A fixed-point approach to control problems for Kolmogorov type second-order equations and systems

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Abstract

In this paper, the second-order differential equations and systems of Kolmogorov type are defined. With reference to population dynamics models, unlike the first-order equations which give the expression of the per capita rate, in the case of the second-order equations, the law of change of the per capita rate is given. Several control problems with fixed final time and fixed final state, with additive and multiplicative control, are studied. Their controllability is proved with fixed-point methods, the theorems of Banach, Schauder, Krasnoselskii, Avramescu and Perov.

Authors

Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology Babes-Bolyai University, Cluj-Napoca Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Alexandru Hofman
Faculty of Mathematics and Computer Science Babes-Bolyai University, Cluj-Napoca Romania

Keywords

Kolmogorov system; Lotka–Volterra system; control problem; fixed point; matrix convergent to zero; Volterra–Fredholm integral equation.

Paper coordinates

Al. Hofman, R. Precup, A fixed-point approach to control problems for Kolmogorov type second-order equations and systems, 27 (2025), art. no. 7, https://doi.org/10.1007/s11784-024-01160-5

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Journal

Journal of Fixed Point Theory and Applications

Publisher Name

Springer Nature

DOI

https://doi.org/10.1007/s11784-024-01160-5

Print ISSN

1661-7738

Online ISSN

1661-7746

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A fixed point approach to control problems for Kolmogorov type second order equations and systems

Alexandru Hofman, Radu Precup A. Hofman, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania alexandru.hofman@ubbcluj.ro R. Precup, Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania r.precup@ictp.acad.ro

Abstract.

In this paper the second order differential equations and systems of Kolmogorov type are defined. With reference to population dynamics models, unlike the first order equations which give the expression of the per capita rate, in the case of the second order equations, the law of change of the per capita rate is given. Several control problems with fixed final time and fixed final state, with additive and multiplicative control are studied. Their controllability is proved with fixed point methods, the theorems of Banach, Schauder, Krasnoselskii, Avramescu and Perov.

Key words and phrases:

Kolmogorov system, Lotka-Volterra system, control problem, fixed point, matrix convergent to zero, Volterra-Fredholm integral equation.

1991 Mathematics Subject Classification:

34H05, 37N25, 34A12, 34K35

1. Introduction and Preliminaries

The well-known Lotka-Volterra system for the dynamics of two species in competition (prey-predator),

\begin{cases}x^{\prime}=x\left(\alpha-\beta y\right)\\ y^{\prime}=-y\left(\beta-\gamma x\right),\end{cases}

finds its generalization in the form of Kolmogorov’s system [11, 27]

(1.1)

\begin{cases}x^{\prime}=xf(x,y)\\ y^{\prime}=yg(x,y).\end{cases}

The particularity of this system consists in the explicit expressions $f\left(x,y\right)$ and $g\left(x,y\right)$ of the growth rates per capita $\frac{x^{\prime}}{x}$ and $\frac{y^{\prime}}{y}$ of the two species. Systems of this type, with rates $f$ and $g$ given by explicit expressions, arise by modeling from numerous problems in ecology, biology, medicine, engineering and economics (see, e.g., [1, 3, 6, 7, 19]), and their analysis covers a series of mathematical aspects such as the existence of solutions, periodicity, permanence, extinction, stability, bifurcation and control [6, 7, 14, 15, 16, 17, 28].

Through variable changes $x=e^{u}$ and $y=e^{v},$ system (1.1) turns into the normal form

\begin{cases}u^{\prime}=f\left(e^{u},e^{v}\right)\\ v^{\prime}=g\left(e^{u},e^{v}\right).\end{cases}

In this paper, by a first-order Kolmogorov equation we mean an equation of the form $\ x^{\prime}=xf(t,x).\$ As before, by changing the variable $x=e^{u}$ it becomes $u^{\prime}=f\left(t,e^{u}\right)$ .

By analogy we say that a second-order equation is a second-order Kolmogorov equation if it has the form

\left(\frac{x^{\prime}}{x}\right)^{\prime}=f(t,x),

or equivalently

x^{\prime\prime}-\frac{1}{x}x^{\prime^{2}}=xf(t,x).

For these equations, in the language of population dynamics, $f(t,x)$ gives the change in the per capita rate $\frac{x^{\prime}}{x}.$ More generally, we call Kolmogorov equations of order $n,$ the equations of the form

\left(\frac{x^{\prime}}{x}\right)^{(n-1)}=f(t,x).

All these equations have the property that by changing the variable $x=e^{u}$ they become

u^{\prime\prime}=f\left(t,e^{u}\right)

and

u^{(n)}=f\left(t,e^{u}\right),

respectively.

Generally speaking (see [4]), from a practical point of view, it is important that a mathematical model that describes a certain process can be controlled so that its solution representing the state of the process satisfies a certain requirement, called the controllability condition. The control, which can be a constant, a vector or a function of time, is most often exercised over the parameters of the model (see [20]). For Kolmogorov equations and systems, in case that the rate of change $f$ is given and no parameters are highlighted, one can exercise the control on $f$ itself. We can perform the control by changing this rate $f$ either additively as $f-\lambda,$ or multiplicatively as $\lambda f.$ One of the common approaches in dealing with the controllability of models is based on fixed point theory (see, e.g., [4, 13, 18, 24, 25]).

In this paper we deal with control problems for second-order equations and systems of Kolmogorov type with additive and multiplicative controls. In case of one equation, the requirement is that, starting from a known initial state, say $x(0)=a,\ x^{\prime}(0)=0,$ the state variable reaches at a moment of time $T,$ a desired level $x_{T}$ , i.e., $x\left(T\right)=x_{T}.$ For a two-dimensional system, the controllability condition will be $x\left(T\right)=x_{T},\ y\left(T\right)=y_{T}.$ The controllability is proved via the general controllability principle of fixed point equations [24] and the fixed point theorems of Banach, Schauder, Krasnoselskii, Perov and Avramescu. For systems we use the vector approach based on matrices (see [22]), which allows the formulation of the conditions on the functions $f$ and $g$ cumulatively. This work is a continuation of our previous papers [24, 9, 8, 10, 20].

Working with general Kolmogorov equations and systems, our results can be applied to numerous particular cases, subject to satisfying the required conditions for the functions $f,\ g.$

We conclude this introductory section by recalling some notions and results in fixed point theory that are probably less well known, such as the Krasnoseskii, Avramescu, and Perov theorems (see [2, 5, 12, 21, 22, 26, 23]).

Theorem 1.1 (Krasnoselskii).

Let $D$ be a closed bounded convex subset of a Banach space $X,\ A:D\rightarrow X$ a contraction and $B:D\rightarrow X$ a continuous mapping with $B\left(D\right)$ relatively compact. If

A\left(x\right)+B\left(y\right)\in D\ \ \ \text{for every\ \ }x,y\in D,

then the mapping $A+B$ has at least one fixed point.

Theorem 1.2 (Avramescu).

Let $\left(D_{1},\ d\right)$ be a complete metric space, $D_{2}$ a closed convex subset of a normed space $Y,$ and let $N_{i}:D_{1}\times D_{2}\rightarrow D_{i},$ $i=1,2$ be continuous mappings. Assume that the following conditions are satisfied:

(a):

There is a constant $L\in[0,1)$ such that

d\left(N_{1}\left(x,y\right),\ N_{1}\left(\overline{x},y\right)\right)\leq Ld% \left(x,\ \overline{x}\right)

for all $x,\overline{x}\in D_{1}$ and $y\in D_{2};$

(b):

$N_{2}\left(D_{1}\times D_{2}\right)$ is a relatively compact subset of $Y.$

Then there exists $\left(x,y\right)\in D_{1}\times D_{2}$ with

N_{1}\left(x,y\right)=x,\ \ \ N_{2}\left(x,y\right)=y.

