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New criteria for the existence of non-trivial fixed points in cones

Abstract

We give new criteria for the existence of nontrivial fixed points on cones assuming some monotonicity of the operator on a suitable conical shell. Moreover, we give an application to the existence of multiple solutions for a nonlocal boundary value problem that models the displacement of a beam subject to some feedback controllers.

MSC:47H10, 34B10, 34B18.

1 Introduction and preliminaries

The classical cone compression-expansion fixed point theorem of Krasnosel’skiĭ (see Theorem 1.2 below) and the monotone iterative technique (see Theorem 1.1 below) are among the most popular and fruitful tools to deal with the existence of solutions for nonlinear problems. Following earlier ideas of Persson [1] valid in the finite-dimensional setting, both methods were combined in [2] to obtain the existence of a fixed point, assuming the operator T to be monotone non-decreasing with some conditions on the set of supersolutions. This result was improved in [3] by relaxing the monotonicity condition. More recently, in [4] the authors were able to present a refinement of the results of [2, 3], by allowing a comparison between a point and a boundary, instead of on two boundaries as in Krasnosel’skiĭ’s theorem. This approach, which required a monotonicity assumption on the operator on a conical shell, has proved to be well suited to establish multiplicity results. Our aim in this paper is to pursue this line of research by obtaining new fixed point theorems, valid not only for non-decreasing (Section 3) but also for non-increasing operators (Section 2). We point out that this type of theorems can be combined in the applications to obtain the existence of multiple non-trivial solutions. This fact is illustrated in Section 4, where the existence of multiple positive solutions for a nonlocal boundary value problem modeling the displacement of a beam is discussed.

We now recall some definitions that will be useful in the sequel. A subset K of a real Banach space N is a cone if and only if it is closed, K+KK, λKK for all λ0 and K(K)={0}. A cone K defines the partial ordering in N given by xy if and only if yxK. The notation x<y means xy and yx. The cone K is called normal with a normal constant c1 if and only if xcy for all x,yN with 0xy. Whenever int(K), the symbol xy means yxint(K) and the cone is said to be solid. ∂K denotes the boundary of K and d(x,K) is the distance of x to the boundary of K.

If T:KK satisfies the conditions

Txxfor all xK with x=R

and

Txxfor all xK with x= R ¯ ,

then it is called a cone compression when 0<R< R ¯ and a cone expansion when 0< R ¯ <R.

The closed ball of center x 0 N and radius r>0 is denoted by

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

and for x,yN, with xy, we define the interval

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

Now we recall the two classical fixed point results mentioned above. The first one is known as the monotone iterative method (see, for example, [[5], Theorem 7.A] or [6]).

Theorem 1.1 Let N be a real Banach space with a normal order cone K. Suppose that there exist αβ such that T:[α,β]NN is a completely continuous monotone nondecreasing operator with αTα and Tββ. 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].

The second one, which is widely used for the search of positive fixed points, is due to Krasnosel’skiĭ (see [[5], Theorem 13.D]).

Theorem 1.2 Let N be a real Banach space with order cone K. Suppose that the operator T:KK is completely continuous and either a cone compression or expansion. Then T has a fixed point x on K and

min{R, R ¯ }<x<max{R, R ¯ }.

2 Non-increasing operators

Firstly, we present a result under the assumption of the existence of a lower solution in a solid cone.

Theorem 2.1 Let N be a real Banach space, K be a solid cone and T:KK be a completely continuous operator. Assume that

  1. (1)

    there exist α 1 K, with α 1 T α 1 , and R 1 >0 such that B[ α 1 , R 1 ]K;

  2. (2)

    the map T is monotone non-increasing in the set

    K 2 = { x K : R 1 x α 1 } ;
  3. (3)

    there exists r 1 >0, with r 1 R 1 , such that Txx for all xK with x= r 1 .

Then the map T has at least one non-zero fixed point x 1 K{xK;x= r 1 } such that

min{ r 1 , R 1 } x 1 max{ r 1 , R 1 }.

