# A positive fixed point theorem with applications to systems of Hammerstein integral equations

- Alberto Cabada
^{1}, - José Ángel Cid
^{2}and - Gennaro Infante
^{3}Email author

**2014**:254

https://doi.org/10.1186/s13661-014-0254-8

© Cabada et al.; licensee Springer. 2014

**Received: **21 October 2014

**Accepted: **24 November 2014

**Published: **11 December 2014

## Abstract

We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of positive solutions for systems of nonlinear Hammerstein integral equations. An example is also presented to show the applicability of our results.

**MSC:** 47H10, 34B10, 34B18, 45G15, 47H30.

## Keywords

## 1 Introduction

In this manuscript we pursue the line of research developed in the recent papers [1]–[5] in order to deal with fixed point theorems on cones that mix monotonicity assumptions and conditions in one boundary, instead of imposing conditions on two boundaries as in the celebrated cone compression/expansion fixed point theorem of Krasnosel’skiĭ. In order to do this we employ the well-known monotone iterative method, combined with the classical fixed point index. In Section 2 we prove two results concerning non-decreasing and non-increasing operators in a shell, in presence of an upper or of a lower solution; in Remark 2.4 we present a comparison with previous results in this direction.

*et al.*, in order to show the existence of positive solutions of the fourth-order boundary value problem (BVP)

*k*is precisely the Green’s function associated to the BVP (1.1). Having defined the constant

the main result in [3], regarding the BVP (1.1), is the following.

### Theorem 1.1

*Assume that*${lim}_{s\to \mathrm{\infty}}\frac{f(s)}{s}=+\mathrm{\infty}$

*and there exists*$B\in [0,+\mathrm{\infty}]$

*such that*

*f*

*is non*-

*decreasing on*$[0,B)$.

*If*

(*with the obvious meaning when*$f(s)=0$), *then the BVP* (1.1) *has at least a positive solution*.

*specific*Green’s function. On the other hand the existence of nonnegative solutions for systems of Hammerstein integral equations has been widely studied; see for example [6]–[22] and references therein. In Section 3 we give an extension of Theorem 1.1 to the context of systems of Hammerstein integral equations of the type

providing, under suitable assumptions on the kernels and the nonlinearities, the existence of a positive solution.

computing all the constants that occur in our theory. We also prove that the system (1.4) has a solution for every ${\lambda}_{1},{\lambda}_{2}>0$. A similar result has been proven recently, in the context of one equation subject to nonlinear boundary conditions, by Goodrich [23].

## 2 Two fixed point theorems in cones

*K*of a real Banach space

*X*is a

*cone*if it is closed, $K+K\subset K$, $\lambda K\subset K$ for all $\lambda \ge 0$, and $K\cap (-K)=\{\theta \}$. A cone

*K*defines the partial ordering in

*X*given by

The cone *K* is *normal* if there exists $d>0$ such that for all $x,y\in X$ with $0\u2aafx\u2aafy$ then $\parallel x\parallel \le d\parallel y\parallel $.

We recall a well-known result of fixed point theory, known as the monotone iterative method (see, for example, [24], Theorem 7.A] or [25]).

### Theorem 2.1

*Let* *N* *be a real Banach space with normal order cone* *K*. *Suppose that there exist*$\alpha \le \beta $*such that*$T:[\alpha ,\beta ]\subset N\to N$*is a completely continuous monotone non*-*decreasing operator with*$\alpha \le T\alpha $*and*$T\beta \le \beta $. *Then* *T* *has a fixed point and the iterative sequence*${\alpha}_{n+1}=T{\alpha}_{n}$, *with*${\alpha}_{0}=\alpha $, *converges to the greatest fixed point of* *T* *in*$[\alpha ,\beta ]$, *and the sequence*${\beta}_{n+1}=T{\beta}_{n}$, *with*${\beta}_{0}=\beta $, *converges to the smallest fixed point of* *T* *in*$[\alpha ,\beta ]$.

In the next proposition we recall the main properties of the fixed point index of a completely continuous operator relative to a cone, for more details see [26], [27]. In the sequel the closure and the boundary of subsets of *K* are understood to be relative to *K*.

