## Abstract

The aim of this paper is to draw attention to a general principle for solving control problems for operator equations with the help of fixed point techniques. Three distinct applications are presented: a control problem related to the Lotka–Volterra system, the nonlinear Stokes system, and radial solutions of the Neumann problem for φ-Laplace equations. Only for the first application, the control is an explicit external one, while for the next two applications, the control is dependent on the models and arises from the necessity to conform to the actual modeled process or to a certain boundary condition. From the perspective of those interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method thus paving the way for new applications. From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators.

## Authors

**Radu Precup**

Institute of Advanced Studies in Science and Technology, Babes-Bolyai University, Cluj-Napoca, Romania

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

## Keywords

## Paper coordinates

R. Precup, *On some applications of the controllability principle for fixed point equations*, Results Appl. Math., 13 (2022), art. no. 100236, https://doi.org/10.1016/j.rinam.2021.100236

## About this paper

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Results in Applied Mathematics

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Elsevier

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25900374

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# On some applications of the controllability principle for fixed point equations

###### Abstract

The aim of this paper is to draw attention to a general principle for solving control problems for operator equations with the help of fixed point techniques. Three distinct applications are presented: a control problem related to the Lotka-Volterra system, the nonlinear Stokes system, and radial solutions of the Neumann problem for $\varphi $-Laplace equations. Only for the first application, the control is an explicit external one, while for the next two applications, the control is dependent on the models and arises from the necessity to conform to the actual modeled process or to a certain boundary condition. From the perspective of those interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method thus paving the way for new applications. From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators.

Keywords: control problem, fixed point, boundary value problem, Stokes system, radial solution, Lotka-Volterra system

Subject Classification: 34H05, 34K35, 35Q30, 37N25

## 1 Introduction

For the field of differential equations, “the general problem of control theory is that to reconstituting a differential system (as a matter of fact some of its parameters viewed as control variables) from certain properties of solution” ([2, p. 34]). In [7] we have introduced a controllability principle for a general control problem related to operator equations, that we reproduce here for the convenience of the reader. It consists in finding $(w,\lambda ),$ a solution to the following system

$$\{\begin{array}{c}w=N(w,\lambda )\\ w\in W,\lambda \in \mathrm{\Lambda},(w,\lambda )\in D\end{array}$$ | (1.1) |

associated to the fixed point equation $w=N(w,\lambda ).$ Here $w$ is the state variable, $\lambda $ is the control variable, $W$ is the domain of the states, $\mathrm{\Lambda}$ is the domain of controls and $D$ is the controllability domain, usually given by means of some condition/property imposed to $w,$ or to both $w$ and $\lambda .$ Notice the very general formulation of the control problem, in terms of sets, where $W,\mathrm{\Lambda}$ and $D\subset W\times \mathrm{\Lambda}$ are not necessarily structured sets and $N$ is any mapping from $W\times \mathrm{\Lambda}$ to $W.$

In this context, we say that the equation $w=N(w,\lambda )$ is controllable in $W\times \mathrm{\Lambda}$ with respect to $D,$ providing that problem (1.1) has a solution $(w,\lambda )$. If the solution is unique we say that the equation is uniquely controllable.

Let $\mathrm{\Sigma}$ be the set of all possible solutions $(w,\lambda )$ of the fixed point equation and ${\mathrm{\Sigma}}_{1}$ be the set of all $w$ that are first components of some solutions of the fixed point equation, that is

$\mathrm{\Sigma}$ | $=$ | $\{(w,\lambda )\in W\times \mathrm{\Lambda}:w=N(w,\lambda )\},$ | ||

${\mathrm{\Sigma}}_{1}$ | $=$ | $\{w\in W:\text{there is}\lambda \in \mathrm{\Lambda}\text{with}(w,\lambda )\in \mathrm{\Sigma}\}.$ |

Clearly, the set of all solutions of the control problem (1.1) is given by $\mathrm{\Sigma}\cap D.$

Consider the set-valued map $F:{\mathrm{\Sigma}}_{1}\to \mathrm{\Lambda}$ defined as

$$F\left(w\right)=\{\lambda \in \mathrm{\Lambda}:\text{}(w,\lambda )\in \mathrm{\Sigma}\cap D\}.$$ |

Roughly speaking, $F$ gives the ‘expression’ of the control variable in terms of the state variable.