Finally, we recall some notions about positive matrices and the vector analog of Banach contraction theorem, namely Perov’s fixed point theorem. For a square matrix $M$ $\in\mathcal{M}_{n\times n}\left(\mathbb{R}_{+}\right),$ the spectral radius $\rho\left(M\right)$ is the maximum among the absolute values of its eigenvalues, and the following statements are equivalent (see, e.g., [22]):

(a):: $\rho\left(M\right)<1;$
(b):: $M^{k}\rightarrow 0$ as $k\rightarrow\infty$ (where $0$ stands for the zero matrix of the same order as $M$ );
(c):: $I-M$ is nonsingular and inverse-positive, i.e., $(I-M)^{-1}$ has nonnegative entries (here $I$ stands for the unit matrix of the same order as $M$ ).

A matrix $M$ having these properties is said to be convergent to zero.

We note that a vector-matrix inequality $x\leq Mx+y$ for a matrix which is convergent to zero and two column vectors $x,y\in\mathbf{\mathbb{R}}_{+}^{n},$ equivalently $\ \left(I-M\right)x\leq y,$ can be multiplied by matrix $(I-M)^{-1}$ without changing the inequality to become equivalently $x\leq\left(I-M\right)^{-1}y.$

In addition we note that the entries from the diagonal of a matrix that is convergent to zero are strictly less than one and for $n=2,$ the necessary and sufficient condition for a matrix $M=\left[a_{ij}\right]_{1\leq i,j\leq 2}\$ with nonnegative entries to be convergent to zero is

a_{11}+a_{22}<\min\left\{2,\ 1+\text{det\ }M\right\}.

A matrix that is convergent to zero replaces the contraction constant from Banach contraction principle when dealing with mappings $N$ from $X^{m}$ to $X^{m},$ where $X$ is a complete metric space with metric $d.$ More exactly, Perov’s vector analog of Banach’s contraction principle is the following.

Theorem 1.3 (Perov).

Assume that there are numbers $a_{ij}\in\mathbf{\mathbb{R}}_{+}$ $\left(i,j=1,2,...,m\right)$ with

d\left(N_{i}\left(x\right),N_{i}\left(y\right)\right)\leq\sum\limits_{j=1}^{m}% a_{ij}d\left(x_{j},y_{j}\right),\ \ i=1,2,...,m,

for all $x,y\in X^{m},\ x=\left(x_{1},\ x_{2},\ ...,\ x_{m}\right),$ $y=(y_{1},$ $y_{2},$ $...,$ $y_{m}).$ If $\rho\left(M\right)<1,$ where $M=\left[a_{ij}\right]_{1\leq i,j\leq m},$ then $N$ has a unique fixed point in $X^{m}.$ In addition, the fixed point is the limit of the sequence of successive approximations $\left(N^{k}\left(x\right)\right)$ starting from any point $x$ of $X^{m}.$

Using Perov’s theorem, the nonlinear terms of a system of equations can behave as independently as possible with respect to the variables of the system.

2. Control of second order Kolmogorov equations

2.1. Problems with additive control

We consider the following control problem of a second-order Kolmogorov equation

(2.1)

\begin{cases}\left(\frac{x^{\prime}(t)}{x(t)}\right)^{\prime}=f(t,x(t))-% \lambda\\ x(0)=a,\ \ \ x^{\prime}(0)=0\\ x>0\text{ on\ }\left[0,T\right],\ \ \ x\left(T\right)=x_{T},\end{cases}

where $a,\ x_{T}>0,$ and the additive control $\lambda$ is scalar.

Our first result is an existence and uniqueness theorem of the solution $\left(x,\lambda\right)$ of the control problem with $x$ in a ball of a given radius $\rho$ of the space $C\left[0,T\right]$ endowed with the Chebyshev norm $\left\|x\right\|_{\infty}=\max_{t\in\left[0,T\right]}\left|x\left(t\right)% \right|.$ Denote

\alpha=\ln a,\ \ \ u_{T}=\ln x_{T},\ \ \ \ \gamma=2\frac{\alpha-u_{T}}{T^{2}},% \ \ \ R=\ln\rho

and for a nonnegative number $L,$ in case that $L=0,$ by $\frac{1}{L}$ let it mean $+\infty.$

Theorem 2.1.

Let $L\in\mathbb{R}_{+}$ and $\rho>0.$ Assume that

(2.2)

\exp\left(\max\left\{\left|\alpha\right|,\left|u_{T}\right|\right\}+1\right)% \leq\rho<\frac{2}{LT^{2}}

and the function $f:\left[0,T\right]\times[0,\rho]\rightarrow\mathbb{R}$ is continuous, $f\left(\cdot,0\right)\equiv 0$ and satisfies the Lipschitz condition

(2.3)

\left|f(t,v)-f(t,\overline{v})\right|\leq L|v-\overline{v}|,

for all $t\in[0,T]$ , $v,\overline{v}\in[0,\rho].$ Then the control problem has a unique solution $(x^{\ast},\lambda^{\ast})$ with $x^{\ast}>0,$ $\left\|x^{\ast}\right\|_{\infty}\leq\rho,$ and

(2.4)

\lambda^{\ast}=\frac{2}{T^{2}}\left(\alpha-\ln x_{T}+\int_{0}^{T}\int_{0}^{% \tau}f(s,x^{\ast}(s))dsd\tau\right).

Proof.

We look for positive $x$ of the form $x=e^{u},$ whence $u=\ln\ x.$ The initial conditions give $e^{u(0)}=a,$ whence $u(0)=\ln\ a=\alpha,$ and $u^{\prime}(0)e^{u(0)}=0,$ hence $u^{\prime}(0)=0$ . Also, the controllability condition yields $u\left(T\right)=u_{T}.$ Substituting $x=e^{u}$ in (2.1) gives the second-order equation $u^{\prime\prime}=f\left(t,e^{u(t)}\right)-\lambda.$ Integrating two times we obtain the following equation

(2.5)

u(t)=\alpha+\int_{0}^{t}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau-\frac{\lambda t^{2% }}{2}.

Using the controllability condition $u(T)=u_{T},$ gives the expression of the control parameter in terms of the variable $u$ , namely

(2.6)

\lambda=\frac{2}{T^{2}}\left(\alpha-u_{T}+\int_{0}^{T}\int_{0}^{\tau}f(s,e^{u(% s)})dsd\tau\right).

Substituting $\lambda$ into (2.5) we obtain an integral equation of Volterra-Fredholm type

(2.7)

u(t)=\alpha-\frac{\gamma t^{2}}{2}+\int_{0}^{t}\int_{0}^{\tau}f(s,e^{u(s)})dsd% \tau-\frac{t^{2}}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau.

