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

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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.

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.

In [3] Cid et al., in order to show the existence of positive solutions of the fourth-order boundary value problem (BVP)

u ( 4 ) = λ g ( t ) f ( u ) , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 = u ( 0 ) = u ( 1 ) ,
(1.1)

where λ>0, studied the associated Hammerstein integral equation

u(t)=λ 0 1 k(t,s)g(s)f ( u ( s ) ) ds,
(1.2)

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

γ = max t [ 0 , 1 ] 0 1 k(t,s)g(s)ds,

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

Theorem 1.1

Assume that lim s f ( s ) s =+and there existsB[0,+]such that f is non-decreasing on[0,B). If

0<λ< sup s ( 0 , B ) s γ f ( s )

(with the obvious meaning whenf(s)=0), then the BVP (1.1) 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 [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

u 1 ( t ) = λ 1 a b k 1 ( t , s ) g 1 ( s ) f 1 ( u 1 ( s ) , u 2 ( s ) ) d s , u 2 ( t ) = λ 2 a b k 2 ( t , s ) g 2 ( s ) f 2 ( u 1 ( s ) , u 2 ( s ) ) d s ,
(1.3)

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 1 ( t ) + λ 1 f 1 ( u 1 ( t ) , u 2 ( t ) ) = 0 , t ( 0 , 1 ) , u 2 ( t ) + λ 2 f 2 ( u 1 ( t ) , u 2 ( t ) ) = 0 , t ( 0 , 1 ) , u 1 ( 0 ) = 0 , u 1 ( 1 ) + u 1 ( 1 ) = 0 , u 2 ( 0 ) = 0 , u 2 ( 1 ) ξ u 2 ( η ) = 0 , η ( 0 , 1 ) , 0 < ξ < 1 ,
(1.4)

computing all the constants that occur in our theory. We also prove that the system (1.4) has a solution for every λ 1 , λ 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

A subset K of a real Banach space X is a cone if it is closed, K+KK, λKK for all λ0, and K(K)={θ}. A cone K defines the partial ordering in X given by

xyif and only ifyxK.

We reserve the symbol ‘≤’ for the usual order on the real line. For x,yX, with xy, we define the ordered interval

[x,y]={zX:xzy}.

The cone K is normal if there exists d>0 such that for all x,yX with 0xy then xdy.

We denote the closed ball of center x 0 X and radius r>0 as

B[ x 0 ,r]= { x X : x x 0 r } ,

and the intersection of the cone with the open ball centered at the origin and radius r>0 as

K r =K { x X : x < r } .

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 existαβsuch thatT:[α,β]NNis a completely continuous monotone non-decreasing operator withαTαandTββ. Then T has a fixed point and the iterative sequence α n + 1 =T α n , with α 0 =α, converges to the greatest fixed point of T in[α,β], and the sequence β n + 1 =T β n , with β 0 =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 [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 with0 D K and D ¯ K K, where D K =DK. Assume thatT: D ¯ K Kis a completely continuous operator such thatxTxforx D K . Then the fixed point index i K (T, D K )has the following properties:

  1. (i)

    If there exists eK{0} such that xTx+λe for all x D K and all λ>0, then i K (T, D K )=0.

For example (i) holds ifTxxforx D K .

  1. (ii)

    If Txx for x D K , then i K (T, D K )=0.

  2. (iii)

    If Txλx for all x D K and all λ>1, then i K (T, D K )=1.

For example (iii) holds if eitherTxxforx D K orTxxforx D K .

  1. (iv)

    Let D 1 be open in X such that D 1 ¯ 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 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 constantd1and nonempty interior (i.e. solid) andT:KKa completely continuous operator.

Assume that

(1):there existβK, withTββ, andR>0such thatB[β,R]K,

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

P= { x K : x β  and  R d x } ,

(3):there exists a (relatively) open bounded setVKsuch that i K (T,V)=0and either K ¯ R Vor V ¯ K R .

Then the map T has at least one non-zero fixed point x 1 in K such that

either belongs to P or belongs to  { V K ¯ R , in case  K ¯ R V , K R V ¯ , in case  V ¯ K R .