We have the following general principle for solving the control problem (1.1).

###### Proposition 1

If for some extension $\stackrel{~}{F}:W\to \mathrm{\Lambda}$ of $F$ from ${\mathrm{\Sigma}}_{1}$ to $W,$ there exists a fixed point $w\in W$ of the set-valued map

$$\stackrel{~}{N}\left(w\right):=N(w,\stackrel{~}{F}\left(w\right)),$$ |

i.e.,

$$w=N(w,\lambda ),$$ | (1.2) |

for some $\lambda \in \stackrel{~}{F}\left(w\right),$ then the couple $(w,\lambda )$ is a solution of the control problem *(1.1)*.

Proof. Clearly $(w,\lambda )\in \mathrm{\Sigma}.$ Hence $w\in {\mathrm{\Sigma}}_{1}$ and so $\stackrel{~}{F}\left(w\right)=F\left(w\right).$ Then $\lambda \in F\left(w\right)$ and from the definition of $F,$ it follows that $(w,\lambda )\in D.$ Therefore $(w,\lambda )$ solves (1.1).

Note that $F$ and $\stackrel{~}{F}$ can in particular be single-valued maps and in many cases the extension $\stackrel{~}{F}$ can be done using the expression of $F.$

Two applications for a system modeling cell dynamics related to leukemia have been included in [7]. The aim of this paper is to present some other problems that can be treated by using this principle, hoping this way to pave the way for other further applications.

## 2 The applications

### 2.1 A control problem related to the Lotka-Volterra system

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 ${x}_{T}>0:$

$$\{\begin{array}{c}{x}^{\prime}=ax\left(1-\lambda by\right)\hfill \\ {y}^{\prime}=-cy\left(1-dx\right)\hfill \\ x\left(0\right)={x}_{0},y\left(0\right)={y}_{0},x\left(T\right)={x}_{T},x,y>0.\hfill \end{array}$$ |

###### Theorem 2

Let $$ The system is controllable on a short time interval $[0,T]$ with

$T$ | $$ | $\mathrm{min}\{{\displaystyle \frac{1}{4a}},{\displaystyle \frac{1}{c}},{\displaystyle \frac{2{x}_{0}-{x}_{T}}{4a{x}_{0}}}\},$ | ||

$1-2aT$ | $>$ | $2cT\left(1-2aT+d{x}_{T}\right).$ |

Proof. Assume without loss of generality that $b=1.$ Integrating gives

$$\{\begin{array}{c}x\left(t\right)={x}_{0}+a{\int}_{0}^{t}x\left(s\right)\left(1-\lambda y\left(s\right)\right)\mathit{d}s\hfill \\ y\left(t\right)={y}_{0}-c{\int}_{0}^{t}y\left(s\right)\left(1-dx\left(s\right)\right)\mathit{d}s.\hfill \end{array}$$ |

This is our fixed point equation with $w=(x,y),$ $W=C([0,T];{(0,+\mathrm{\infty})}^{2}),$ $\mathrm{\Lambda}=\mathbb{R}$ and $D=\{(x,y,\lambda ):x\left(T\right)={x}_{T}\}.$ Using the controllability condition $x\left(T\right)={x}_{T},$ we obtain the necessary form of the control parameter $\lambda ,$ namely

$$\lambda =F(x,y):=\frac{a{\int}_{0}^{T}x\left(s\right)\mathit{d}s-{x}_{T}+{x}_{0}}{a{\int}_{0}^{T}x\left(s\right)y\left(s\right)\mathit{d}s}.$$ |

This expression can be used to define the extension $\stackrel{~}{F}$ of $F$ from ${\mathrm{\Sigma}}_{1}$ to the whole $W.$ Thus we have to find a solution $(x,y)$ of the system

$$\{\begin{array}{c}x\left(t\right)=A(x,y):={x}_{0}+a{\int}_{0}^{t}x\left(s\right)\left(1-\stackrel{~}{F}(x,y)y\left(s\right)\right)\mathit{d}s\hfill \\ y\left(t\right)=B(x,y):={y}_{0}-c{\int}_{0}^{t}y\left(s\right)\left(1-dx\left(s\right)\right)\mathit{d}s.\hfill \end{array}$$ |

