Abstract
In this paper we obtain nonlinear alternatives of LeraySchauder and Monch type for nonself vectorvalued operators, under hybrid conditions of Perov contraction and compactness.
Thus, we give vector versions of the theorems of Krasnosel’skii, Avramescu, BurtonKirk and GaoLi Zhang. An application is given to a boundary value problem for a system of second order differential equations in which some of the equations are implicit.
Authors
Veronica Ilea
Babes–Bolyai University, Faculty of Mathematics and Computer Science, ClujNapoca, Romania
Adela Novac
Technical University of ClujNapoca, Department of Mathematics, ClujNapoca, Romania
Diana Otrocol
Technical University of ClujNapoca, Department of Mathematics, ClujNapoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Radu Precup
Department of Mathematics BabesBolyai University, ClujNapoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Keywords
Nonlinear operator; nonself map; fixed point; Perov contraction; nonlinear boundary value problem.
Paper coordinates
V. Ilea, A. Novac, D. Otrocol, R. Precup, Nonlinear alternatives of hybrid type for nonself vectorvalued maps and application, Fixed Point Theory, 24 (2023) no. 1, 221232, http://doi.org/10.24193/fptro.2023.1.11
About this paper
Journal
Fixed Point Theory
Publisher Name
Casa Cărţii de Ştiinţă ClujNapoca
(House of the Book of Science ClujNapoca)
Print ISSN
15835022
Online ISSN
20669208
google scholar link
Nonlinear alternatives of hybrid type for nonself vectorvalued maps and application
Abstract
In this paper we obtain nonlinear alternatives of LeraySchauder and Mönch type for nonself vectorvalued operators, under hybrid conditions of Perov contraction and compactness. Thus we give vector versions of the theorems of Krasnosel’skii, Avramescu, BurtonKirk and GaoLiZhang. An application is given to a boundary value problem for a system of second order differential equations in which some of the equations are implicit.
Keywords: Nonlinear operator; nonself map; fixed point; Perov contraction; nonlinear boundary value problem.
MSC: 47H10, 34B15
1 Introduction
1.1 Krasnosel’skii type results for self maps
Any study in operator equations with hybrid conditions must begin with Krasnosel’skii’s theorem for the sum of two operators.
Theorem 1.1 (Krasnosel’skii).
[12] Let $D$ be a closed bounded convex subset of a Banach space $X,A:D\to X$ a contraction and $B:D\to X$ a continuous mapping with $B\left(D\right)$ relatively compact. If
$$A\left(x\right)+B\left(y\right)\in D\phantom{\rule{1.5em}{0ex}}\text{for every}x,y\in D,$$  (1) 
then the map $N:=A+B$ has at least one fixed point.
The hybrid character of Krasnosel’skii’s theorem lies in the decomposition of the operator $N$ as a sum of two maps $A$ and $B$ with different properties. Condition (1) shows that $N$ is a self map of $D$ as in the fixed point theorems of Banach and Schauder which Krasnosel’skii’s theorem uses together in the proof.
An other possibility for a hybrid approach arises in case of systems, when the domain of $N$ splits as a Cartesian product, say $X\times Y,$ and correspondingly the operator $N$ splits as a couple $({N}_{1},{N}_{2}),$ where ${N}_{1},{N}_{2}$ take their values in $X$ and $Y,$ respectively. A typical result in this direction is the following vector version of Krasnosel’skii’s theorem, due to Avramescu [1], which we state here in a slightly modified form and whose proof is reproduced in [4].
Theorem 1.2 (Avramescu).
[1] Let $({D}_{1},d)$ be a complete metric space, ${D}_{2}$ a closed convex subset of a normed space $Y$ and let ${N}_{i}:{D}_{1}\times {D}_{2}\to {D}_{i},$ $i=1,2$ be continuous mappings. Assume that the following conditions are satisfied:
 (a)

There is a constant $l\in [0,1)$ such that
$$d({N}_{1}(x,y),{N}_{1}(\overline{x},y))\le ld(x,\overline{x})$$ 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 $(x,y)\in {D}_{1}\times {D}_{2}$ with
$${N}_{1}(x,y)=x,{N}_{2}(x,y)=y.$$ 
In this regard, let us note the conclusion of the paper [17] according to which any theorem of continuous dependence of the fixed point on parameters can be associated with a fixed point existence result for operators of the form $N:X\times Y\to X\times Y.$
In the paper [6] it went further and added even more heterogeneity by mixing together the metrical topology of a complete metric space with the normtopology and weak topology of a Banach space. Thus, results were obtained that combine the BanachPerov, Schauder and Arino–GautierPenot fixed point theorems and the BanachPerov theorem with the strong and weaktopology versions of Mönch’s fixed point theorem. These results are dealing with operators of the form
$$N=({N}_{1},\mathrm{\dots},{N}_{n}),\text{}{N}_{i}:V\to {V}_{i}\phantom{\rule{1em}{0ex}}\text{for}i=1,\mathrm{\dots},n,$$ 
on a Cartesian product space
$$V={V}_{1}\times \mathrm{\dots}\times {V}_{n},$$ 
where a number $p(0\le p\le n)$ of spaces, ${V}_{1},\mathrm{\dots},{V}_{p}$ are complete metric spaces endowed with the metrics ${d}_{i},$ $i=1,\mathrm{\dots},p,$ a number $q(0\le q\le np)$ are Banach spaces considered with their strong topologies, and the remaining ones are Banach spaces with their weak topologies. Let
$$X=\prod _{i=1}^{p}{V}_{i},Y=\prod _{i=p+1}^{p+q}{V}_{i},Z=\prod _{i=p+q+1}^{n}{V}_{i}$$ 
and
$$F=({N}_{1},..,{N}_{p}),G=({N}_{p+1},\mathrm{\dots},{N}_{p+q}),H=({N}_{p+q+1},\mathrm{\dots},{N}_{n}),$$ 
be defined respectively if $p>0,$ $q>0$ and $n(p+q)>0.$ We note that if $p=0$ then the set $X$ does not appear; analogously for the set $Y$ when $q=0,$ and for $Z$ if $p+q=n.$
Denote by $d$ the vectorvalued metric $d:X\to {\mathbb{R}}_{+}^{p}$ given by
$$d(x,\overline{x})={({d}_{1}({x}_{1},{\overline{x}}_{1}),\mathrm{\dots},{d}_{p}({x}_{p},{\overline{x}}_{p}))}^{T},$$ 
where $x=({x}_{1},\mathrm{\dots},{x}_{p}),$ $\overline{x}=({\overline{x}}_{1},\mathrm{\dots},{\overline{x}}_{p}),$ and use the notation
$d(F(x,y,z),F(\overline{x},y,z))$  
$=$  ${({d}_{1}({N}_{1}(x,y,z),{N}_{1}(\overline{x},y,z)),\mathrm{\dots},{d}_{p}({N}_{p}(x,y,z),{N}_{p}(\overline{x},y,z)))}^{T},$ 
for $x,\overline{x}\in X,$ $y\in Y$ and $z\in Z.$
Thus we have a first generalization of Avramescu’s theorem for self maps.