Proof First, note that 0< R 1 d( α 1 ,K) α 1 . Since B[ α 1 , R 1 ]K, it is clear that if xK with x= R 1 , then x α 1 . Since T is non-increasing in K 2 and x, α 1 K 2 , we have that

TxT α 1 α 1 xfor all xK with x= R 1 .

If Tx=x, we have a fixed point such that x= R 1 , on the contrary, we deduce that Txx for all x= R 1 , which together with (3) implies, by Theorem 1.2, that there exists a non-zero fixed point x 1 with the desired localization property. □

Remark 2.2 Note that if T α 1 α 1 , in the proof of the previous result, it is showed that the map T has at least one non-zero fixed point x 1 K such that

min{ r 1 , R 1 }< x 1 <max{ r 1 , R 1 }.

As a direct consequence of Theorem 2.1, we obtain the following ‘dual result’ for a non-increasing operator in a solid cone with an upper solution.

Corollary 2.3 Let N be a real Banach space, K be a solid cone and T:KK be a completely continuous operator. Assume that condition (3) is fulfilled together with

  1. (4)

    there exist β 1 K, with T β 1 β 1 , and R 1 >0 such that B[T β 1 , R 1 ]K;

  2. (5)

    the map T is monotone non-increasing in the set

    K 5 = { x K : R 1 x max { T β 1 , β 1 } } .

Then the map T has at least one non-zero fixed point x 1 in x 1 K{xK;x= r 1 }, and

min{ r 1 , R 1 } x 1 max{ r 1 , R 1 }.

Proof As in the proof of Theorem 2.1, we deduce that T β 1 R 1 .

If β 1 =T β 1 β 1 T β 1 R 1 , then T β 1 β 1 B[T β 1 , R 1 ]K, which implies that T β 1 β 1 0, i.e., β 1 is a fixed point of T. Obviously, if this is the case, β 1 =T β 1 = R 1 and the result holds.

On the other hand, if β 1 > R 1 , we know that β 1 ,T β 1 K 5 and, since T is non-increasing in K 5 and T β 1 belongs to the interior of K, we deduce that 0T β 1 T(T β 1 ), i.e., T β 1 K is a non-zero lower solution of the operator T.

As consequence, all the conditions of Theorem 2.1 are fulfilled and the result holds. □

Now, we give a fixed point result for a non-increasing operator under the assumption of the existence of an upper solution in a cone not necessarily solid, but it verifies an extra condition.

Theorem 2.4 Let N be a real Banach space, K be a cone that satisfies the following condition:

there exists σ1 such that for x,yK with x=σy we have xy,
(2.1)

and T:KK is a completely continuous operator. Assume that

  1. (6)

    there exists β 1 K, β 1 0, such that β 1 T β 1 ;

  2. (7)

    the map T is monotone non-increasing in the set

    K 7 = { x K : β 1 x σ β 1 } ;
  3. (8)

    there exists r 1 >0, with r 1 R 1 :=σ β 1 , such that Txx for all xK with x= r 1 .

Then the map T has at least one non-zero fixed point x 1 K{xK;x= r 1 } such that

min{ r 1 , R 1 } x 1 max{ r 1 , R 1 }.

Proof By the property (2.1), we have that if xK with x= R 1 , then x β 1 . The definition of R 1 says that x, β 1 K 7 ; so, since T is non-increasing on K 7 , we have

TxT β 1 β 1 xfor all xK with x= R 1 .

If Tx=x, we have a fixed point such that x= R 1 ; on the contrary, Txx for all xK such that x= R 1 which, together with (8), implies by Theorem 1.2 the existence of a fixed point with the desired localization property. □

Remark 2.5 (i) Note that if T β 1 β 1 , in the proof of the two previous results, it is showed that the map T has at least one non-zero fixed point x 1 K such that

min{ r 1 , R 1 }< x 1 <max{ r 1 , R 1 }.