We shall look for a solution in the closed convex and bounded set

$${D}_{0}:=\{(x,y)\in C([0,T];{\mathbb{R}}^{2}):{m}_{x}\le x\le {M}_{x},{m}_{y}\le y\le {M}_{y}\},$$ |

where

$$\begin{array}{cc}{m}_{x}=\frac{2{x}_{0}\left(1-2aT\right)-{x}_{T}}{1-2aT},\hfill & {M}_{x}=\frac{{x}_{T}}{1-2aT},\hfill \\ {m}_{y}={y}_{0}\frac{1-2cT\left(1+d{M}_{x}\right)}{1-cT\left(1+d{M}_{x}\right)},\hfill & {M}_{y}=\frac{{y}_{0}}{1-cT\left(1+d{M}_{x}\right)}.\hfill \end{array}$$ |

One can check that

$$ |

Let $\stackrel{~}{N}=(A,B).$ We have that $\stackrel{~}{N}\left({D}_{0}\right)\subset {D}_{0}.$ Indeed,

$\left|A(x,y)\left(t\right)-{x}_{0}\right|$ | $\le $ | $aT{M}_{x}+\left|a{\displaystyle {\int}_{0}^{t}}x\mathit{d}s-{x}_{T}+{x}_{0}\right|$ | ||

$\le $ | $2aT{M}_{x}+{x}_{T}-{x}_{0}.$ |

Hence

$${m}_{x}=2{x}_{0}-{x}_{T}-2aT{M}_{x}\le A(x,y)\left(t\right)\le 2aT{M}_{x}+{x}_{T}={M}_{x}.$$ |

Also

$$\left|B(x,y)\left(t\right)-{y}_{0}\right|\le cT{M}_{y}\left(1+d{M}_{x}\right).$$ |

Hence

$${m}_{y}={y}_{0}-cT{M}_{y}\left(1+d{M}_{x}\right)\le B(x,y)\left(t\right)\le {y}_{0}+cT{M}_{y}\left(1+d{M}_{x}\right)={M}_{y}.$$ |

In addition, the Arzelà–Ascoli theorem guarantees that the operator $\stackrel{~}{N}$ is compact on ${D}_{0}.$ Thus Schauder’s fixed point theorem applies and the proof is finished.

### 2.2 The nonlinear stationary Stokes system

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])

$$\{\begin{array}{c}-\mathrm{\Delta}u+\nabla p=h\left(x\right)\phantom{\rule{1em}{0ex}}\text{in}\mathrm{\Omega}\hfill \\ \text{div}u=0\hfill \\ u=0\phantom{\rule{1em}{0ex}}\text{on}\partial \mathrm{\Omega}.\hfill \end{array}$$ | (2.3) |

Here $\mathrm{\Omega}\subset {\mathbb{R}}^{n}$ is a bounded open set, $u=({u}_{1},..,{u}_{n})$ and $h=({h}_{1},..,{h}_{n})\in {H}^{-1}(\mathrm{\Omega};{\mathbb{R}}^{n}).$ 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

$u$ | $=$ | ${\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}h,$ | ||

$q$ | $:$ | $=\nabla p=h+\mathrm{\Delta}u=h+\mathrm{\Delta}{\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}h,$ | ||

$p$ | $\in $ | ${L}^{2}\left(\mathrm{\Omega}\right),$ |

where $V=\{u\in {H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n}):\text{div}u=0\}.$ 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 $\mathrm{\Phi}\left(u\right),$ $\mathrm{\Phi}:{H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n})\to {H}^{-1}(\mathrm{\Omega};{\mathbb{R}}^{n}),$ namely the problem

$$\{\begin{array}{c}-\mathrm{\Delta}u+\nabla p=\mathrm{\Phi}\left(u\right)\phantom{\rule{1em}{0ex}}\text{in}\mathrm{\Omega}\hfill \\ \text{div}u=0\hfill \\ u=0\phantom{\rule{1em}{0ex}}\text{on}\partial \mathrm{\Omega},\hfill \end{array}$$ | (2.4) |

or equivalently

$$\{\begin{array}{c}u={\left(-\mathrm{\Delta}\right)}^{-1}\left(\mathrm{\Phi}\left(u\right)-\nabla p\right),u\in {H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n})\hfill \\ \text{div}u=0.\hfill \end{array}$$ |

