Existence and multiplicity results for some generalized Hammerstein equations with a parameter

López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 https://doi.org/10.1186/s13662-019-2359-y RESEARCH Open Access Existence and multiplicity results for some generalized Hammerstein equations with a parameter Lucía López-Somoza1and Feliz Minhós2,3* *Correspondence: [email protected] 2Departamento de Matemática, Escola de Ciências e Tecnologia, Universidade de Évora, Évora, Portugal 3Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada, Universidade de Évora, Évora, Portugal Full list of author information is available at the end of the article Abstract This paper considers the existence and multiplicity of fixed points for the integral operator Tu(t)=λT 0 k(t,s)f(s,u(s),u(s),...,u(m)(s))ds,t∈[0,T]≡I, where λ> 0 is a positive parameter, k:I×I→Ris a kernel function such that k∈Wm,1(I×I), mis a positive integer with m≥1, and f:I×Rm+1 →[0,+∞[isan L1-Carathéodory function. The existence of solutions for these Hammerstein equations is obtained by fixed point index theory on new type of cones. Therefore some assumptions must hold only for, at least, one of the derivatives of the kernel or, even, for the kernel on a subset of the domain. Assuming some asymptotic conditions on the nonlinearity f, we get sufficient conditions for multiplicity of solutions. Two examples will illustrate the potentialities of the main results, namely the fact that the kernel function and/or some derivatives may only be positive on some subintervals, which can degenerate to a point. Moreover, an application of our method to general Lidstone problems improves the existent results in the literature in this field. MSC: 34B08; 34B10; 34B15; 34B18; 34B27 Keywords: Hammerstein equations; Nonlinear boundary value problems; Parameter dependence; Degree theory; Fixed points in cones 1 Introduction In this work we will study the existence and multiplicity of fixed points of the integral operator Tu(t)=λT 0k(t,s)fs,u(s),u(s),...,u(m)(s)ds,t∈[0,T]≡I,(1) where •λ>0is a positive parameter, •k:I×I→Ris a kernel function such that k∈Wm,1(I×I), ©The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 2 of 20 •mis a positive integer with m≥1,and •f:I×Rm+1 →[0,+∞[is an L1-Carathéodory function. Thesolvabilityofthistypeofintegralequations,knownasHammersteinequations(see [12]),hasbeenconsideredbymanyauthors.Infacttheyhavebecomebothageneralization ofdifferential equationsandboundaryvalueproblemsandamainfieldforapplicationsof methods and techniques of nonlinear analysis, as it can be seen, for instance, in [1–3,7, 11,13–17,21]. In [4], the authors consider a third order three-point boundary value problem, whose solutions are the fixed points of the integral operator Tu(t)=λ1 0G(t,s)fs,u(s),u(s),u(s)ds,t∈[0,1], where G(t,s) is an explicit Green’s function, verifying some adequateproperties such that G(t,s)and∂G ∂t(t,s) are bounded and nonnegative in the square [0,1]×[0,1], but ∂2G ∂t2(t,s) could change sign, being nonnegative in a subset of the square. In [8], a generalized Hammerstein