Let $B_{R}$ be the closed ball of the space $C\left[0,T\right]$ , centered at the origin and of radius $R.$ We look for a fixed point $u\in B_{R}$ of the operator $A$ given by

$\displaystyle A(u)(t)$	$\displaystyle=$	$\displaystyle\alpha-\frac{\gamma t^{2}}{2}+\int_{0}^{t}\int_{0}^{\tau}f(s,e^{u% (s)})dsd\tau-\frac{t^{2}}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau$
	$\displaystyle=$	$\displaystyle\left(1-\frac{t^{2}}{T^{2}}\right)\alpha+\frac{t^{2}}{T^{2}}u_{T}% +\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau$
		$\displaystyle-\frac{t^{2}}{T^{2}}\int_{t}^{T}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau.$

We apply Banach’s fixed point theorem to the operator $A$ on the closed ball $B_{R}$ . First we prove that $A$ is a contraction. Let $u,\ \overline{u}\in B_{R}.$ Using the Lipschitz condition on $f$ and arguments related to convex combinations, we obtain the following estimate

			$\displaystyle\|A(u)(t)-A(\overline{u})(t)\|$
		$\displaystyle\leq$	$\displaystyle\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}\|f(s% ,e^{u(s)})-f(s,e^{\overline{u}(s)})\|dsd\tau+\frac{t^{2}}{T^{2}}\int_{t}^{T}% \int_{0}^{\tau}\|f(s,e^{u(s)})-f(s,e^{\overline{u}(s)})\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\int_{0}^{T}\int_{0}^{\tau}\|f(s,e^{u(s)})-f(s,e^{\overline{u}(s)}% )\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle L\int_{0}^{T}\int_{0}^{\tau}\|e^{u(s)}-e^{\overline{u}(s)}\|dsd\tau.$

Now using Lagrange’s mean value theorem we have $\left|e^{u(s)}-e^{\overline{u}(s)}\right|\leq\rho\left|u\left(s\right)-% \overline{u}(s)\right|.$ Then

|A(u)(t)-A(\overline{u})(t)|\leq\frac{T^{2}}{2}L\rho||u-\overline{u}||_{\infty}.

Taking the the maximum for $t\in[0,T]$ gives

||A(u)-A(\overline{u})||_{\infty}\leq\frac{T}{2}^{2}L\rho||u-\overline{u}||_{% \infty}.

By our assumption one has $\ \frac{T^{2}}{2}L\rho<1,$ thus $A$ is a contraction on the ball $B_{R}.$ We now prove that the operator $A$ maps $B_{R}$ into itself, that is

||u||_{\infty}\leq R\ \ \ \text{implies}\ \ \ ||A(u)||_{\infty}\leq R.

Using the expression of $\gamma,$ the inequality $\left(1-\sigma\right)z_{1}+\sigma z_{2}\leq\max\left\{z_{1},z_{2}\right\}\ \ % \left(z_{1},z_{2}\in\mathbb{R}_{+},\ \sigma\in\left[0,1\right]\right),$ and the inequality $\left|f\left(t,v\right)\right|=\left|f\left(t,v\right)-f\left(0,v\right)\right% |\leq L\left|v\right|,$ we find that

			$\displaystyle\|A(u)(t)\|$
		$\displaystyle\leq$	$\displaystyle\left\|\left(1-\frac{t^{2}}{T^{2}}\right)\alpha+\frac{t^{2}}{T^{2}% }u_{T}\right\|+\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}% \left\|f(s,e^{u(s)})\right\|dsd\tau$
			$\displaystyle+\frac{t^{2}}{T^{2}}\int_{t}^{T}\int_{0}^{\tau}\left\|f(s,e^{u(s)}% )\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\int_{0% }^{T}\int_{0}^{\tau}\left\|f(s,e^{u(s)})\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+L\rho% \frac{T^{2}}{2}$
		$\displaystyle<$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+1.$

By our assumptions, $\max\left\{\left|\alpha\right|,\left|u_{T}\right|\right\}+1\leq e^{\rho}=R,$ thus $\left|A(u)(t)\right|\leq R$ for all $t\in\left[0,T\right],$ whence $\left\|A\left(u\right)\right\|_{\infty}\leq R$ as wished. Therefore the Banach contraction theorem applies and together with (2.6) gives the conclusion.

We note the case $L=0$ is trivial since then $f=0$ and using (2.6) and (2.7) we immediately obtain the solution $\left(x^{\ast},\lambda^{\ast}\right),$ where

x^{\ast}\left(t\right)=\exp\left(\alpha-\left(\frac{t}{T}\right)^{2}\left(% \alpha-u_{T}\right)\right),\ \ \ \lambda^{\ast}=\frac{2}{T^{2}}\left(\alpha-u_% {T}\right).

∎

Remark 2.1.

If we do not require for the solution to satisfy $\left\|x^{\ast}\right\|_{\infty}\leq\rho,$ that is, the radius $\rho$ is not a priori given, but we assume however that $f:\left[0,T\right]\times[0,+\infty)\rightarrow\mathbb{R}$ is continuous and satisfies the Lipschitz condition (2.3) for all $t\in[0,T]$ , $v,\overline{v}\in[0,+\infty)$ , for some $L$ with

\exp\left(\max\left\{\left|\alpha\right|,\left|u_{T}\right|\right\}+1\right)<% \frac{2}{LT^{2}},

then we may conclude that the control problem has a unique solution $\left(x^{\ast},\lambda^{\ast}\right)$ with $x^{\ast}>0$ and

\left\|x^{\ast}\right\|_{\infty}<\frac{2}{LT^{2}}.

Indeed, for the existence it suffices to take any number $\rho$ as in (2.2) and apply the previous result. For uniqueness, assume that $x^{\ast\ast}$ comes from an other solution. Then $\left\|x^{\ast}\right\|_{\infty}\leq\rho<\left\|x^{\ast\ast}\right\|_{\infty}<% \frac{2}{LT^{2}},$ which contradicts the conclusion of Theorem 2.1 applied to $\rho^{\prime}=\left\|x^{\ast\ast}\right\|_{\infty}$ instead of $\rho.$

Using Schauder’s fixed point theorem, we do not need $f$ to satisfy a Lipschitz condition. Instead we will assume a logarithmic growth condition.

Theorem 2.2.

Assume that the function $f:\left[0,T\right]\times[0,\rho]\rightarrow\mathbb{R}$ is continuous and satisfies the growth condition

(2.9)

\left|f(t,v))\right|\leq l_{1}\left|\ln v\right|+l_{2},

for all $t\in[0,T]$ , $v\in(0,\rho]$ and some constants $l_{1},l_{2}\in\mathbb{R}_{+}$ with $l_{1}<\frac{2}{T^{2}}.$ In addition assume that

(2.10)

\rho\geq\exp\left(\frac{2\max\left\{\left|\alpha\right|,\left|u_{T}\right|% \right\}+T^{2}l_{2}}{2-T^{2}l_{1}}\right).

Then the control problem has at least one solution $(x^{\ast},\lambda^{\ast})$ with $x^{\ast}>0,$ $\left\|x^{\ast}\right\|_{\infty}\leq\rho,$ and $\lambda^{\ast}$ given by (2.4).

Proof.

Proceeding analogously as in the proof of the previous theorem, we obtain the fixed point equation $u=A(u),$ $u\in B_{R}$ and the expression of operator $A$ given by (2.1). By standard arguments based on the Arzelà–Ascoli theorem, one has that $A:B_{R}\rightarrow C\left[0,T\right]$ is completely continuous. We next deal with the invariance of $B_{R}$ . For any $u\in B_{R},$ we have

			$\displaystyle\|A(u)(t)\|$
		$\displaystyle\leq$	$\displaystyle\left\|\left(1-\frac{t^{2}}{T^{2}}\right)\alpha+\frac{t^{2}}{T^{2}% }u_{T}\right\|+\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}% \left\|f(s,e^{u(s)})\right\|dsd\tau$
			$\displaystyle+\frac{t^{2}}{T^{2}}\int_{t}^{T}\int_{0}^{\tau}\left\|f(s,e^{u(s)}% )\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\int_{0% }^{T}\int_{0}^{\tau}\left\|f(s,e^{u(s)})\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\int_{0% }^{T}\int_{0}^{\tau}(l_{1}\|u(s)\|+l_{2})\ dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\frac{T% ^{2}}{2}\left(l_{1}R+l_{2}\right).$

One easily can see that (2.10) yields

\max\left\{\left|\alpha\right|,\left|u_{T}\right|\right\}+\frac{T^{2}}{2}\left% (l_{1}R+l_{2}\right)\leq R.