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

$W$ | $=$ | ${H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n}),\mathrm{\Lambda}={H}^{-1}(\mathrm{\Omega};{\mathbb{R}}^{n}),D=V\times {H}^{-1}(\mathrm{\Omega};{\mathbb{R}}^{n}),$ | ||

$N(u,q)$ | $=$ | ${\left(-\mathrm{\Delta}\right)}^{-1}\left(\mathrm{\Phi}\left(u\right)-q\right).$ |

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

$u$ | $\in $ | $V,\text{}u={\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}\mathrm{\Phi}\left(u\right)\phantom{\rule{1em}{0ex}}\text{and}$ | ||

$q$ | $=$ | $q\left(u\right)=\nabla p\left(u\right)=\mathrm{\Phi}\left(u\right)+\mathrm{\Delta}u=\mathrm{\Phi}\left(u\right)+\mathrm{\Delta}{\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}\mathrm{\Phi}\left(u\right).$ |

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,

$$F\left(u\right)=\mathrm{\Phi}\left(u\right)+\mathrm{\Delta}{\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}\mathrm{\Phi}\left(u\right),u\in {\mathrm{\Sigma}}_{1}.$$ |

Obviously, we can extend $F$ from ${\mathrm{\Sigma}}_{1}$ to the whole space ${H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n})$ by using the same expression. Thus

$$\stackrel{~}{F}\left(u\right)=\mathrm{\Phi}\left(u\right)+\mathrm{\Delta}{\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}\mathrm{\Phi}\left(u\right),u\in {H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n}).$$ |

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

$N(u,\stackrel{~}{F}\left(u\right))$ | $=$ | ${(-\mathrm{\Delta})}^{-1}(\mathrm{\Phi}\left(u\right)-\stackrel{~}{F}\left(u\right))={(-\mathrm{\Delta})}^{-1}(\mathrm{\Phi}\left(u\right)-\mathrm{\Phi}\left(u\right)-\mathrm{\Delta}{\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}\mathrm{\Phi}\left(u\right))$ | ||

$=$ | ${\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}\mathrm{\Phi}\left(u\right).$ |

Thus, the solvability of the nonlinear Stokes system reduces to finding a fixed point $u\in $ ${H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n})$ of the operator $N(u,\stackrel{~}{F}\left(u\right))={\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}\mathrm{\Phi}\left(u\right),$ that is to solving the equation

$$u={\mathrm{Pr}}_{V}{(-\mathrm{\Delta})}^{-1}\mathrm{\Phi}\left(u\right)\phantom{\rule{2em}{0ex}}\text{in}{H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n}).$$ |

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.

###### Theorem 3

If there exists $\theta \in [0,1)$ such that

$${\left|\mathrm{\Phi}\left(u\right)-\mathrm{\Phi}\left(v\right)\right|}_{{H}^{-1}}\le \theta {\left|u-v\right|}_{{H}_{0}^{1}}$$ |

for all $u,v\in {H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n}),$ then problem *(2.4)* has a unique solution.

Proof. For any $u,v\in {H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n}),$ one has

${\left|N(u,\stackrel{~}{F}\left(u\right))-N(v,\stackrel{~}{F}\left(v\right))\right|}_{{H}_{0}^{1}}$ | $\le $ | ${\left|{\left(-\mathrm{\Delta}\right)}^{-1}\left(\mathrm{\Phi}\left(u\right)-\mathrm{\Phi}\left(v\right)\right)\right|}_{{H}_{0}^{1}}$ | ||

$=$ | ${\left|\mathrm{\Phi}\left(u\right)-\mathrm{\Phi}\left(v\right)\right|}_{{H}^{-1}}$ | |||

$\le $ | $\theta {\left|u-v\right|}_{{H}_{0}^{1}}.$ |

Thus the operator $N(u,\stackrel{~}{F}\left(u\right))$ is a contraction on ${H}_{0}^{1}(\mathrm{\Omega};{\mathbb{R}}^{n}).$