Proof

Since B[β,R]K, if xK with x=R, then xβ.

Suppose first that we can choose αK with α=R and Tαα. Since αβ and due to the normality of the cone K we have [α,β]P, which implies that T is non-decreasing on [α,β]. Then we can apply the Theorem 2.1 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 α does not exist. Thus Txx for all xK with x=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 K ¯ R (when K ¯ R V) or to the K R V ¯ (when V ¯ 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 P is contained in the set {xK: R d xdβ}. 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) andT:KKa completely continuous operator.

Assume that

(1′):there existαK, withTαα, and0<R<αsuch thatB[α,R]K,

(2′):the map T is non-increasing in the set

P ˜ = { x K : R x α } ,

(3′):there exists a (relatively) open bounded setVKsuch that i K (T,V)=1and either K ¯ R Vor V ¯ K R .

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

either belongs to  P ˜  or belongs to  { V K ¯ R , in case  K ¯ R V , K R V ¯ , in case  V ¯ K R .

Proof

Let xK be such that x=R. Then by (1′) we have xα and since x,α P ˜ it follows from (2′) that

TxTααx.

Now, if for some x K R is the case that Txx then we are done. If not, Txx for all x 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

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 1 ( t ) = λ 1 a b k 1 ( t , s ) g 1 ( s ) f 1 ( u 1 ( s ) , u 2 ( s ) ) d s : = T 1 ( u 1 , u 2 ) ( t ) , u 2 ( t ) = λ 2 a b k 2 ( t , s ) g 2 ( s ) f 2 ( u 1 ( s ) , u 2 ( s ) ) d s : = T 2 ( u 1 , u 2 ) ( t ) ,
(3.1)

where we assume the following assumptions:

(H1): λ i >0, for i=1,2.

(H2): k i :[a,b]×[a,b][0,+) is continuous, for i=1,2.

(H3): g i :[a,b][0,+) is continuous, g i (s)>0 for all s[a,b], for i=1,2.

(H4): f i :[0,+)×[0,+)[0,+) is continuous, for i=1,2.

(H5): There exist continuous functions Φ i :[a,b][0,+) and constants 0< c i <1, a a i < b i b such that for every i=1,2,

k i (t,s) Φ i (s)for t,s[a,b]and c i Φ i (s) k i (t,s)for t[ a i , b i ] and s[a,b],

and

γ i , := min t [ a i , b i ] a i b i g i (s) k i (t,s)ds>0.

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

( u 1 , u 2 ) :=max { u 1 , u 2 } ,

where w := max t [ a , b ] |w(t)|.

Set c=min{ c 1 , c 2 } and let us define

K i ˜ := { w C [ a , b ] : w ( t ) 0  for all  t [ a , b ]  and  min t [ a i , b i ] w ( t ) c w } ,

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

K:= { ( u 1 , u 2 ) K 1 ˜ × K 2 ˜ } ,

which is a normal cone with d=1.

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

T( u 1 , u 2 )(t):= ( T 1 ( u 1 , u 2 ) ( t ) , T 2 ( u 1 , u 2 ) ( t ) )

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 (H1)-(H5) hold and moreover:

(H6):There exist constants B 1 , B 2 >0such that for everyi=1,2, f i (,)is non-decreasing on[0, B 1 ]×[0, B 2 ] (that is, if( u 1 , u 2 ),( v 1 , v 2 ) R 2 with0 u i v i B i fori=1,2, then f i ( u 1 , u 2 ) f i ( v 1 , v 2 )fori=1,2).

(H7):For everyM>0there existsρ=ρ(M)>0such that, for everyi=1,2,

inf { f 1 ( u , v ) ρ : ( u , v ) [ ρ , ρ / c ] × [ 0 , ρ / c ] } > M , inf { f 2 ( u , v ) ρ : ( u , v ) [ 0 , ρ / c ] × [ ρ , ρ / c ] } > M .

Then the system (3.1) has at least one positive solution in K provided that

0< λ i < sup r 1 ( 0 , B 1 ) , r 2 ( 0 , B 2 ) ( 1 c ) r i f i ( r 1 , r 2 ) γ i ,
(3.2)

where

γ i := max t [ a , b ] a b g i (s) k i (t,s)ds>0,for i=1,2.