It follows that $\left\|A\left(u\right)\right\|_{\infty}\leq R$ as desired. Thus Schauder’s fixed point theorem applies and gives the conclusion. ∎

Remark 2.2.

Here again, if we do not require for the solution to satisfy $\left\|x^{\ast}\right\|_{\infty}\leq\rho,$ that is, the radius $\rho$ is not a priori given, but we assume however that $f:\left[0,T\right]\times(0,+\infty)\rightarrow\mathbb{R}$ is continuous and satisfies the growth condition (2.9) for all $t\in[0,T]$ , $v\in(0,+\infty)$ and some constants $l_{1},l_{2}\in\mathbb{R}_{+}$ with $l_{1}<\frac{2}{T^{2}},$ then the control problem has at least one solution. This statement is obvious if we use the above result for any $\rho\geq\rho_{0}$ where

\rho_{0}=\exp\left(\frac{2\max\left\{\left|\alpha\right|,\left|u_{T}\right|% \right\}+T^{2}l_{2}}{2-T^{2}l_{1}}\right).

Taking in particular $\rho=\rho_{0},$ we find a solution with $\left\|x^{\ast}\right\|_{\infty}\leq\rho_{0}.$

The next result combines the two previous ones assuming that $f$ splits as $f=f_{1}+f_{2},$ where $f_{1}$ satisfies a Lipschitz condition while $f_{2}$ satisfies a logaritmic growth condition. The result is based on Krasnoselskii’s fixed point theorem for a sum of two operators. Here sign $L=1$ if $L>0$ and sign $L=0$ if $L=0.$

Theorem 2.3.

Assume that the function $f:\left[0,T\right]\times[0,\rho]\rightarrow\mathbb{R}$ is continuous and splits as $f=f_{1}+f_{2},$ where $f_{1}$ is like in Theorem 2.1 and $f_{2}$ is like in Theorem 2.2. In addition assume that

(2.11)

\rho\geq\exp\left(\frac{2\max\left\{\left|\alpha\right|,\left|u_{T}\right|% \right\}+T^{2}l_{2}+2\text{sign\ }L}{2-T^{2}l_{1}}\right).

Then the control problem has at least one solution $(x^{\ast},\lambda^{\ast})$ with $x^{\ast}>0,\ \left\|x^{\ast}\right\|_{\infty}\leq\rho,$ and $\lambda^{\ast}$ given by (2.4).

Proof.

Now the operator $A$ can be decomposed as $A=A_{1}+A_{2},$ where

A_{1}\left(u\right)\left(t\right)=\alpha-\frac{\gamma t^{2}}{2}+\int_{0}^{t}% \int_{0}^{\tau}f_{1}(s,e^{u(s)})dsd\tau-\frac{t^{2}}{T^{2}}\int_{0}^{T}\int_{0% }^{\tau}f_{1}(s,e^{u(s)})dsd\tau

and

A_{2}\left(u\right)\left(t\right)=\int_{0}^{t}\int_{0}^{\tau}f_{2}(s,e^{u(s)})% dsd\tau-\frac{t^{2}}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}f_{2}(s,e^{u(s)})dsd\tau.

The operator $A_{1}$ is a contraction on $B_{R}$ with the contraction constant $\frac{T}{2}^{2}L\rho<1$ and $A_{2}$ is completely continuous. It remains to check Krasnoselskii’s strong invariance condition

A_{1}\left(u\right)+\ A_{2}\left(v\right)\in B_{R}\text{ \ for all \ }u,v\in B% _{R}.

As above one has

$\displaystyle\left\|A_{1}\left(u\right)\left(t\right)+\ A_{2}\left(v\right)% \left(t\right)\right\|$	$\displaystyle\leq$	$\displaystyle\left\|A_{1}\left(u\right)\left(t\right)\right\|+\left\|A_{2}\left(v% \right)\left(t\right)\right\|$
	$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+L\rho% \frac{T^{2}}{2}+\frac{T^{2}}{2}\left(l_{1}R+l_{2}\right)$
	$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\text{% sign\ }L+\frac{T^{2}}{2}\left(l_{1}R+l_{2}\right)$

From (2.11) we easily see that

\max\left\{\left|\alpha\right|,\left|u_{T}\right|\right\}+\text{sign\ }L+\frac% {T^{2}}{2}\left(l_{1}R+l_{2}\right)\leq R,

as desired. ∎

We note that Theorem 2.3 reduces to Theorem 2.1 if $L>0$ and $f_{2}=0$ (when sign $\ L=1$ and $l_{1}=l_{2}=0$ ); it reduces to Theorem 2.2 if $f_{1}=0$ (when $L=0$ and sign $L=0$ ).

2.2. Problem with a multiplicative control

We consider the following control problem

(2.12)

\begin{cases}\left(\frac{x^{\prime}(t)}{x(t)}\right)^{\prime}=\lambda f(t,x(t)% )\\ x(0)=a,\ \ \ x^{\prime}(0)=0\\ x>0\text{ on\ }\left[0,T\right],\ \ \ x\left(T\right)=x_{T},\end{cases}

with the multiplicative control parameter $\lambda.$

We have the following result on the unique controllability of the problem under a given bound of the positive solution of the equation.

Theorem 2.4.

Let $\rho\geq\exp\left(\left|\alpha\right|+\left|u_{T}-\alpha\right|\right)$ and $f:\left[0,T\right]\times[0,\rho]\rightarrow\left(0,+\infty\right)$ a continuous function satisfying the Lipschitz condition

\left|f\left(t,x\right)-f\left(t,y\right)\right|\leq L\left|x-y\right|

for all $t\in\left[0,T\right]$ and $x,y\in(0,\rho].$ If

L<\frac{f_{\rho}}{\rho T^{2}\left|u_{T}-\alpha\right|},

where $f_{\rho}:=\int_{0}^{T}\int_{0}^{\tau}\min_{x\in\left[\frac{1}{\rho},\rho\right% ]}f(s,x)dsd\tau.$ Then there exists a unique solution $(x^{\ast},\lambda^{\ast})$ of the control problem (2.12) with $x^{\ast}>0,\$ $\left\|x^{\ast}\right\|_{\infty}\leq\rho,$ and

\lambda^{\ast}=\frac{u_{T}-\alpha}{\int_{0}^{T}\int_{0}^{\tau}f(s,x^{\ast})dsd% \tau}.

Proof.

As above, we look for a positive solution $x$ in the form $x=e^{u},$ the initial and controllability conditions become $u(0)=\alpha:=\ln\ a$ and $u(T)=u_{T}:=\ln\ x_{T},$ respectively, while the equation reads as follows

u^{\prime\prime}=\lambda f\left(t,e^{u(t)}\right).

Integration leads to

(2.13)

u(t)=\alpha+\lambda\int_{0}^{t}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau

and using the controllability condition we obtain that

\lambda=\frac{u_{T}-\alpha}{\int_{0}^{T}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau}.

