Gabriel McCracken, 2020/07/26 21:13, 2020/07/26 21:16
Dear all,
I think I figured out how to deal with solution theory for real Hilbert spaces (see Lemma 2.22, Cor. 4.8 in Overleaf). The Picard-type Theorem still holds for real Hilbert spaces H but we must adjust the condition on the material law:
Instead of the material law itself, we have to work with the restriction of a material law M valued in L(HC), where HC is the complexification of H. Hence we need as an additional condition that M[H]⊆H.
The coercivity condition is imposed on M and not its restriction.
This is nice, because it means that I do not have to modify the examples that naturally live on real Hilbert spaces.
One could also extend a maximal monotone relation A on a real Hilbert to a maximal monotone relation on its complexification via Lemma 2.20 and apply complex solution theory, but that's not as nice.
Best,
Gabriel
Gabriel McCracken, 2020/06/16 14:31, 2020/06/16 14:32
Hi,
I've made the observation that Theorem 4.1 (in the Overleaf notes) holds even if zM(z)−c is monotone only for z with Rez=ν>sb(M), not ≥ν. Consequently, existence of solutions in the main Theorem 5.6 is guaranteed also in this case.
I suppose that Rez≥ν becomes important if you compare solutions to different ν, which we unfortunately didn't have the time to do. It's something I'd like to look into after the workshop (see my proposed Proposition 5.8).
If we're asked a question about Eventual Independence of ν after the presentation, I'd just honestly say that we were looking into that, but didn't have the time to understand it completely and present it. Is that ok with you?
Best,
Gabriel
Gabriel McCracken, 2020/05/23 15:31, 2020/05/23 15:35
Hi,
I don't yet have a conclusive idea how to solve our difficulties with Lemma 5.5 and Prop 5.6 (in our Overleaf project) so I changed my strategy to going through everything before in detail. Everything up to 3.10 should be correct, and the lengthy estimate that x_n is a Cauchy sequence in the proof of 3.11 is also correct, but after that I'm struggling to understand rest of the proof of 3.11.
Also, if I see it correctly, 3.11 corresponds to Prop 2.4. in Sascha's paper [Tr13], where he states “In particular, A+B is maximal monotone”. I think this is wrong, however, but it is right if we additionally assume that the estimate C(y):=supλ>0‖Bλxλ‖<∞ holds for ally∈H, because then 1+A+B is surjective. Or do you think there is another argument to avoid needing C(y)<∞?
Moreover, I think one can replace supλ>0‖Bλxλ‖<∞ by limsupλ→0‖Bλxλ‖<∞, i.e. ‖Bλxλ‖ is eventually bounded.
Best,
Gabriel
Sahiba Arora, 2020/05/23 19:33
Dear Gabriel
I think you have already edited the proof of Proposition 3.11 so I'm not sure what part after proving (xλn) is Cauchy was unclear.
I think in Proposition 2.4 in [Tr13], it is clear that C(y)<∞ for all y∈H. Without that of course we can't appeal to Minty to show A+B is maximal monotone.
Regards,
Sahiba
Gabriel McCracken, 2020/05/23 21:19
I thought of and proved Lemma 2.7, and with its help the remaining part of 3.11 becomes easy. Please have a look at it and check whether my proof really works.
I cite a short version here:
Let A be monotone and (xn,yn) a sequence of points in A such that either
xn→x strongly and yn⇀y weakly, or
yn→y strongly and xn⇀x weakly.
Then A∪{(x,y)} is monotone.
Sahiba Arora, 2020/05/23 22:23
Yeah it seems right to me. Honestly, I don't remember what the previous version of the proof was to recall where it failed.
Gabriel McCracken, 2020/05/19 11:38, 2020/05/19 11:39
Dear all,
as mentioned in the Overleaf project chat, I added Minty's Theorem and its proof in our Overleaf project, giving some more details here and there.