Theorem 1.3.
[6] Let ${K}_{i}\subset {V}_{i},$ $i=p+1,\mathrm{\dots},n,$ be convex such that ${K}_{i}$ is compact for $i=p+1,\mathrm{\dots},p+q,$ and weakly compact for $i=p+q+1,\mathrm{\dots},n.$ Let ${K}_{Y}=\prod _{i=p+1}^{p+q}{K}_{i},{K}_{Z}=\prod _{i=p+q+1}^{n}{K}_{i},$and $N=({N}_{1},\mathrm{\dots},{N}_{n}):X\times {K}_{Y}\times {K}_{Z}\to V$ a map with
$${N}_{i}:X\times {K}_{Y}\times {K}_{Z}\to {V}_{i},i=1,\mathrm{\dots},p;{N}_{i}:X\times {K}_{Y}\times {K}_{Z}\to {K}_{i},i=p+1,\mathrm{\dots},n.$$ 
On the maps $F,G,H$ one assumes that
(i) for each point $x\in X,$ $F(x,.,.)$ is sequentially continuous from ${K}_{Y}\times {K}_{Z}$ to $X$ with respect to the strong topology on ${K}_{Y}$ and weak topology on ${K}_{Z},$ and there exists a square matrix $M$ of size $p$ having nonnegative entries and with the spectral radius $$ such that
$$d(F(x,y,z),F(\overline{x},y,z))\le Md(x,\overline{x})$$ 
for all $x,\overline{x}\in X,$ $y\in {K}_{Y}$ and $z\in {K}_{Z}.$
(ii) $G$ and $H$ are sequentially continuous from $X\times {K}_{Y}\times {K}_{Z}$ to ${K}_{Y}$ and ${K}_{Z},$ respectively, with respect to the strong topology on ${K}_{Y}$ and weak topology on ${K}_{Z}.$
Then there exists ${v}^{\ast}=({x}^{\ast},{y}^{\ast},{z}^{\ast})\in X\times {K}_{Y}\times {K}_{Z}$ with $N\left({v}^{\ast}\right)={v}^{\ast}.$
A second generalization of Avramescu’s theorem uses together the BanachPerov fixed point theorem and the strong and weak versions of Mönch’s fixed point theorem for self maps.
Theorem 1.4.
[6] Let ${D}_{i}\subset {V}_{i}$ be a closed convex set for $i=p+1,\mathrm{\dots},n,$ ${D}_{Y}=\prod _{i=p+1}^{p+q}{D}_{i},{D}_{Z}=\prod _{i=p+q+1}^{n}{D}_{i},$and let $N=({N}_{1},\mathrm{\dots},{N}_{n}):X\times {D}_{Y}\times {D}_{Z}\to V$ be a map with
$${N}_{i}:X\times {D}_{Y}\times {D}_{Z}\to {V}_{i},i=1,\mathrm{\dots},p;{N}_{i}:X\times {D}_{Y}\times {D}_{Z}\to {D}_{i},i=p+1,\mathrm{\dots},n.$$ 
Assume that the maps $F,G,H$ satisfy the following conditions:
(i) for each point $x\in X,$ $F(x,.,.)$ is sequentially continuous from ${D}_{Y}\times {D}_{Z}$ to $X$ with respect the strong topology on ${D}_{Y}$ and the weak topology on ${D}_{Z},$ and there exists a square matrix $M$ of size $p$ having nonnegative entries and spectral radius less than $1$ such that
$$d(F(x,y,z),F(\overline{x},y,z))\le Md(x,\overline{x})$$ 
for all $x,\overline{x}\in X$, $y\in {D}_{Y}$, and $z\in {D}_{Z};$
(ii) $G$ and $H$ are sequentially continuous from $X\times {D}_{Y}\times {D}_{Z}$ to ${D}_{Y}$ and ${D}_{Z},$ respectively, with respect the strong topology on ${D}_{Y}$ and the weak topology on ${D}_{Z};$
(iii) for some points $({x}_{p+1},\mathrm{\dots},{x}_{p+q})\in {D}_{Y}$ and $({x}_{p+q+1},\mathrm{\dots},{x}_{n})\in {D}_{Z},$ one has
$$\begin{array}{c}{C}_{i}\subset {V}_{i}\text{countable for}i=1,..,p,\text{}\hfill \\ {C}_{i}\subset {D}_{i}\text{countable for}i=p+1,\mathrm{\dots},n,C={\prod}_{i=1}^{n}{C}_{i},\hfill \\ \overline{{C}_{i}}=\overline{\text{conv}}\left(\left\{{x}_{i}\right\}\cup {N}_{i}\left(C\right)\right),\text{}i=p+1,\mathrm{\dots},p+q,\hfill \\ {\overline{{C}_{i}}}^{w}=\overline{\text{conv}}\left(\left\{{x}_{i}\right\}\cup {N}_{i}\left(C\right)\right),\text{}i=p+q+1,\mathrm{\dots},n,\hfill \\ \text{implies}\hfill \\ \overline{{C}_{i}}\text{is strongly compact for}i=p+1,\mathrm{\dots},p+q\text{and}\hfill \\ {\overline{{C}_{i}}}^{w}\text{is weakly compact for}i=p+q+1,\mathrm{\dots},n.\hfill \end{array}$$ 
Then there exists ${v}^{\ast}=({x}^{\ast},{y}^{\ast},{z}^{\ast})\in X\times {D}_{Y}\times {D}_{Z}$ with $N\left({v}^{\ast}\right)={v}^{\ast}.$
1.2 Krasnosel’skii type results for nonself maps
Several extensions of Krasnosel’skii’s theorem to nonself maps have been given. One is due to O’Regan and is based on the fact that the map $A+B$ in Krasnosel’skii’s theorem is condensing and thus the LeraySchauder continuation principle for condensing maps can be used. Stated as a continuation theorem, O’Regan’s result reads as follows.