Substituting $\lambda$ into (2.13) we obtain an integral equation of the Volterra-Fredholm type

u(t)=\alpha+(u_{T}-\alpha)\frac{\int_{0}^{t}\int_{0}^{\tau}f(s,e^{u(s)})dsd% \tau}{\int_{0}^{T}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau},

that is a fixed point equation $u=A(u)$ for the operator $A$ defined by the right hand site of the equation on the closed ball $B_{R}$ of the space $C[0,T],$ with center at the origin and radius $R.$ We apply Banach’s contraction theorem. Let $u,\ \overline{u}\in B_{R}.$ In the following estimates, for simplicity, we make use of the notation

J(t,u):=\int_{0}^{t}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau.

We have the following estimate

|A(u)(t)-A(\overline{u})(t)|\leq|u_{T}-\alpha|\left|\frac{J(t,u)}{J(T,u)}-% \frac{J(t,\overline{u})}{J(T,\overline{u})}\right|.

Furthermore

	$\displaystyle\left\|\frac{J(t,u)}{J(T,u)}-\frac{J(t,\overline{u})}{J(T,% \overline{u})}\right\|$	$\displaystyle\leq\left\|\frac{J(t,u)}{J(T,u)}-\frac{J(t,\overline{u})}{J(T,u)}% \right\|+\left\|\frac{J(t,\overline{u})}{J(T,u)}-\frac{J(t,\overline{u})}{J(T,% \overline{u})}\right\|$
		$\displaystyle=\frac{\|J(t,u)-J(t,\overline{u})\|}{J(T,u)}+J(t,\overline{u})\frac% {\|J(T,u)-J(T,\overline{u})\|}{J(T,u)J(T,\overline{u})}$
		$\displaystyle\leq\frac{1}{J(T,u)}\left(\left\|J(t,u)-J(t,\overline{u})\right\|+\|% J(T,u)-J(T,\overline{u})\|\right).$

Next, from

\left|J(t,u)-J(t,\overline{u})\right|,\ \ \ \left|J(T,u)-J(T,\overline{u})% \right|\leq L\rho\frac{T^{2}}{2}||u-\overline{u}||_{\infty},

and $J(T,u)\geq f_{\rho},$ we derive

\left|A(u)(t)-A(\overline{u})(t)\right|\leq\frac{L\rho T^{2}}{f_{\rho}}\left|u% _{T}-\alpha\right|\left\|u-\overline{u}\right\|_{\infty}.

It follows that

||A(u)-A(\overline{u})||_{\infty}\leq\frac{L\rho T^{2}}{f_{\rho}}\left|u_{T}-% \alpha\right|\left\|u-\overline{u}\right\|_{\infty},

which in view of our assumption shows that $A$ is a contraction on $B_{R}.$ Next we show that $A\left(B_{R}\right)\subset B_{R}.$ Indeed, for any $u\in B_{R}$ one has

			$\displaystyle\left\|A(u)(t)\right\|\leq\left\|\alpha\right\|+\left\|u_{T}-\alpha% \right\|\frac{\int_{0}^{t}\int_{0}^{\tau}f(s,e^{u(s)})dsd\tau}{\int_{0}^{T}\int% _{0}^{\tau}f(s,e^{u(s)})dsd\tau}$
		$\displaystyle\leq$	$\displaystyle\left\|\alpha\right\|+\left\|u_{T}-\alpha\right\|\leq R$

since $\rho\geq\exp\left(\left|\alpha\right|+\left|u_{T}-\alpha\right|\right).$ Hence $\left\|A\left(u\right)\right\|_{\infty}\leq R$ as desired. The conclusion follows now from Banach’s fixed point theorem. ∎

3. Control of second order Kolmogorov systems

We consider the following control second-order Kolmogorov system

(3.1)

\begin{cases}\left(\frac{x^{\prime}(t)}{x(t)}\right)^{\prime}=f(x(t),y(t))-% \lambda\\ \left(\frac{y^{\prime}(t)}{y(t)}\right)^{\prime}=g(x(t),y(t))-\mu\\ x(0)=a,\ x^{\prime}(0)=0,\ \ y(0)=b,\ y^{\prime}(0)=0\\ x,\ y>0\ \ \text{on\ }\left[0,T\right],\ \ \ x\left(T\right)=x_{T},\ y\left(T% \right)=y_{T},\end{cases}

where $a,b,\ x_{T},\ y_{T}>0$ and the controls $\lambda$ and $\mu$ are constant.

Denote

	$\displaystyle R$	$\displaystyle:$	$\displaystyle=\ln\rho,\ \ \ u_{T}=\ln x_{T},\ \ \ v_{T}=\ln y_{T},\ \ \ \alpha% =\ln a,\ \ \ \beta=\ln b,$
	$\displaystyle\gamma$	$\displaystyle=$	$\displaystyle\frac{2}{T^{2}}\left(\alpha-u_{T}\right),\ \ \ \theta=\frac{2}{T^% {2}}\left(\beta-v_{T}\right).$

The next theorem guarantees the unique controllability of the system in a given ball.

Theorem 3.1.

Let

(3.2)

\rho\geq\exp\left(1+\max\left\{\left|\alpha\right|,\left|u_{T}\right|,\left|% \beta\right|,\left|v_{T}\right|\right\}\right)

and assume that the functions $f,g:\left[0,\rho\right]^{2}\rightarrow\mathbb{R}$ satisfy $f\left(0,\cdot\right)\equiv 0,\ g\left(\cdot,0\right)\equiv 0$ and the Lipschitz conditions

	$\displaystyle\left\|f(x,y)-f(\overline{x},\overline{y})\right\|$	$\displaystyle\leq$	$\displaystyle a_{11}\|x-\overline{x}\|+a_{12}\|y-\overline{y}\|,$
	$\displaystyle\left\|g(x,y)-g(\overline{x},\overline{y})\right\|$	$\displaystyle\leq$	$\displaystyle a_{21}\|x-\overline{x}\|+a_{22}\|y-\overline{y}\|,$

for all $x,y,\overline{x},\overline{y}\in[0,\rho]$ and some nonnegative constants $a_{ij}\ \left(i,j=1,2\right),$ and that the matrix

M=\frac{\rho T^{2}}{2}\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}

is convergent to zero. Then the control problem has a unique solution $\left(x^{\ast},y^{\ast},\lambda^{\ast},\mu^{\ast}\right)$ with $x^{\ast},y^{\ast}>0$ and $\left\|x^{\ast}\right\|_{\infty},\ \left\|y^{\ast}\right\|_{\infty}\leq\rho.$

Proof.

Making the change of variables $x=e^{u}$ and $y=e^{v}$ leads to the system

\begin{cases}u^{\prime\prime}=f\left(e^{u(t)},e^{v(t)}\right)-\lambda\\ v^{\prime\prime}=g\left(e^{u(t)},e^{v(t)}\right)-\mu,\end{cases}

the initial conditions become $u\left(0\right)=\alpha,\ u^{\prime}\left(0\right)=0,\ v\left(0\right)=\beta$ and $v\left(0\right)=0.$ Also the controllability conditions read as $u\left(T\right)=u_{T}$ and $v\left(T\right)=v_{T}.$

Successive integrations lead to the integral system

(3.3)

\begin{cases}u(t)=\alpha+\int_{0}^{t}\int_{0}^{\tau}f(e^{u(s)},e^{v(s)})dsd% \tau-\frac{\lambda t^{2}}{2}\\ v(t)=\beta+\int_{0}^{t}\int_{0}^{\tau}g(e^{u(s)},e^{v(s)})dsd\tau-\frac{\mu t^% {2}}{2}.\end{cases}

Using the controllability conditions $u(T)=u_{T},$ $v(T)=v_{T}$ gives the expression of the control parameters in terms of the state variables $u,v,$ namely