Also, I completed the proof of Cor.3.5 (Cor 2.3 in [Tr12]), because there was some argument missing to show that ¯A is surjective and came across some small insight that might be useful somewhere else: If A−c is monotone, then the estimate
‖x−y‖≤1c‖f−g‖
not only gives us that A is injective, but that if (xn,fn) is a sequence in A such that fn is a Cauchy sequence, then also xn is a Cauchy sequence.
Lastly, I'm still a bit confused about Hendrik's proposed proof or rather how to implement it into the Overleaf project without having too much overlap with section 5. In fact, I think the proof of Prop 5.6 that Sahiba wrote works also for f∈dom(∂t,ν) if you have a look at Lemma 5.5. I think the h<0 should rather be 0<h<1 and then the right hand side of the estimate becomes
suph∈(0,1)‖τhf−fh‖ instead supt≥0‖f′(t)‖
What do you think, could the proof work like this?
Best,
Gabriel
Sahiba Arora, 2020/05/19 13:02
Dear Gabriel
Currently, the proof of Lemma 5.5 that I wrote is incorrect and I'm not sure how to do it for the half-line. Maybe as Hendrik suggested we should stick to the whole line.
Regards
Sahiba
Gabriel McCracken, 2020/05/20 00:19
Hi Sahiba,
I'm a bit confused where you see an error. Is it that applying τ−h to ϕ might not yield the wanted result because it corresponds to a difference quotient for a left-sided derivative, but we're dealing with the positive half-line so we're getting problems near 0? So the proof might work on the whole real line, right?
Best,
Gabriel
Sahiba Arora, 2020/05/20 00:49, 2020/05/20 00:49
Dear Gabriel
The problem in the proof of Lemma 5.5 is that the lim sup in the 2nd displayed formula might not be finite. Basically the proof lacks justification why taking f∈C1c(R≥0) is sufficient.
Best,
Sahiba
Sahiba Arora, 2020/04/23 11:56
Dear all,
I have a doubt in trostoff12.pdf. Is ∂ν in Section 4.2 same as ∂ν,0 in the previous section?
Proposition 4.9 in trostoff12.pdf is the same as Theorem 4.1.2 in the ISem lectures which says that functions in the domain of ∂t,ν are continuous. So I think ∂ν in trostoff12.pdf is just ∂t,ν in the ISem lectures which is same as ∂ν,0 in trostoff12.pdf?
If yes, then why change the notation mid-way?
Regards, Sahiba.
Hendrik Vogt, 2020/04/23 13:30
Dear Sahiba,
ah, that's a good question! Only now I see that there's a major difference between trostorff12.pdf and trostorff13.pdf: In the latter paper, the time derivative is considered for functions defined on all of R (as was the case in Picard's theorem 6.2.1), whereas in the earlier paper the functions are defined on [0,∞). In this case, ∂ν is a proper extension of ∂ν,0! Have a look at page 5854:
The index 0 serves as a reminder for the implicit initial condition u(0+)=0 for elements u∈D(∂ν,0).
I'm not sure yet how to deal with this, but maybe we should concentrate on the case closer to Picard's theorem.
Best wishes, Hendrik
Sahiba Arora, 2020/04/23 16:29
Dear Hendrik
Oh, so that means if we have u∈D(∂ν), then u∈D(∂t,ν)⟺u(0)=0?
Best wishes, Sahiba
Hendrik Vogt, 2020/04/25 19:05, 2020/04/25 19:05
Dear Sahiba,
a late answer: yes, this is Remark 4.11 in trostorff12.pdf.
Best wishes, Hendrik
Sahiba Arora, 2020/04/13 19:27
Dear Hendrik
After reading the basics of Maximal monotone relations and Minty's Theorem, do you suggest that one should first read trostorff12.pdf till Theorem 4.19 and then trostorff13.pdf till Theorem 3.7 (omitting Section 2.3 for now) or is there another way we should be going about this?
