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

Cabada et al. Boundary Value Problems 2014, 2014:254http://www.boundaryvalueproblems.com/content/2014/1/254

R E S E A R C H Open Access

A positive fixed point theorem withapplications to systems of Hammersteinintegral equationsAlberto Cabada1, José Ángel Cid2 and Gennaro Infante3*

*Correspondence:[email protected] di Matematica eInformatica, Università dellaCalabria, Arcavacata di Rende,Cosenza, 87036, ItalyFull list of author information isavailable at the end of the article

AbstractWe present new criteria on the existence of fixed points that combine somemonotonicity assumptions with the classical fixed point index theory. As anillustrative application, we use our theoretical results to prove the existence ofpositive solutions for systems of nonlinear Hammerstein integral equations. Anexample is also presented to show the applicability of our results.MSC: Primary 47H10; secondary 34B10; 34B18; 45G15; 47H30

Keywords: cone; boundary value problem; fixed point index; positive solution;nonlocal boundary condition; system

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

In [] Cid et al., in order to show the existence of positive solutions of the fourth-orderboundary value problem (BVP)

u() = λg(t)f (u), t ∈ (, ),

u() = u() = = u′′() = u′′(),(.)

where λ > , studied the associated Hammerstein integral equation

u(t) = λ

∫

k(t, s)g(s)f

(u(s)

)ds, (.)

© 2014 Cabada et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly credited.

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where k is precisely the Green’s function associated to the BVP (.). Having defined theconstant

γ ∗ = maxt∈[,]

∫

k(t, s)g(s) ds,

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

Theorem . Assume that lims→∞ f (s)s = +∞ and there exists B ∈ [, +∞] such that f is

non-decreasing on [, B). If

< λ < sups∈(,B)

sγ ∗f (s)

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

Note that the above theorem is valid for a specific Green’s function. On the other handthe existence of nonnegative solutions for systems of Hammerstein integral equations hasbeen widely studied; see for example [–] and references therein. In Section we givean extension of Theorem . to the context of systems of Hammerstein integral equationsof the type

u(t) = λ

∫ b

ak(t, s)g(s)f

(u(s), u(s)

)ds,

u(t) = λ

∫ b

ak(t, s)g(s)f

(u(s), u(s)

)ds,

(.)

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

In order to show the applicability of our results, we discuss the following system ofsecond-order ODEs, subject to local and nonlocal boundary conditions, which generatestwo different kernels:

u′′ (t) + λf

(u(t), u(t)

)= , t ∈ (, ),

u′′(t) + λf

(u(t), u(t)

)= , t ∈ (, ),

u′() = , u() + u′

() = ,

u′() = , u() – ξu(η) = , η ∈ (, ), < ξ < ,

(.)

computing all the constants that occur in our theory. We also prove that the system (.)has a solution for every λ,λ > . A similar result has been proven recently, in the contextof one equation subject to nonlinear boundary conditions, by Goodrich [].

2 Two fixed point theorems in conesA subset K of a real Banach space X is a cone if it is closed, K + K ⊂ K , λK ⊂ K for allλ ≥ , and K ∩ (–K) = {θ}. A cone K defines the partial ordering in X given by

x y if and only if y – x ∈ K .



We reserve the symbol ‘≤’ for the usual order on the real line. For x, y ∈ X, with x y, wedefine the ordered interval

[x, y] = {z ∈ X : x z y}.

The cone K is normal if there exists d > such that for all x, y ∈ X with x y then‖x‖ ≤ d‖y‖.

We denote the closed ball of center x ∈ X and radius r > as

B[x, r] ={

x ∈ X : ‖x – x‖ ≤ r}

,

and the intersection of the cone with the open ball centered at the origin and radius r > as

Kr = K ∩ {x ∈ X : ‖x‖ < r

}.

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

Theorem . Let N be a real Banach space with normal order cone K . Suppose thatthere exist α ≤ β such that T : [α,β] ⊂ N → N is a completely continuous monotone non-decreasing operator with α ≤ Tα and Tβ ≤ β . Then T has a fixed point and the iterativesequence αn+ = Tαn, with α = α, converges to the greatest fixed point of T in [α,β], andthe sequence βn+ = Tβn, with β = β , converges to the smallest fixed point of T in [α,β].

In the next proposition we recall the main properties of the fixed point index of a com-pletely continuous operator relative to a cone, for more details see [, ]. In the sequelthe closure and the boundary of subsets of K are understood to be relative to K .

Proposition . Let D be an open bounded set of X with ∈ DK and DK = K , where DK =D ∩ K . Assume that T : DK → K is a completely continuous operator such that x = Tx forx ∈ ∂DK . Then the fixed point index iK (T , DK ) has the following properties:

(i) If there exists e ∈ K \ {} such that x = Tx + λe for all x ∈ ∂DK and all λ > , theniK (T , DK ) = .