$\displaystyle\lambda$	$\displaystyle=$	$\displaystyle\frac{2}{T^{2}}\left(\alpha-u_{T}\right)+\frac{2}{T^{2}}\int_{0}^% {T}\int_{0}^{\tau}f(e^{u(s)},e^{v(s)})dsd\tau$
	$\displaystyle=$	$\displaystyle\gamma+\frac{2}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}f(e^{u(s)},e^{v(% s)})dsd\tau,$
$\displaystyle\mu$	$\displaystyle=$	$\displaystyle\frac{2}{T^{2}}\left(\beta-v_{T}\right)+\frac{2}{T^{2}}\int_{0}^{% T}\int_{0}^{\tau}g(e^{u(s)},e^{v(s)})dsd\tau$
	$\displaystyle=$	$\displaystyle\theta+\frac{2}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}g(e^{u(s)},e^{v(% s)})dsd\tau.$

Replacing in (3.3) we arrive to the Volterra-Fredholm integral system

\begin{cases}u(t)=\alpha-\frac{\gamma t^{2}}{2}+\int_{0}^{t}\int_{0}^{\tau}f(e% ^{u(s)},e^{v(s)})dsd\tau-\frac{t^{2}}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}f(e^{u(% s)},e^{v(s)})dsd\tau\\ u(t)=\beta-\frac{\theta t^{2}}{2}+\int_{0}^{t}\int_{0}^{\tau}g(e^{u(s)},e^{v(s% )})dsd\tau-\frac{t^{2}}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}g(e^{u(s)},e^{v(s)})% dsd\tau,\end{cases}

which can be seen as a fixed point equation for the operator $N=(A,B),$ where

	$\displaystyle A\left(u,v\right)\left(t\right)$	$\displaystyle=$	$\displaystyle\alpha-\frac{\gamma t^{2}}{2}+\int_{0}^{t}\int_{0}^{\tau}f(e^{u(s% )},e^{v(s)})dsd\tau-\frac{t^{2}}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}f(e^{u(s)},e% ^{v(s)})dsd\tau$
		$\displaystyle=$	$\displaystyle\alpha-\frac{\gamma t^{2}}{2}+\left(1-\frac{t^{2}}{T^{2}}\right)% \int_{0}^{t}\int_{0}^{\tau}f(e^{u(s)},e^{v(s)})dsd\tau-\frac{t^{2}}{T^{2}}\int% _{t}^{T}\int_{0}^{\tau}f(e^{u(s)},e^{v(s)})dsd\tau,$

	$\displaystyle B\left(u,v\right)\left(t\right)$	$\displaystyle=$	$\displaystyle\beta-\frac{\theta t^{2}}{2}+\int_{0}^{t}\int_{0}^{\tau}g(e^{u(s)% },e^{v(s)})dsd\tau-\frac{t^{2}}{T^{2}}\int_{0}^{T}\int_{0}^{\tau}g(e^{u(s)},e^% {v(s)})dsd\tau$
		$\displaystyle=$	$\displaystyle\beta-\frac{\theta t^{2}}{2}+\left(1-\frac{t^{2}}{T^{2}}\right)% \int_{0}^{t}\int_{0}^{\tau}g(e^{u(s)},e^{v(s)})dsd\tau-\frac{t^{2}}{T^{2}}\int% _{t}^{T}\int_{0}^{\tau}g(e^{u(s)},e^{v(s)})dsd\tau.$

We shall apply Perov’s fixed point theorem in the set

D_{R}:=\{(u,v)\in C([0,T];\mathbb{R}^{2}):\ ||u||_{\infty}\leq R,\ ||v||_{% \infty}\leq R\}.

First we show that $N$ is a Perov contraction. Let $(u,v),(\overline{u},\overline{v})\in D_{R}.$ Using the Lipschitz conditions and arguments related to convex combinations, we have

			$\displaystyle\|A(u,v)(t)-A(\overline{u},\overline{v})(t)\|$
		$\displaystyle\leq$	$\displaystyle\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}\|f(e% ^{u},e^{v})-f(e^{\overline{u}},e^{\overline{v}})\|dsd\tau+\frac{t^{2}}{T^{2}}% \int_{t}^{T}\int_{0}^{\tau}\|f(e^{u},e^{v})-f(e^{\overline{u}},e^{\overline{v}}% )\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\int_{0}^{T}\int_{0}^{\tau}(a_{11}\|e^{u}-e^{\overline{u}}\|+a_{12}% \|e^{v}-e^{\overline{v}}\|)dsd\tau.$

Furthermore, using Lagrange’s mean theorem one has $\left|e^{u}-e^{\overline{u}}\right|\leq\rho\left|u-\overline{u}\right|$ and $\left|e^{v}-e^{\overline{v}}\right|\leq\rho\left|v-\overline{v}\right|$ and then

			$\displaystyle\int_{0}^{T}\int_{0}^{\tau}(a_{11}\|e^{u}-e^{\overline{u}}\|+a_{12}% \|e^{v}-e^{\overline{v}}\|)dsd\tau$
		$\displaystyle\leq$	$\displaystyle\frac{\rho T^{2}}{2}\left(a_{11}\left\\|u-\overline{u}\right\\|_{% \infty}+a_{12}\left\\|v-\overline{v}\right\\|_{\infty}\right).$

It follows that

\left\|A(u,v)-A(\overline{u},\overline{v})\right\|_{\infty}\leq\frac{\rho T^{2% }}{2}\left(a_{11}\left\|u-\overline{u}\right\|_{\infty}+a_{12}\left\|v-% \overline{v}\right\|_{\infty}\right).

Similarly

\left\|B(u,v)-B(\overline{u},\overline{v})\right\|_{\infty}\leq\frac{\rho T^{2% }}{2}\left(a_{21}\left\|u-\overline{u}\right\|_{\infty}+a_{22}\left\|v-% \overline{v}\right\|_{\infty}\right).

These two inequalities can be written in the vector form

\left[\begin{array}[]{c}||A(u,v)-A(\bar{u},\overline{v})||_{\infty}\\ ||B(u,v)-B(\bar{u},\overline{v})||_{\infty}\end{array}\right]\leq M\left[% \begin{array}[]{c}||u-\bar{u}||_{\infty}\\ ||v-\overline{v}||_{\infty}\end{array}\right].

Since the matrix $M$ is assumed to be convergent to zero, the operator $N=\left(A,B\right)$ is a Perov contraction on $D_{R}.\$ It remains to prove the invariance of the set $D_{R}$ , that is,

||u||_{\infty}\leq R,\ ||v||_{\infty}\leq R\ \ \ \text{imply}\ \ ||A(u,v)||_{% \infty}\leq R,\ ||B(u,v)||_{\infty}\leq R.

Since $\alpha-\frac{\gamma t^{2}}{2}=\left(1-\frac{t^{2}}{T^{2}}\right)\alpha+\frac{t% ^{2}}{T^{2}}u_{T}$ and

\left|f\left(e^{u(s)},e^{v(s)}\right)\right|=\left|f\left(e^{u(s)},e^{v(s)}% \right)-f\left(0,e^{v\left(s\right)}\right)\right|\leq a_{11}e^{u\left(s\right% )}\leq\rho a_{11},

we have

			$\displaystyle\|A(u,v)(t)\|$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\left(1% -\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}\left\|f(e^{u(s)},e^{v(s)% })\right\|dsd\tau+\frac{t^{2}}{T^{2}}\int_{t}^{T}\int_{0}^{\tau}\left\|f(e^{u(s)% },e^{v(s)})\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\int_{0% }^{T}\int_{0}^{\tau}\left\|f(e^{u(s)},e^{v(s)})\right\|dsd\tau.$

Since $f\left(0,\cdot\right)\equiv 0,$ one has

\left|f\left(e^{u(s)},e^{v(s)}\right)\right|=\left|f\left(e^{u(s)},e^{v(s)}% \right)-f\left(0,e^{v\left(s\right)}\right)\right|\leq a_{11}e^{u\left(s\right% )}\leq\rho a_{11}.