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

$$\left|f(x,u,v)-f(x,\overline{u},\overline{v})\right|\le a\left|u-\overline{u}\right|+b\left|v-\overline{v}\right|$$ |

for all $u,\overline{u}\in {\mathbb{R}}^{n};$ $v,\overline{v}\in {\mathbb{R}}^{{n}^{2}}$ 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

${\left|\mathrm{\Phi}\left(u\right)-\mathrm{\Phi}\left(v\right)\right|}_{{H}^{-1}}$ | $=$ | $|f(.,u,\nabla u)-f(.,v,\nabla v){|}_{{H}^{-1}}$ | ||

$\le $ | $\frac{1}{\sqrt{{\lambda}_{1}}}}|f(.,u,\nabla u)-f(.,v,\nabla v){|}_{{L}^{2}$ | |||

$\le $ | $\frac{1}{\sqrt{{\lambda}_{1}}}}\left(a{\left|u-v\right|}_{{L}^{2}}+b{\left|\nabla \left(u-v\right)\right|}_{{L}^{2}}\right)$ | |||

$=$ | $\frac{1}{\sqrt{{\lambda}_{1}}}}\left(a{\left|u-v\right|}_{{L}^{2}}+b{\left|u-v\right|}_{{H}_{0}^{1}}\right)$ | |||

$\le $ | $\left({\displaystyle \frac{a}{{\lambda}_{1}}}+{\displaystyle \frac{b}{\sqrt{{\lambda}_{1}}}}\right){\left|u-v\right|}_{{H}_{0}^{1}}.$ |

### 2.3 Radial solutions of the Neumann problem for $\varphi $-Laplace equations

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

$$\{\begin{array}{cc}L\left(v\right)\left(r\right):=-{\left({r}^{n-1}\varphi \left({v}^{\prime}\right)\right)}^{\prime}+\epsilon {r}^{n-1}v={r}^{n-1}h\left(r\right)\hfill & \text{in}({R}_{0},R)\hfill \\ {v}^{\prime}\left({R}_{0}\right)={v}^{\prime}(R)=0.\hfill & \end{array}$$ | (2.5) |

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:

$$v\left(r\right)=\lambda +{\int}_{{R}_{0}}^{r}{\varphi}^{-1}\left({s}^{1-n}{\int}_{s}^{R}{\tau}^{n-1}\left(h-\epsilon v\right)\mathit{d}\tau \right)\mathit{d}s\phantom{\rule{2em}{0ex}}(r\in [{R}_{0},R])$$ | (2.6) |

$${\int}_{{R}_{0}}^{R}{r}^{n-1}\left(h-\epsilon v\right)\mathit{d}r=0,$$ | (2.7) |

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

$W$ | $=$ | $C[{R}_{0},R],\mathrm{\Lambda}=\mathbb{R},D={D}_{0}\times \mathbb{R},$ | ||

${D}_{0}$ | $=$ | $\{v\in C[{R}_{0},R]:v\text{satisfies (}\text{2.7}\text{)}\},$ | ||

$N(v,\lambda )\left(r\right)$ | $=$ | $\lambda +{\displaystyle {\int}_{{R}_{0}}^{r}}{\varphi}^{-1}\left({s}^{1-n}{\displaystyle {\int}_{s}^{R}}{\tau}^{n-1}\left(h-\epsilon v\right)\mathit{d}\tau \right)\mathit{d}s.$ |

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

$${\int}_{{R}_{0}}^{R}{r}^{n-1}\left(h-\epsilon \lambda -\epsilon {\int}_{{R}_{0}}^{r}{\varphi}^{-1}\left({s}^{1-n}{\int}_{s}^{R}{\tau}^{n-1}\left(h-\epsilon v\right)\mathit{d}\tau \right)\mathit{d}s\right)\mathit{d}r=0,$$ |

whence the expression of $\lambda ,$ namely

$\lambda $ | $=$ | $F\left(v\right)={\displaystyle {\int}_{{R}_{0}}^{R}}{r}^{n-1}\left(h-\epsilon {\displaystyle {\int}_{{R}_{0}}^{r}}{\varphi}^{-1}\left({s}^{1-n}{\displaystyle {\int}_{s}^{R}}{\tau}^{n-1}\left(h-\epsilon v\right)\mathit{d}\tau \right)\mathit{d}s\right)\mathit{d}r/\left(\epsilon {\displaystyle {\int}_{{R}_{0}}^{R}}{r}^{n-1}\mathit{d}r\right)$ | ||