Theorem 1.5 (O’Regan).
[13] Let $U$ be an open set in a closed, convex set $C$ of a Banach space $(X,.{}_{X})$. Assume $0\in U$ and $N:\overline{U}\to C$ is given by $N=A+B,$ where $A:\overline{U}\to C$ is a $\varphi $contraction, i.e., there exists a continuous nondecreasing function $\varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ satisfying $$ for $t>0$, such that ${\leftA\left(x\right)A\left(y\right)\right}_{X}\le \phi \left({\leftxy\right}_{X}\right)$ for all $x,y\in \overline{U},$ and $B:\overline{U}\to C$ is completely continuous. Then either,
(i) $N$ has a fixed point in $\overline{U},$ or
(ii) there is a point $u\in \partial U$ and $\lambda \in (0,1)$ with $u=\lambda N\left(u\right).$
Using the original idea of the proof of Krasnosel’skii’s theorem and Schaefer’s fixed point theorem instead of Schauder’s one, we have the following result due to Burton and Kirk, also presented as a continuation theorem.
Theorem 1.6 (BurtonKirk).
[5] Let $X$ be a Banach space, $A,B:X\to X,A$ a contraction and $B$ completely continuous. Then either
(i) $x=\lambda A(\frac{x}{\lambda})+\lambda Bx$ has a solution in $X$ for $\lambda =1,$ or
(ii) the set of all such solutions, $$ is unbounded.
A similar result is given in [9] by replacing the equation $x=\lambda A(x/\lambda )+\lambda Bx$ with $x=A(x)+\lambda Bx.$
Theorem 1.7 (GaoLiZhang).
[9] Let $X$ be a Banach space, $A,B:X\to X,A$ a contraction and $B$ completely continuous. Then either
(i) $x=A(x)+\lambda B\left(x\right)$ has a solution in $V$ for $\lambda =1,$ or
(ii) the set of all such solutions, $$ is unbounded.
In the proofs, one uses the homotopy $\lambda {\left(IA\right)}^{1}B$ in case of Theorem 1.6, in contrast to Theorem 1.7 where the homotopy ${\left(IA\right)}^{1}\lambda B$ is used.
For other extensions of Krasnosel’skii’s fixed point theorem we refer to [2], [3], [8], [10], [11], [14], [15], [18] and [19]. A variational version of Avramescu’s theorem is given in [4].
The aim of this paper is to obtain nonlinear alternatives of LeraySchauder and Mönch type for nonself vectorvalued operators, under hybrid conditions of Perov contraction and compactness. Thus we shall extend the theorems of Krasnosel’skii, Avramescu, BurtonKirk and GaoLiZhang. An application is given to a boundary value problem for a system of second order differential equations in which some of the equations are implicit.
2 Nonlinear alternatives
2.1 Avramescu type principle for nonself maps
Consider a system of two operator equations
$$\{\begin{array}{c}{N}_{1}(x,y)=x\hfill \\ {N}_{2}(x,y)=y.\hfill \end{array}$$  (2) 
We have the following general topological principle in terms of fixed point index (see [7, Section 2.1]).
Theorem 2.1.
Let $Y$ be a Banach space, $K\subset Y$ a retract of $Y$ and $U\subset K$ open in $K.$ Let $\mathrm{\Lambda}$ be a topological space and
${N}_{1}$  $:$  $\mathrm{\Lambda}\times \overline{U}\to \mathrm{\Lambda},$  
${N}_{2}$  $:$  $\mathrm{\Lambda}\times \overline{U}\to K$ 
be two mappings such that the following conditions are satisfied:
(a) For each $y\in \overline{U},$ there is a unique $x=:S\left(y\right)\in \mathrm{\Lambda}$ with
$${N}_{1}(S\left(y\right),y)=S\left(y\right);$$ 
(b) There is a compact map $H:\overline{U}\times [0,1]\to K,$ ${H}_{\lambda}:=H(\cdot ,\lambda ),$ with
$$i({H}_{0},U,K)\ne 0\phantom{\rule{1.5em}{0ex}}\text{and}{H}_{1}={N}_{2}(S(.),.).$$ 
Then either
(i) the system (2) has a solution $(x,y)\in \mathrm{\Lambda}\times \overline{U}$, or
(ii) there is a point $y\in {\partial}_{K}U$ and $\lambda \in (0,1)$ with $y=H(y,\lambda ).$
Proof.