Proof

Due to (3.2) we can fix β i (0, B i ), i=1,2, such that

β i λ i γ i f i ( β 1 , β 2 )>c β i ,i=1,2.
(3.3)

On the other hand, for M>max{ 1 λ 1 γ 1 , , 1 λ 2 γ 2 , } let ρ=ρ(M)>0 as in (H7) and fix R<min{ 1 c 1 + c β 1 , 1 c 1 + c β 2 ,ρ}.

Let us check that the assumptions of Theorem 2.3 are satisfied with

β(t)=( β 1 , β 2 )for all t[a,b],

and

V= { ( u 1 , u 2 ) K : min t [ a 1 , b 1 ] u 1 ( t ) < ρ  and  min t [ a 2 , b 2 ] u 2 ( t ) < ρ } .

Claim 1. B[β,R]K and Tββ .

Since β is constant and R<min{ 1 c 1 + c β 1 , 1 c 1 + c β 2 } a direct computation shows that B[β,R]K. Now, from (3.3) it follows for each t[a,b] and i=1,2

[ T i β](t)= λ i a b k i (t,s) g i (s) f i ( β 1 , β 2 )ds λ i γ i f i ( β 1 , β 2 )< β i .

Moreover, since β i T i β β i , i=1,2, and taking into account (3.3) we have for t[ a i , b i ] and i=1,2,

β i [ T i β ] ( t ) = β i λ i a b k i ( t , s ) g i ( s ) f i ( β 1 , β 2 ) d s β i λ i γ i f i ( β 1 , β 2 ) > c β i c β i T i β .

As a consequence, we have Tββ, and the claim is proven.

Claim 2. T is non-decreasing on the set{xK:xβ}.

Let u=( u 1 , u 2 ),v=( v 1 , v 2 )K be such that 0 u i (t) v i (t) β i for all t[a,b] and i=1,2. Since f is non-decreasing in [0, β 1 ]×[0, β 2 ] we have for all t[a,b] and i=1,2,

[ T i v](t)[ T i u](t)= λ i a b k i (t,s) g i (s) [ f i ( v ( s ) ) f i ( u ( s ) ) ] ds0.

Moreover, for all t[ a i , b i ], r[0,1] and i=1,2,

[ T i v ] ( t ) [ T i u ] ( t ) = λ i a b k i ( t , s ) g i ( s ) [ f i ( v ( s ) ) f i ( u ( s ) ) ] d s λ i a b c Φ i ( s ) g i ( s ) [ f i ( v ( s ) ) f i ( u ( s ) ] d s c λ i a b k i ( r , s ) g i ( s ) [ f i ( v ( s ) ) f i ( u ( s ) ) ] d s = c ( [ T i v ] ( r ) [ T i u ] ( r ) ) ,

therefore min t [ a i , b i ] ([ T i v](t)[ T i u](t))c T i v T i u , i=1,2, so TuTv, and since P{xK:xβ}, T is also non-decreasing on P.

Claim 3. K ¯ R Vand i K (T,V)=0.

Firstly, note that since R<ρ then we have K ¯ R K ρ V.

Now let e(t)1 for t[a,b]. Then (e,e)K and we are going to prove that

( u 1 , u 2 )T( u 1 , u 2 )+μ(e,e)for ( u 1 , u 2 )V and μ0.

If not, there exist ( u 1 , u 2 )V and μ0 such that ( u 1 , u 2 )=T( u 1 , u 2 )+μ(e,e).

Without loss of generality, we can assume that for all t[ a 1 , b 1 ] we have

ρ u 1 (t)ρ/c, min t [ a 1 , b 1 ] u 1 (t)=ρand0 u 2 (t)ρ/c.

Then, for t[ a 1 , b 1 ], we obtain

u 1 ( t ) = λ 1 a b k 1 ( t , s ) g 1 ( s ) f 1 ( u 1 ( s ) , u 2 ( s ) ) d s + μ e ( t ) λ 1 a 1 b 1 k 1 ( t , s ) g 1 ( s ) f 1 ( u 1 ( s ) , u 2 ( s ) ) d s + μ λ 1 M ρ γ 1 , + μ > ρ + μ .

