studying trafficnetwork relative equilibrium -ows are transformed into a new problem, that is to findfixed points. (3)On the base of equivalent condition, we endow a new norm to theEuclidean product space, so that it is also a complete space. We study the fixed point ofthe function that was suitably made in the product space. To get the aim, we give thedefinitions ofθ- strongly monotone and L-Lipschitz continuous. Existence and unique-ness of the solution to the equilibrium problem are guaranteed when the cost functionisθ- strongly monotone and L-Lipschitz continuous on its variable. Specially, whenθand L are constants, a more general result can be achieved.