A vector function defined on, for some, is called a sub (or super) solution of ( 1.1 )–( 1.3 ), if all the following hold: (1) ; (2) for, and for almost all ; (3) (2.1) .
It can be easily verified that k φ 1 ( x ) is a super-solution of (1.1 - 1.3 1.1 - 1.3iciently small k > 0, where φ 1 ( x ) is the unique positive solution ofor2.1).
Choose (L> frac{1}{r_{1}} Mmax vert h_{1}(psi,tilde{psi })vert ), we then have (mathcal{L}(bar{v} geq 0) and (bar{v} x,t)) is a super-solution of (2.6) for (tleq 0).
So z is a super-solution of problem (4.1).
Therefore, u ¯ is a super-solution of problem (1.1).
It can be verified that, for the case m = q < r < 1, a sufficiently large constant L is a super-solution of (1.1).
With the development of set-valued analysis, some researchers studied the super efficient solutions for vector optimization problems with set-valued maps.
Here, we construct some super-solution to the problem, which is bounded for any (T>0).
Similarly, (u_{infty}(x)) is a super-solution to (3.6).
Furthermore, 1 l ≤ u l ≤ u k for k < l, and a super-solution (sub-solution) comparison theory holds for (2.2) (see [19]).
There are some methods to deal with these problems, for example, sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree.