The same conclusion holds if in condition (2.1) we assume that the constant σ>1. To verify this, it is enough to take into account that if there is a fixed point xK with x= R 1 , then TxT β 1 β 1 x and, as a consequence, σ β 1 =x= β 1 .

(ii) An example of a cone satisfying condition (2.1) is, for instance, the one used in [7]

K= { f C [ 0 , 1 ] : f ( t ) = c t 2 , c 0 } .

Lemma 2.6 Condition (2.1) is equivalent to the following one:

there exists σ1 such that for x,yK with xσy we have xy.
(2.2)

Proof Obviously, if condition (2.2) is fulfilled, then (2.1) also holds.

Suppose now that (2.1) is satisfied, and let x,yK be such that xσy; in consequence, there is λ1 such that x/λ=σy. Condition (2.1) shows that xλyy, i.e., condition (2.2) holds. This proves the result. □

In an analogous way to Corollary 2.3, we arrive at the following ‘dual result’.

Corollary 2.7 Let N be a real Banach space, K be a cone that satisfies the condition (2.1) and T:KK be a completely continuous operator. Assume that

  1. (9)

    there exists α 1 K, α 1 0 such that α 1 T α 1 ;

  2. (10)

    the map T is monotone non-increasing in the set

    K 10 = { x K : min { α 1 , T α 1 } x σ T α 1 } ;
  3. (11)

    there exists r 1 >0, with r 1 R 1 :=σT α 1 , such that Txx for all xK with x= r 1 .

Then the map T has at least one non-zero fixed point x 1 K{xK;x= r 1 } such that

min{ r 1 , R 1 } x 1 max{ r 1 , R 1 }.

Proof By the property (2.1), we have that if xK with x= R 1 , then xT α 1 .

Suppose now that α 1 σT α 1 . From Lemma 2.6, we have that α 1 T α 1 , which implies that α 1 is a fixed point with α 1 = R 1 and σ=1, so the result holds.

When α 1 <σT α 1 , by the definition of R 1 , it is obvious that α 1 ,T α 1 K 10 ; so, since T is non-increasing on K 10 , we have

T(T α 1 )T( α 1 )>0,

and the results holds from Theorem 2.4. □

Remark 2.8 Note that if T α 1 α 1 or σ>1, in the proof of the previous result, it is showed that the map T has at least one non-zero fixed point x 1 K such that

min{ r 1 , R 1 }< x 1 <max{ r 1 , R 1 }.

3 Non-decreasing operators

For non-decreasing operators, in the case of an upper solution, it was proved in [4] the following result which is an improvement of those in [2, 3].

Theorem 3.1 [4]

Let N be a real Banach space, K be a normal solid cone with a normal constant c1 and T:KK be a completely continuous operator. Assume that

  1. (12)

    there exist β 1 K, with T β 1 β 1 , and R 1 >0 such that B[ β 1 , R 1 ]K;

  2. (13)

    the map T is monotone non-decreasing in the set

    K 13 = { x K : R 1 c x c β 1 } ;
  3. (14)

    there exists r 1 >0, with r 1 R 1 , such that Txx for all xK with x= r 1 .

Then the map T has at least one non-zero fixed point x 1 in K that either belongs to K 13 or is such that

min{ r 1 , R 1 }< x 1 <max{ r 1 , R 1 }.

In the sequel, we prove a fixed point result in this direction for a not necessarily normal cone that satisfies (2.1).

Theorem 3.2 Let N be a real Banach space, K be a solid cone that satisfies condition (2.1), and T:KK be a completely continuous operator. Assume that conditions (12) and (14) hold and

(13′) the map T is monotone non-decreasing in the set

K 13 = { x K : R 1 x σ β 1 } .

Then the map T has at least one non-zero fixed point x 1 in K that either belongs to K 13 or is such that

min{ r 1 , R 1 }< x 1 <max{ r 1 , R 1 }.

Proof Since B[ β 1 , R 1 ]K, it is clear that if xK with x= R 1 , then x β 1 .

Suppose first that there is α 1 K with α 1 = R 1 and T α 1 α 1 . Notice that in this case α 1 T α 1 T β 1 β 1 .