For example (i) holds if Tx � x for x ∈ ∂DK .(ii) If ‖Tx‖ ≥ ‖x‖ for x ∈ ∂DK , then iK (T , DK ) = .

(iii) If Tx = λx for all x ∈ ∂DK and all λ > , then iK (T , DK ) = .For example (iii) holds if either Tx � x for x ∈ ∂DK or ‖Tx‖ ≤ ‖x‖ for x ∈ ∂DK .

(iv) Let D be open in X such that D ⊂ DK . If iK (T , DK ) = and iK (T , DK ) = , then T

has a fixed point in DK \ DK . The same holds if iK (T , DK ) = and iK (T , D

K ) = .

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

Theorem . Let X be a real Banach space, K a normal cone with normal constant d ≥ and nonempty interior (i.e. solid) and T : K → K a completely continuous operator.

Assume that

() there exist β ∈ K , with Tβ β , and R > such that B[β , R] ⊂ K ,



() the map T is non-decreasing in the set

P ={

x ∈ K : x β andRd

≤ ‖x‖}

,

() there exists a (relatively) open bounded set V ⊂ K such that iK (T , V ) = and eitherKR ⊂ V or V ⊂ KR.

Then the map T has at least one non-zero fixed point x in K such that

either belongs to P or belongs to

{V \ KR, in case KR ⊂ V ,KR \ V , in case V ⊂ KR.

Proof Since B[β , R] ⊂ K , if x ∈ K with ‖x‖ = R, then x β .Suppose first that we can choose α ∈ K with ‖α‖ = R and Tα � α. Since α β and due

to the normality of the cone K we have [α,β] ⊂P , which implies that T is non-decreasingon [α,β]. Then we can apply the Theorem . to ensure the existence of a fixed point of Ton [α,β], which, in particular, is a non-trivial fixed point.

Now suppose that such α does not exist. Thus Tx � x for all x ∈ K with ‖x‖ = R, whichby Proposition .(iii) implies that iK (T , KR) = . Since, by assumption, iK (T , V ) = we getthe existence of a non-trivial fixed point x belonging to the set V \ KR (when KR ⊂ V ) orto the KR \ V (when V ⊂ KR). �

Remark . We note that we can use either Proposition .(i), or Proposition .(ii), inorder to check the assumption () in Theorem .. We also stress that P is contained inthe set {x ∈ K : R

d ≤ ‖x‖ ≤ d‖β‖}. Therefore Theorem . is a genuine generalization of theprevious fixed point theorems obtained in [–]. Moreover, we show in the applicationsthat in many cases it is useful to apply Theorem . with a set V different from Kr .

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

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

Assume that

(′) there exist α ∈ K , with Tα α, and < R < ‖α‖ such that B[α, R] ⊂ K ,(′) the map T is non-increasing in the set

P ={

x ∈ K : R ≤ ‖x‖ ≤ ‖α‖},

(′) there exists a (relatively) open bounded set V ⊂ K such that iK (T , V ) = and eitherKR ⊂ V or V ⊂ KR.

Then the map T has at least one non-zero fixed point such that

either belongs to P or belongs to

{V \ KR, in case KR ⊂ V ,KR \ V , in case V ⊂ KR.



Proof Let x ∈ K be such that ‖x‖ = R. Then by (′) we have x α and since x,α ∈ P itfollows from (′) that

Tx � Tα � α � x.

Now, if for some x ∈ ∂KR is the case that Tx x then we are done. If not, Tx � x for allx ∈ ∂KR which by Proposition . implies that iK (T , KR) = . This result together with (′)gives the existence of a non-zero fixed point with the desired localization property. �

3 An application to a system of Hammerstein integral equationsWe now apply the results of the previous section in order to prove the existence of positivesolutions of the system of integral equations

u(t) = λ

∫ b

ak(t, s)g(s)f

(u(s), u(s)

)ds := T(u, u)(t),

u(t) = λ

∫ b

ak(t, s)g(s)f

(u(s), u(s)

)ds := T(u, u)(t),

(.)

where we assume the following assumptions:

(H) λi > , for i = , .(H) ki : [a, b] × [a, b] → [, +∞) is continuous, for i = , .(H) gi : [a, b] → [, +∞) is continuous, gi(s) > for all s ∈ [a, b], for i = , .(H) fi : [, +∞) × [, +∞) → [, +∞) is continuous, for i = , .(H) There exist continuous functions i : [a, b] → [, +∞) and constants < ci < , a ≤

ai < bi ≤ b such that for every i = , ,

ki(t, s) ≤ i(s) for t, s ∈ [a, b] and ci ·i(s) ≤ ki(t, s) for t ∈ [ai, bi] and s ∈ [a, b],

and

γi,∗ := mint∈[ai ,bi]

∫ bi

ai

gi(s)ki(t, s) ds > .