equation is studied u(t)=1 0k(t,s)g(s)fs,u(s),u(s),...,u(m)(s)ds,(2) with k: [0,1]2→Ra kernel function such that k∈Wm,1([0,1]2), m≥1isinteger, g: [0,1] →[0,∞) is nonnegative almost everywhere in t∈[0,1] nonnegative, and f: [0,1]×Rm+1 →[0,∞)isanL ∞-Carathéodory function. Moreover, both the kernel k(t,s) and its derivatives ∂ik ∂ti(t,s), for i=1,...,m, are bounded and nonnegative on the square [0,1]×[0,1]. Ourworkgeneralizestheexistingresultsintheliteratureintroducinganewtypeofcone: K=u∈Cm(I,R):u(i)(t)≥0,t∈[mi,ni],i∈J; min t∈[aj,bj]u(j)(t)≥ξju(j)[cj,dj],j∈J1, where u(j)[cj,dj]:= max t∈[cj,dj]u(j)(t), J≡{0,1,...,m}and J1⊂J0⊂J,J1=∅. We note that the non-negativeness of the functions and their derivatives may happen onlyonasubinterval,possiblydegenerate(thatisreducedtoapoint),and,asJ1⊂J,J1=∅, thesecondpropertycanhold,locally,onlyforarestrictednumberofderivatives,including the function itself. This way, it is not required, as it was usual, that k(t,s)and∂ik ∂ti(t,s)have constant sign on the square of definition. Another important novelty is that, in the second property of the cone, we are considering the norm of the functions on a subset of the domain and not on the whole interval [0,T].Moreover,aswewillsee,thetwosubintervalsinvolvedinthiscondition,[aj,bj]and [cj,dj],musthavenonemptyintersectionbutarenotrequiredtosatisfyanyotherinclusion property (that is, it may occur that [aj,bj]⊂[cj,dj]and[cj,dj]⊂[aj,bj]). López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 3 of 20 In particular, the second property in the cone (which ensures that the minimum of the function and the derivatives on some interval is bigger than its norm on another interval) willbegivenbycertaininequalities(introducedin(H4))thatboththekernelanditsderivativesmusthold.However,contrarytorecentreferencessuchas[8],theboundsmusthold only for, at least, one of the derivatives of the kernel or, even, for the kernel on a subset of the domain. Thispaperisorganizedinthefollowingway:Sect.2containsthemainassumptions,the definition of the new cone and some properties on the integral operator. In Sect. 3,the existence results are obtained with several asymptotic assumptions on fof the sublinear or superlinear type, near 0 or +∞.Section4presents existence and multiplicity results applyingfixedpointindextheory.Section5hastwoexamplestoillustrateourmainresults and,moreover,toemphasizetheimportancethat(H4)holdsonlyforsomederivativesand thatthe subsets couldbe reduced to apoint. Last section contains anapplication to 2n-th order Lidstone problems, giving new sufficient conditions for the solvability of this kind ofproblems.Infact,ourmethodallowsthatthenonlinearitiesmaydependonderivatives of even and odd order, which is new in the literature on this type of problems, as it can be seen, for instance, in [5,18,23,26].In this way, our results fill some gaps and improve the study of Lidstone and complementary Lidstone problems. 