In virtute of (a), $(x,y)\in \mathrm{\Lambda}\times \overline{U}$ is a solution of (2) if and only if $x=S\left(y\right)$ and $y$ is a fixed point of ${H}_{1}.$ Assume that (ii) does not hold. Hence ${H}_{\lambda}$ is fixed point free on ${\partial}_{K}U$ for $\lambda \in (0,1).$ Obviously, condition $i({H}_{0},U,K)\ne 0$ makes necessary that ${H}_{0}$ is fixed point free on ${\partial}_{K}U$ too. If ${H}_{1}$ has a fixed point in ${\partial}_{K}U,$ then (i) holds and we are finished. Otherwise, $H$ is an admissible homotopy joining ${H}_{1}$ with ${H}_{0}$ and from the homotopy invariance of the fixed point index we should have $i({H}_{1},U,K)=i({H}_{0},U,K)\ne 0,$ which guarantees that ${H}_{1}$ has a fixed point in $U,$ thus again condition (i) is satisfied. ∎
Theorem 2.1 gives in particular hybrid results for nonself maps of Krasnosel’kii, BurtonKirk and GaoLiZhang types. Instead the common contraction property, we consider its vector analogue, the Perov contraction (for this notion and related topics of vector analysis, we refer the reader to [16, Chapter 10]). To this aim, consider $({X}_{i},.{}_{i}),$ $i=1,\mathrm{\dots},n$ Banach spaces and $X={X}_{1}\times \mathrm{\dots}\times {X}_{n}$ the product space endowed with the norm
$${\leftx\right}_{X}={\left{x}_{1}\right}_{1}+\mathrm{\dots}+{\left{x}_{n}\right}_{n}.$$ 
All topological notions such as continuity, compactness, boundary of a set, related to the product space $X$ will be considered with respect to the norm $.{}_{X}.$ Also, on $X$ we consider the vectorvalued norm
$$\Vert x\Vert ={({\left{x}_{1}\right}_{1},\mathrm{\dots},{\left{x}_{n}\right}_{n})}^{T}.$$ 
With respect to the vectorvalued metric $\Vert xy\Vert ,$ the space $X$ is a complete generalized metric space.
In such a product space $X$ we have:
Corollary 2.2.
Let $U\subset X$ be open with $0\in U$,$A:X\to X$ a Perov contraction and $B:\overline{U}\to X$ compact. Then either
(i) the map $A+B$ has a fixed point in $\overline{U},$ or
(ii) there is a point $x\in \partial U$ and $\lambda \in (0,1)$ with $x=\lambda \left(A\left(x\right)+B\left(x\right)\right).$
Proof.
Apply Theorem 2.1. Here $K=Y=\mathrm{\Lambda}=X,{N}_{1}(x,y)=A\left(x\right)+B\left(y\right),{N}_{2}(x,y)=x,$ $S={\left(IA\right)}^{1}B$ and $H(x,\lambda )={\left(I\lambda A\right)}^{1}\lambda B.$ ∎
A similar result is the vector version of BurtonKirk theorem:
Corollary 2.3.
Let $U\subset X$ be open with $0\in U$,$A:X\to X$ a Perov contraction and $B:\overline{U}\to X$ compact. Then either
(i) the map $A+B$ has a fixed point in $\overline{U},$ or
(ii) there is a point $x\in \partial U$ and $\lambda \in (0,1)$ with $x=\lambda A\left(\frac{x}{\lambda}\right)+\lambda B\left(x\right).$
Proof.
Apply Theorem 2.1. Here $K=Y=\mathrm{\Lambda}=X,{N}_{1}(x,y)=A\left(x\right)+B\left(y\right),{N}_{2}(x,y)=x,$ $S={\left(IA\right)}^{1}B$ and $H(x,\lambda )=\lambda {\left(IA\right)}^{1}B.$ ∎
Theorem 2.1 also yields the vector version of GaoLiZhang:
Corollary 2.4.
Let $U\subset X$ be open with $0\in U$,$A:X\to X$ a Perov contraction and $B:\overline{U}\to X$ compact. Then either
(i) the map $A+B$ has a fixed point in $\overline{U},$ or
(ii) there is a point $x\in \partial U$ and $\lambda \in (0,1)$ with $x=A\left(x\right)+\lambda B\left(x\right).$
Proof.
Apply Theorem 2.1. Here $K=Y=\mathrm{\Lambda}=X,{N}_{1}(x,y)=A\left(x\right)+B\left(y\right),{N}_{2}(x,y)=x,$ $S={\left(IA\right)}^{1}B$ and $H(x,\lambda )={\left(IA\right)}^{1}\lambda B.$ ∎
2.2 AvramescuMönch theorem for nonself maps
We finish this section by an AvramescuMönch type result for nonself maps. It involves a compactness condition which does not make possible the use of the fixed point index.
Theorem 2.5.
Let $\mathrm{\Lambda}$ be a complete generalized metric space endowed with the vectorvalued metric $d,$ $Y$ a Banach space, $U\subset Y$ open with $0\in U,$ ${N}_{1}:\mathrm{\Lambda}\times \overline{U}\to \mathrm{\Lambda}$ and ${N}_{2}:\mathrm{\Lambda}\times \overline{U}\to Y$ two maps such that
(a) ${N}_{1}(.,y)$ is a Perov contraction with the same Lipschitz matrix for $y\in \overline{U};$
(b) ${N}_{2}$ is continuous and a Mönch map, i.e., if $C={C}_{1}\times {C}_{2}\subset \mathrm{\Lambda}\times \overline{U}$ is countable and ${C}_{2}\subset \overline{\text{conv}}\left(\left\{0\right\}\cup {N}_{2}\left(C\right)\right),$ then ${\overline{C}}_{2}$ is compact.
Then either
(i) the system (2) has a solution $(x,y)\in \mathrm{\Lambda}\times \overline{U}$, or
(ii) there is a point $(x,y)\in \mathrm{\Lambda}\times \partial U$ and $\lambda \in (0,1)$ with
$$\{\begin{array}{c}{N}_{1}(x,y)=x\hfill \\ \lambda {N}_{2}(x,y)=y.\hfill \end{array}$$  (3) 
Proof.