### Proposition 2.2

*Let*

*D*

*be an open bounded set of*

*X*

*with*$0\in {D}_{K}$

*and*${\overline{D}}_{K}\ne K$,

*where*${D}_{K}=D\cap K$.

*Assume that*$T:{\overline{D}}_{K}\to K$

*is a completely continuous operator such that*$x\ne Tx$

*for*$x\in \partial {D}_{K}$.

*Then the fixed point index*${i}_{K}(T,{D}_{K})$

*has the following properties*:

- (i)
*If there exists*$e\in K\setminus \{0\}$*such that*$x\ne Tx+\lambda e$*for all*$x\in \partial {D}_{K}$*and all*$\lambda >0$,*then*${i}_{K}(T,{D}_{K})=0$.

*For example*(i)

*holds if*$Tx\u22e0x$

*for*$x\in \partial {D}_{K}$.

- (ii)
*If*$\parallel Tx\parallel \ge \parallel x\parallel $*for*$x\in \partial {D}_{K}$,*then*${i}_{K}(T,{D}_{K})=0$. - (iii)
*If*$Tx\ne \lambda x$*for all*$x\in \partial {D}_{K}$*and all*$\lambda >1$,*then*${i}_{K}(T,{D}_{K})=1$.

*For example*(iii)

*holds if either*$Tx\u22e1x$

*for*$x\in \partial {D}_{K}$

*or*$\parallel Tx\parallel \le \parallel x\parallel $

*for*$x\in \partial {D}_{K}$.

- (iv)
*Let*${D}^{1}$*be open in**X**such that*$\overline{{D}^{1}}\subset {D}_{K}$.*If*${i}_{K}(T,{D}_{K})=1$*and*${i}_{K}(T,{D}_{K}^{1})=0$,*then**T**has a fixed point in*${D}_{K}\setminus \overline{{D}_{K}^{1}}$.*The same holds if*${i}_{K}(T,{D}_{K})=0$*and*${i}_{K}(T,{D}_{K}^{1})=1$.

We state our first result on the existence of non-trivial fixed points.

### Theorem 2.3

*Let* *X* *be a real Banach space*, *K* *a normal cone with normal constant*$d\ge 1$*and nonempty interior* (*i*.*e*. *solid*) *and*$T:K\to K$*a completely continuous operator*.

*Assume that*

(1):*there exist*$\beta \in K$, *with*$T\beta \u2aaf\beta $, *and*$R>0$*such that*$B[\beta ,R]\subset K$,

*the map*

*T*

*is non*-

*decreasing in the set*

(3):*there exists a* (*relatively*) *open bounded set*$V\subset K$*such that*${i}_{K}(T,V)=0$*and either*${\overline{K}}_{R}\subset V$*or*$\overline{V}\subset {K}_{R}$.

*Then the map*

*T*

*has at least one non*-

*zero fixed point*${x}_{1}$

*in*

*K*

*such that*

### Proof

Since $B[\beta ,R]\subset K$, if $x\in K$ with $\parallel x\parallel =R$, then $x\u2aaf\beta $.

Suppose first that we can choose $\alpha \in K$ with $\parallel \alpha \parallel =R$ and $T\alpha \u2ab0\alpha $. Since $\alpha \u2aaf\beta $ and due to the normality of the cone *K* we have $[\alpha ,\beta ]\subset \mathcal{P}$, which implies that *T* is non-decreasing on $[\alpha ,\beta ]$. Then we can apply the Theorem 2.1 to ensure the existence of a fixed point of *T* on $[\alpha ,\beta ]$, which, in particular, is a non-trivial fixed point.

