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Interpret $x_r$ as route $r$'s sending rate, $w_r>0$ as its demand weight, and ${\kappa }_r>0$ as its response speed. Resource $j$ sees aggregate load $y_j={\sum }_{s:j\in s}{x}_s$ and returns the resource congestion price ${\mu }_j=p_j(y_j)$. Route $r$ sums these feedback signals to its route congestion price $P_r={\sum }_{j\in r}{\mu }_j$. Its rate increases when $x_r{P}_r<w_r$ and decreases when $x_r{P}_r>w_r$. At equilibrium, each user balances weighted logarithmic utility against the summed resource prices: $x_r{P}_r=w_r$.

Take finite route and resource sets, nonempty routes, positive $w_r,{\kappa }_r$, and initially positive rates. The standard interpretation of increasing prices supplies a price positive somewhere on each route; strictly increasing nonnegative prices automatically do so. More generally, nondecreasing prices suffice provided every route contains at least one resource with a nonzero price function. Define the primal congestion potentialBecause $p_j$ is continuous and nondecreasing, its integral is a convex function. The positive-weight logarithmic sum is strictly concave; subtracting the convex load costs makes $U$ a strictly concave function on the positive orthant. Its gradient satisfiesThus this is gradient flow with diagonal mobility, ascending the utility-minus-cost potential.

We must also prove boundedness and existence of an optimizer. For each route choose a resource on it whose price is positive somewhere. Monotonicity supplies constants $c_r>0,a_r<\infty$ such that this resource's price is at least $c_r$ for loads at least $a_r$. Its integrated cost is therefore at least $c_r{x}_r-d_r$. A resource may be chosen by several routes, but at most $|R|$ times. Since all costs are nonnegative,for a finite constant $D$. Each scalar expression on the right is bounded above and tends to minus infinity as $x_r\to \infty$. Consequently a superlevel set $U(x)\ge U(x(0))$ is bounded. On a bounded set, if any $x_r\downarrow 0$, its logarithmic term tends to minus infinity while the remaining terms are bounded above. The superlevel set therefore stays away from every zero coordinate and is compact inside the positive orthant. The same argument gives an attained maximum $x^{\ast }$; strict concavity makes it unique, and its interior first-order equations giveAlong every trajectory,with equality exactly at $x^{\ast }$. Thus the trajectory remains in its compact superlevel set, exists for all positive time, and $U(x(t))$ increases to a finite limit. The nonnegative dissipation on the right has a finite time integral. It is also uniformly continuous: it is a continuous function on the trapped compact set, and the continuous vector field is bounded there, making $x(t)$ uniformly Lipschitz in time. The result uniformly continuous integrable functions vanish at infinity gives $dU/dt\to 0$. Every subsequential limit is therefore a critical point, hence $x^{\ast }$. Compactness then proves the full convergenceContinuity, rather than differentiability, of the prices is enough. For completeness, square-root coordinates for primal congestion control also establish uniqueness of the underlying trajectories: with $x_r={\kappa }_r{z}_r^2/4$, the system becomes $\dot{z}=\nabla V(z)$ for $V(z)=U(x(z))$. Each resource cost is a convex nondecreasing function composed with a convex quadratic load, while the logarithmic terms remain strictly concave. Hence $V$ is concave, and for two trajectories $\frac{1}{2}\frac{d}{dt}\Vert z-\tilde{z}{\Vert }^2\le 0$ by monotonicity of its gradient. Identical initial data therefore give identical solutions without assuming Lipschitz price functions. If some initial rates are zero, ${\dot{x}}_r={\kappa }_r{w}_r>0$ at that boundary, so the nonnegative physical trajectory enters the positive orthant and the same proof applies from any positive time.

There is a genuine qualification if the word increasing is meant merely nondecreasing without a nontrivial-price assumption. With one route, $w=\kappa =1$ and $p(y)\equiv 0$, the displayed model has $\dot{x}=1$, so $x(t)=x(0)+t$ and no equilibrium. Positive weights, nonempty priced routes, and positive adjustment rates are necessary model assumptions, not consequences of an arbitrary nonnegative continuous price function. Under the standard positive-data, nontrivially priced interpretation, every physical trajectory converges to the unique point proved above.