Then

|A(u,v)(t)|\leq\max\left\{\left|\alpha\right|,\left|u_{T}\right|\right\}+\frac% {\rho T^{2}}{2}a_{11}.

Similarly

\left|B\left(u,v\right)\left(t\right)\right|\leq\max\left\{\left|\beta\right|,% \left|v_{T}\right|\right\}+\frac{\rho T^{2}}{2}a_{22}.

Since the elements from the diagonal of a convergent to zero matrix are less than one, we have

\frac{\rho T^{2}}{2}a_{11}<1\ \ \text{and\ \ }\frac{\rho T^{2}}{2}a_{22}<1.

Then according to (3.2) we deduce

\left\|A\left(u,v\right)\right\|_{\infty},\ \ \left\|B\left(u,v\right)\right\|% _{\infty}\leq 1+\max\left\{\left|\alpha\right|,\left|u_{T}\right|,\left|\beta% \right|,\left|v_{T}\right|\right\}\leq R.

Therefore, the operator $N=(A,B)$ maps $D_{R}$ into itself and thus Perov’s fixed point theorem applies and guarantees the existence of a unique fixed point $(u^{\ast},v^{\ast})\in D_{R}.$ Finally, $x^{\ast}=e^{u^{\ast}},$ $y^{\ast}=e^{v^{\ast}}$ and $\lambda^{\ast},$ $\mu^{\ast}$ calculated according to (3) give the desired solution of control problem (3.1). ∎

Here again, if instead of the Lipschitz conditions, $f$ and $g$ only have a logarithmic growth, then one can prove the existence of a least one solution of the control problem.

Theorem 3.2.

Let $f,g:\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}$ be continuous and satisfy the logarithmic growth conditions

(3.5)		$\displaystyle\left\|f(x,y)\right\|$	$\displaystyle\leq$	$\displaystyle a_{11}\left\|\ln x\right\|+a_{12}\left\|\ln y\right\|+b_{1},$
	$\displaystyle\left\|g(x,y)\right\|$	$\displaystyle\leq$	$\displaystyle a_{21}\left\|\ln x\right\|+a_{22}\left\|\ln y\right\|+b_{2},\$

for all $x,y\in\left(0,\infty\right)$ and some constants $a_{ij},\ b_{i}\in\mathbb{R}_{+}\ \left(i,j=1,2\right).$ Then for each $T>0$ for which the matrix

M=\frac{T^{2}}{2}\left[a_{ij}\right]_{1\leq i,j\leq 2}

converges to zero, the control problem (3.1) has at least one solution $(x^{\ast},y^{\ast},\lambda^{\ast},\mu^{\ast})$ with $x^{\ast}>0$ and $y^{\ast}>0.$

Proof.

We shall apply Schauder’s fixed point theorem to the operator $\left(A,B\right)$ in a bounded closed convex set $D$ of the form

D=B_{R_{1}}\times B_{R_{2}},

where

B_{R_{i}}=\left\{w\in C\left[0,T\right]:\ \left\|w\right\|_{\infty}\leq R_{i}% \right\}\ \left(i=1,2\right).

We need to prove that one can find two positive numbers $R_{1}$ and $R_{2}$ such that the following invariance condition is satisfied:

\left\|u\right\|_{\infty}\leq R_{1},\ \left\|v\right\|_{\infty}\leq R_{2}\ \ % \ \text{imply\ \ }\left\|A\left(u,v\right)\right\|_{\infty}\leq R_{1},\ \ % \left\|B\left(u,v\right)\right\|_{\infty}\leq R_{2}.

Using (3.5) we have

$\displaystyle\|A(u,v)(t)\|$	$\displaystyle\leq$	$\displaystyle\left\|\left(1-\frac{t^{2}}{T^{2}}\right)\alpha+\frac{t^{2}}{T^{2}% }u_{T}\right\|+\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}% \left\|f(e^{u(s)},e^{v(s)})\right\|dsd\tau$
		$\displaystyle+\frac{t^{2}}{T^{2}}\int_{t}^{T}\int_{0}^{\tau}\left\|f(e^{u(s)},e% ^{v(s)})\right\|dsd\tau$
	$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\int_{0% }^{T}\int_{0}^{\tau}\left\|f(e^{u(s)},e^{v(s)})\right\|dsd\tau$
	$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\frac{T% ^{2}}{2}\left(a_{11}R_{1}+a_{12}R_{2}+b_{1}\right)$

A similar estimate holds for $B.\$ Hence,

	$\displaystyle\left\\|A(u,v)\right\\|_{\infty}$	$\displaystyle\leq$	$\displaystyle\frac{T^{2}}{2}(a_{11}R_{1}+a_{12}R_{2})+\eta_{1},$
	$\displaystyle\left\\|B(u,v)\right\\|_{\infty}$	$\displaystyle\leq$	$\displaystyle\frac{T^{2}}{2}(a_{21}R_{1}+a_{22}R_{2})+\eta_{2},$

where

\eta_{1}=\max\left\{\left|\alpha\right|,\left|u_{T}\right|\right\}+\frac{T^{2}% }{2}b_{1},\ \ \ \eta_{2}=\max\left\{\left|\beta\right|,\left|v_{T}\right|% \right\}+\frac{T^{2}}{2}b_{2}.

We write the two inequalities in the vector form

\left[\begin{array}[]{c}\left\|A(u,v)\right\|_{\infty}\\ \left\|B(u,v)\right\|_{\infty}\end{array}\right]\leq M\left[\begin{array}[]{c}% R_{1}\\ R_{2}\end{array}\right]+\left[\begin{array}[]{c}\eta_{1}\\ \eta_{2}\end{array}\right].

Thus, for the desired invariance property, we would like to have

M\left[\begin{array}[]{c}R_{1}\\ R_{2}\end{array}\right]+\left[\begin{array}[]{c}\eta_{1}\\ \eta_{2}\end{array}\right]\leq\left[\begin{array}[]{c}R_{1}\\ R_{2}\end{array}\right],

equivalently

\left[\begin{array}[]{c}\eta_{1}\\ \eta_{2}\end{array}\right]\leq(I-M)\left[\begin{array}[]{c}R_{1}\\ R_{2}\end{array}\right].

Since the matrix $M$ converges to zero, one has $(I-M)^{-1}\in\mathcal{M}_{2\times 2}\left(\mathbb{R}_{+}\right)$ and thus we can multiply by $(I-M)^{-1}$ without changing the inequality. It turns out that

(I-M)^{-1}\left[\begin{array}[]{c}\eta_{1}\\ \eta_{2}\end{array}\right]\leq\left[\begin{array}[]{c}R_{1}\\ R_{2}\end{array}\right].

This inequality allows the choice of the radii $R_{1},R_{2}>0$ to guarantee the invariance property. Thus Schauder’s fixed point theorem can be applied to $N=\left(A,B\right)$ in $B_{R_{1}}\times B_{R_{2}}$ . ∎

Our last result is an application of Avramescu’s fixed point theorem to control problem (3.1) when $f$ satisfies a Lipschitz condition with respect to the first variable only, and $g$ has a logarithmic growth in the last variable.

Theorem 3.3.

