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

Cabada et al. Boundary Value Problems 2014, 2014:254 http://www.boundaryvalueproblems.com/content/2014/1/254 R E S E A R C H Open Access A positive fixed point theorem with applications to systems of Hammerstein integral equations Alberto Cabada1, José Ángel Cid2and Gennaro Infante3* *Correspondence: gennaro[email protected] 3Dipartimento di Matematica e Informatica, Università della Calabria, Arcavacata di Rende, Cosenza, 87036, Italy Full list of author information is available at the end of the article 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: Primary 47H10; secondary 34B10; 34B18; 45G15; 47H30 Keywords: cone; boundary value problem; fixed point index; positive solution; nonlocal boundary condition; system 1 Introduction In 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 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’ski˘ ı. In order to do this we employ the well-known monotone iterative method, combined with the classical fixed point index. In Section  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 . we present a comparison with previous results in this direction. In []Cidet al., in order to show the existence of positive solutions of the fourth-order boundary 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 Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Cabada et al. Boundary Value Problems 2014, 2014:254 Page 2 of 10 http://www.boundaryvalueproblems.com/content/2014/1/254 where kis precisely the Green’s function associated to the BVP (.). Having defined the constant γ∗=max t∈[,]   k(t,s)g(s)ds, themainresultin[], 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<sup s∈(,B) s γ∗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 hand the existence of nonnegative solutions for systems of Hammerstein integral equations has been widely studied; see for example [–] and references therein. In Section we give an extension of Theorem . to the context of systems of Hammerstein integral equations of the type u(t)=λb a k(t,s)g(s)fu(s),u(s)ds, u(t)=λb a k(t,s)g(s)fu(s),u(s)ds, (.) providing, under suitable assumptions on the kernels and the nonlinearities, the existence of a positive solution. In order to show the applicability of our results, we discuss the following system of second-order ODEs, subject to local and nonlocal boundary conditions, which generates two different kernels: u (t)+λfu(t),u(t)=, t∈(,), u (t)+λf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 context of one equation subject to nonlinear boundary conditions, by Goodrich []. 2 Two fixed point theorems in cones AsubsetKof a real Banach space Xis a cone if it is closed, K+K⊂K,λK⊂Kfor all λ≥, and K∩(–K)={θ}.AconeKdefines the partial ordering in Xgiven by xyif and only if y–x∈K. Cabada et al. Boundary Value Problems 2014, 2014:254 Page 3 of 10 http://www.boundaryvalueproblems.com/content/2014/1/254 We reserve the symbol ‘≤’ for the usual order on the real line. For x,y∈X,withxy,we define the ordered interval [x,y]={z∈X:xzy}. The cone Kis normal if there exists d>  such that for all x,y∈Xwith  xythen x≤dy. We denote the closed ball of center x∈Xand 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 iterative method (see, for example, [, Theorem .A] or []). Theorem . Let N be a real Banach space with normal order cone K.Suppose that there exist α≤βsuch that T :[α,β]⊂N→N is a completely continuous monotone nondecreasing operator with α≤Tαand Tβ≤β.Then T has a fixed point and the iterative sequence αn+ =Tαn,with α=α,converges to the greatest fixed point of T in [α,β], and the 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 completely continuous operator relative to a cone, for more details see [,]. In the sequel the closure and the boundary of subsets of Kare