We work in the space C[a, b] × C[a, b] endowed with the norm

∥∥(u, u)∥∥ := max

{‖u‖∞,‖u‖∞}

,

where ‖w‖∞ := maxt∈[a,b] |w(t)|.Set c = min{c, c} and let us define

Ki :={

w ∈ C[a, b] : w(t) ≥ for all t ∈ [a, b] and mint∈[ai ,bi]

w(t) ≥ c‖w‖∞}

,

and consider the cone K in C[a, b] × C[a, b] defined by

K :={

(u, u) ∈ K × K}

,

which is a normal cone with d = .



Under our assumptions it is routine to check that the integral operator

T(u, u)(t) :=(T(u, u)(t), T(u, u)(t)

)

leaves K invariant and is completely continuous.Now we present our main result concerning the existence of positive solutions for the

system (.).

Theorem . Assume that the assumptions (H)-(H) hold and moreover:

(H) There exist constants B, B > such that for every i = , , fi(·, ·) is non-decreasing on[, B] × [, B] (that is, if (u, u), (v, v) ∈ R with ≤ ui ≤ vi ≤ Bi for i = , , thenfi(u, u) ≤ fi(v, v) for i = , ).

(H) For every M > there exists ρ = ρ(M) > such that, for every i = , ,

inf

{f(u, v)

ρ: (u, v) ∈ [ρ,ρ/c] × [,ρ/c]

}> M,

inf

{f(u, v)

ρ: (u, v) ∈ [,ρ/c] × [ρ,ρ/c]

}> M.

Then the system (.) has at least one positive solution in K provided that

< λi < supr∈(,B),r∈(,B)

( – c)ri

fi(r, r)γ ∗i

, (.)

where

γ ∗i := max

t∈[a,b]

∫ b

agi(s)ki(t, s) ds > , for i = , .

Proof Due to (.) we can fix βi ∈ (, Bi), i = , , such that

βi – λiγ∗i fi(β,β) > cβi, i = , . (.)

On the other hand, for M > max{ λγ,∗ ,

λγ,∗ } let ρ = ρ(M) > as in (H) and fix R <min{ –c

+c · β, –c+c · β,ρ}.

Let us check that the assumptions of Theorem . are satisfied with

β(t) = (β,β) for all t ∈ [a, b],

and

V ={

(u, u) ∈ K : mint∈[a,b]

u(t) < ρ and mint∈[a,b]

u(t) < ρ}

.

Claim . B[β , R] ⊂ K and Tβ β .Since β is constant and R < min{ –c

+c · β, –c+c · β} a direct computation shows that

B[β , R] ⊂ K . Now, from (.) it follows for each t ∈ [a, b] and i = ,

[Tiβ](t) = λi

∫ b

aki(t, s)gi(s)fi(β,β) ds ≤ λiγ

∗i fi(β,β) < βi.



Moreover, since ‖βi – Tiβ‖∞ ≤ βi, i = , , and taking into account (.) we have fort ∈ [ai, bi] and i = , ,

βi – [Tiβ](t) = βi – λi

∫ b

aki(t, s)gi(s)fi(β,β) ds

≥ βi – λiγ∗i fi(β,β) > cβi ≥ c‖βi – Tiβ‖∞.

As a consequence, we have Tβ β , and the claim is proven.Claim . T is non-decreasing on the set {x ∈ K : x β}.Let u = (u, u), v = (v, v) ∈ K be such that ≤ ui(t) ≤ vi(t) ≤ βi for all t ∈ [a, b] and

i = , . Since f is non-decreasing in [,β] × [,β] we have for all t ∈ [a, b] and i = , ,

[Tiv](t) – [Tiu](t) = λi

∫ b

aki(t, s)gi(s)

[fi(v(s)

)– fi

(u(s)

)]ds ≥ .

Moreover, for all t ∈ [ai, bi], r ∈ [, ] and i = , ,

[Tiv](t) – [Tiu](t) = λi

∫ b

aki(t, s)gi(s)

[fi(v(s)

)– fi

(u(s)

)]ds

≥ λi

∫ b

aci(s)gi(s)

[fi(v(s)

)– fi(u(s)

]ds

≥ cλi

∫ b

aki(r, s)gi(s)

[fi(v(s)

)– fi

(u(s)

)]ds

= c([Tiv](r) – [Tiu](r)

),

therefore mint∈[ai ,bi]([Tiv](t) – [Tiu](t)) ≥ c‖Tiv – Tiu‖∞, i = , , so Tu Tv, and sinceP ⊂ {x ∈ K : x β}, T is also non-decreasing on P .