2 Hypothesis and auxiliary results LetusconsiderE=Cm(I,R)andthenormu=max{u(i)∞,i∈J},withv∞=sup t∈I|v(t)|. It is clear that (E,·)isaBanachspace. Throughout this paper we will make the following assumptions: (H1)The kernel function k:I×I→Ris such that k∈Wm,1(I×I),withm≥1.Moreover, for i=0,...,m–1, it holds that, for every ε>0and every fixed τ∈I,there exists some δ>0such that |t–τ|<δimplies that  ∂ik ∂ti(t,s)–∂ik ∂ti(τ,s)<εfor a.e. s∈I. Finally, forthe mthderivative of the kernel, itholds that, forevery ε>0andevery fixed τ∈I,thereexistasetZτ∈Iwith measure equal to zero and some δ>0such that |t–τ|<δimplies that  ∂mk ∂tm(t,s)–∂mk ∂tm(τ,s)<ε,∀s∈I\Zτsuch that s<min{t,τ}or s>max{t,τ}. (H2)For each i∈J0⊂J,J0=∅, there exists a subinterval [mi,ni]such that ∂ik ∂ti(t,s)≥0for all t∈[mi,ni],s∈I. It is possible that the interval is degenerated, that is, mi=ni. (H3)For all i∈J, there exist positive functions hi∈L1(I)such that  ∂ik ∂ti(t,s)≤hi(s)for all t∈Iand a.e. s∈I. López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 4 of 20 (H4)For each j∈J1⊂J0,J1=∅,thereexistsubintervals[aj,bj]⊂[mj,nj]and[cj,dj],with [aj,bj]∩[cj,dj]=∅,positivefunctionsφj:I→[0,∞),andconstantsξj∈(0,1) such that  ∂jk ∂tj(t,s)≤φj(s)for all t∈[cj,dj]and a.e. s∈I, and ∂jk ∂tj(t,s)≥ξjφj(s)for all t∈[aj,bj]and a.e. s∈I. Moreover, φj∈L1(I)satisfies that bj ajφj(s)ds>0. (H5)There exists i0∈J0such that either [ci0,di0]≡Ior [mi0,ni0]≡Iand, moreover, {0,1,...,i0}⊂J0. (H6)The nonlinearity f:I×Rm+1 →[0,∞)satisfies L1-Carathéodory conditions, that is, •f(·,x0,...,xm)is measurable for each (x0,...,xm)fixed. •f(t,·,...,·)is continuous for a.e. t∈I. •Foreachr>0,thereexistsϕr∈L1(I)such that f(t,x0,...,xm)≤ϕr(t), ∀(x0,...,xm)∈(–r,r)m+1,a.e.t∈I. (H7)Functions hidefined in (H3)and ϕrdefined in (H6)are such that hiϕr∈L1(I)for every i∈Jand r>0. We will look for fixed points of operator Ton a suitable cone on the Banach space E. We recall that a cone Kis a closed and convex subset of Esatisfying the two following properties: •Ifx∈K,thenλx∈Kfor all λ≥0. •K∩(–K)={0}. In particular, taking into account the properties satisfied by the kernel k,wedefine K=u∈Cm(I,R):u(i)(t)≥0,t∈[mi,ni],i∈J0; min t∈[aj,bj]u(j)(t)≥ξju(j)[cj,dj],j∈J1. Lemma1 Hypothesis (H5)warrants that K is a cone in E. Proof WeneedtoverifythatKisaclosedandconvexsubsetofCm(I,R)andthatitsatisfies the two properties which characterize cones in a Banach space. First of all, from the definition of K, it is clear that it is closed. We will see that it is convex. For u,v∈Kand λ∈(0,1), it is clear that (1–λ)u(i)(t)+λv(i)(t)≥0fort∈[mi,ni],i∈J0. López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 5 of 20 In addition, for j∈J1, min t∈[aj,bj](1–λ)u(j)(t)+λv(j)(t)≥(1–λ)min t∈[aj,bj]u(j)(t)+λmin t∈[aj,bj]v(j)(t) ≥(1–λ)ξju(j)[cj,dj]+λξjv(j)[cj,dj] =ξj(1–λ)u(j)[cj,dj]+λv(j)[cj,dj] ≥ξj(1–λ)u(j)+λv(j)[cj,dj]. Thus, (1– λ)u+λv∈K. Moreover, from the definition of K,itistrivialtocheckthatifx∈K,thenλx∈Kfor all λ≥0. Now, to prove that K∩(–K)={0}, we will distinguish between two different cases: (I) There exists i0∈J0such that [mi0,ni0]≡I. Suppose that u,–u∈K.Thenu(i0)(t)≥0and –u(i0)(t)≥0for all t∈I,which implies that u(i0)≡0on I.Ifi0≥1,u(i0–1) is constant on I. Now, we have that u(i0–1)(t)≥0and –u(i0–1)(t)≥0for all