Now suppose that such *α* does not exist. Thus $Tx\u22e1x$ for all $x\in K$ with $\parallel x\parallel =R$, which by Proposition 2.2(iii) implies that ${i}_{K}(T,{K}_{R})=1$. Since, by assumption, ${i}_{K}(T,V)=0$ we get the existence of a non-trivial fixed point ${x}_{1}$ belonging to the set $V\setminus {\overline{K}}_{R}$ (when ${\overline{K}}_{R}\subset V$) or to the ${K}_{R}\setminus \overline{V}$ (when $\overline{V}\subset {K}_{R}$). □

### Remark 2.4

We note that we can use either Proposition 2.2(i), or Proposition 2.2(ii), in order to check the assumption (3) in Theorem 2.3. We also stress that $\mathcal{P}$ is contained in the set $\{x\in K:\frac{R}{d}\le \parallel x\parallel \le d\parallel \beta \parallel \}$. Therefore Theorem 2.3 is a genuine generalization of the previous fixed point theorems obtained in [1]–[4]. Moreover, we show in the applications that in many cases it is useful to apply Theorem 2.3 with a set *V* different from ${K}_{r}$.

We observe that, following some ideas introduced in [2], Theorem 2.1], it is possible to modify the assumptions of Theorem 2.3 in order to deal with non-increasing operators. The next result describes precisely this situation.

### Theorem 2.5

*Let* *X* *be a real Banach space*, *K* *a cone with nonempty interior* (*i*.*e*. *solid*) *and*$T:K\to K$*a completely continuous operator*.

*Assume that*

(1′):*there exist*$\alpha \in K$, *with*$T\alpha \u2aaf\alpha $, *and*$0<R<\parallel \alpha \parallel $*such that*$B[\alpha ,R]\subset K$,

*the map*

*T*

*is non*-

*increasing in the set*

(3′):*there exists a* (*relatively*) *open bounded set*$V\subset K$*such that*${i}_{K}(T,V)=1$*and either*${\overline{K}}_{R}\subset V$*or*$\overline{V}\subset {K}_{R}$.

*Then the map*

*T*

*has at least one non*-

*zero fixed point such that*

### Proof

Now, if for some $x\in \partial {K}_{R}$ is the case that $Tx\u2aafx$ then we are done. If not, $Tx\u22e0x$ for all $x\in \partial {K}_{R}$ which by Proposition 2.2 implies that ${i}_{K}(T,{K}_{R})=0$. This result together with (3′) gives the existence of a non-zero fixed point with the desired localization property. □

## 3 An application to a system of Hammerstein integral equations

where we assume the following assumptions:

(H_{1}):${\lambda}_{i}>0$, for $i=1,2$.

(H_{2}):${k}_{i}:[a,b]\times [a,b]\to [0,+\mathrm{\infty})$ is continuous, for $i=1,2$.

(H_{3}):${g}_{i}:[a,b]\to [0,+\mathrm{\infty})$ is continuous, ${g}_{i}(s)>0$ for all $s\in [a,b]$, for $i=1,2$.

(H_{4}):${f}_{i}:[0,+\mathrm{\infty})\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is continuous, for $i=1,2$.

_{5}): There exist continuous functions ${\mathrm{\Phi}}_{i}:[a,b]\to [0,+\mathrm{\infty})$ and constants $0<{c}_{i}<1$, $a\le {a}_{i}<{b}_{i}\le b$ such that for every $i=1,2$,

where ${\parallel w\parallel}_{\mathrm{\infty}}:={max}_{t\in [a,b]}|w(t)|$.

*K*in $C[a,b]\times C[a,b]$ defined by

which is a normal cone with $d=1$.

leaves *K* invariant and is completely continuous.

Now we present our main result concerning the existence of positive solutions for the system (3.1).

### Theorem 3.1

*Assume that the assumptions* (H_{1})-(H_{5}) *hold and moreover*:

(H_{6}):*There exist constants*${B}_{1},{B}_{2}>0$*such that for every*$i=1,2$, ${f}_{i}(\cdot ,\cdot )$*is non*-*decreasing on*$[0,{B}_{1}]\times [0,{B}_{2}]$ (*that is*, *if*$({u}_{1},{u}_{2}),({v}_{1},{v}_{2})\in {\mathbb{R}}^{2}$*with*$0\le {u}_{i}\le {v}_{i}\le {B}_{i}$*for*$i=1,2$, *then*${f}_{i}({u}_{1},{u}_{2})\le {f}_{i}({v}_{1},{v}_{2})$*for*$i=1,2$).