Each user maintains a transmission rate $x_r$. Resource $j$ measures its aggregate load $y_j={\sum }_{r:j\in r}{x}_r$ and returns the resource congestion price ${\mu }_j=p_j(y_j)$. The user adds these feedback signals to form its route congestion price $c_r={\sum }_{j\in r}{\mu }_j$. Its increase term is the constant ${\kappa }_r{w}_r$, while its decrease term ${\kappa }_r{x}_r{c}_r$ is proportional to both its current rate and the congestion signal. This is a distributed congestion control algorithm: resources need only local loads, and users need only prices along their own routes.

There is a needed distinction in the word increasing. If it permits nondecreasing price functions, the all-zero prices are allowed. Then ${\dot{x}}_r={\kappa }_r{w}_r>0$, so there is no equilibrium. The intended convergence statement holds when each nonempty route meets at least one resource whose price is positive at some load. This is automatic if every price is strictly increasing and nonnegative. We prove it under the more general nondecreasing assumption with this explicit condition, for finite sets of nonempty routes and nonnegative initial rates.

Write $P_j(y)={\int }_0^y{p}_j(u)du$. Each $P_j$ is convex and nondecreasing, and the primal congestion potential isOn positive rates this is strictly concave: the sum of positive-weight logarithms is strictly concave, while the resource term is convex. Its derivatives areConsequently the gradient flow with diagonal mobility satisfieswith equality exactly at an equilibrium.

We must also prove trapping and existence of the equilibrium. For each route choose a resource on it and constants $M_r,c_r^0>0$ such that its price is at least $c_r^0$ beyond load $M_r$. Finiteness gives common constants $M,c>0$. If $X={\max }_r{x}_r$ is large, a resource on a maximizing route has load at least $X$, so the resource penalty is at least $c(X-M)$. With $W={\sum }_r{w}_r$, therefore,Once all rates are bounded above, a rate tending to zero also makes $U\to -\infty$ through its positive-weight logarithm. Thus every nonempty superlevel set of $U$ is compact inside the positive orthant. A maximum exists by restricting to such a superlevel set, and strict concavity makes it unique, say $x^{\ast }$. It is interior and satisfies ${\partial }_{r}U(x^{\ast })=0$, equivalentlyConversely every positive equilibrium is a stationary point of the strictly concave function $U$ and hence is this maximum.

Only continuity of the price functions is given, so it is useful to justify trajectories without silently assuming differentiability. Put $z_r=2\sqrt{x_r/{\kappa }_r}$ and $V(z)=U(({\kappa }_r{z}_r^2/4{)}_r)$. The dynamics become ordinary gradient flow ascent, $\dot{z}=\nabla V(z)$. The function $V$ is strictly concave on the positive orthant: its user terms are $2w_r\log z_r$ plus constants, and each penalty is a convex nondecreasing function $P_j$ composed with the convex load ${\sum }_{r:j\in r}{\kappa }_r{z}_r^2/4$. Hence the supporting-tangent inequalities giveThe squared distance between two solutions is nonincreasing, proving uniqueness even though the vector field need only be continuous.

Nonnegative initial rates enter the positive orthant immediately: along a solution, variation of constants in ${\dot{x}}_r+{\kappa }_r{c}_r(t)x_r={\kappa }_r{w}_r$ expresses $x_r(t)$ as a positive exponential times $x_r(0)$ plus a strictly positive integral. Also ${\dot{x}}_r\le {\kappa }_r{w}_r$, so rates cannot become infinite in finite time. The distance argument applies after any positive time and, by taking that time down to zero, also proves uniqueness for initial zero rates.

For a positive starting time, the increasing $U$ traps the trajectory in a compact superlevel set. Its derivative has finite integral since $U$ is bounded above there. It is uniformly continuous in time, being a continuous function of the state on a compact set with bounded velocity. By uniformly continuous integrable functions vanish at infinity, $\dot{U}(t)\to 0$. Every limit point consequently has ${\partial }_{r}U=0$ for every $r$, since all rates are bounded away from zero. There is only the limit point $x^{\ast }$, soThis establishes global convergence of primal congestion control with the needed nonzero-price condition, and supplies the zero-price counterexample when increasing is read in the weak sense.

For the alternative increasing-supply resource-price dynamics, consider the dual congestion potentialIts domain is ${\mu }_j\ge 0$ with every $c_r>0$. Each integrated supply is strictly convex, because $q_j$ is strictly increasing, and each negative logarithm of a positive linear function is convex. Thus $H$ is strictly convex. Its sublevel sets are bounded: for each $j$, $q_j(1)>0$ gives a linear lower bound on the supply primitive at large ${\mu }_j$, whereas the negative logarithmic terms grow only logarithmically in ${\max }_j{\mu }_j$. A route-price sum tending to zero makes $H\to +\infty$ on a bounded set. A minimum therefore exists in this domain.