Regards
Sahiba
Hendrik Vogt, 2020/04/13 22:19
Dear Sahiba,
I do suggest browsing through those parts of the papers. My plan is to make more suggestions about what details to concentrate on. For now let me only say a few things:
- For the moment you can ignore the minimal sections and Proposition 2.4(c) in trostorff12.pdf.
- Likewise, first ignore Lemma 3.2 in trostorff13.pdf.
- A crucial ingredient is Proposition 3.6 in trostorff13.pdf, which actually even holds for f∈dom(∂t,ν). I suggest to use the following nice characterisation of dom(∂t,ν) for the proof of this more general result: an element f∈L2,ν(R;H) belongs to dom(∂t,ν) if and only if (1h(τhf−f))0<h<1 is bounded. In the proof of this equivalence you can try and use the relative weak compactness of bounded subsets of H and the distributional description of dom(∂t,ν) from Proposition 4.1.1 in the ISem lectures.
Best wishes, Hendrik
Hendrik Vogt, 2020/04/16 16:49, 2020/04/17 17:37
Dear all,
I just realised that I need to make the “nice characterisation” above more precise: in addition to the mentioned equivalence one obtains
∂t,νf=limh→01h(τhf−f)(f∈dom(∂t,ν))
(with convergence in L2,ν(R;H); please check if this claim is correct!).
In case you wonder what I want with this characterisation - it easily implies the following: if R is an autonomous Lipschitz continuous mapping from L2,ν(R;H) to itself, with Lipschitz constant L, then R maps dom(∂t,ν) to itself, and
‖∂t,ν(Rf)‖L2,ν(R;H)≤L‖∂t,νf‖L2,ν(R;H)(f∈dom(∂t,ν)).
Please have a look yourself as to why I find this very useful!
Best wishes, Hendrik
Sahiba Arora, 2020/04/22 22:57
Dear Hendrik,
I have written down the proof of the characterization in the notes in Overleaf (Lemma 5.9).
Best wishes,
Sahiba
Hendrik Vogt, 2020/04/23 13:15
Dear Sahiba,
ah, I see that what I wrote was ambiguous. What I meant:
f∈dom(∂t,ν)⟺(1h(τhf−f))0<h<1 is bounded,
and 1h(τhf−f)→∂t,νf if one of the (equivalent) properties holds. In particular: The convergence follows from the boundedness; this is important!
One more thing: You have to explain in more detail why it suffices to consider f∈C1c.
Best wishes, Hendrik.
Sahiba Arora, 2020/04/23 15:32
Dear Hendrik
Oh, so I misunderstood the claim. I have made the required changes, I hope it is correct now.
Best wishes, Sahiba.
Hendrik Vogt, 2020/04/25 19:10
Dear Sahiba,
in the 2nd displayed formula in the proof, why should the limsup be finite?
Let me emphasise one thing: I only thought about the whole real line here; note that I wrote R in my original proof. I didn't think about what the correct version for the half line is!
Best wishes, Hendrik
Hendrik Vogt, 2020/04/13 12:05
Dear all,
so let's move the discussion here
As a start, I'm posting the things regarding monotone relations that I already sent by e-mail.
I will use the property that A−c is monotone for some c>0 (which one might call “strictly monotone”):
1. Instead of Proposition 3.4 in trostorff13.pdf I'd formulate
and prove a more general (simple) result: If A is a relation and
A−c is monotone, for some c>0, then
‖u−v‖≤1c‖f−g‖((u,f),(v,g)∈A).
2. The above result can be used to show: If A is a surjective
relation and A−c is monotone, for some c>0, then A is maximal
monotone (and hence also A−c is maximal monotone).
3. One can use 1. and 2. to show: If A is a relation with dense
range, and A−c is monotone, for some c>0, then the closure of A
is maximal monotone.
4. As a corollary of Minty's theorem one can see:
If A−c is maximal monotone, for some c>0, then A is surjective.