Solving (2) reduces to the fixed point equation $y={N}_{2}(S\left(y\right),y)$ in $\overline{U},$ where $S:\overline{U}\to \mathrm{\Lambda}$ is defined by ${N}_{1}(S\left(y\right),y)=S\left(y\right)$ and whose welldefinition and continuity are guarantees by (a). Next we use Mönch’s fixed point theorem for nonself maps ([7, Theorem 18.1]). To this end we first need to check Mönch’s compactness condition for the operator $T:=$ ${N}_{2}(S(.),.).$ If $C\subset \overline{U}$ is countable and $C\subset \overline{\text{conv}}\left(\left\{0\right\}\cup T\left(C\right)\right),$ then $C\subset \overline{\text{conv}}\left(\left\{0\right\}\cup {N}_{2}(S\left(C\right),C)\right),$ whence in view of (b) implies that ${\overline{C}}_{2}$ is compact. Next observe that if a point $(x,y)\in \mathrm{\Lambda}\times \partial U$ and $\lambda \in (0,1)$ satisfy (3) then $x=S\left(y\right)$ and $\lambda {N}_{2}(S\left(y\right),y)=y,$ that is $\lambda T\left(y\right)=y,$ which means that the LeraySchauder boundary condition does not hold. Thus Mönch’s fixed point theorem applies and gives the result. ∎
3 Application
We present an application of the vector version of BurtonKirk theorem, Corollary 2.3, to the following boundary value problem for a system of $n$ equations
${u}_{i}^{\prime \prime}$  $=$  ${f}_{i}(t,{V}_{i}u)+{g}_{i}(t,{V}_{0}u),\text{a.e.}t\in (0,1)$  (4)  
${u}_{i}\left(0\right)$  $=$  ${u}_{i}\left(1\right)=0,i=1,2,\mathrm{\cdots},n,$ 
where ${V}_{0}u$ and ${V}_{i}u$ denote the vectors
$${V}_{0}u=(u,{u}^{\prime})=({u}_{1},\mathrm{\cdots},{u}_{n},{u}_{1}^{\prime},\mathrm{\cdots},{u}_{n}^{\prime}),{V}_{i}u=({V}_{0}u,{u}_{i}^{\prime \prime})=(u,{u}^{\prime},{u}_{i}^{\prime \prime}).$$ 
Also the mappings ${f}_{i}$ and ${g}_{i}$ are assumed to satisfy the Carathéodory conditions and some additional conditions given below.
Note that the equations are implicit due to the dependence on ${u}_{i}^{\prime \prime}$ of the terms ${f}_{i}(t,{V}_{i}u)$.
Denoting $Lu={u}^{\prime \prime}$ and letting ${x}_{i}=L{u}_{i},$ the system is equivalent to
${x}_{i}$  $=$  ${f}_{i}(t,{V}_{i}{L}^{1}x)+{g}_{i}(t,{V}_{0}{L}^{1}x),\text{a.e.}t\in (0,1)$  
${x}_{i}$  $\in $  ${L}^{2}(0,1),i=1,2,\mathrm{\cdots},n.$ 
Let
${A}_{i}\left(x\right)$  $:$  $={f}_{i}(t,{V}_{i}{L}^{1}x)$  
${B}_{i}\left(x\right)$  $:$  $={g}_{i}(t,{V}_{0}{L}^{1}x).$ 
Thus our problem appears as a fixed point equation
$$x=A\left(x\right)+B\left(x\right),x\in {L}^{2}(0,1;{\mathbb{R}}^{n}).$$ 
We shall apply the vector version of BurtonKirk theorem in the space ${L}^{2}(0,1;{\mathbb{R}}^{n}).$ Before checking the conditions of this theorem, let us recall:
0) Some basic results on Sobolev spaces in one dimension.
One denotes by ${H}_{0}^{1}(0,1)$ the space of all absolutely continuous functions on $[0,1]$ vanishing at $0$ and $1$ and whose derivative belongs to ${L}^{2}(0,1).$ This is a Hilbert space under the scalar product and norm
$${\u27e8x,y\u27e9}_{{H}_{0}^{1}}={\int}_{0}^{1}{x}^{\prime}{y}^{\prime}\mathit{d}t,{\leftx\right}_{{H}_{0}^{1}}={\left{x}^{\prime}\right}_{{L}^{2}}={\left({\int}_{0}^{1}{x}^{\prime 2}\mathit{d}t\right)}^{1/2}.$$ 
By ${H}^{1}(0,1)$ one denotes the dual of ${H}_{0}^{1}(0,1)$ and the following embeddings hold: ${H}_{0}^{1}(0,1)\subset {L}^{2}(0,1)\subset {H}^{1}(0,1).$ According to Poincaré’s inequality one has
${\leftx\right}_{{L}^{2}}$  $\le $  $\frac{1}{\pi}}{\leftx\right}_{{H}_{0}^{1}}\phantom{\rule{2em}{0ex}}\left(x\in {H}_{0}^{1}(0,1)\right),$  
${\leftx\right}_{{H}^{1}}$  $\le $  $\frac{1}{\pi}}{\leftx\right}_{{L}^{2}}\phantom{\rule{2.33em}{0ex}}\left(x\in {L}^{2}(0,1)\right).$ 
The operator $Lx={x}^{\prime \prime}$ is an isometry between ${H}_{0}^{1}(0,1)$ and ${H}^{1}(0,1),$ so ${\left{L}^{1}x\right}_{{H}_{0}^{1}}={\leftx\right}_{{H}^{1}}$ $\left(x\in {H}^{1}(0,1)\right).$
Also note that the number ${\mu}_{1}={\pi}^{2}$ is the first eigenvalue of the operator ${x}^{\prime \prime}$ under the Dirichlet boundary condition $x\left(0\right)=x\left(1\right)=0.$
1) We now guarantee that $A$ is a Perov contraction on ${L}^{2}(0,1;{\mathbb{R}}^{n}).$ To this aim assume that ${f}_{i}$ are Lipschitz continuous, more exactly
$$\left{f}_{i}(t,x,y,z){f}_{i}(t,\overline{x},\overline{y},\overline{z})\right\le \sum _{j=1}^{n}\left({a}_{ij}\left{x}_{j}{\overline{x}}_{j}\right+{b}_{ij}\left{y}_{j}{\overline{y}}_{j}\right\right)+{c}_{i}\leftz\overline{z}\right$$  (5) 
for all $x,\overline{x},y,\overline{y}\in {\mathbb{R}}^{n};$ $z,\overline{z}\in \mathbb{R}$ and a.e. $t\in (0,1).$
Then for any $x,y\in {L}^{2}(0,1;{\mathbb{R}}^{n}),$ one has