If T α 1 = α 1 T α 1 α 1 R 1 , then α 1 T α 1 B[T α 1 , R 1 ]K, which implies that α 1 T α 1 , i.e., α 1 K 13 is a fixed point of T and the result is fulfilled.

Suppose now that T α 1 > R 1 . If T α 1 σ β 1 we know, by condition (2.1), that T α 1 β 1 and, as a consequence, β 1 K 13 ( β 1 R 1 ) is a fixed point of T.

So, if R 1 <T α 1 <σ β 1 we have that T α 1 T 2 α 1 and, arguing as before, we have that either α 1 or β 1 are fixed points of T, or R 1 < T 2 α 1 <σ β 1 .

By recurrence we verify that the set P= { T n α 1 } n N K 13 .

Since T is monotone non-decreasing on K 13 , we have that the same holds in P. Moreover, T(P)[ α 1 , β 1 ]. Now, the completely continuous character of the operator T implies that T ( P ) ¯ is a compact set of N. We can ensure the existence of a fixed point on P K 13 from [[8], Proposition 1.1.7].

Now suppose that Txx for all xK with x= R 1 . By (14) there exists r 1 >0 such that Txx for all xK with x= r 1 . Therefore by Theorem 1.2 there exists a non-zero fixed point. □

Remark 3.3 An example of a solid cone satisfying condition (2.1) is the following one:

K= { c f ( t ) : f L ( [ 0 , 1 ] ) , f 0  a.e.  t [ 0 , 1 ] , f 1 > 0 , c 0 } .

Now, we give a result under the assumption of the existence of a lower solution.

Theorem 3.4 Let N be a real Banach space, K be a normal cone (not necessarily solid) with a normal constant c1 that satisfies condition (2.1), and T:KK be a completely continuous operator. Assume that there is a lower solution as in (9), and

  1. (15)

    the map T is monotone non-decreasing in the set

    K 15 = { x K : α 1 c x c σ α 1 } ;
  2. (16)

    there exists r 1 >0, with r 1 R 1 :=σ α 1 , such that Txx for all xK with x= r 1 .

Then the map T has at least one non-zero fixed point x 1 in K that either belongs to K 15 or is such that

min{ r 1 , R 1 }< x 1 <max{ r 1 , R 1 }.

Proof By the property (2.1), we have that if xK with x= R 1 , then x α 1 .

Suppose first that we can choose β 1 K with β 1 = R 1 and T β 1 β 1 . Since α 1 β 1 and due to the normality of the cone K, we have that [ α 1 , β 1 ] K 15 , which implies that T is nondecreasing on [ α 1 , β 1 ]. Then we can apply Theorem 1.1 to ensure the existence of the extremal fixed points of T on [ α 1 , β 1 ].

Now suppose that Txx for all xK with x= R 1 . By (16) there exists r 1 >0 such that Txx for all xK with x= r 1 . Therefore by Theorem 1.2, there exists a non-zero fixed point x 1 in the required set. □

Remark 3.5 We stress that the above theorems can be combined to prove the existence of multiple fixed points. The idea is to use a nesting argument similar to those utilized, for example, in [9, 10], where the authors used the classical fixed point index, and in [11, 12], where Theorem 1.2 was used. In the next section we do this in the case of the existence of two non-trivial fixed points, and we refer to Theorem 3.4 of [4] to give an idea of the type of results that may be stated in the case of n fixed points.

4 Applications to a nonlocal BVP

We now discuss the existence of positive solutions of the nonlocal boundary value problem (BVP)

u ( 4 ) (t)=λg(t)f ( u ( t ) ) ,t(0,1),
(4.1)
u (0)=0, θ 1 u (1)+u( ξ 1 )=0,
(4.2)
u (0)=0, θ 2 u (1)+ u ( ξ 2 )=0,
(4.3)

where, for i=1,2, θ i R and ξ i [0,1], g L 1 [0,1], g0 a.e. and f:[0,)[0,) is continuous. BVP (4.1)-(4.3) that has been studied in [13] models the displacement of a beam with feedback controllers; in particular, the boundary conditions mean that the shear force and the angular attitude vanish at t=0, and in t=1 they are related to the displacement and to the bending moment registered in other points of the beam.

