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For every $z\in B$, $z_3+3>2$ and $|z_1+z_2|<2$. Consequently each of the three partial derivatives in the preceding gradient has absolute value below $1/2$, soThe segment joining $x,y\in B$ lies inside $B$. The mean value theorem and Cauchy-Schwarz inequality therefore giveFor $x=y$ both sides vanish. This proves the required Lipschitz continuity uniformly on the ball.

A metric space is sequentially compact if every sequence has a convergent subsequence whose limit belongs to the space. A limit point $\ell$ of $S$ is a point for which every open ball about $\ell$ contains a point of $S$ other than $\ell$.

Let $(x_n)$ lie in a compact metric space. If one value occurs infinitely often, take a constant subsequence. Otherwise its set of values $S$ is infinite. If $S$ had no limit point, every point of the space would have a neighborhood containing at most one member of $S$. A finite subcover of these neighborhoods would force $S$ finite, a contradiction. Thus $S$ has a limit point; the stated subsequence result supplies a convergent subsequence. This proves sequential compactness.

Assume now that $X$ is nonempty, as is necessary for the requested minimum and fixed point. The function $r(x)=d(f(x),x)$ is continuous, as a composition of $x\mapsto (f(x),x)$ with the metric. The extreme value theorem on a compact space implies that $r$ attains its infimum, say at $p$. If $f(p)\ne p$, strict decrease of distances givescontradicting minimality. Thus $f(p)=p$. If $p,q$ were distinct fixed points, then $d(p,q)=d(f(p),f(q))<d(p,q)$, another contradiction. Hence there is exactly one fixed point. This is the fixed point theorem for a strictly distance-decreasing map on a compact metric space.

Compactness is essential. On $\mathbb{R}$, takeHere $f(x)-x=\log (1+e^{-x})>0$, so there is no fixed point. Nevertheless $0<f^{\prime }(x)=e^x/(1+e^x)<1$ everywhere, and the mean value theorem gives $|f(x)-f(y)|<|x-y|$ for distinct $x,y$. There is no uniform Lipschitz constant less than $1$, since $f^{\prime }(x)\to 1$ as $x\to \infty$.

The partial derivatives satisfy a global bound of $1$ in absolute value. For example,because $2|x|\le 1+x^2$ and $1+x^2\ge 1$; the same argument applies to ${\partial }_{y}f$. Apply the one-dimensional mean value theorem first along a horizontal segment and then along a vertical segment:Thus $f$ is Lipschitz with constant $1$ for the coordinate-sum norm.

The drift coefficient $b(z)=rz$ and diffusion coefficient $a(z)=\sigma z$ are Lipschitz continuity and have linear growth. The standard existence and pathwise uniqueness theorem for stochastic differential equations therefore supplies a unique global strong solution of a stochastic differential equation, adapted to the driving Brownian filtration.

Apply Itô's formula to $x\exp ((r-{\sigma }^2/2)t+\sigma B_t)$. Its time derivative contributes $(r-{\sigma }^2/2)X_t$ and its Itô correction contributes ${\sigma }^2{X}_t/2$, giving the required equation and initial value. HenceIn particular the solution stays positive. For $t>s$, factor out $X_s$ and use independence and the normal law of $B_t-B_s$:The exponential moment exists, so this conditional expectation is well-defined for every finite $s,t$.

For continuous $G$ and $f$, restrict to the closed supremum-norm ball of radius $R$ in $C([0,1])$. Boundedness on the compact kernel and nonlinearity boxes makes the integral map preserve this ball for sufficiently small positive $\mu$. Uniform continuity of $f$ proves continuity of the map; uniform continuity of $G$ gives an equicontinuous bounded image. Arzelà-Ascoli makes that image relatively compact, and Schauder gives a solution. No global Lipschitz hypothesis or uniqueness conclusion is needed.

A second-order ordinary differential equation involves a second derivative as its highest derivative. In explicit form $y^{\prime \prime }=f(x,y,y^{\prime })$, introducing $v=y^{\prime }$ converts it into the first-order system $y^{\prime }=v$, $v^{\prime }=f(x,y,v)$. Under local regularity and Lipschitz continuity in the dependent variables, an initial value problem is specified by both $y(x_0)$ and $y^{\prime }(x_0)$. The linear case is a second-order linear differential equation.

For a globally Lipschitz nonlinearity of constant $ϵ$, the map taking $v$ to the zero-Dirichlet Poisson solution with load $f(v)$ is a contraction in the gradient norm if the displayed inequality holds. Poincaré gives the factor $C_P$ from the Sobolev input to $L^2$, and testing the linear Poisson equation gives another factor $C_P$ back to the gradient norm. The condition yields a unique weak solution without requiring $f(0)=0$. With the optimal Poincaré constant it reads $ϵ<{\lambda }_1$.

A positive measurable function $\ell$ is slowly varying at infinity iffor every fixed $u>0$. It has regular variation of index zero. “Slowly” concerns relative change under multiplication of the argument; it does not mean that the function must be bounded, have a small derivative, or converge to a finite value.

Every positive constant is a slowly varying function. So are $\log t$, $1/\log t$ and $(\log t{)}^a$ for any fixed real number $a$, on $t>1$. Indeed,The first grows without bound, the second tends to zero, and the constant does neither. Another unbounded example is $\ell (t)=e^{\sqrt{\log t}}$:This function can grow much faster than $\log t$ while retaining regular variation of index zero.