${\left{A}_{i}\left(x\right){A}_{i}\left(y\right)\right}_{{L}^{2}}$  
$\le $  $\sum _{j=1}^{n}}\left({a}_{ij}{\left{L}^{1}\left({x}_{j}{y}_{j}\right)\right}_{{L}^{2}}+{b}_{ij}{\left{L}^{1}{\left({x}_{j}{y}_{j}\right)}^{\prime}\right}_{{L}^{2}}\right)+{c}_{i}{\left{x}_{i}{y}_{i}\right}_{{L}^{2}$  
$\le $  $\sum _{j=1}^{n}}\left({\displaystyle \frac{{a}_{ij}}{{\pi}^{2}}}+{\displaystyle \frac{{b}_{ij}}{\pi}}\right){\left{x}_{j}{y}_{j}\right}_{{L}^{2}}+{c}_{i}{\left{x}_{i}{y}_{i}\right}_{{L}^{2}},$ 
where we used
$${\left{L}^{1}\left({x}_{j}{y}_{j}\right)\right}_{{L}^{2}}\le \frac{1}{\pi}{\left{L}^{1}\left({x}_{j}{y}_{j}\right)\right}_{{H}_{0}^{1}}=\frac{1}{\pi}{\left{x}_{j}{y}_{j}\right}_{{H}^{1}}\le \frac{1}{{\pi}^{2}}{\left{x}_{j}{y}_{j}\right}_{{L}^{2}}.$$ 
Therefore $A$ is a Perov contraction if
$$  (7) 
where $\rho \left(M\right)$ is the spectral radius of the matrix $M={\left[{m}_{ij}\right]}_{1\le i,j\le n}$ whose entries are
$${m}_{ij}=\frac{{a}_{ij}}{{\pi}^{2}}+\frac{{b}_{ij}}{\pi}\phantom{\rule{1em}{0ex}}\text{for}j\ne i,{m}_{ii}=\frac{{a}_{ii}}{{\pi}^{2}}+\frac{{b}_{ii}}{\pi}+{c}_{i}.$$ 
2) A priori bounds for the solutions of the equations
$$x=\lambda A\left(\frac{1}{\lambda}x\right)+\lambda B\left(x\right),\lambda \in (0,1).$$  (8) 
Step 1: Bounds for ${\left{x}_{i}\right}_{{H}^{1}}.$
We have
$${x}_{i}=\lambda {A}_{i}(t,\frac{1}{\lambda}{V}_{i}{L}^{1}x)+\lambda {B}_{i}(t,{V}_{0}{L}^{1}x).$$  (9) 
Now we introduce a sign condition on ${g}_{i},$ namely
$${x}_{i}{g}_{i}(t,x,y)\le 0\phantom{\rule{2em}{0ex}}\text{for every}x,y\in {\mathbb{R}}^{n}\text{and a.e.}t\in (0,1).$$  (10) 
Now we multiply by ${L}^{1}{x}_{i}$ in (9) and integrate over $[0,1]$ and we observe that ${L}^{1}{x}_{i}{g}_{i}(t,{V}_{0}{L}^{1}x)\le 0,$ we obtain
${\u27e8{x}_{i},{L}^{1}{x}_{i}\u27e9}_{{L}^{2}}$  
$=$  ${\left{L}^{1}{x}_{i}\right}_{{H}_{0}^{1}}^{2}={\left{x}_{i}\right}_{{H}^{1}}^{2}\le \lambda {\displaystyle {\int}_{0}^{1}}{L}^{1}{x}_{i}{A}_{i}(t,{\displaystyle \frac{1}{\lambda}}{V}_{i}{L}^{1}x)\mathit{d}t$  
$\le $  ${\int}_{0}^{1}}\left{L}^{1}{x}_{i}\right\left({\displaystyle \sum _{j=1}^{n}}\left({a}_{ij}\left{L}^{1}{x}_{j}\right+{b}_{ij}\left{\left({L}^{1}{x}_{j}\right)}^{\prime}\right\right)+{c}_{i}\left{x}_{i}\right+\left{f}_{i}(t,0)\right\right)\mathit{d}t$  
$\le $  $\sum _{j=1}^{n}}\left({a}_{ij}{\left{L}^{1}{x}_{j}\right}_{{L}^{2}}+{b}_{ij}{\left{\left({L}^{1}{x}_{j}\right)}^{\prime}\right}_{{L}^{2}}\right){\left{L}^{1}{x}_{i}\right}_{{L}^{2}}+{\varphi}_{i}{\left{L}^{1}{x}_{i}\right}_{{L}^{2}$  
$+{c}_{i}{\displaystyle {\int}_{0}^{1}}\left{L}^{1}{x}_{i}\right\left{x}_{i}\right\mathit{d}t,$ 
where ${\varphi}_{i}={\left{f}_{i}(\cdot ,0)\right}_{{L}^{2}}.$ We have
$${\left{\left({L}^{1}{x}_{j}\right)}^{\prime}\right}_{{L}^{2}}={\left{L}^{1}{x}_{j}\right}_{{H}_{0}^{1}},{\left{L}^{1}{x}_{j}\right}_{{L}^{2}}\le \frac{1}{\pi}{\left{L}^{1}{x}_{j}\right}_{{H}_{0}^{1}}=\frac{1}{\pi}{\left{x}_{j}\right}_{{H}^{1}}.$$ 
Also
$${\int}_{0}^{1}\left{L}^{1}{x}_{i}\right\left{x}_{i}\right\mathit{d}t={\u27e8{x}_{i},\sigma {L}^{1}{x}_{i}\u27e9}_{{L}^{2}},$$ 
where $\sigma \left(t\right)$ gives the sign of the function ${x}_{i}\left(t\right)\left({L}^{1}{x}_{i}\right)\left(t\right).$ Furthermore
$${\u27e8{x}_{i},\sigma {L}^{1}{x}_{i}\u27e9}_{{L}^{2}}={\left{x}_{i}\right}_{{H}^{1}}{\left\sigma {L}^{1}{x}_{i}\right}_{{H}_{0}^{1}}={\left{x}_{i}\right}_{{H}^{1}}{\left{L}^{1}{x}_{i}\right}_{{H}_{0}^{1}}={\left{x}_{i}\right}_{{H}^{1}}^{2}$$ 
Then
$${\left{x}_{i}\right}_{{H}^{1}}^{2}\le \sum _{j=1}^{n}\left(\frac{{a}_{ij}}{{\pi}^{2}}+\frac{{b}_{ij}}{\pi}\right){\left{x}_{i}\right}_{{H}^{1}}{\left{x}_{j}\right}_{{H}^{1}}+{c}_{i}{\left{x}_{i}\right}_{{H}^{1}}^{2}+\frac{{\varphi}_{i}}{\pi}{\left{x}_{i}\right}_{{H}^{1}},$$ 
whence
$${\left{x}_{i}\right}_{{H}^{1}}\le \sum _{j=1}^{n}\left(\frac{{a}_{ij}}{{\pi}^{2}}+\frac{{b}_{ij}}{\pi}\right){\left{x}_{j}\right}_{{H}^{1}}+{c}_{i}{\left{x}_{i}\right}_{{H}^{1}}+\frac{{\varphi}_{i}}{\pi}.$$ 
These can be put under the matrix form
$${\Vert x\Vert}_{{H}^{1}}\le M{\Vert x\Vert}_{{H}^{1}}+\overline{\varphi},$$ 
with columns ${\Vert x\Vert}_{{H}^{1}}={\left[{\left{x}_{i}\right}_{{H}^{1}}\right]}_{1\le i\le n}^{T}$ and $\overline{\varphi}=\frac{1}{\pi}{\left[{\varphi}_{i}\right]}_{1\le i\le n}^{T}.$ Since the spectral radius of $M$ is less than one, we may multiply by ${\left(IM\right)}^{1}$ and obtain
$${\Vert x\Vert}_{{H}^{1}}\le {\left(IM\right)}^{1}\overline{\varphi}$$ 
which proves that ${\left{x}_{i}\right}_{{H}^{1}}$ are bounded.