I'm looking forward to your comments!
Best wishes, Hendrik
Sahiba Arora, 2020/04/17 16:04, 2020/04/17 19:41
Dear Hendrik
I have added the proof of 1, 2, 3 and 4 to the notes in Overleaf.
I have used Minty's theorem to prove 2, but I think you have a simpler proof in mind using 1. Am I right?
Regards
Sahiba
Hendrik Vogt, 2020/04/17 23:37
Dear Sahiba,
it's great that you started to work on this! You are right, I thought that one can show 2 by simply using 1. However, now it seems to me that one does actually need Minty's theorem. My mistake was this: 2 should actually read
2'. If A is a surjective relation and A−c is monotone, for some c>0, then A−c is maximal monotone.
I hope that this is a simple consequence of 1. And, this is important: 2' gives us one of the implications of Minty's theorem (which I'd call the “trivial” implication).
Now, if A−c is maximal monotone, then A is maximal monotone as well, but this implication I can only prove with your argument, using the full power of Minty's theorem. I hope I didn't make another mistake!
Best wishes, Hendrik
Sahiba Arora, 2020/04/18 01:16, 2020/04/18 01:17
Dear Hendrik
I don't think I understand you correctly. If A is surjective and A−c is monotone, then I+1c(A−c) is surjective and hence A−c is maximal monotone (by the “trivial implication” of Minty).
I still don't see where to use 1 .
Doesn't 1 tell us that A is injective as well or does it give something more that I'm missing?
Regards
Sahiba
Hendrik Vogt, 2020/04/18 10:05
Dear Sahiba,
yes, 1 tells us that A is injective - and that's the crucial point! Any extension B of A with the property that B−c is monotone is still injective, but a surjective relation cannot have a proper injective extension. Does this help?
Best wishes, Hendrik
Sahiba Arora, 2020/04/18 14:15
Dear Hendrik
Ah yes, this is more direct. I have added this proof to the notes now and also written the prove of 2 using 2' and Minty's theorem. However, I have also used contraction mapping principle. I hope that is not an overkill.
Best wishes,
Sahiba
Hendrik Vogt, 2020/04/18 16:06
Dear Sahiba,
ah, yes, the proof of 2' (in Corollary 2.2) is exactly what I meant You can make it even simpler, avoiding the contradiction, by omitting “proper” in the first line and omitting “∖A” in the second line. Then you obtain B⊆A and hence B=A. (I write this because I try to avoid contractictions whenever possible, as a matter of principle.)
As for the proof of 2, didn't you have a much simpler proof before? By Minty's theorem, maximal monotonicity of A−c implies that (c+1)+(A−c)=1+A is surjective. Then 2' (or Minty) shows that A is maximal monotone. I thought your initial argument looked similar to this. Or am I mistaken?
Best wishes, Hendrik
Sahiba Arora, 2020/04/18 16:28
Dear Hendrik
In contrast, I like proofs by contradictions .
I don't know why I changed the proof for 2 and complicated it. I have fixed it back.
Best wishes, Sahiba.
discussion/project_l.txt · Last modified: 2020/03/07 15:30 by matcs
Discussion on Project L
Dear all,
I think I figured out how to deal with solution theory for real Hilbert spaces (see Lemma 2.22, Cor. 4.8 in Overleaf). The Picard-type Theorem still holds for real Hilbert spaces H but we must adjust the condition on the material law:
This is nice, because it means that I do not have to modify the examples that naturally live on real Hilbert spaces.
One could also extend a maximal monotone relation A on a real Hilbert to a maximal monotone relation on its complexification via Lemma 2.20 and apply complex solution theory, but that's not as nice.
Best, Gabriel
Hi,
I've made the observation that Theorem 4.1 (in the Overleaf notes) holds even if zM(z)−c is monotone only for z with Rez=ν>sb(M), not ≥ν. Consequently, existence of solutions in the main Theorem 5.6 is guaranteed also in this case.