The derivatives areIf a used resource had ${\mu }_j=0$ at the minimum, then $q_j(0)=0$ but $y_j>0$, so increasing that coordinate would strictly lower $H$. Therefore the minimizer has strictly positive prices at every used resource. An unused resource has its unique minimizing price zero. At used resources its equations areThe multiplicative dynamics are ${\dot{\mu }}_j=-{\kappa }_j{\mu }_j{\partial }_{j}H$. On strictly positive used-resource prices, equilibrium is equivalent to these equations, and strict convexity proves existence and uniqueness of the positive equilibrium. No full-row-rank assumption on $A$ is needed here: the strictly increasing individual supplies already give strict convexity.

If zero prices at used resources are allowed, the unqualified uniqueness assertion is false because of boundary equilibria of multiplicative resource prices. For one unit-weight route using two resources and $q_1(u)=q_2(u)=u$, all three price vectorsare equilibria. At $(1,0)$ the user rate is one; the first resource has demand equal to supply, while the second has zero derivative because its price is zero. The last two vectors do not minimize $H$. Thus the positive-price domain, or a boundary rule preventing a used resource from remaining at zero price, is essential.

Finally, the primal and dual equilibrium equations agree when $q_j$ and $p_j$ are inverse functions on the relevant ranges. For example, continuous strictly increasing prices with $p_j(0)=0$ and $p_j(y)\to \infty$ have inverses satisfying the stated supply assumptions. Then ${\mu }_j=p_j(y_j)$ is equivalent to $y_j=q_j({\mu }_j)$, while both systems have $x_r=w_r/c_r$. With inverse price and supply functions, the two equilibria describe the same rates and prices. Without this relation they need not agree: for one route and one resource with $w=1$, $p(y)=y$ and $q(\mu )=2\mu$, the primal equilibrium is $(x,\mu )=(1,1)$ while the dual equilibrium is $(\sqrt{2},1/\sqrt{2})$.

For the increasing-supply resource-price dynamics, assume a finite network, positive weights $w_r$, nonempty routes, nonnegative prices, and initially positive prices on every used resource. Resource $j$ announces price ${\mu }_j$, and a route sees total price $s_r={\sum }_{j\in r}{\mu }_j$. Maximizing $w_r\log x-s_{r}x$ over $x>0$ gives $x_r=w_r/s_r$: this is the route's weighted proportional fairness demand. The supply $q_j({\mu }_j)$ increases with its price, and the multiplicative resource-price dynamics raise a positive price when total demand exceeds supply and lower it otherwise.

The boundary equilibria of multiplicative resource prices show why the positivity qualification is necessary. With two resources, one route using both, $w=1$, ${\kappa }_1={\kappa }_2=1$ and $q_j(u)=u$, the prices $(0,1)$ and $(1,0)$ are distinct equilibria with well-defined route price one. The positive equilibrium is $(1/\sqrt{2},1/\sqrt{2})$. A zero coordinate stays zero because it multiplies its own derivative. Thus the printed claim about all trajectories is false if these boundary initial conditions are admitted. We now prove the intended convergence for positive initial prices; unused resources may start at zero.

For positive-price convergence for increasing resource supplies, define $Q_j(u)={\int }_0^u{q}_j(\eta )d\eta$. Strict increase of $q_j$ makes $Q_j$ strictly convex, even without differentiability of $q_j$. On the convex domain $\mu \ge 0$ with every $s_r>0$, the supplied function $V={\sum }_r{w}_r\log s_r-{\sum }_j{Q}_j({\mu }_j)$ is strictly concave, since each logarithmic term is concave and the negative integral terms give strictness in every changed coordinate. Its gradient isIt is also coercive toward large prices: for fixed $a>0$, $Q_j(u)\ge q_j(a)(u-a)$ when $u\ge a$, while the positive logarithmic terms grow only logarithmically in the largest coordinate. If any route price tends to zero with the other prices bounded, $V\to -\infty$. Consequently every nonempty superlevel set is compact and bounded away from zero route prices. $V$ attains a unique maximum ${\mu }^{\ast }$.

For a used resource, the derivative at a zero coordinate is ${\sum }_{r:j\in r}{w}_r/s_r>0$, since $q_j(0)=0$. Such a coordinate cannot vanish at the maximum. An unused resource has its unique maximizing coordinate at zero. Thus ${\mu }^{\ast }$ is positive on used resources and satisfies demand equal to supply there. It is the unique equilibrium in that positive domain.