understood to be relative to K. Proposition . Let D be an open bounded set of X with ∈DKand DK=K,where DK= D∩K.Assume that T :DK→K is a completely continuous operator such that x =Tx for x∈∂DK.Then the fixed point index iK(T,DK)has the following properties: (i) If there exists e∈K\{}such that x=Tx +λefor all x∈∂DKand all λ>,then iK(T,DK)=. For example (i) holds if Tx xfor x∈∂DK. (ii) If Tx≥xfor x∈∂DK,then iK(T,DK)=. (iii) If Tx =λxfor all x∈∂DKand all λ>,then iK(T,DK)=. For example (iii) holds if either Tx xfor x∈∂DKor Tx≤xfor x∈∂DK. (iv) Let Dbe open in Xsuch that D⊂DK.If iK(T,DK)=and iK(T,D K)=,then T has a fixed point in DK\D K.ThesameholdsifiK(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, Cabada et al. Boundary Value Problems 2014, 2014:254 Page 4 of 10 http://www.boundaryvalueproblems.com/content/2014/1/254 () the map Tis non-decreasing in the set P=x∈K:xβand R d≤x, () there exists a (relatively)open bounded set V⊂Ksuch that iK(T,V)=and either KR⊂Vor V⊂KR. Then the map T has at least one non-zero fixed point xin K such that either belongs to Por belongs to V\KR,in case KR⊂V, KR\V,in case V⊂KR. Proof Since B[β,R]⊂K,ifx∈Kwith x=R,thenxβ. Supposefirstthatwecanchooseα∈Kwith α=Rand Tαα.Sinceαβand due to the normality of the cone Kwe have [α,β]⊂P, which implies that Tis non-decreasing on [α,β]. Then we can apply the Theorem . to ensure the existence of a fixed point of T on [α,β], which, in particular, is a non-trivial fixed point. Now suppose that such αdoes not exist. Thus Tx xfor all x∈Kwith x=R,which by Proposition .(iii) implies that iK(T,KR) = . Since, by assumption, iK(T,V)=weget the existence of a non-trivial fixed point xbelonging to the set V\KR(when KR⊂V)or to the KR\V(when V⊂KR).  Remark . We note that we can use either Proposition .(i), or Proposition .(ii), in order to check the assumption () in Theorem ..WealsostressthatPis contained in the set {x∈K:R d≤x≤dβ}. Therefore Theorem . is a genuine generalization of the previous fixed point theorems obtained in [–]. Moreover, we show in the applications thatinmanycasesitisusefultoapplyTheorem. with a set Vdifferent from Kr. We observe that, following some ideas introduced in [,Theorem.],itispossibleto modifytheassumptionsofTheorem. 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 Tis non-increasing in the set  P=x∈K:R≤x≤α, ()there exists a (relatively)open bounded set V⊂Ksuch that iK(T,V)=and either KR⊂Vor V⊂KR. Then the map T has at least one non-zero fixed point such that either belongs to  Por belongs to V\KR,in case KR⊂V, KR\V,in case V⊂KR. Cabada et al. Boundary Value Problems 2014, 2014:254 Page 5 of 10 http://www.boundaryvalueproblems.com/content/2014/1/254 Proof Let x∈Kbe such that x=R.Thenby( )wehavexαand since x,α∈ Pit follows from ()that Tx Tααx. Now, if for some x∈∂KRisthecasethatTx xthen we are done. If not, Tx xfor all x∈∂KRwhich 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 equations We now apply the results of the previous section in order to prove the existence of positive solutions of the system of integral equations u(t)=λb a k(t,s)g(s)fu(s),u(s)ds := T(u,u)(t), u(t)=λb a k(t,s)g(s)fu(s),u(s)ds := T(u,u)(t), (.) whereweassumethefollowingassumptions: (H)λi>,fori=,. (H)ki:[a,b]×[a,b]→[, +∞)is continuous, for i=,. (H)gi:[a,b]→[, +∞)is continuous, gi(s)>for all s∈[a,b],fori=,. (H)fi:[,+∞)×[, +∞)→[, +∞)is continuous, for i=,. (H) There exist continuous functions i:[a,b]→[, +∞)and constants <ci<,a≤ ai<bi≤bsuch 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,∗:= min t∈[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 min t∈[ai,bi]w(t)≥cw∞, and consider the cone Kin C[a,b]×C[a,b]definedby K:= (u,u)∈˜ Kט K, which is a normal cone with d=. Cabada et al. Boundary Value Problems 2014, 2014:254 Page 6 of 10 