25.5. SUM OF PSEUDOMONOTONE OPERATORS 859

then you could apply the above argument and obtain a further subsequence for which theliminf condition would hold for the sum. Thus, we must have for this new subsequence,

lim supn→∞

⟨wn,un−u⟩ ≥ 0.

Then, using this subsequence,

0≥ lim supn→∞

⟨zn +wn,un−u⟩ ≥ δ + lim supn→∞

⟨wn,un−u⟩ ≥ δ

which is a contradiction. Thus the liminf condition must hold for some subsequence. Incase both are bounded and pseudomonotone, things are easier. You don’t have to take asubsequence.

It is not entirely clear whether the sum of modified bounded pseudomonotone operatorsis modified bounded pseudomonotone. This is because when you go to a subsequence,the limsup gets smaller and so it is not entirely clear whether the subsequence for A willcontinue to yield the limit condition if a further subsequence is taken.

In fact, you can add a bounded pseudomonotone to a generalized bounded pseudomono-tone and get a generalized bounded pseudomonotone. The proof is just like the above andis given next.

Theorem 25.5.3 Suppose A,B : X →P (X ′). If A is bounded pseudomonotone and B isgeneralized bounded pseudomonotone, then A+B is generalized bounded pseudomono-tone.

Proof: It is clear that Ax+Bx is closed and convex because this is true of both of thesets in the sum. It is also bounded because both terms in the sum are bounded. It onlyremains to verify the limit condition. Suppose then that

un→ u weakly

Will the limit condition hold for A+B when applied to this further subsequence? Supposezn ∈ Axn,wn ∈ Bxn and

lim supn→∞

⟨zn +wn,un−u⟩ ≤ 0 (25.5.27)

If v is given, is there a subsequence such that the liminf condition holds? From the above,

lim supn→∞

⟨zn +wn,un−u⟩ ≤ lim supn→∞

⟨zn,un−u⟩+ lim supn→∞

⟨wn,un−u⟩ (25.5.28)

and so, if the second term ≤ 0, since B is modified bounded pseudomonotone, there is asubsequence, still denoted with n for which

lim infn→∞⟨wn,un− v⟩ ≥ ⟨w(v) ,u− v⟩ , w(v) ∈ B(u) (*)

lim infn→∞⟨wn,un−u⟩ ≥ ⟨w(u) ,u−u⟩= 0

You just get a subsequence which works for v and note that the limsup condition is onlystrengthened for the subsequence and then obtain a further subsequence which goes with