Step 2: ${\left{B}_{i}\left(x\right)\right}_{{L}^{2}}\le {\overline{C}}_{i}.$ To this aim we impose a second condition to ${g}_{i},$ namely the growth condition
$$\left{g}_{i}(t,x,y)\right\le \sum _{j=1}^{n}\left({\alpha}_{ij}{\left{x}_{j}\right}^{p}+{\beta}_{ij}\left{y}_{j}\right\right)+{\gamma}_{i}\left(t\right)\phantom{\rule{2.5em}{0ex}}\left(x,y\in {\mathbb{R}}^{n}\right),$$  (11) 
where ${\alpha}_{ij},{\beta}_{ij}$ are nonnegative, $p\ge 1$ and ${\gamma}_{i}\in {L}^{2}(0,1;{\mathbb{R}}_{+}).$ Notice the large generality of this growth condition with respect to ${x}_{j}$ since no restriction on the exponent $p$ is required.
Then we have
$${\left{B}_{i}\left(x\right)\right}_{{L}^{2}}\le \sum _{j=1}^{n}\left({\alpha}_{ij}{\left{L}^{1}{x}_{j}\right}_{{L}^{2p}}^{p}+{\beta}_{ij}{\left{\left({L}^{1}{x}_{j}\right)}^{\prime}\right}_{{L}^{2}}\right)+{\left{\gamma}_{i}\right}_{{L}^{2}}$$ 
Furthermore, since ${H}_{0}^{1}(0,1)\subset {L}^{2p}(0,1)$ continuously, there is a constant ${\eta}_{1}$ such that
${\left{L}^{1}{x}_{j}\right}_{{L}^{2p}}^{p}$  $\le $  ${\eta}_{1}{\left{L}^{1}{x}_{j}\right}_{{H}_{0}^{1}}^{p}={\eta}_{1}{\left{x}_{j}\right}_{{H}^{1}}^{p},$  
${\left{\left({L}^{1}{x}_{j}\right)}^{\prime}\right}_{{L}^{2}}$  $=$  ${\left{L}^{1}{x}_{j}\right}_{{H}_{0}^{1}}={\left{x}_{j}\right}_{{H}^{1}}.$ 
Hence
$${\left{B}_{i}\left(x\right)\right}_{{L}^{2}}\le \sum _{j=1}^{n}\left({\overline{\alpha}}_{ij}{\left{x}_{j}\right}_{{H}^{1}}^{p}+{\beta}_{ij}{\left{x}_{j}\right}_{{H}^{1}}\right)+{\left{\gamma}_{i}\right}_{{L}^{2}}$$ 
which in view of the result from Step 1 is bounded.
Step 3: ${\left{x}_{i}\right}_{{L}^{2}}\le {C}_{i}.$Indeed, using (3) we have
${\left{x}_{i}\right}_{{L}^{2}}$  $\le $  $\lambda {\left{A}_{i}\left({\displaystyle \frac{1}{\lambda}}x\right)\right}_{{L}^{2}}+\lambda {\left{B}_{i}\left(x\right)\right}_{{L}^{2}}$  
$\le $  $\sum _{j=1}^{n}}{m}_{ij}{\left{x}_{j}\right}_{{L}^{2}}+{\varphi}_{i}+{\overline{C}}_{i},$ 
or, under the matrix form
$${\Vert x\Vert}_{{L}^{2}}\le M{\Vert x\Vert}_{{L}^{2}}+\stackrel{~}{\varphi},$$ 
where $\Vert x\Vert ,$ $\stackrel{~}{\varphi}$ are the column vectors $\Vert x\Vert ={\left[{\left{x}_{i}\right}_{{L}^{2}}\right]}_{1\le i\le n}^{T}$ and $\stackrel{~}{\varphi}={\left[{\varphi}_{i}+{\overline{C}}_{i}\right]}_{1\le i\le n}^{T}.$ Then
$${\Vert x\Vert}_{{L}^{2}}\le {\left(IM\right)}^{1}\stackrel{~}{\varphi}$$ 
which proves our claim.
3) Complete continuity of $B.$ The linear operator ${L}^{1}$ is compact from ${L}^{2}(0,1)$ to ${C}^{1}[0,1],$ while due to the growth property (11), the Nemytskii’s operator associated to ${g}_{i}$ is continuous and bounded (maps bounded sets into bounded sets) from ${C}^{1}$ to ${L}^{2}.$ Consequently, as the composition of the previous two operators, ${B}_{i}$ is completely continuous from ${L}^{2}(0,1;{\mathbb{R}}^{n})$ to ${L}^{2}(0,1).$
4) Application of Corollary 2.3. The set of all solutions of the equations (8) being bounded it can be included in an open ball $U$ of ${L}^{2}(0,1;{\mathbb{R}}^{n})$ centered at the origin and of a sufficiently large radius. Then all the assumptions of Corollary 2.3 are fulfilled.