Along any trajectory,The initial superlevel set traps the trajectory and supplies a global upper bound $M$ on every price, and a lower bound on every route price. On any finite time interval $g_j$ is bounded, so ${\mu }_j(t)={\mu }_j(0)\exp ({\kappa }_j{\int }_0^t{g}_j(\mu (u))du)$ stays positive if initially positive. The bounded trajectory therefore exists for all positive times.

More strongly, each used resource has a uniform positive demand boundChoose ${\delta }_j>0$ with $q_j({\delta }_j)<c_j/2$. While $0<{\mu }_j\le {\delta }_j$, its derivative is positive, so ${\mu }_j(t)\ge \min ({\mu }_j(0),{\delta }_j)>0$. Unused prices satisfy ${\dot{\mu }}_j=-{\kappa }_j{\mu }_j{q}_j({\mu }_j)$ and converge to zero.

The nonnegative function $dV/dt$ has finite integral, since $V$ is increasing and bounded above. It is uniformly continuous along the trajectory: on the compact trapped set, the vector field is bounded, the trajectory is Lipschitz, and ${\sum }_j{\kappa }_j{\mu }_j{g}_j^2$ is a continuous, hence uniformly continuous function of $\mu$. The criterion uniformly continuous integrable functions vanish at infinity applies here: otherwise separated intervals around positive peaks would give an infinite integral. Therefore $dV/dt\to 0$. Since used prices stay bounded away from zero, $g_j(\mu (t))\to 0$ on every used resource. Every limit point is thus the unique maximizer ${\mu }^{\ast }$, with zero prices on unused resources. Compactness now implies $\boxed{\mu (t)\to {\mu }^{\ast }}$. Only continuity and strict increase of the supply functions were used; differentiability or an unbounded supply was not required. The boundary counterexample remains a necessary qualification of the literal source assertion.

For an integer $s\ge 0$, the Sobolev space iswhere $D^{\beta }$ is a weak derivative. For $p<\infty$ one may use the Sobolev norm $({\sum }_{|\beta |\le s}\Vert D^{\beta }u{\Vert }_p^p{)}^{1/p}$; for $p=\infty$ use the maximum of the finitely many essential-supremum norms.

For $1\le p<d$, the Sobolev inequality is $\Vert u{\Vert }_{p^{\ast }}\le C_{d,p}\Vert \nabla u{\Vert }_p$, with Sobolev conjugate exponent $p^{\ast }=dp/(d-p)$. For $d<p<\infty$, Morrey's inequality supplies a continuous representative satisfyingAt $p=\infty$ the representative is Lipschitz continuous. At the critical exponent $p=d>1$, first-order Sobolev regularity gives every finite $L^q$ embedding for $q\ge d$, with an inhomogeneous norm, but generally no $L^{\infty }$ embedding. The one-dimensional endpoint $W^{1,1}(\mathbb{R})\hookrightarrow L^{\infty }(\mathbb{R})$ is an exception.

For the proof of Morrey's inequality, start with a smooth $u$ and write $u_{B(x,r)}$ for its average on a ball. Averaging the fundamental theorem of calculus along a line segment and changing radial variables givesThe last step is the Holder inequality; integrability of the kernel to power $p^{\prime }$ is exactly $(d-1)p^{\prime }<d$. For $r=|x-y|$, translate the averaging ball along the segment from $x$ to $y$. The fundamental theorem of calculus along a line segment and the Holder inequality giveCombining the two point-to-average bounds and this average-to-average bound proves the required Hölder estimate. The point-to-average bound with $r=1$, together with $|u_{B(x,1)}|\le |B_1{|}^{-1/p}\Vert u{\Vert }_p$, gives the supremum estimate. Density of smooth functions in a Sobolev space then gives a uniformly convergent sequence of smooth representatives, preserving both bounds. For $p=\infty$, mollification gives the Lipschitz version.

For the decay conclusion assume $d<p<\infty$. The representative is uniformly continuous. If $|u(x_j)|\ge 2\theta >0$ along points escaping to infinity, the Hölder bound gives a radius $\rho >0$, independent of $j$, on which $|u|\ge \theta$. A subsequence has disjoint radius-$\rho$ balls, each contributing at least ${\theta }^p|B_{\rho }|$ to $\Vert u{\Vert }_p^p$, a contradiction. This is uniformly continuous integrable functions vanish at infinity.