t∈[ci0–1,di0–1],thatis, u(i0–1) ≡0on [ci0–1,di0–1]. Then, since u(i0–1) is constant on I,wededucethat u(i0–1) ≡0on I. Using the same argument repeatedly, we conclude that u≡0on I.Inthisway,we have proved that K∩(–K)={0}. (II) There exists i0∈J0such that [ci0,di0]≡I. Suppose again that u,–u∈K. Then, from the fact that min t∈[ai0,bi0]u(i0)(t)≥ξi0u(i0)Iand min t∈[ai0,bi0]–u(i0)(t)≥ξi0u(i0)I, it is deduced that u(i0)I=0, which implies that u(i0)≡0on I. Now, following the same arguments as in Case (I), we deduce the result.  In the next section, we will ensure the existence of fixed points of operator T.Before that, we need to prove that operator Tis compact. Lemma2 If (H1)–(H7)hold,then operator Tmaps the cone into itself and is completely continuous. Proof Usingstandardtechniques,itis easytoprovethatTiswelldefined, T(K)⊂K,and Tis continuous in Cm(I,R). We will see now that Tis completely continuous. Consider B=u∈E;u≤r. It is obvious that T(B)isuniformlyboundedinCm(I). López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 6 of 20 Let us prove that T(B) is equicontinuous in Cm(I). Let t2∈Ibe fixed. Then, for every ε>0,takeδ>0givenin(H1), and for i=0,...,m–1,itholdsthat|t1–t2|<δimplies that (Tu)(i)(t1)–(Tu)(i)(t2)≤λT 0 ∂ik ∂ti(t1,s)–∂ik ∂ti(t2,s)fs,u(s),...,u(m)(s)ds ≤λT 0 ∂ik ∂ti(t1,s)–∂ik ∂ti(t2,s)ϕr(s)ds≤ελT 0ϕr(s)ds, and, since ϕr∈L1(I), it is clear that there exists a positive constant κ1such that (Tu)(i)(t1)–(Tu)(i)(t2)<κ1ε for all u∈B. On the other hand, for the mth derivative, for every ε>0,takeδ>0givenin(H1)and |t1–t2|<δ,t1<t2,impliesthat (Tu)(m)(t1)–(Tu)(m)(t2)≤λT 0 ∂mk ∂tm(t1,s)–∂mk ∂tm(t2,s)fs,u(s),...,u(m)(s)ds ≤λT 0 ∂mk ∂tm(t1,s)–∂mk ∂tm(t2,s)ϕr(s)ds =λt1 0 ∂mk ∂tm(t1,s)–∂mk ∂tm(t2,s)ϕr(s)ds +λt2 t1 ∂mk ∂tm(t1,s)–∂mk ∂tm(t2,s)ϕr(s)ds +λT t2 ∂mk ∂tm(t1,s)–∂mk ∂tm(t2,s)ϕr(s)ds. From (H1), it is clear that first and third integrals in the last term of previous expression can be arbitrarily small when |t1–t2|<δ.Moreover,|∂mk ∂tm(t1,·)–∂mk ∂tm(t2,·)|ϕr(·)∈L1[t1,t2], and so there exists some δ>0suchthat λt2 t1 ∂mk ∂tm(t1,s)–∂mk ∂tm(t2,s)ϕr(s)ds<ε when |t1–t2|<δ. Therefore it is clear that, for |t1–t2|<min{δ,δ},t1<t2, there exists a positive constant κ2such that (Tu)(m)(t1)–(Tu)(m)(t2)<κ2ε for all u∈B. Analogously, when |t1–t2|<δ,t1>t2, thereexists some some positive constant κ3such that (Tu)(m)(t1)–(Tu)(m)(t2)<κ3ε for all u∈B. López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 7 of 20 Wehaveprovedthepointwise equicontinuityon I.Moreover,sinceIiscompact,pointwise equicontinuity is equivalent to uniform equicontinuity. This way, we can affirm that T(B) is equicontinuous in Cm(I). FromAscoli–Arzelàtheorem,wededucethatT(B)isrelativelycompactinCm(I).Therefore, Tis completely continuous.  