Claim . KR ⊂ V and iK (T , V ) = .Firstly, note that since R < ρ then we have KR ⊂ Kρ ⊂ V .Now let e(t) ≡ for t ∈ [a, b]. Then (e, e) ∈ K and we are going to prove that

(u, u) = T(u, u) + μ(e, e) for (u, u) ∈ ∂V and μ ≥ .

If not, there exist (u, u) ∈ ∂V and μ ≥ such that (u, u) = T(u, u) + μ(e, e).Without loss of generality, we can assume that for all t ∈ [a, b] we have

ρ ≤ u(t) ≤ ρ/c, mint∈[a,b]

u(t) = ρ and ≤ u(t) ≤ ρ/c.

Then, for t ∈ [a, b], we obtain

u(t) = λ

∫ b

ak(t, s)g(s)f

(u(s), u(s)

)ds + μe(t)

≥ λ

∫ b

a

k(t, s)g(s)f(u(s), u(s)

)ds + μ ≥ λMργ,∗ + μ > ρ + μ.

Thus, we obtain ρ = mint∈[a,b] u(t) > ρ + μ ≥ ρ , a contradiction.Therefore by Proposition . we have iK (T , V ) = and the proof is finished. �



Remark . The following condition, similar to the one given in [], implies (H) and itis easier to check.

(H)∗ For every i = , , limui→+∞ fi(u,u)ui

= +∞, uniformly w.r.t. uj ∈ [,∞), j = i.

Remark . In order to deal with negative kernels ki(t, s) < we can require conditions(H), (H), and (H) on the absolute value of the kernel such that |ki(t, s)| > and condi-tions (H), (H), and (H) on sgn(ki) · fi.

As an illustrative example, we apply our results to the system of ODEs

u′′ (t) + λf

(u(t), u(t)

)= , t ∈ (, ),

u′′(t) + λf

(u(t), u(t)

)= , t ∈ (, ),

(.)

with the BCs

u′() = , u() + u′

() = ,

u′() = , u() = ξu(η), η, ξ ∈ (, ).

(.)

To the system (.)-(.) we associate the system of integral equations

u(t) = λ

∫

k(t, s)f

(u(s), u(s)

)ds,

u(t) = λ

∫

k(t, s)f

(u(s), u(s)

)ds,

(.)

where the Green’s functions are given by

k(t, s) =

⎧⎨⎩

– t, s ≤ t,

– s, s > t,(.)

and

k(t, s) =

– ξ( – s) –

⎧⎨⎩

ξ

–ξ(η – s), s ≤ η,

, s > η–

⎧⎨⎩

t – s, s ≤ t,

, s > t.(.)

The Green’s function k was studied in [] were it was shown that we may take (with ournotation)

(s) = ( – s), γ ∗ =

.

The choice of [a, b] = [, ] gives

c =

, γ,∗ = .



The kernel k was extensively studied in [, ] and is more complicated to be dealtwith, due to the presence of the nonlocal term in the BCs. In this case we may take

(s) = k(, s) =

⎧⎨⎩

–s–ξ

, if η < s ≤ ,–s–ξ (η–s)

–ξ, if ≤ s ≤ η,

γ ∗ =

– ξη

( – ξ ).

The choice, as in [], of [a, b] = [, b], where

b =

⎧⎨⎩

–ξη

(–ξ ) , if + ξη ≤ η,

(–ξ ) , if + ξη > η,

leads to

c = – ξη – ( – ξ )b

– ξη, γ,∗ =

⎧⎨⎩

b, if + ξη ≤ η,

–ξη+ξη

(–ξ )(–ξ ) , if + ξη > η.

We now fix, as in [], η = /, ξ = /. This gives b = / and

γ ∗ =

, c =

, γ,∗ =

.

Furthermore take

f(u, u) =( + sin(u)

)u

, f(u, u) =( + sin(u)

)u

. (.)

In the case of the nonlinearities (.), we can choose B = B = π/. We observe that con-dition (H)∗ holds, we note that c = min{c, c} = / and that

supr∈(,π/),r∈(,π/)

ri

fi(r, r)γ ∗i

= +∞, for every i.

As a consequence, by means of Theorem ., we obtain a non-zero solution of the system(.)-(.) for every λ,λ ∈ (,∞).

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll the authors contributed equally and significantly in writing this article. All the authors read and approved the finalmanuscript.

Author details1Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiagode Compostela, 15782, Spain. 2Departamento de Matemáticas, Universidade de Vigo, Pabellón 3, Campus de Ourense,Ourense, 32004, Spain. 3Dipartimento di Matematica e Informatica, Università della Calabria, Arcavacata di Rende,Cosenza, 87036, Italy.

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

Received: 21 October 2014 Accepted: 24 November 2014



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