This BVP can be rewritten as a Hammerstein integral equation of the form

u(t)= 0 1 k(t,s)λg(s)f ( u ( s ) ) ds:=Tu(t),
(4.4)

where the Green’s function k(t,s) and its properties are given in the following result.

Lemma 4.1 [13]

Let θ 1 + ξ 1 >1 and θ 2 + ξ 2 >1. The Green’s function k(t,s) for the linear fourth-order boundary value problem

u ( 4 ) ( t ) = y ( t ) , t ( 0 , 1 ) , u ( 0 ) = 0 , u ( 0 ) = 0 , θ 1 u ( 1 ) + u ( ξ 1 ) = 0 , θ 2 u ( 1 ) + u ( ξ 2 ) = 0 ,
(4.5)

is given by

k ( t , s ) = θ 2 ( θ 1 + ξ 1 2 2 t 2 2 ) θ 1 2 ( 1 s ) 2 + { ( θ 1 + ξ 1 2 2 t 2 2 ) ( ξ 2 s ) , s ξ 2 , 0 , s > ξ 2 , { 1 6 ( ξ 1 s ) 3 , s ξ 1 , 0 , s > ξ 1 , + { 1 6 ( t s ) 3 , s t , 0 , s > t .
(4.6)

Moreover, for (t,s)[0,1]×[0,1], we have

c ˆ k(0,s)k(t,s)k(0,s):=Φ(s),

where

c ˆ :=1 1 2 θ 1 + ξ 1 2 .

Note that θ 1 + ξ 1 >1 implies that 2 θ 1 + ξ 1 2 >1 and so 0< c ˆ <1. Now, with the above conditions, it is routine to prove that T:C[0,1]C[0,1] leaves invariant the cone

K= { u C [ 0 , 1 ] : min t [ 0 , 1 ] u ( t ) c ˆ u } ,

where in C[0,1] we are considering the supremum norm u=sup{u(t):t[0,1]}. It is also known that K is a normal solid cone with a normal constant c=1.

We will make use of the numbers

γ = inf t [ 0 , 1 ] 0 1 k(t,s)g(s)ds, γ = sup t [ 0 , 1 ] 0 1 k(t,s)g(s)ds,

note that γ =1/M and γ =1/m, in the notation of [13].

Now we present the main result of this section.

Theorem 4.2 Assume that the hypotheses in Lemma 4.1 hold. Moreover, let β 1 , α 2 , R 1 , R 2 (0,+) be such that

β 1 < c ˆ R 2 , α 2 R 2 ( 2 ( 2 θ 1 + ξ 1 2 ) 1 ) and β 1 R 1 ( 2 ( 2 θ 1 + ξ 1 2 ) 1 ) .

Assume that g satisfies that γ >0 and, moreover,

  1. (i)

    f is non-decreasing on [ c ˆ R 1 , β 1 ];

  2. (ii)

    f is non-increasing on [ c ˆ R 2 , α 2 ];

  3. (iii)

    lim u 0 + f ( u ) u =+ and lim u + f ( u ) u =0.

Then BVP (4.1)-(4.3) has at least two positive solutions for any λ>0 satisfying

α 2 γ f ( α 2 ) λ β 1 γ f ( β 1 ) .
(4.7)

Proof The main idea in the proof is to apply Theorems 2.1 and 3.1 in two disjoint conical shells in order to get two different non-trivial fixed points. Firstly, we are going to check that the conditions of Theorem 2.1 are satisfied.

  1. (1.a)

    α 2 T α 2 and there exists R 2 >0 such that B[ α 2 , R 2 ]K .