The finite-$p$ restriction is necessary. If the printed range includes $p=\infty$, its decay assertion is false: $u\equiv 1$ belongs to $W^{1,\infty }$ but does not tend to zero.

Work with real-valued functions. The screened sine-Gordon energy is well defined: $0\le 1-\cos u\le u^2/2$, and $fu$ is integrable by the Cauchy-Schwarz inequality. It is coercive, becauseA minimizing sequence is bounded in the Hilbert space $H^1$. Weak sequential compactness of bounded sequences in a reflexive Banach space supplies a weakly convergent subsequence. The squared H1 space norm is weakly lower semicontinuous, the source pairing is weakly continuous, and the nonlinear term is covered by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. The direct method in the calculus of variations therefore gives a minimizer $\varphi$.

Taking its first variation in any $v\in H^1$ givesThus the Euler-Lagrange equation isas a weak solution, equivalently in distributions when tested against smooth compactly supported functions. The derivative of the nonlinear term is justified by $|\sin \varphi |\le |\varphi |$ and the second-order remainder bound $|\cos |\le 1$.

Now $G=f-\sin \varphi \in L^2$, so the supplied elliptic regularity estimate puts $\varphi$ in $H^2$. Applying the Sobolev inequality to $\varphi$ and each first weak derivative gives $\varphi \in W^{1,6}({\mathbb{R}}^3)$. Morrey's inequality and uniformly continuous integrable functions vanish at infinity prove that its continuous representative tends to zero.

The final printed supremum estimate is false in general. The maximum bound for a monotone reaction term involves $h(s)=s+\sin s$, which is odd and strictly increasing: $h^{\prime }=1+\cos s\ge 0$, and its zeros are isolated. If $M=\Vert f{\Vert }_{\infty }<\infty$, a positive maximum $m$ of $\varphi$ satisfies $h(m)\le f\le M$; apply the same argument to $-\varphi$. The valid general estimate isThe bound by $M$ is valid if $M\le \pi$, but $h(s)$ can be smaller than $s$ for larger positive $s$.

For an explicit failure of the source-size bound for the screened sine-Gordon equation, take $a=3\pi /2$, $L=10$, $\varphi (x)=a{e}^{-|x{|}^2/L^2}$ and $f=-\Delta \varphi +\varphi +\sin \varphi$. These are smooth functions in the required spaces. With $q=|x{|}^2/L^2$,Hence $\Vert f{\Vert }_{\infty }\le a-1+6a/L^2<a=\Vert \varphi {\Vert }_{\infty }$. Moreover $V(s)=s^2/2+1-\cos s$ has $V^{\prime \prime }=1+\cos s\ge 0$, so the energy is convex and this critical point is a minimizer. The example therefore satisfies even the minimizing and $C^2$ hypotheses of the printed claim.

For finite nonempty routes and positive weights, the potential of increasing-supply resource-price dynamics has a unique maximum. Increasing supplies give a linear penalty at large prices, so its superlevel sets are compact, and logarithmic route terms exclude zero route totals. Used resources have uniformly positive demand on a trapped set; a sufficiently small positive price therefore increases, giving a positive lower bound. The potential derivative is ${\sum }_j{\kappa }_j{\mu }_j({\partial }_{j}V{)}^2$. It has finite integral and is uniformly continuous, so it tends to zero by uniformly continuous integrable functions vanish at infinity. Every limit point then satisfies the unique stationary equations. Unused resource prices decay to zero. Only continuity and strict increase of the supply functions are needed.

For real $f\in L^2({\mathbb{R}}^3)$, this energy functional on $H^1({\mathbb{R}}^3)$ is coercive and attains its minimum by the direct method in the calculus of variations. The nonnegative potential term is weakly lower semicontinuous by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. Its Euler-Lagrange equation is $-\Delta u+u+\sin u=f$ in the weak solution sense. The extra linear restoring term screens the Sine-Gordon equation nonlinearity. The scalar potential $V(s)=s^2/2+1-\cos s$ is convex because $V^{\prime \prime }(s)=1+\cos s\ge 0$; hence any weak critical point is a minimizer. The elliptic regularity estimate for $-\Delta +1$, together with $|\sin u|\le |u|$, gives $u\in H^2$. The Sobolev inequality then puts $u$ in $W^{1,6}$, so Morrey's inequality and uniformly continuous integrable functions vanish at infinity give a continuous representative tending to zero.