Even convergence is unnecessary. The bounded functionoscillates between $e^{-1}$ and $e$. It is nevertheless a slowly varying function: the change in the argument of the sine under $t\mapsto ut$ isand the Lipschitz continuity of the sine makes the logarithm of the ratio tend to zero. The sequences $t_k=\exp (\exp (\pi /2+2\pi k))$ and $s_k=\exp (\exp (3\pi /2+2\pi k))$ give the two values $e$ and $e^{-1}$ for every nonnegative integer $k$, so there is no limit at infinity.

Contrast this with $h(t)=2+\sin (\log t)$, which is positive and bounded but is not a slowly varying function. Fix $u=e^{\pi /2}$. Along $t_k=e^{2\pi k}$ its rescaling ratio is $h(u{t}_k)/h(t_k)=3/2$; along $s_k=e^{\pi +2\pi k}$ the ratio is $1/2$. The persistent oscillations prevent even a limit for this one fixed rescaling. It is not regularly varying with any finite index either.

The useful distinction is whether a fixed proportional change of input eventually produces a negligible proportional change of output. A slowly varying function can therefore preserve the scaling exponent of a power law while changing its magnitude substantially. The decomposition in regular variation explains why a logarithmic factor can matter in numerical estimates without changing the limiting exponent.

For the autonomous differential equation $v^{\prime }=\sqrt{1-v^2}$, the real solutions are nondecreasing and lie in $[-1,1]$. On each nonconstant interval, separation of variables gives $\arcsin v=t-C$, so the solution crosses from $-1$ to $1$ in exactly time $\pi$. Every nonconstant solution on the entire real time axis is consequentlyfor some real $a$. The two constant solutions $v\equiv \pm 1$ complete the list; solutions on shorter intervals are restrictions of these. At the joins both the derivative and the right-hand side are zero, so these are continuously differentiable functions. The right-hand side is continuous but fails Lipschitz continuity at both endpoints, allowing arbitrary waiting time at $-1$ before departure. Once $v$ reaches $1$, monotonicity forces it to stay there. The sine representation without the nonnegative-cosine restriction would incorrectly introduce decreasing pieces.

For an explicit Runge-Kutta method with coefficients $A,b$, put $c=A1$ and let $c^2$ mean componentwise squaring. The conditions $b^{T}1=1$, $b^{T}c=1/2$, $b^T{c}^2=1/3$ and $b^{T}Ac=1/6$ match the exact flow expansion through degree three: the two cubic differentials are $f^{\prime \prime }[f,f]$ and $(f^{\prime }{)}^{2}f$. For sufficiently smooth vector fields these give local error $O(h^4)$ and, under the usual finite-interval Lipschitz continuity assumptions, global error $O(h^3)$.

For the autonomous differential equation $y^{\prime }=-|y-y_{\ast }{|}^p$, the displayed family stays at $y_{\ast }$ until time $c$ and then decreases below it. Here $[z{]}_{+}=\max (z,0)$. Direct differentiation verifies the equation on each side; since $1/(1-p)>1$, the derivative tends to zero at departure, making the joined function continuously differentiable. Distinct departure times give different solutions from the same equilibrium initial condition. The ratio $|f(y_{\ast }+\epsilon )-f(y_{\ast })|/\epsilon ={\epsilon }^{p-1}$ is unbounded, so the local Lipschitz continuity hypothesis of the Picard-Lindelöf theorem fails.

A solution starting above $y_{\ast }$ reaches it in finite time. It can then use any of these continuations. For $p=1/2$, $y_{\ast }=-1$, and initial value $y(0)=0$, arrival occurs at time two; departure at $c\ge 2$ gives $y=-1-(t-c{)}^2/4$ afterwards. Waiting-time solutions of a sublinear Bernoulli equation give a related mechanism for departures in the opposite direction.

Let q be continuous and strictly positive, with a constant a. A nonnegative solution of a differential equation may stay zero until any prescribed point $x_0$ and become positive afterwards. For $x>x_0$, defineand set $y=0$ before $x_0$. The Bernoulli power substitution verifies the equation after $x_0$. Continuity of q gives $y=O((x-x_0{)}^{1/(1-p)})$, so the joining derivative is zero and the equation also holds at the join. These different waiting times give nonunique solutions from zero. The right side lacks Lipschitz continuity as a function of y near zero, so the uniqueness conclusion of the Picard-Lindelöf theorem does not apply there.

A standard Weierstrass function is a uniformly convergent lacunary cosine series, here parameterized by $a>1$ and an integer frequency multiplier $b>1$. Its continuity follows from the Weierstrass M-test. In the range $1<a<b$, its modulus of continuity has the fractional-power scale ${\delta }^{\log a/\log b}$ for the positive odd-frequency model; the critical case $a=b$ can have the larger scale $\delta \log (1/\delta )$. These examples distinguish small approximation error from ordinary Lipschitz continuity.

The diffusion coefficient obtained from a scale transform for an additive-noise diffusion need only be defined on an open interval. If it is Lipschitz, its limit at a finite endpoint is zero: a positive limit would bound the derivative $1/h$ of the inverse scale and prevent the inverse from diverging. Extending it by zero beyond finite endpoints therefore preserves global Lipschitz continuity.