I suppose that Rez≥ν becomes important if you compare solutions to different ν, which we unfortunately didn't have the time to do. It's something I'd like to look into after the workshop (see my proposed Proposition 5.8).
If we're asked a question about Eventual Independence of ν after the presentation, I'd just honestly say that we were looking into that, but didn't have the time to understand it completely and present it. Is that ok with you?
Best, Gabriel
Hi,
I don't yet have a conclusive idea how to solve our difficulties with Lemma 5.5 and Prop 5.6 (in our Overleaf project) so I changed my strategy to going through everything before in detail. Everything up to 3.10 should be correct, and the lengthy estimate that x_n is a Cauchy sequence in the proof of 3.11 is also correct, but after that I'm struggling to understand rest of the proof of 3.11.
Also, if I see it correctly, 3.11 corresponds to Prop 2.4. in Sascha's paper [Tr13], where he states “In particular, A+B is maximal monotone”. I think this is wrong, however, but it is right if we additionally assume that the estimate C(y):=supλ>0‖Bλxλ‖<∞ holds for all y∈H, because then 1+A+B is surjective. Or do you think there is another argument to avoid needing C(y)<∞?
Moreover, I think one can replace supλ>0‖Bλxλ‖<∞ by limsupλ→0‖Bλxλ‖<∞, i.e. ‖Bλxλ‖ is eventually bounded.
Best, Gabriel
Dear Gabriel
I think you have already edited the proof of Proposition 3.11 so I'm not sure what part after proving (xλn) is Cauchy was unclear.
I think in Proposition 2.4 in [Tr13], it is clear that C(y)<∞ for all y∈H. Without that of course we can't appeal to Minty to show A+B is maximal monotone.
Regards,
Sahiba
I thought of and proved Lemma 2.7, and with its help the remaining part of 3.11 becomes easy. Please have a look at it and check whether my proof really works. I cite a short version here:
Let A be monotone and (xn,yn) a sequence of points in A such that either
Then A∪{(x,y)} is monotone.
Yeah it seems right to me. Honestly, I don't remember what the previous version of the proof was to recall where it failed.
Dear all,
as mentioned in the Overleaf project chat, I added Minty's Theorem and its proof in our Overleaf project, giving some more details here and there.
Also, I completed the proof of Cor.3.5 (Cor 2.3 in [Tr12]), because there was some argument missing to show that ¯A is surjective and came across some small insight that might be useful somewhere else: If A−c is monotone, then the estimate
‖x−y‖≤1c‖f−g‖
not only gives us that A is injective, but that if (xn,fn) is a sequence in A such that fn is a Cauchy sequence, then also xn is a Cauchy sequence.
Lastly, I'm still a bit confused about Hendrik's proposed proof or rather how to implement it into the Overleaf project without having too much overlap with section 5. In fact, I think the proof of Prop 5.6 that Sahiba wrote works also for f∈dom(∂t,ν) if you have a look at Lemma 5.5. I think the h<0 should rather be 0<h<1 and then the right hand side of the estimate becomes
suph∈(0,1)‖τhf−fh‖ instead supt≥0‖f′(t)‖
What do you think, could the proof work like this?
Best, Gabriel
Dear Gabriel
Currently, the proof of Lemma 5.5 that I wrote is incorrect and I'm not sure how to do it for the half-line. Maybe as Hendrik suggested we should stick to the whole line.
Regards Sahiba
Hi Sahiba,
I'm a bit confused where you see an error. Is it that applying τ−h to ϕ might not yield the wanted result because it corresponds to a difference quotient for a left-sided derivative, but we're dealing with the positive half-line so we're getting problems near 0? So the proof might work on the whole real line, right?
Best, Gabriel
Dear Gabriel
The problem in the proof of Lemma 5.5 is that the lim sup in the 2nd displayed formula might not be finite. Basically the proof lacks justification why taking f∈C1c(R≥0) is sufficient.