http://www.boundaryvalueproblems.com/content/2014/1/254 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 Kinvariant 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≤Bifor i=,,then fi(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<sup r∈(,B),r∈(,B) ( – c)ri fi(r,r)γ∗ i ,(.) where γ∗ i:= max t∈[a,b]b a gi(s)ki(t,s)ds >, for i =,. Proof Due to (.)wecanfixβi∈(, Bi), i=,,suchthat βi–λiγ∗ ifi(β,β)>cβi,i= ,. (.) On the other hand, for M>max{ λγ,∗, λγ,∗}let ρ=ρ(M)>asin(H )andfixR< 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:min t∈[a,b]u(t)<ρand min t∈[a,b]u(t)<ρ. Claim . B[β,R]⊂KandTββ. Since βis 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]andi=, [Tiβ](t)=λib a ki(t,s)gi(s)fi(β,β)ds ≤λiγ∗ ifi(β,β)<βi. Cabada et al. Boundary Value Problems 2014, 2014:254 Page 7 of 10 http://www.boundaryvalueproblems.com/content/2014/1/254 Moreover, since βi–Tiβ∞≤βi,i= ,, and taking into account (.)wehavefor t∈[ai,bi]andi=,, βi–[Tiβ](t)=βi–λib a ki(t,s)gi(s)fi(β,β)ds ≥βi–λiγ∗ ifi(β,β)>cβi≥cβi–Tiβ∞. As a consequence, we have Tββ,andtheclaimisproven. Claim . T is non-decreasing on the set {x∈K:xβ}. Let u=(u,u), v=(v,v)∈Kbe such that  ≤ui(t)≤vi(t)≤βifor all t∈[a,b]and i=,.Sincefis non-decreasing in [,β]×[,β] we have for all t∈[a,b]andi=,, [Tiv](t)–[Tiu](t)=λib a ki(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 a ki(t,s)gi(s)fiv(s)–fiu(s)ds ≥λib a ci(s)gi(s)fiv(s)–fi(u(s)ds ≥cλib a ki(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=,,soTu Tv, and since P⊂{x∈K:xβ},Tis also non-decreasing on P. Claim . KR⊂Vandi K(T,V)=. Firstly, note that since R<ρthen we have KR⊂Kρ⊂V. Now let e(t)≡fort∈[a,b]. Then (e,e)∈Kand we are going to prove that (u,u)=T(u,u)+μ(e,e)for(u,u)∈∂Vand μ≥. If not, there exist (u,u)∈∂Vand μ≥suchthat(u,u)=T(u,u)+μ(e,e). Without loss of generality, we can assume that for all t∈[a,b]wehave ρ≤u(t)≤ρ/c,min t∈[a,b]u(t)=ρand  ≤u(t)≤ρ/c. Then, for t∈[a,b], we obtain u(t)=λb a k(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≥λMργ,∗+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.  Cabada et al. Boundary Value Problems 2014, 2014:254 Page 8 of 10 http://www.boundaryvalueproblems.com/content/2014/1/254 Remark . The following condition, similar to the one given in [], implies (H)andit is easier to check. (H)∗For every i=,,limui→+∞fi(u,u) ui=+∞,uniformlyw.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 conditions (H), (H), and (H)onsgn(ki)·fi. As an illustrative example, we apply our results to the system of ODEs u (t)+λfu(t),u(t)=, t∈(,), u (t)+λf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, (.) wheretheGreen’sfunctionsaregivenby 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 []wereitwasshownthatwemaytake(withour notation) (s)=(–s), γ∗ = . Thechoiceof[a,b]=[,]gives c= ,γ,∗=. Cabada et al. Boundary Value Problems 2014, 2014:254 Page 9 of 10 http://www.boundaryvalueproblems.com/content/2014/1/254 The kernel kwas extensively studied in [,] and is more complicated to be dealt with, due to the presence of the nonlocal term in the BCs. In this case we may take (s)=k(,s)=⎧ ⎨ ⎩ –s –ξ,ifη<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+ξη (–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 condition (H)∗holds, we note that c=min{c,c}= / and that sup r∈(,π/),r∈(,π/) ri fi(r,r)γ∗ i =+∞,foreveryi. As a consequence, by means of Theorem ., we obtain a non-zero solution of the system (.)-(.)foreveryλ,λ∈(, ∞). Competing interests The authors declare that they have no competing interests. Authors’ contributions All the authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript. Author details 1Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de 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. 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. Received: 21 October 2014 Accepted: 24 November 2014