Let us consider the classical Lotka-Volterra system (see [1, 8]) modeling the prey-predator dynamics, with a control parameter $\lambda$ aimed to modify the rate of attack on the prey population in a such way that after a time period $T,$ the prey population reaches a desired level $x_T>0:$

A mathematical model for the steady-state flow of an incompressible fluid in a given domain and with null velocity on its boundary is given by the well-known Stokes system (see [4, 5, 9, 14, 15])

Here $\mathrm{\Omega }\subset {ℝ}^n$ is a bounded open set, $u=(u_1,..,u_n)$ and $h=(h_1,..,h_n)\in H^{-1}(\mathrm{\Omega };{ℝ}^n).$ Physically, there are relevant the cases $n=2$ and $n=3,$ when $h\left(x\right)$ stands for the external force, the unknown functions $u$ and $p$ are the velocity and pressure, respectively, while the condition $\text{div }u=0$ means that the fluid is incompressible.

It is well-known that the problem has a unique solution

where $V=\{u\in H_0^1(\mathrm{\Omega };{ℝ}^n):\text{div }u=0\}.$ Note that the representation of $q$ as a gradient of a function $p$ is guaranteed by Rham’s Lemma.

Consider now the case of a reaction force $\mathrm{\Phi }\left(u\right),$ $\mathrm{\Phi }:H_0^1(\mathrm{\Omega };{ℝ}^n)\to H^{-1}(\mathrm{\Omega };{ℝ}^n),$ namely the problem

or equivalently

Denoting $q:=\nabla p$, it appears to be a control problem for a fixed point equation, where $u$ is the state variable, $q$ is the control variable and $\text{div }u=0$ is the controllability condition. Notice the specificity of this problem for which the control is of the process itself, in order to keep the fluid incompressibility, and not an external one. Here

If $(u,q)$ is a solution of the problem, then

The last equality gives the necessary expression of the control variable $q$ in terms of the state variable $u.$ Thus, with the notation in Section 1,

Obviously, we can extend $F$ from ${\mathrm{\Sigma }}_1$ to the whole space $H_0^1(\mathrm{\Omega };{ℝ}^n)$ by using the same expression. Thus

Now the map $N(u,\tilde{F}\left(u\right))$ from the new fixed point equation is in this case

Thus, the solvability of the nonlinear Stokes system reduces to finding a fixed point $u\in$ $H_0^1(\mathrm{\Omega };{ℝ}^n)$ of the operator $N(u,\tilde{F}\left(u\right))={\mathrm{Pr}}_V{(-\mathrm{\Delta })}^{-1}\mathrm{\Phi }\left(u\right),$ that is to solving the equation

The projection operator ${\mathrm{Pr}}_V$ being nonexpansive, the fixed point approach for elliptic equations (see [11]) can be easily adapted to this case. Here is an existence and uniqueness result based on Banach’s contraction principle, showing the unique controllability of the above problem.

For example, the Nemytskii operator $\mathrm{\Phi }\left(u\right)=f(x,u,\nabla u),$ where $f:\mathrm{\Omega }\times {ℝ}^n\times {ℝ}^{n^2}\to {ℝ}^n$ is a Carathéodory function such that $f(.,0,0)\in L^2(\mathrm{\Omega };{ℝ}^n)$ and

for all $u,\overline{u}\in {ℝ}^n;$ $v,\overline{v}\in {ℝ}^{n^2}$ and some $a,b\ge 0,$ satisfies the condition of Theorem 3 provided that Here ${\lambda }_1$ is the first eigenvalue of the Dirichlet problem for $-\mathrm{\Delta }.$ Indeed, using two times Poincaré’s inequality, one has

Some other existence results for problem (2.4) can be obtained via Schauder’s and Leray-Schauder’s fixed point theorems, as in [11], if the global Lipschitz condition on $\mathrm{\Phi }$ is relaxed.

In paper [12], it is discussed the existence and localization of radial solutions of the Neumann problem for general $\varphi$-Laplace equations with a state-dependent source term. The starting assumption was that for each  $h\in C[R_0,R],$ there exists at least one solution to the problem

Here is an increasing homeomorphism with $\varphi \left(0\right)=0,$ $\epsilon >0$ and Notice that this happens for singular homeomorphisms, i.e., for as shows Corollary 2.4 in [3].

A double integration leads (see [12, Lemma 2.1]) to the conclusion that a function $v$ is a solution of (2.5) if and only if it satisfies the following two conditions:

where $\lambda =v\left(R_0\right).$ Relation (2.6) is a fixed point equation in the unknown $v\in C[R_0,R],$ $\lambda$ appears as a control parameter and (2.7) can be seen as the controllability condition. Hence, under the notations in Section 1, we have

Assuming that $(v,\lambda )$ is a solution and replacing it in (2.7) we have

whence the expression of $\lambda ,$ namely

We use the same expression for the extension $\tilde{F}$ of $F$ to the whole space $C[R_0,R]$ and we consider the operator

Clearly, $\tilde{N}\left(D_0\right)\subset D_0$ and $\tilde{N}$ is completely continuous. We think that the existence of a fixed point of $\tilde{N}$ could be obtained via Schauder’s fixed point theorem. The set $D_0$ is closed and convex but unbounded. Hence, in order that Schauder’s theorem apply we need to find a ball of a sufficiently large radius which is invariated by $\tilde{N}$ . Such a ball exists if we consider that the function ${\varphi }^{-1}$ has only a linear growth, i.e., $\left|{\varphi }^{-1}\left(s\right)\right|\le \theta \left|s\right|+C$ for all $s\in ℝ$ and Indeed, one has

and

Hence

Since we can choose the radius $\rho$ of the ball sufficiently large such that

which guarantee that the intersection $D_{\rho }$ of $D_0$ with the closed ball centered at the origin and of radius $\rho$ is nonempty ($h/\epsilon$ belongs to both of them) and $\tilde{N}\left(D_{\rho }\right)\subset D_{\rho }.$ Hence Schauder’s fixed point applies. Thus we have the following result.

Notice that for a singular homeomorphism, i.e., when condition (2.8) holds with $\theta =0$ and $C=a.$ Thus Theorem 4 applies for every $\epsilon >0$.

Also note that if ${\varphi }^{-1}$ is $\theta$-Lipschitz on $ℝ$ and (2.9) holds, then for any $h,$ problem (2.5) has a unique solution. In this case, the operator $\tilde{N}$ is a contraction on $D_0$ and the result follows from Banach’s contraction principle.

Finally, note that if instead of $h\left(r\right)$ we consider more generaly $f(r,v\left(r\right)),$ and we assume that $\left|f(r,s)\right|\le \mu \left|s\right|+\gamma$ for all $s\in ℝ,$ then

Hence, if there exists $\rho >0$ with

then Schauder’s theorem applies in the ball of radius $\rho .$ This sufficient condition holds if and $\varphi$ is singular or more general with ${lim}_{s\to +\mathrm{\infty }}{\varphi }^{-1}\left(s\right)/s=0$. The case $\mu =0$ and $\varphi$ singular is covered by Corollary 2.4 in [3]. More general, the invariance of a large ball is obtained if where $l={lim}_{s\to +\mathrm{\infty }}{\varphi }^{-1}\left(s\right)/s.$