_{7}):

*For every*$M>0$

*there exists*$\rho =\rho (M)>0$

*such that*,

*for every*$i=1,2$,

*Then the system*(3.1)

*has at least one positive solution in*

*K*

*provided that*

*where*

### Proof

On the other hand, for $M>max\{\frac{1}{{\lambda}_{1}{\gamma}_{1,\ast}},\frac{1}{{\lambda}_{2}{\gamma}_{2,\ast}}\}$ let $\rho =\rho (M)>0$ as in (H_{7}) and fix $R<min\{\frac{1-c}{1+c}\cdot {\beta}_{1},\frac{1-c}{1+c}\cdot {\beta}_{2},\rho \}$.

*Claim 1.*
$B[\beta ,R]\subset K$
*and*
$T\beta \u2aaf\beta $
*.*

*β*is constant and $R<min\{\frac{1-c}{1+c}\cdot {\beta}_{1},\frac{1-c}{1+c}\cdot {\beta}_{2}\}$ a direct computation shows that $B[\beta ,R]\subset K$. Now, from (3.3) it follows for each $t\in [a,b]$ and $i=1,2$

As a consequence, we have $T\beta \u2aaf\beta $, and the claim is proven.

*Claim 2.* *T* *is non-decreasing on the set*$\{x\in K:x\u2aaf\beta \}$.

*f*is non-decreasing in $[0,{\beta}_{1}]\times [0,{\beta}_{2}]$ we have for all $t\in [a,b]$ and $i=1,2$,

therefore ${min}_{t\in [{a}_{i},{b}_{i}]}([{T}_{i}v](t)-[{T}_{i}u](t))\ge c{\parallel {T}_{i}v-{T}_{i}u\parallel}_{\mathrm{\infty}}$, $i=1,2$, so $Tu\u2aafTv$, and since $\mathcal{P}\subset \{x\in K:x\u2aaf\beta \}$, *T* is also non-decreasing on $\mathcal{P}$.

*Claim 3.*${\overline{K}}_{R}\subset V$*and*${i}_{K}(T,V)=0$.

Firstly, note that since $R<\rho $ then we have ${\overline{K}}_{R}\subset {K}_{\rho}\subset V$.

If not, there exist $({u}_{1},{u}_{2})\in \partial V$ and $\mu \ge 0$ such that $({u}_{1},{u}_{2})=T({u}_{1},{u}_{2})+\mu (e,e)$.

Thus, we obtain $\rho ={min}_{t\in [{a}_{1},{b}_{1}]}u(t)>\rho +\mu \ge \rho $, a contradiction.

Therefore by Proposition 2.2 we have ${i}_{K}(T,V)=0$ and the proof is finished. □

### Remark 3.2

The following condition, similar to the one given in [7], implies (H_{7}) and it is easier to check.

${({\mathrm{H}}_{7})}^{\ast}$: For every $i=1,2$, ${lim}_{{u}_{i}\to +\mathrm{\infty}}\frac{{f}_{i}({u}_{1},{u}_{2})}{{u}_{i}}=+\mathrm{\infty}$, uniformly w.r.t. ${u}_{j}\in [0,\mathrm{\infty})$, $j\ne i$.

### Remark 3.3

In order to deal with negative kernels ${k}_{i}(t,s)<0$ we can require conditions (H_{2}), (H_{3}), and (H_{5}) on the absolute value of the kernel such that $|{k}_{i}(t,s)|>0$ and conditions (H_{4}), (H_{6}), and (H_{7}) on $sgn({k}_{i})\cdot {f}_{i}$.

As a consequence, by means of Theorem 3.1, we obtain a non-zero solution of the system (3.4)-(3.5) for every ${\lambda}_{1},{\lambda}_{2}\in (0,\mathrm{\infty})$.

## Declarations

### Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments, which have improved the correctness and the presentation of the manuscript. A Cabada was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Projects MTM2010-15314 and MTM2013-43014-P, JA Cid was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2013-43404-P and G Infante was partially supported by G.N.A.M.P.A. - INdAM (Italy). This paper was partially written during a visit of G Infante to the Departamento de Análise Matemática of the Universidade de Santiago de Compostela. G Infante is grateful to the people of the aforementioned Departamento for their kind and warm hospitality.

## Authors’ Affiliations

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