Let $\rho$ be such that

(3.6)

\rho\geq\max\left\{\exp\left(1+\max\left\{\left|\alpha\right|,\left|u_{T}% \right|\right\}\right),\ \exp\frac{2\max\left\{\left|\beta\right|,\left|v_{T}% \right|\right\}+T^{2}c}{2-T^{2}b}\right\}

and $f,g:\left[0,\rho\right]^{2}\rightarrow\mathbb{R}$ be continuous and $f\left(0,\cdot\right)\equiv 0.$ Assume that

	$\displaystyle\left\|f\left(x,y\right)-f\left(\overline{x},y\right)\right\|$	$\displaystyle\leq$	$\displaystyle a\left\|x-\overline{x}\right\|\ \ \ \text{for all }x,\overline{x},% y\in\left[0,\rho\right],$
	$\displaystyle\left\|g\left(x,y\right)\right\|$	$\displaystyle\leq$	$\displaystyle b\left\|\ln y\right\|+c\ \ \ \text{for all }x\in\left[0,\rho\right% ],\ y\in(0,\rho],$

where $a<\frac{2}{\rho T^{2}}$ and $b<\frac{2}{T^{2}}.$ Then problem (3.1) has at least one solution $\left(x^{\ast},y^{\ast},\lambda^{\ast},\mu^{\ast}\right)$ with $x^{\ast},y^{\ast}>0$ and $\left\|x^{\ast}\right\|_{\infty},\left\|y^{\ast}\right\|_{\infty}\leq\rho.$

Proof.

As in the proof of Theorem 3.1, one has

\left\|A(u,v)-A(\overline{u},v)\right\|_{\infty}\leq\frac{\rho T^{2}}{2}a\left% \|u-\overline{u}\right\|_{\infty}.

Since $\frac{\rho T^{2}}{2}a<1,$ the operator $A\left(\cdot,v\right)$ is a contraction in $B_{R}$ with a Lipschitz constant independent of $v.$ By standard arguments, one has that $B$ is completely continuous on $B_{R}\times B_{R}.$ In order to apply Avramescu’s fixed point theorem, it remains to prove that $A\left(B_{R}\times B_{R}\right),\ B\left(B_{R}\times B_{R}\right)\subset B_{R}.$

As in the proof of Theorem 3.1, based on (3.6), we have

\left\|A\left(u,v\right)\right\|_{\infty}\leq 1+\max\left\{\left|\alpha\right|% ,\left|u_{T}\right|\right\}\leq R,

while from the proof of Theorem 3.2, one has

\left\|B(u,v)\right\|_{\infty}\leq\max\left\{\left|\beta\right|,\left|v_{T}% \right|\right\}+\frac{T^{2}}{2}\left(bR+c\right)\leq R.

Thus Avramescu’s theorem applies and gives the result. ∎

Acknowledgements

The authors are very thankful to reviewers for their valuable comments and remarks that led to an improved version of the paper.

References

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			$\displaystyle\|A(u)(t)-A(\overline{u})(t)\|$
		$\displaystyle\leq$	$\displaystyle\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}\|f(s% ,e^{u(s)})-f(s,e^{\overline{u}(s)})\|dsd\tau+\frac{t^{2}}{T^{2}}\int_{t}^{T}% \int_{0}^{\tau}\|f(s,e^{u(s)})-f(s,e^{\overline{u}(s)})\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\int_{0}^{T}\int_{0}^{\tau}\|f(s,e^{u(s)})-f(s,e^{\overline{u}(s)}% )\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle L\int_{0}^{T}\int_{0}^{\tau}\|e^{u(s)}-e^{\overline{u}(s)}\|dsd\tau.$

			$\displaystyle\|A(u)(t)\|$
		$\displaystyle\leq$	$\displaystyle\left\|\left(1-\frac{t^{2}}{T^{2}}\right)\alpha+\frac{t^{2}}{T^{2}% }u_{T}\right\|+\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}% \left\|f(s,e^{u(s)})\right\|dsd\tau$
			$\displaystyle+\frac{t^{2}}{T^{2}}\int_{t}^{T}\int_{0}^{\tau}\left\|f(s,e^{u(s)}% )\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\int_{0% }^{T}\int_{0}^{\tau}\left\|f(s,e^{u(s)})\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+L\rho% \frac{T^{2}}{2}$
		$\displaystyle<$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+1.$

			$\displaystyle\|A(u)(t)\|$
		$\displaystyle\leq$	$\displaystyle\left\|\left(1-\frac{t^{2}}{T^{2}}\right)\alpha+\frac{t^{2}}{T^{2}% }u_{T}\right\|+\left(1-\frac{t^{2}}{T^{2}}\right)\int_{0}^{t}\int_{0}^{\tau}% \left\|f(s,e^{u(s)})\right\|dsd\tau$
			$\displaystyle+\frac{t^{2}}{T^{2}}\int_{t}^{T}\int_{0}^{\tau}\left\|f(s,e^{u(s)}% )\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\int_{0% }^{T}\int_{0}^{\tau}\left\|f(s,e^{u(s)})\right\|dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\int_{0% }^{T}\int_{0}^{\tau}(l_{1}\|u(s)\|+l_{2})\ dsd\tau$
		$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\frac{T% ^{2}}{2}\left(l_{1}R+l_{2}\right).$

$\displaystyle\left\|A_{1}\left(u\right)\left(t\right)+\ A_{2}\left(v\right)% \left(t\right)\right\|$	$\displaystyle\leq$	$\displaystyle\left\|A_{1}\left(u\right)\left(t\right)\right\|+\left\|A_{2}\left(v% \right)\left(t\right)\right\|$
	$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+L\rho% \frac{T^{2}}{2}+\frac{T^{2}}{2}\left(l_{1}R+l_{2}\right)$
	$\displaystyle\leq$	$\displaystyle\max\left\{\left\|\alpha\right\|,\left\|u_{T}\right\|\right\}+\text{% sign\ }L+\frac{T^{2}}{2}\left(l_{1}R+l_{2}\right)$

	$\displaystyle\left\|\frac{J(t,u)}{J(T,u)}-\frac{J(t,\overline{u})}{J(T,% \overline{u})}\right\|$	$\displaystyle\leq\left\|\frac{J(t,u)}{J(T,u)}-\frac{J(t,\overline{u})}{J(T,u)}% \right\|+\left\|\frac{J(t,\overline{u})}{J(T,u)}-\frac{J(t,\overline{u})}{J(T,% \overline{u})}\right\|$
		$\displaystyle=\frac{\|J(t,u)-J(t,\overline{u})\|}{J(T,u)}+J(t,\overline{u})\frac% {\|J(T,u)-J(T,\overline{u})\|}{J(T,u)J(T,\overline{u})}$
		$\displaystyle\leq\frac{1}{J(T,u)}\left(\left\|J(t,u)-J(t,\overline{u})\right\|+\|% J(T,u)-J(T,\overline{u})\|\right).$

A fixed-point approach to control problems for Kolmogorov type second-order equations and systems

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A fixed point approach to control problems for Kolmogorov type second order equations and systems

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction and Preliminaries

Theorem 1.1 (Krasnoselskii).

Theorem 1.2 (Avramescu).

Theorem 1.3 (Perov).

2. Control of second order Kolmogorov equations

2.1. Problems with additive control

Theorem 2.1.

Proof.

Remark 2.1.

Theorem 2.2.

Proof.

Remark 2.2.

Theorem 2.3.

Proof.

2.2. Problem with a multiplicative control

Theorem 2.4.

Proof.

3. Control of second order Kolmogorov systems

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

Theorem 3.3.

Proof.

Acknowledgements

References

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