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[the probleme is on page 156][1] Benchohra, Mouffak; Hamani, Samira, Boundary value problems for differential inclusions with fractional order, Discuss. Math., Differ. Incl. Control Optim. 28, 147-164 (2008). ZBL1181.26012.I'm having technical problems or just lack of knowledge problems, so I would appreciate your help. the problem:: let $v_{*}\in F$, and for every $w \in F$, we have $$|v_{n}-v_{*}| \leq |v_{n}-w|+|w-v_{*}|...(1)$$

Then, $$|v_{n}-v_{*}| \leq d(v_{n},F)...(2)$$ and $d(x,A):=\inf\lbrace|x-y|,~~y \in A\rbrace$

So the problem I'm having is how to get from (1) to (2). I don't know what to do to get to (2).

I'm really desperate and I would be more than thankful for any idea.

[the problem is on page 156][1]: https://scholar.google.com/scholar?hl=fr&as_sdt=0%2C5&q=M.%20Benchohra%20and%20S.%20Hamani%2C%20Boundary%20value%20problems%20for%20differential%20inclusions%20with%20fractional%20order&btnG=

Jude
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  • The first equation always holds in metric spaces, it is the triangular inequality. – Dog_69 Mar 12 '19 at 16:34
  • @Dog_69 And? How is that relevant? This problem isn't limited to metric spaces. – Rushabh Mehta Mar 12 '19 at 16:36
  • the first one is clear, I'm asking how to get from (1) to (2). – Jude Mar 12 '19 at 16:40
  • I'm new here @Dog_69 . so please pardon me if i did any mistake. – Jude Mar 12 '19 at 16:42
  • What are $F$ and $v_n$ ? – ippiki-ookami Mar 12 '19 at 17:08
  • Equation $(1)$ doesn't imply equation $(2)$, as you've stated the problem. Does $v_$ have any other defining condition than being an element of $F$? Otherwise the only statement you could make is that $d(v_n,F) \leq |v_n-v_|$ (the reverse inequality of $(2)$), by the definition of $\inf$. Is the reverse inequality what you meant? – jawheele Mar 12 '19 at 17:12
  • @ jawheele yep that what i thought too ... but i found it as it is in an artical .. and i dont know how or why they have concluded that...F is a multivalued map (i.e.$F(t,y_{*}(t))$). – Jude Mar 12 '19 at 17:18
  • @ippiki-ookami F is a multivalued map (i.e.F(t,y_{*}(t))). $F : J \times \mathbb{R} \rightarrow \mathcal{P}(\mathbb{R})$. – Jude Mar 12 '19 at 17:21
  • Could you give a reference to this article ? – ippiki-ookami Mar 12 '19 at 17:24
  • @DonThousan If it is a general property it can't imply any special relation, otherwise the latter would be true always. – Dog_69 Mar 12 '19 at 17:31
  • @ippiki-ookami yes I do its title is: M. Benchohra and S. Hamani, Boundary values problems for differential inclusions with fractional order – Jude Mar 12 '19 at 17:42
  • @ippiki-ookami here is the URL of the artical ::https://scholar.google.com/scholar?hl=fr&as_sdt=0%2C5&q=M.+Benchohra+and+S.+Hamani%2C+Boundary+value+problems+for+differential+inclusions+with+fractional+order&btnG= – Jude Mar 12 '19 at 17:43
  • So I took a look, but am as confused as you are. – ippiki-ookami Mar 12 '19 at 18:24
  • @ippiki-ookami hhh well i can relate to that ..and thank you very much for trying. – Jude Mar 12 '19 at 18:25
  • @hanaachan Usually, mathematicians don't have any problem answer question, don't matter who asks. My advice is to write an email to one of the authors explaining your doubt. I'm sure they will be glad. – Dog_69 Mar 12 '19 at 20:45
  • @hanaachan yes maybe I should heed your advice and ask them .. it doesn't hurt to try right. thank u. – Jude Mar 12 '19 at 21:28

1 Answers1

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From (1) there is no way to get (2). This is probably just a mistake. Since $w \in F$ difference $|v_n-w|$ almost more than infinum. To this is added positive number limited by the size of the set $F$. $d(v_n,F)$ is lower limit.

Александр Карасёв
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