$=$ | $\frac{n}{\left({R}^{n}-{R}_{0}^{n}\right)\epsilon}}{\displaystyle {\int}_{{R}_{0}}^{R}}{r}^{n-1}\left(h-\epsilon {\displaystyle {\int}_{{R}_{0}}^{r}}{\varphi}^{-1}\left({s}^{1-n}{\displaystyle {\int}_{s}^{R}}{\tau}^{n-1}\left(h-\epsilon v\right)\mathit{d}\tau \right)\mathit{d}s\right)\mathit{d}r.$ |

We use the same expression for the extension $\stackrel{~}{F}$ of $F$ to the whole space $C[{R}_{0},R]$ and we consider the operator

$\stackrel{~}{N}\left(v\right)$ | $=$ | $N(v,\stackrel{~}{F}\left(v\right))=\stackrel{~}{F}\left(v\right)+{\displaystyle {\int}_{{R}_{0}}^{r}}{\varphi}^{-1}\left({s}^{1-n}{\displaystyle {\int}_{s}^{R}}{\tau}^{n-1}\left(h-\epsilon v\right)\mathit{d}\tau \right)\mathit{d}s$ | ||

$=$ | $\stackrel{~}{F}\left(v\right)+{\displaystyle {\int}_{{R}_{0}}^{r}}{\varphi}^{-1}\left(-{s}^{1-n}{\displaystyle {\int}_{{R}_{0}}^{s}}{\tau}^{n-1}\left(h-\epsilon v\right)\mathit{d}\tau \right)\mathit{d}s.$ |

Clearly, $\stackrel{~}{N}\left({D}_{0}\right)\subset {D}_{0}$ and $\stackrel{~}{N}$ is completely continuous. We think that the existence of a fixed point of $\stackrel{~}{N}$ 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 $\stackrel{~}{N}$ . 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 $s\in \mathbb{R}$ and $$ Indeed, one has

$\left|{\displaystyle {\int}_{{R}_{0}}^{r}}{\varphi}^{-1}\left(-{s}^{1-n}{\displaystyle {\int}_{{R}_{0}}^{s}}{\tau}^{n-1}\left(h-\epsilon v\right)\mathit{d}\tau \right)\mathit{d}s\right|$ | $\le $ | ${\int}_{{R}_{0}}^{R}}\left(\theta {s}^{1-n}{\displaystyle {\int}_{{R}_{0}}^{s}}{\tau}^{n-1}\left|h-\epsilon v\right|\mathit{d}\tau +C\right)\mathit{d}s$ | ||

$\le $ | ${\left(R-{R}_{0}\right)}^{2}\theta \left({\left|h\right|}_{\mathrm{\infty}}+\epsilon {\left|v\right|}_{\mathrm{\infty}}\right)+\left(R-{R}_{0}\right)C$ |

and

$$\left|\stackrel{~}{F}\left(v\right)\right|\le \frac{{\left|h\right|}_{\mathrm{\infty}}}{\epsilon}+{\left(R-{R}_{0}\right)}^{2}\theta \left({\left|h\right|}_{\mathrm{\infty}}+\epsilon {\left|v\right|}_{\mathrm{\infty}}\right)+\left(R-{R}_{0}\right)C.$$ |

Hence

$$\left|\stackrel{~}{N}\left(v\right)\left(r\right)\right|\le \frac{{\left|h\right|}_{\mathrm{\infty}}}{\epsilon}+2{\left(R-{R}_{0}\right)}^{2}\theta \left({\left|h\right|}_{\mathrm{\infty}}+\epsilon {\left|v\right|}_{\mathrm{\infty}}\right)+2\left(R-{R}_{0}\right)C.$$ |

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

$\frac{{\left|h\right|}_{\mathrm{\infty}}}{\epsilon}}+2{\left(R-{R}_{0}\right)}^{2}\theta \left({\left|h\right|}_{\mathrm{\infty}}+\epsilon \rho \right)+2\left(R-{R}_{0}\right)C$ | $\le $ | $\rho ,$ | ||

$\rho $ | $>$ | $\frac{{\left|h\right|}_{\mathrm{\infty}}}{\epsilon}},$ |

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 $\stackrel{~}{N}\left({D}_{\rho}\right)\subset {D}_{\rho}.$ Hence Schauder’s fixed point applies. Thus we have the following result.

