In communication with Mehmet Cankaya, we consider the subject.
There is known result of D. Gale and H. Nikaido, The Jacobian matrix and the global univalence of mappings,
Mathematischen Annalen 159 (1965) 81-93.
For simplicity we will consider $2$- dimensional case. Let $D=(a,b) \times (c,d)$ and suppose that $F=(f_1,f_2)$ is $C^1$ on $D$.
Suppose that (i) $D_1f_1= f_{11}\neq 0 $ and $det F' \neq 0$ on $D$.
We outline a proof that (i) implies (I) $F$ is injective on $D$.
Let $c\in F(D)$ and let $F(a_1,b_1)=c$. Find solutions of equation $F(x,y)=c=(c_1,c_2)$ on $D$.
Consider (2) $f_1(x,y)=c_1$. We will show that the set $f_1^{-1}(c_1)$ is graph of a function. Since $f_{11}=D_1f_1 \neq 0 $ on $D=(a,b) \times (c,d)$ we can suppose that $f_{11}> 0 $. Then (a1) the function $f_1(x,y)$ is increasing in $x$ for every fixed $y$; then there exists an open set $U$ containing $a_1$, an open set $V$ containing $b_1$, and a unique continuously differentiable function $g: V \rightarrow U$ such that $f_1(g(y),y)=c_1$, $y \in V$. Using continuity we can extend $g$ on $(c,d)$ such that $g \in C^1(c,d)$.
Thus there is a function $x=g(y)$, in a
$y \in (c,d)$ such that
$f_1(g(y),y)=c_1$, $y \in (c,d)$. By (a1), $\Gamma_g= f_1^{-1}(c_1)$, where $\Gamma_g$ is the graph of $g$ over $(c,d)$.
Hence (3) $f_{11}(g(y),y) g'(y) + f_{12}(g(y),y) =0$, $y \in (c,d)$.
Let us show that equation $h_2(y)=c_2$ has a unique solution on $(c,d)$. Contrary suppose that $h_2(y_1)=h_2(y_2)$ for $y_1\neq y_2$. Then there is
$y_0$ such that $h_2'(y_0)=0$. Hence (4) $f_{21}(g(y_0),y_0) g'(y_0) + f_{22}(g(y_0),y_0) =0$. Since $det F' \neq 0$, from (3) and (4) it follows that $g'(y_0)=0$ and
therefore $f_{12}(g(y_0),y_0) = f_{22}(g(y_0),y_0) =0$, which is a contradiction.
It seems that using a modification of the above proof we can get a corresponding version of this result for convex sets in n-dimensional space.
I can approve your proof with an example. Please look at Existence and Uniqueness of the solutions of (4.7) and (4.8) at p.138.
Hence (3) $f_{11}(g(y),y) g'(y) + f_{12}(g(y),y) =0$, $y \in (c,d)$.
Hence (4) $f_{21}(g(y_0),y_0) g'(y_0) + f_{22}(g(y_0),y_0) =0$
What are they?, Determinant or not?
If you know the estimating equations, you can realize that we do not need the convex function (or set?). The convex function and convex set are same thing, are they? The book attached is an example for this. My curisoity is that how we can say thhe uniqueness for them.
My answer involve only the monotony, so that it solves a weaker related similar problem. (i) Let f be a C^2 real function defined in an open convex subset D of R^n, such that all the determinants in the upper left corner of the Hessian are strictly positive at any point of D. Then the second derivative f " of f is positive definite. By Taylor's formula, we infer that f(y) - f(x) > f ' (x)(y-x) for x, y distinct points in D. Assume additionally that with respect to the usual product order relation on R^n (defined by the convex cone of all vectors having non negative components), the first derivative f ' =grad f is non negative at any point in D. Then for x < y, one deduces
f(y)-f(x) > f ' (x)(y-x)>= 0, which implies f(y)>f(x) (strictl monotony). Hence such functions are strictly monotone. (ii) Going back to the case of convex differentiable mappings g from the open convex subset D contained in R^n, into R^n, using the same order relation (on the components), from convexity we infer that
g(y)-g(x) >= g ' (x)(y - x ) in the sense of the product-order relation. Assume that x < y (strict inequality), and the Jacobi matrix representing g ' (x) has all the entries positive numbers. Then g(y)-g(x) >= g ' (x)(y - x )>0. Hence the transformation g is strictly increasing. Notice that in both above examples we cannot deduce the injectivity, since there exist not comparable elements in R^n, n>=2.
Dear Mehmet Niyazi Cankaya and Octav Olteanu,
I need to read carefully Mehmet question
''The convex function and convex set are same thing, are they?''
We can consider non convex function on convex sets.
I will answer in more details as soon as possible.
Dear Miodrag Mateljevic, of course, the case of convex mappings is a particular one. Moreover, the injectivity does nof follow from the preceding remarks. However, the strictly increasing monotony holds via a an elementary and simple proof. Unfortunately, strict monotony does not imply injectivity, since there exist non comparable elements.
Dear Miodrag Mateljevic, may I ask you to prepare your answer as an attached file, by using mathematical notations? Your first answer is very difficult to be followed..
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