    We have α 2 T α 2 because of the inequality α 2 γ f ( α 2 ) λ. On the other hand, the inequality α 2 R 2 (2(2 θ 1 + ξ 1 2 )1) implies that B[ α 2 , R 2 ]K. Indeed, let uB[ α 2 , R 2 ], that is,

    α 2 R 2 u(t) α 2 + R 2 for all t[0,1].

    Since 2 θ 1 + ξ 1 2 >1, we have α 2 > R 2 and then u(t)>0 for all t[0,1]. Moreover, it is easy to check that

    min t [ 0 , 1 ] u(t) α 2 R 2 c ˆ ( α 2 + R 2 ) c ˆ u,

    which means that uK.

  2. (1.b)

    The map T is monotone non-increasing in the set

    K 2 = { x K : R 2 x α 2 } .

    This fact is a consequence of the assumption (ii).

  3. (1.c)

    There exists r 2 >0 , with r 2 R 2 , such that Txx for all xK with x= r 2 .

    By the second part of the assumption (iii), we have this result for r 2 > R 2 large enough.

So, Theorem 2.1 implies the existence of a solution x 2 K such that

R 2 x 2 < r 2 .

Now, we are going to check that the conditions of Theorem 3.1 are satisfied.

  1. (2.a)

    T β 1 β 1 and there exists R 1 >0 such that B[ β 1 , R 1 ]K .

    We have T β 1 β 1 because of the inequality λ β 1 γ f ( β 1 ) . Again, β 1 R 1 (2(2 θ 1 + ξ 1 2 )1) implies that B[ β 1 , R 1 ]K by reasoning as in the proof of claim (1.a).

  2. (2.b)

    The map T is monotone non-decreasing in the set

    K 1 = { x K : R 1 x β 1 = β 1 } .

    This fact is a consequence of the assumption (i).

  3. (2.c)

    There exists r 1 >0 , with r 1 R 1 , such that Txx for all xK with x= r 1 .

    By the first part of the assumption (iii), we have this result for 0< r 1 < R 1 small enough.

Then, applying Theorem 3.1, we get the existence of a solution x 1 K such that r 1 < x 1 β 1 . Since β 1 < c ˆ R 2 , we have that x 1 x 2 and the theorem is proven. □

The following example illustrates our previous theorem.

Example 4.3 We consider the BVP

u ( 4 ) (t)=λf ( u ( t ) ) ,t(0,1),
(4.8)
u (0)= u (0)= 5 6 u (1)+u ( 1 5 ) = 2 7 u (1)+ u ( 4 5 ) =0,
(4.9)

with

f(u)= u + 300 e e 20 ( 2 u ) ( tan 1 ( 21 u ) + π 2 ) π .

In Figures 1, 2 and 3 you can see the behavior of the function f on different intervals.

Figure 1
figure 1

Graph of f on [0,1] .

Figure 2
figure 2

Graph of f on [1,3] .

Figure 3
figure 3

Graph of f on [3,30] .

In this example g(t)=1, and by [13] we know that γ = 3 , 677 31 , 500 and γ = 23 , 809 63 , 000 . Moreover, since θ 1 = 5 6 and ξ 1 = 1 5 , we have c ˆ = 53 128 .

Now, it is easy to check that all the assumptions of Theorem 4.2 are satisfied by taking β 1 = 3 2 , α 2 =20, R 1 = 1 2 and R 2 =8. So, by Theorem 4.2, BVP (4.8)-(4.9) has at least two positive solutions provided that

0.74665λ3.24075.

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Acknowledgements

Alberto Cabada and José Ángel Cid were partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2010-15314. This paper was partially written during the visit of Gennaro Infante to the Departamento de Análise Matemática of the Universidade de Santiago de Compostela. Gennaro Infante is grateful to the people of the aforementioned Departamento for their kind and warm hospitality.

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Cabada, A., Cid, J.Á. & Infante, G. New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl 2013, 125 (2013). https://doi.org/10.1186/1687-1812-2013-125

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Keywords

  • Krasnosel’skiĭ fixed point theorem
  • positive solutions
  • conical shells
  • multiplicity