Best, Sahiba
Dear all,
I have a doubt in trostoff12.pdf. Is ∂ν in Section 4.2 same as ∂ν,0 in the previous section?
Proposition 4.9 in trostoff12.pdf is the same as Theorem 4.1.2 in the ISem lectures which says that functions in the domain of ∂t,ν are continuous. So I think ∂ν in trostoff12.pdf is just ∂t,ν in the ISem lectures which is same as ∂ν,0 in trostoff12.pdf?
If yes, then why change the notation mid-way?
Regards, Sahiba.
Dear Sahiba,
ah, that's a good question! Only now I see that there's a major difference between trostorff12.pdf and trostorff13.pdf: In the latter paper, the time derivative is considered for functions defined on all of R (as was the case in Picard's theorem 6.2.1), whereas in the earlier paper the functions are defined on [0,∞). In this case, ∂ν is a proper extension of ∂ν,0! Have a look at page 5854:
I'm not sure yet how to deal with this, but maybe we should concentrate on the case closer to Picard's theorem.
Best wishes, Hendrik
Dear Hendrik
Oh, so that means if we have u∈D(∂ν), then u∈D(∂t,ν)⟺u(0)=0?
Best wishes, Sahiba
Dear Sahiba,
a late answer: yes, this is Remark 4.11 in trostorff12.pdf.
Best wishes, Hendrik
Dear Hendrik
After reading the basics of Maximal monotone relations and Minty's Theorem, do you suggest that one should first read trostorff12.pdf till Theorem 4.19 and then trostorff13.pdf till Theorem 3.7 (omitting Section 2.3 for now) or is there another way we should be going about this?
Regards Sahiba
Dear Sahiba,
I do suggest browsing through those parts of the papers. My plan is to make more suggestions about what details to concentrate on. For now let me only say a few things:
- For the moment you can ignore the minimal sections and Proposition 2.4(c) in trostorff12.pdf.
- Likewise, first ignore Lemma 3.2 in trostorff13.pdf.
- A crucial ingredient is Proposition 3.6 in trostorff13.pdf, which actually even holds for f∈dom(∂t,ν). I suggest to use the following nice characterisation of dom(∂t,ν) for the proof of this more general result: an element f∈L2,ν(R;H) belongs to dom(∂t,ν) if and only if (1h(τhf−f))0<h<1 is bounded. In the proof of this equivalence you can try and use the relative weak compactness of bounded subsets of H and the distributional description of dom(∂t,ν) from Proposition 4.1.1 in the ISem lectures.
Best wishes, Hendrik
Dear all,
I just realised that I need to make the “nice characterisation” above more precise: in addition to the mentioned equivalence one obtains ∂t,νf=limh→01h(τhf−f)(f∈dom(∂t,ν)) (with convergence in L2,ν(R;H); please check if this claim is correct!).
In case you wonder what I want with this characterisation - it easily implies the following: if R is an autonomous Lipschitz continuous mapping from L2,ν(R;H) to itself, with Lipschitz constant L, then R maps dom(∂t,ν) to itself, and ‖∂t,ν(Rf)‖L2,ν(R;H)≤L‖∂t,νf‖L2,ν(R;H)(f∈dom(∂t,ν)). Please have a look yourself as to why I find this very useful!
Best wishes, Hendrik
Dear Hendrik,
I have written down the proof of the characterization in the notes in Overleaf (Lemma 5.9).
Best wishes, Sahiba
Dear Sahiba,
ah, I see that what I wrote was ambiguous. What I meant: f∈dom(∂t,ν)⟺(1h(τhf−f))0<h<1 is bounded, and 1h(τhf−f)→∂t,νf if one of the (equivalent) properties holds. In particular: The convergence follows from the boundedness; this is important!
One more thing: You have to explain in more detail why it suffices to consider f∈C1c.