Thus we can state following conclusion result.
References
 [1] C. Avramescu, On a fixed point theorem (in Romanian), St. Cerc. Mat. 22 (1970), no. 2, 215–221.
 [2] C.S. Barroso and E.V. Teixeira, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Anal. 60 (2005), 625–650.
 [3] I. Basoc and T. Cardinali, A hybrid nonlinear alternative theorem and some hybrid fixed point theorems for multimaps, J. Fixed Point Theory Appl. 17 (2015), 413–424.
 [4] I. Benedetti, T. Cardinali and R. Precup, Fixed point–critical point hybrid theorems and application to systems with partial variational structure, J. Fixed Point Theory Appl. 23 (2021), 63, 1–19.
 [5] T. A. Burton and C. Kirk, A fixed point theorem of KrasnoselskiiSchaefer type, Math. Nachr. 189 (1998), 23–31.
 [6] T. Cardinali, R. Precup and P. Rubbioni, Heterogeneous vectorial fixed point theorems, Mediterr. J. Math. 14 (2017), 83, 1–12.
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 [9] H. Gao, Y. Li and B. Zhang, A fixed point theorem of KrasnoselskiiSchaefer type and its applications in control and periodicity of integral equations, Fixed Point Theory 12 (2011), 91–112.
 [10] L. Górniewicz and A. Ouahab, Some fixed point theorems of a Krasnosel’skii type and application to differential inclusions, Fixed Point Theory 17 (2016), 85–92.
 [11] G.L. Karakostas, An extension of Krasnoselsk’s fixed point theorem for contractions and compact mappings, Topol. Methods Nonlinear Anal. 22 (2003), 181–191.
 [12] M.A. Krasnosel’skii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl. Ser. 2 10 (1958), 345–409.
 [13] D. O’Regan, Fixedpoint theory for the sum of two operators, Appl. Math. Lett. 9 (1996), 1–8.
 [14] S. Park, Generalizations of the Krasnoselskii fixed point theorem, Nonlinear Anal. 67 (2007), 3401–3410.
 [15] I.R. Petre and A. Petruşel, Krasnoselskii’s theorem in generalized Banach spaces and applications, Electron. J. Qual. Theory Differ. Equ. 2012, 85, 1–20.
 [16] R. Precup, Methods in Nonlinear Integral Equations, Springer Science + Business Media, Dordrecht, 2011.
 [17] I.A. Rus, On the fixed points of mappings defined on a Cartesian product. III (in Romanian), Studia Univ. Babeş–Bolyai Math. 24 (1979), no. 2, 55–56.
 [18] A. Ouahab, Some Perov’s and Krasnosel’skii type fixed point results and application, Comm. Appl. Anal. 19 (2015), 623–642.
 [19] T. Xiang and R. Yuan, Critical type of Krasnosel’skii fixed point theorem, Proc. Amer. Math. Soc. 139 (2007), 1033–1044.
[1] C. Avramescu, On a fixed point theorem (in Romanian), St. Cerc. Mat., 22(1970), no. 2, 215–221.
[2] C.S. Barroso, E.V. Teixeira, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Anal., 60(2005), 625–650.
[3] I. Basoc, T. Cardinali, A hybrid nonlinear alternative theorem and some hybrid fixed point theorems for multimaps, J. Fixed Point Theory Appl., 17(2015), 413–424.
[4] I. Benedetti, T. Cardinali, R. Precup, Fixed point–critical point hybrid theorems and application to systems with partial variational structure, J. Fixed Point Theory Appl., 23(2021), 63, 1–19.
[5] T.A. Burton, C. Kirk, A fixed point theorem of KrasnoselskiiSchaefer type, Math. Nachr., 189(1998), 23–31.
[6] T. Cardinali, R. Precup, P. Rubbioni, Heterogeneous vectorial fixed point theorems, Mediterr. J. Math., 14(2017), 83, 1–12.
[7] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985
[8] B.C. Dhage, Local fixed point theory for the sum of two operators in Banach spaces, Fixed Point Theory, 4(2003), 49–60.
[9] H. Gao, Y. Li, B. Zhang, A fixed point theorem of KrasnoselskiiSchaefer type and its applications in control and periodicity of integral equations, Fixed Point Theory, 12(2011), 91–112.
[10] L. Gorniewicz, A. Ouahab, Some fixed point theorems of a Krasnosel’skii type and application to differential inclusions, Fixed Point Theory, 17(2016), 85–92.
[11] G.L. Karakostas, An extension of Krasnoselsk’s fixed point theorem for contractions and compact mappings, Topol. Methods Nonlinear Anal., 22(2003), 181–191.
[12] M.A. Krasnosel’skii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl. Ser. 2, 10(1958), 345–409.
[13] D. O’Regan, Fixedpoint theory for the sum of two operators, Appl. Math. Lett., 9(1996), 1–8.
[14] A. Ouahab, Some Perov’s and Krasnosel’skii type fixed point results and application, Comm. Appl. Anal., 19(2015), 623–642.
[15] S. Park, Generalizations of the Krasnoselskii fixed point theorem, Nonlinear Anal., 67(2007), 3401–3410.
[16] I.R. Petre, A. Petru¸sel, Krasnoselskii’s theorem in generalized Banach spaces and applications, Electron. J. Qual. Theory Differ. Equ., 2012, 85, 1–20.
[17] R. Precup, Methods in Nonlinear Integral Equations, Springer Science + Business Media, Dordrecht, 2011.
[18] I.A. Rus, On the fixed points of mappings defined on a Cartesian product, III (in Romanian), Studia Univ. Babes–Bolyai Math., 24(1979), no. 2, 55–56.
[19] T. Xiang, R. Yuan, Critical type of Krasnosel’skii fixed point theorem, Proc. Amer. Math. Soc., 139(2007), 1033–1044.