3 Existence results In this section, we present two different methods to prove the existence of fixed points of operator (1). As we will see with the examples given in the next section, our two methods are not comparable, but complementary, in the sense that some examples can be solved using one of the methods (but not the other one) and vice-versa. The first method is based on the fixed point index theory on arbitrary open sets which might be, in particular, unbounded (see [9] for details). To develop this method, we need to introduce some notation Λi:=T 0hi(s)ds,Λi:=bi ai ξiφi(s)ds, ¯ Λ:=(m+1)maxΛi:i∈Jand Λ:=max{ξiΛi:i∈J1}. Moreover, we will compute the following limits at 0 and ∞: f0:= liminf |x0|,...,|xm|→0min t∈I f(t,x0,...,xm) |x0|+···+|xm|,f∞:= limsup |x0|,...,|xm|→∞ max t∈I f(t,x0,...,xm) |x0|+···+|xm|. Our main result is the following one. Theorem3 Let (H1)–(H7)hold and assume that ¯ Λf∞<Λf0.Then,for all λ∈(1 Λf0,1 ¯ Λf∞), operator Thas a fixed point in the cone K,which is not a trivial solution. Proof Fix λ∈(1 Λf0,1 ¯ Λf∞)andlet0<ε<f0be such that 1 Λ(f0–ε)≤λ≤1 ¯ Λ(f∞+ε). Then there exists some δ>0suchthat,whenu≤δ, the following inequality holds: ft,u(t),...,u(m)(t)>(f0–ε)u(t)+···+u(m)(t),∀t∈I. Define now Ωδ={u∈K;u<δ}and let u∈∂Ωδ.WewillprovethatTuu. Indeed, for t∈[ai,bi]andj∈J1, the following inequalities are satisfied: (Tu)(j)(t)=λT 0 ∂jk ∂tj(t,s)fs,u(s),...,u(m)(s)ds ≥λbj aj ∂jk ∂tj(t,s)fs,u(s),...,u(m)(s)ds ≥λbj aj ξjφj(s)fs,u(s),...,u(m)(s)ds >λbj aj ξjφj(s)(f0–ε)u(s)+···+u(m)(s)ds López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 8 of 20 ≥λ(f0–ε)ξju(j)[aj,bj]bj aj ξjφj(s)ds =λ(f0–ε)ξju(j)[aj,bj]Λj≥λ(f0–ε)Λjξju(j)(t). Taking into account the definition of Λ,wededucethat,forsomej∈J1,(Tu)(j)(t)>u(j)(t) for all t∈[aj,bj]. Therefore Tuuand, from [10, Theorem 2.3.3], we deduce that iK(T,Ωδ)=0. By the same token, there is some value ˜ C>0suchthatifmin{|u(i)(t)|:i∈J}≥˜ C,then ft,u(t),...,u(m)(t)≤f∞+εu(t)+···+u(m)(t)≤(m+1)f∞+εu,∀t∈I. Take now C>{δ,˜ C}and let ΩC=m i=0{u∈K:mint∈I|u(i)(t)|<C}.SinceΩCis an unbounded subset of the cone K, we need to check that the fixed point index of Twith respect to ΩC,iK(T,ΩC), can be defined. In particular, iK(T,ΩC) is well defined only if the set of fixed points of Tin ΩC,thatis,(I–T)–1({0})∩ΩC,iscompact(see[9]). Clearly, the continuity of (I–T)impliesthat(I–T)–1({0})∩ΩCis closed. On the other hand, we observe that if (I–T)–1({0})∩ΩCis unbounded, thenwe would have an infinite number of fixed points of Tin ΩC, which would be, in particular, fixed pointsofTintheconeK.Thus,itcanbesupposedtheexistenceofsomepositiveconstant Msuch that u<Mfor all u∈(I–T)–1({0})∩ΩC. Finally,theproofthat(I–T)–1({0})∩ΩCisequicontinuousisanalogoustotheonemade for Lemma 2. Now, we will prove that Tu≤ufor all u∈∂ΩC.Takeu∈∂ΩC,inotherwords, u∈Ksatisfying that min{mint∈I|u(i)(t)|:i∈J}=C.Then,fori∈J, (Tu)(i)(t)≤λT 0 ∂ik ∂ti(t,s)fs,u(s),...,u(m)(s)ds ≤λT 0hi(s)fs,u(s),...,u(m)(s)ds ≤(m+1)λT 0hi(s)f∞+εuds =(m+1)λf∞+εuΛi≤λf∞+εu¯ Λ≤u. As a consequence, Tu≤uand so, from [10, Theorem 2.3.3], we deduce that iK(T,ΩC)=1. Therefore, we can ensure the existence of a fixed point of operator Tin ¯ ΩC\Ωδ. We present now another approach to prove both existence and multiplicity of fixed points of the integral operator T. This method, which is not comparable with the previous one, is based on the fixed point index theory on bounded sets. First of all, we need to introduce the following sets: Kρ=u∈K;u<ρ, Vρ=u∈K:min t∈[ai,bi]u(i)(t)<ρ,i∈J2,u(i)∞<ρ,i∈J\J2, where J2={i∈J:[ci,di]≡I}. López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 9 of 20 Moreover, to ensure that the sets Kρand Vρare not the same, we need to change condition (H5)into ( H5)There exists i0∈{0,...,m}such that [ci0,di0]≡Iand, moreover, {0,1,...,i0}⊂J0. In this situation, it occurs that J2=∅and therefore KρVρKρ c,being c=min{ξi:i∈J2}.