Thus, we obtain ρ= min t [ a 1 , b 1 ] u(t)>ρ+μρ, 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 (H7) and it is easier to check.

( H 7 ) : For every i=1,2, lim u i + f i ( u 1 , u 2 ) u i =+, uniformly w.r.t. u j [0,), ji.

Remark 3.3

In order to deal with negative kernels k i (t,s)<0 we can require conditions (H2), (H3), and (H5) on the absolute value of the kernel such that | k i (t,s)|>0 and conditions (H4), (H6), and (H7) on sgn( k i ) f i .

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

u 1 ( t ) + λ 1 f 1 ( u 1 ( t ) , u 2 ( t ) ) = 0 , t ( 0 , 1 ) , u 2 ( t ) + λ 2 f 2 ( u 1 ( t ) , u 2 ( t ) ) = 0 , t ( 0 , 1 ) ,
(3.4)

with the BCs

u 1 ( 0 ) = 0 , u 1 ( 1 ) + u 1 ( 1 ) = 0 , u 2 ( 0 ) = 0 , u 2 ( 1 ) = ξ u 2 ( η ) , η , ξ ( 0 , 1 ) .
(3.5)

To the system (3.4)-(3.5) we associate the system of integral equations

u 1 ( t ) = λ 1 0 1 k 1 ( t , s ) f 1 ( u 1 ( s ) , u 2 ( s ) ) d s , u 2 ( t ) = λ 2 0 1 k 2 ( t , s ) f 2 ( u 1 ( s ) , u 2 ( s ) ) d s ,
(3.6)

where the Green’s functions are given by

k 1 (t,s)={ 2 t , s t , 2 s , s > t ,
(3.7)

and

k 2 (t,s)= 1 1 ξ (1s){ ξ 1 ξ ( η s ) , s η , 0 , s > η { t s , s t , 0 , s > t .
(3.8)

The Green’s function k 1 was studied in [28] were it was shown that we may take (with our notation)

Φ 1 (s)=(2s), γ 1 = 3 2 .

The choice of [ a 1 , b 1 ]=[0,1] gives

c 1 = 1 2 , γ 1 , =1.

The kernel k 2 was extensively studied in [28], [29] 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

Φ 2 (s)= k 2 (0,s)={ 1 s 1 ξ , if  η < s 1 , 1 s ξ ( η s ) 1 ξ , if  0 s η , γ 2 = 1 ξ η 2 2 ( 1 ξ ) .

The choice, as in [29], of [ a 2 , b 2 ]=[0, b 2 ], where

b 2 ={ 1 ξ η 2 ( 1 ξ ) , if  1 + ξ η 2 η , 1 ( 2 ξ ) , if  1 + ξ η > 2 η ,

leads to

c 2 = 1 ξ η ( 1 ξ ) b 2 1 ξ η , γ 2 , ={ b 2 2 , if  1 + ξ η 2 η , 1 2 ξ η 2 + ξ 2 η 2 2 ( 1 ξ ) ( 2 ξ ) , if  1 + ξ η > 2 η .

We now fix, as in [28], η=1/2, ξ=1/4. This gives b 2 =4/7 and

γ 2 = 5 8 , c 2 = 25 49 , γ 2 , = 19 56 .

Furthermore take

f 1 ( u 1 , u 2 )= ( 2 + sin ( u 2 ) ) u 1 2 , f 2 ( u 1 , u 2 )= ( 2 + sin ( u 1 ) ) u 2 2 .
(3.9)

In the case of the nonlinearities (3.9), we can choose B 1 = B 2 =π/2. We observe that condition ( H 7 ) holds, we note that c=min{ c 1 , c 2 }=1/2 and that

sup r 1 ( 0 , π / 2 ) , r 2 ( 0 , π / 2 ) r i 2 f i ( r 1 , r 2 ) γ i =+,for every 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 λ 1 , λ 2 (0,).

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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.

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Correspondence to Gennaro Infante.

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All the authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.

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Keywords

  • cone
  • boundary value problem
  • fixed point index
  • positive solution
  • nonlocal boundary condition
  • system