Let $X,Y$ be Banach spaces, define by $F:X\times Y \rightarrow \mathbb{R}$ be a functional, $F_u,F_v$ be Frechet derivative of $F$ with respect to $u$ and $v$ variables. We show that $F$ is Frechet differentiable. We have \begin{align} P&=\Vert F\left(u+h,v+l\right)-F\left(u,v\right)-F_u\left(u,v\right)h-F_v\left(u,v\right)l \Vert \\ &= \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h+F\left(u,v+l\right)-F\left(u,v\right)-F_v\left(u,v\right)l \Vert \\ &\leq \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h \Vert +\Vert F\left(u,v+l\right)-F\left(u,v\right)-F_v\left(u,v\right)l \Vert \\ &\leq […]

I have tried in vain to search for a closed form solution (involving known functions) to the simple first order equation $\dfrac{dy}{dx}=x^2+y^2, y(0) = 0$. Can anyone help with a solution that does not have the form of an infinite series?

Suppose that $T: M \to M$ is a self map of a nonempty closed set $M$ in a complete metric Space $(X,d)$. Suppose further that $$d(Tx,Ty) \le k(a,b)d(x,y)$$ for all $x,y \in M$ with $0 \lt a \le d(x,y) \le b$ and arbitrary numbers $a,b$.Here $0 \le k(a,b) \lt 1$. Then show that $T$ has […]

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