(3) We formulate sufficient conditions under which the index of the previous sets is either 1or0. Lemma4 Let 1 N=max{supt∈IT 0|∂ik ∂ti(t,s)|ds:i∈J}and fρ=supf(t,x0,...,xm) ρ;t∈I,xi∈[–ρ,ρ],i∈J. If there is some value ρ>0for which λfρ N<1, (I1 ρ) then iK(T,Kρ)=1. Proof Let us see that given u∈∂Kρand μ≥1, it occurs that Tu=μu. Suppose that previous assertion is not true. Then there exist u∈∂Kρand μ≥1for which μu(i)(t)=λT 0 ∂ik ∂ti(t,s)fs,u(s),...,u(m)(s)ds. Now, if we compute the supremum for t∈I, we get the following inequalities: μu(i)∞≤λsup t∈IT 0 ∂ik ∂ti(t,s)fs,u(s),...,u(m)(s)ds ≤λρfρsup t∈IT 0 ∂ik ∂ti(t,s)ds≤λρ fρ N<ρ. Consequently, μρ =μmax{u(i)∞:i∈J}<ρ, which contradicts that μ≥1. Thus (see [9, Corollary 7.4]) iK(T,Kρ)=1.  Lemma5 For i ∈J1,let 1 Mi=inft∈[ai,bi]bi ai ∂ik ∂ti(t,s)ds,and fi ρ=inff(t,x0,...,xm) ρ:t∈[ai,bi],xj∈0, ρ ξj,j∈J2,xk∈[0,ρ],k∈J\J2. If there exist some constant ρ>0and some index i0∈J1such that λfi0 ρ Mi0>1, (I0 ρ) then iK(T,Vρ)=0. López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 16 of 20 In this case, with the notation introduced in Lemma 4,wehavethat 1 N=max5 384,1 24,1 8,1 2=1 2 and fρ2=supt(ex+y2+z2+w2) ρ2:t∈[0,1],x,y,z,w∈[–ρ2,ρ2]=eρ2+3ρ2 2 ρ2, and so (I1 ρ2)holdsforanyλ<2ρ2 eρ2+3ρ2 2. Analogously, with the notation used in Lemma 5, 1 M0=29 7500,1 M1=7 1944, f0 ρ1=inft(ex+y2+z2+w2) ρ1:t∈[0.1,0.9],x∈[0,4ρ1],y∈[0,6ρ1],z,w∈[0,ρ1] =0.1 ρ1, and f1 ρ=inft(ex+y2+z2+w2) ρ1:t∈0, 1 3,x∈[0,4ρ1],y∈[0,6ρ1],z,w∈[0,ρ1]=0, and thus (I0 ρ1)holdsforλ>75,000ρ1 29 . Therefore, as a consequence of (C1)inTheorem6, for any pair of values ρ1,ρ2>0such that ρ1<cρ2=ρ2 6and 75,000ρ1 29 <2ρ2 eρ2+3ρ2 2, problem (5) has at least a nontrivial solution for all λ∈75,000ρ1 29 ,2ρ2 eρ2+3ρ2 2. In particular, there exists at least a nontrivial solution of (5) for all λ∈(0,0.4171). On the other hand, we obtain that f0=liminf |x|,|y|,|z|,|w|→0min t∈[0,1] t(ex+y2+z2+w2) |x|+|y|+|z|+|w|=0, and thus Theorem 3cannot be applied to this example. 5 Application to some 2n-th order problems In this section we contribute to filling some gaps on the study of general 2n-th order Lidstone boundary value problems; for n≥1, usually the nonlinearities may depend only on theevenderivatives(see,forexample,[5,18,23,26]),or generalcomplementary Lidstone López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 17 of 20 problems(see[25]andthereferencestherein).Therefore,weconsiderthefollowingproblem, with a full nonlinearity: ⎧ ⎨ ⎩ u(2n)(t)=f(t,u(t),...,u(2n–1)(t)), t∈[0,1], u(2k)(0)=u(2k)(1)=0, k=0,...,n–1, (6) and denote by G(t,s) the related Green’s function. It can be checked that, for n≥2,g(t,s)=∂2n–4G ∂t2n–4 (t,s)istheGreen’sfunctionrelatedtothe