###### Theorem 4

If $\varphi $ is an unbounded increasing homeomorphism (i.e., $b=+\mathrm{\infty}$) and ${\varphi}^{-1}$ is such that

$$\left|{\varphi}^{-1}\left(s\right)\right|\le \theta \left|s\right|+C$$ | (2.8) |

for every $s\in \mathbb{R}$ and some nonnegative constants $\theta ,C$ with

$$ | (2.9) |

then for each $h\in C[{R}_{0},R],$ problem (2.5) has at least one solution.

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 $\mathbb{R}$ and (2.9) holds, then for any $h,$ problem (2.5) has a unique solution. In this case, the operator $\stackrel{~}{N}$ 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 $s\in \mathbb{R},$ then

$\left|\stackrel{~}{N}\left(v\right)\left(r\right)\right|$ | $\le $ | $\frac{|f(.,v){|}_{\mathrm{\infty}}}{\epsilon}}+2(R-{R}_{0}){\varphi}^{-1}\left((R-{R}_{0})\right|f(.,v)-\epsilon v{|}_{\mathrm{\infty}})$ | ||

$\le $ | $\frac{1}{\epsilon}}\left(\mu {\left|v\right|}_{\mathrm{\infty}}+\gamma \right)+2\left(R-{R}_{0}\right){\varphi}^{-1}\left(\left(R-{R}_{0}\right)\left(\left(\mu +\epsilon \right){\left|v\right|}_{\mathrm{\infty}}+\gamma \right)\right).$ |

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

$$\frac{1}{\epsilon}\left(\mu \rho +\gamma \right)+2\left(R-{R}_{0}\right){\varphi}^{-1}\left(\left(R-{R}_{0}\right)\left(\left(\mu +\epsilon \right)\rho +\gamma \right)\right)\le \rho ,$$ |

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 $l={lim}_{s\to +\mathrm{\infty}}{\varphi}^{-1}\left(s\right)/s.$

## 3 Conclusion

Through this paper we wanted to highlight the advantages of a general method of solving control problems, inside the theory of operator equations, benefiting from fixed point techniques. Three types of apparently distinct applications were considered.

The first one illustrates those problems with an external explicit control through which it intervenes so that the evolution of the investigated process follows the desired path or leads to the desired result. Such kind of controls are frequently imposed in engineering, economics, ecology and medicine. For many control problems associated to various classes of equations, additional techniques of investigation, and related topics, the interested reader can see the excellent Coron’s monograph [6]; some examples from medicine can be found in paper [7].

Surprisingly, a number of seemingly uncontrolled problems can still be treated as control problems. In such situations, the control is a hidden one and acts from inside the model so that it fits the problem under investigation. Such is the case of the Stokes system - the second application of this paper - for which the control is given by the pressure and comes from the necessity to adjust the flow rate of the incompressible fluid through the porous medium. Another example is of a boundary value problem - as in our third application - where the unknown value of the solution at some point takes over the function of a control variable in order for the solution to satisfy a boundary condition.

From the perspective of those readers interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method, thus paving the way for new applications.

## References

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- [7] I.Ş. Haplea, L.G. Parajdi and R. Precup, On the controllability of a system modeling cell dynamics related to leukemia, Symmetry 2021,13, 1867.
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[9] Temam R., Navier–Stokes equations and nonlinear functional analysis, SIAM, Philadelphia (1995), Google Scholar

[10] Precup R., Linear and semilinear partial differential equations, Walter de Gruyter, Berlin-Boston (2013), Google Scholar

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[14] Precup R., Fixed point theorems for decomposable multivalued maps and applications, Z Anal Anwend, 22 (2003), pp. 843-861, View Record in ScopusGoogle Scholar

[15] Precup R., Rodriguez-Lopez J., Fixed point index theory for decomposable multivalued maps and applications to φ-Laplacian problem, Nonlinear Anal, 199 (2020), Article 111958 1–16, ArticleDownload PDFView Record in ScopusGoogle Scholar