Best wishes, Hendrik.
Dear Hendrik
Oh, so I misunderstood the claim. I have made the required changes, I hope it is correct now.
Best wishes, Sahiba.
Dear Sahiba,
in the 2nd displayed formula in the proof, why should the limsup be finite?
Let me emphasise one thing: I only thought about the whole real line here; note that I wrote R in my original proof. I didn't think about what the correct version for the half line is!
Best wishes, Hendrik
Dear all,
so let's move the discussion here
As a start, I'm posting the things regarding monotone relations that I already sent by e-mail.
I will use the property that A−c is monotone for some c>0 (which one might call “strictly monotone”):
1. Instead of Proposition 3.4 in trostorff13.pdf I'd formulate and prove a more general (simple) result: If A is a relation and A−c is monotone, for some c>0, then ‖u−v‖≤1c‖f−g‖((u,f),(v,g)∈A).
2. The above result can be used to show: If A is a surjective relation and A−c is monotone, for some c>0, then A is maximal monotone (and hence also A−c is maximal monotone).
3. One can use 1. and 2. to show: If A is a relation with dense range, and A−c is monotone, for some c>0, then the closure of A is maximal monotone.
4. As a corollary of Minty's theorem one can see: If A−c is maximal monotone, for some c>0, then A is surjective.
I'm looking forward to your comments!
Best wishes, Hendrik
Dear Hendrik
I have added the proof of 1, 2, 3 and 4 to the notes in Overleaf.
I have used Minty's theorem to prove 2, but I think you have a simpler proof in mind using 1. Am I right?
Regards Sahiba
Dear Sahiba,
it's great that you started to work on this! You are right, I thought that one can show 2 by simply using 1. However, now it seems to me that one does actually need Minty's theorem. My mistake was this: 2 should actually read
2'. If A is a surjective relation and A−c is monotone, for some c>0, then A−c is maximal monotone.
I hope that this is a simple consequence of 1. And, this is important: 2' gives us one of the implications of Minty's theorem (which I'd call the “trivial” implication).
Now, if A−c is maximal monotone, then A is maximal monotone as well, but this implication I can only prove with your argument, using the full power of Minty's theorem. I hope I didn't make another mistake!
Best wishes, Hendrik
Dear Hendrik
I don't think I understand you correctly. If A is surjective and A−c is monotone, then I+1c(A−c) is surjective and hence A−c is maximal monotone (by the “trivial implication” of Minty).
I still don't see where to use 1
.
Doesn't 1 tell us that A is injective as well or does it give something more that I'm missing?
Regards
Sahiba
Dear Sahiba,
yes, 1 tells us that A is injective - and that's the crucial point! Any extension B of A with the property that B−c is monotone is still injective, but a surjective relation cannot have a proper injective extension. Does this help?
Best wishes, Hendrik
Dear Hendrik
Ah yes, this is more direct. I have added this proof to the notes now and also written the prove of 2 using 2' and Minty's theorem. However, I have also used contraction mapping principle. I hope that is not an overkill.
Best wishes,
Sahiba
Dear Sahiba,
ah, yes, the proof of 2' (in Corollary 2.2) is exactly what I meant
You can make it even simpler, avoiding the contradiction, by omitting “proper” in the first line and omitting “∖A” in the second line. Then you obtain B⊆A and hence B=A. (I write this because I try to avoid contractictions whenever possible, as a matter of principle.)
As for the proof of 2, didn't you have a much simpler proof before? By Minty's theorem, maximal monotonicity of A−c implies that (c+1)+(A−c)=1+A is surjective. Then 2' (or Minty) shows that A is maximal monotone. I thought your initial argument looked similar to this. Or am I mistaken?
Best wishes, Hendrik
Dear Hendrik
In contrast, I like proofs by contradictions
.
I don't know why I changed the proof for 2 and complicated it. I have fixed it back.
Best wishes, Sahiba.