problem ⎧ ⎨ ⎩ u(4)(t)=0, t∈[0,1], u(0)=u(1)=u(0)= u(1)=0, whose explicit expression has been calculated in Example 8. As a consequence of the calculations made in that example, we know that the following facts hold for n≥2: •∂2n–4G ∂t2n–4 (t,s)=g(t,s)≥0on [0,1]×[0,1] and ∂2n–4G ∂t2n–4 (t,s)=0on the boundary of the square. •∂2n–3G ∂t2n–3 (t,s)=∂g ∂t(t,s)≥0on [0,t2]×[0,1],witht2=1–√3 3. •∂2n–2G ∂t2n–2 (t,s)=∂2g ∂t2(t,s)≤0on [0,1]×[0,1],and∂2n–2G ∂t2n–2 (t,s)=0on the boundary of the square. •∂2n–1G ∂t2n–1 (t,s)=∂3g ∂t3(1,s)≥0for s∈[0,1]. With this information, we can obtain some results about the constant sign both of the derivativesofsmallerorderofGand of the Green’s function itself. 1. Since ∂2n–4G ∂t2n–4 (t,s)≥0,forn≥3, it holds that for each fixed s∈[0,1],∂2n–5G ∂t2n–5 (·,s)is increasing. Assume that it is nonnegative. Then it would occur that ∂2n–6G ∂t2n–6 (·,s)is also increasing and, since from the boundary value conditions it holds that ∂2n–6G ∂t2n–6 (0,s)=∂2n–6G ∂t2n–6 (1,s)=0, we would conclude that ∂2n–6G ∂t2n–6 (t,s)=0on [0,1]×[0,1], which is not possible. The same argument holds if we assume that ∂2n–5G ∂t2n–5 (·,s)is nonpositive. Therefore, necessarily ∂2n–5G ∂t2n–5 (·,s)is sign-changing and, since it is increasing, we know for sure that ∂2n–5G ∂t2n–5 (0,s)<0and ∂2n–5G ∂t2n–5 (1,s)>0for all s∈[0,1]. 2. Now, since ∂2n–5G ∂t2n–5 (·,s)is sign-changing and increasing, ∂2n–6G ∂t2n–6 (·,s)will be first decreasing and then increasing. This, together with the boundary value conditions ∂2n–6G ∂t2n–6 (0,s)=∂2n–6G ∂t2n–6 (1,s)=0,impliesthat∂2n–6G ∂t2n–6 is nonpositive. 3. Since ∂2n–6G ∂t2n–6 is nonpositive, we can follow an analogous argument to the one made in 1. to deduce that ∂2n–7G ∂t2n–7 is sign-changing and decreasing. In particular, this implies that ∂2n–7G ∂t2n–7 (0,s)>0and ∂2n–7G ∂t2n–7 (1,s)<0for all s∈[0,1] and n≥4. 4. Finally, analogously to 2., we can deduce that ∂2n–8G ∂t2n–8 is nonnegative on [0,1]×[0,1] for n≥4. We note that we could repeat all the previous arguments iteratively and this way we could deduce the following sign-criteria for the derivatives of G.So,forn≥k 2: •Ifk≡0( mod 4),then∂2n–kG ∂t2n–k(t,s)≥0on [0,1]×[0,1]. •Ifk≡1( mod 4),then∂2n–kG ∂t2n–k(·,s)is sign-changing and increasing for every s∈[0,1]. In particular, ∂2n–kG ∂t2n–k(0,s)<0and ∂2n–kG ∂t2n–k(1,s)>0for every s∈[0,1]. López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 18 of 20 •Ifk≡2( mod 4),then∂2n–kG ∂t2n–k(t,s)≤0on [0,1]×[0,1]. •Ifk≡3( mod 4),then∂2n–kG ∂t2n–k(·,s)is sign-changing and decreasing for every s∈[0,1]. In particular, ∂2n–kG ∂t2n–k(0,s)>0and ∂2n–kG ∂t2n–k(1,s)<0for every s∈[0,1]. Inparticular,ifniseven,wecoulddeducethatG(t,s)≥0on[0,1]×[0,1]and,fornodd, G(t,s)≤0 on [0,1]×[0,1]. Therefore, the Green’s function and its derivatives satisfy the required hypotheses: (H1)As in Example 8, this condition holds as a direct consequence of the general properties of the Green’s function. (H2)As we have just proved, we could take [m2n–i,n2n–i]=[0,1]for i≡0( mod 4), [m2n–i,n2n–i]={1}for i≡1( mod 4),[m2n–i,n2n–i]={0}for i≡2( mod 4),and [m2n–i,n2n–i]={0}for i≡3( mod 4). (H3)It is enough to take hi(s)=max{|∂iG ∂ti(t,s)|:t∈[0,1]}for i∈J. (H4)For n≥2,wecouldtakeJ1={2n–4,2n–3}. As a consequence of Example 8,we know that  ∂2n–4G ∂t2n–4 (t,s)=∂2n–4G ∂t2n–4 (t,s)=g(t,s)≤φ2n–4(s), with φ2n–4(s)= 1 9√3⎧ ⎨ ⎩ s(1–s2)3 2,0≤s≤1 2, (1–s)(2s–s2)3 2,1 2<s≤1. Moreover, it holds that, for any closed interval [a2n–4,b2n–4]⊂[0,1],thereexistsa constant ξ2n–4(a2n–4,b2n–4)∈(0,1) such that ∂2n–4G ∂t2n–4 (t,s)≥ξ2n–4(a2n–4,b2n–4)φ2n–4(s), for all t∈[a2n–4,b2n–4],s∈[0,1]. Analogously, from Example 8we know that  ∂2n–3G ∂t2n–3 (t,s)= ∂g ∂t(t,s)≤φ2n–3(s)=1 6s(1–s)⎧ ⎨ ⎩ 2–s,0≤s≤1 2, 1+s,1 2<s≤1, for all t∈[0,1],s∈[0,1] and for any interval [0,b2n–3]with b2n–3 <1–√3 3there exists ξ2n–3(b2n3)∈(0,1) such that ∂2n–3G ∂t2n–3 (t,s)≥ξ2n–3(b2n3)φ2n–3(s)for all t∈[0,b2n–3],s∈[0,1]. (H5)As we have already seen, it holds that [m2n–4,n2n–4]=[0,1]. López-Somoza and Minhós Advances in Difference Equations (2019) 2019:423 Page 19 of 20 Then, we could work in the cone for n∈Nsuch that n≥max{2, i 2}, K=u∈C2n–1[0,1],R:u(2n–i)(t)≥0,t∈[0,1],i≡0mod 4; u(2n–i)(1)≥0,i≡1mod 4; u(2n–i)(0)≥0,i≡2mod 4; u(2n–i)(0)≥0,i≡3mod 4; min t∈[a2n–4,b2n–4]u(2n–4)(t)≥ξ2n–4(a2n–4,b2n–4)u(2n–4)[0,1]; min t∈[0,b2n–3]u(2n–3)(t)≥ξ2n–3(b2n–3)u(2n–3)[0,1]. Thus, for any nonlinearity fsatisfying (H6) and either conditions of Theorem 3or those of Theorem 6, it is possible to find nontrivial solution of problem (6). Acknowledgements This paper was mostly written during a stay of Lucía López-Somoza in Évora. Lucía López-Somoza would like to acknowledge her gratitude towards the Department of Mathematics of the University of Évora, and specially towards Professor Feliz Minhós for his kindness and hospitality. Funding First author was partially supported by Xunta de Galicia (Spain), project EM2014/032, AIE Spain and FEDER, grants MTM2013-43014-P, MTM2016-75140-P, and FPU scholarship, Ministerio de Educación, Cultura y Deporte, Spain. F. Minhós was supported by FCT-Fundação para a Ciência e a Tecnologia, via project UID/MAT/ 04674/2019. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. Competing interests There are no competing interests. Authors’ contributions Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript. Author details 1Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain. 2Departamento de Matemática, Escola de Ciências e Tecnologia, Universidade de Évora, Évora, Portugal. 3Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada, Universidade de Évora, Évora, Portugal. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 1 June 2019 Accepted: 30 September 2019 References 1. Agarwal, R.P., O’Regan, D., Wong, P.J.Y.: Eigenvalues of a system of Fredholm integral equations. Math. Comput. Model. 39, 1113–1150 (2004) 2. Benmezai, A., Graef, J.R., Kong, L.: Positive solutions for abstract Hammerstein equations and applications. Commun. Math. Anal. 16, 47–65 (2014) 3. Cabada, A., Infante, G., Tojo, F.A.F.: Nontrivial solutions of Hammerstein integral equations with reflections. Bound. 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