Articles of monotone functions

A contraction that satisfies a weak monotonicity condition but is not monotone

Let $K=[0,1]\times [0,1]$. Find a continuous mapping $F:K\rightarrow \mathbb{R}^2$ satisfying: $\|F(x)-F(y)\|\leq \|x-y\| \quad\forall x,y\in K,$ There exists $\gamma>0$ such that for all $x,y\in K$ $$\left<F(x),y-x\right>\geq 0\Longrightarrow\left<F(y), y-x\right>\geq \gamma\|x-y\|^2,$$ There exist $u,v\in K$ such that $\left<F(u)-F(v), u-v\right><0$. Here, $\|.\|$ is the Euclidean norm and $\left<,.,\right>$ is the scalar product in $\mathbb{R}^2$. Thank you for all comments […]

A strongly pseudomonotone map that is not strongly monotone

Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that $F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that $$ \left<F(y)-F(x), y-x\right>\geq \gamma\|y-x\|^2, \quad \forall x,y\in K. $$ $F$ is strongly pseudomonotone on $K$ if there exists $\gamma>0$ such that $$ \left<F(x), y-x\right>\geq 0 \Longrightarrow \left<F(y), […]

structure and properties of a function inherited from its integrals

Let $f:[0,1]^2 \rightarrow \{0,1\}$, $f_B(b) = \int_0^1 f(b, s)\; ds$ and $f_S(s) = \int_0^1 f(b, s)\; db$, such that $f_B$ is non-decreasing and $f_S$ is non-increasing. What can we infer about $f$ from the monotonicity of $f_B$ and $f_S$? For instance, the following class of functions satisfies the monotonicity constraints \begin{align*} f(b,s) = \begin{cases} 1 […]

Property of Solutions to Two Correlated Optimization Problems

Here are two integral equations that correlated with each other For $w\in[0,w_0]$: \begin{equation} x^*_w\triangleq\begin{cases} &-\infty,~\mbox{if }\nexists~x\geq 0~\mbox{such that }y^*_v+p-v\leq x\leq y^*_v+p-h-w,~\mbox{for some }v\in[0,v_0]\\ &\\ &\displaystyle\underset{x\geq 0}{\arg\max}~G^s(x;w)=\frac{1}{v_0}\int_0^{v_0}\mathbf{1}_{\{v+x-p-y^*_v\geq 0\}}[y^*_v+p-h-x-w]^+dv,~\mbox{otherwise} \end{cases}. \end{equation} and for $v\in[0,v_0]$: \begin{equation} y^*_v\triangleq\begin{cases} &-\infty,~\mbox{if }\nexists~y\geq 0~\mbox{such that }x^*_w-p+h+w\leq y\leq x^*_w-p+v,~\mbox{for some }w\in[0,w_0]\\ &\\ &\displaystyle\underset{y\geq 0}{\arg\max}~G^c(y;v)=\frac{1}{w_0}\int_0^{w_0}\mathbf{1}_{\{y+p-h-x^*_w-w\geq 0\}}[x^*_w+v-p-y]^+dw,~\mbox{otherwise} \end{cases}. \end{equation} where $v_0>w_0>0$ are constants and $p>h>0$ […]

Behavior of the Nth root of N?

Taking the Nth root of some real number $N$ (ie: $R(N) = N^{1/N}$), generally $R(X) > R(Y)$ when $X < Y$. This obviously isn’t the case though when $ X< Y < 3$. Put another way, starting at $N = 0.1$ and increasing $N$ by $0.1$ if we plot each R(N) we see that the […]

How to find the limit of the sequence $x_n =\frac{1}{2}$, if $x_0=0$ and $x_1=1$?

The formula is only applicable on values for $n\geq 2$. I know that the sequence is monotonic with a lower bound at $\frac 1 2$, but I am unsure how to find the supremum of the sequence. EDIT: $x_2 = \frac 1 2, x_3 = \frac 3 4, x_4 = \frac 5 8$. Does that […]

Why is Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, …$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued $\Sigma-$measurable functions, i.e. for every $k\ge 1$ and every $x$ in $X$, $$0 \le f_k(x) \le f_{k+1}(x)$$. Next, set the pointwise limit of the sequence ${f_n}$ to be $f$. That is, […]

Recursive Monotone Decreasing Sequence Proof ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$

I have been looking at this for hours and it isn’t making anymore sense than it did in the first hour. If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit. ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$ What I though I had to […]

Expectation inequality for a set and its subset.

Given that a function $g(x)$ is a monotone increasing function. Its domain is an interval [c,d]. There are two sets $a_j \in A, j= 1,…,J$ and $b_k \in B,k=1,…,K$ on this interval, which satisfy $c_z \in C = B$\A and $c_z \geq a_j$ for all $a_j \in A$. For example, the interval $[c,d]=[0,8]$,$A=\{0,1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5,6,7\}$. […]

Why is a strictly monotonic mapping between intervals continuous?

Consider the following problem: Say $a,b\in\mathbb R$, $a <b$ and $f:[a,b]\rightarrow\mathbb R $ strictly monotonic, let’s say increasing for the sake of simplicity. Let $f ([a,b])=[c,d]$ for some $c,d\in\mathbb R$. Show that $f$ is continuous. This question is part of an old exam in introductory real analysis. Intuitively, I understand that a discontinuity in $f$ […]