A generalization of Krasnosel'skii compression fixed point theorem by using star convex sets

Proceedings of the Royal Society of Edinburgh, page 1 of 27 DOI:10.1017/prm.2018.119 A generalization of Krasnosel’skii compression fixed point theorem by using star convex sets Cristina Lois-Prados and Rosana Rodr´ıguez-L´opez Instituto de Matem´aticas, Facultade de Matem´aticas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain ([email protected]; [email protected]) (MS received 7 November 2017; accepted 18 April 2018) In the framework of fixed point theory, many generalizations of the classical results due to Krasnosel’skii are known. One of these extensions consists in relaxing the conditions imposed on the mapping, working with k-set contractions instead of continuous and compact maps. The aim of this work if to study in detail some fixed point results of this type, and obtain a certain generalization by using star convex sets. Keywords: Krasnosel’skii fixed point theorem; k-set contraction; star convex set 2010 Mathematics subject classification: 34B05; 34B15; 34B18; 47H08; 47H09; 47H10 1. Introduction Fixed point theorems are a fundamental tool to study the existence of solution to a wide range of different problems in mathematics, such as boundary value problems in the framework of differential equations. For T:D⊂X−→ Xa mapping between sets, we say that an element x∈Dis a fixed point of Tif T(x)=x. Here, we consider fixed point results that provide the existence and location of fixed points for mappings and sets satisfying certain hypotheses. The usual technique followed to apply these results to boundary value problems consists in transforming the problem into an integral equation. In this way, it is obtained a mapping whose fixed points are the solutions to the boundary value problem. Our main results generalize some classical fixed point theorems due to Krasnosel’skii. Two of these classical results can be found in [6] (dated in 1960), and they establish that, if (X,|| · ||) is a Banach space, CaconeinX,T:C−→ Ca continuous and compact map such that T(0) = 0, and there exist r, R ∈R+,r<R, satisfying some of the following conditions x−T(x)/∈C, ∀x∈C,||x|| ⩽r, T(x)−x/∈C, ∀x∈C,||x|| ⩾R, (1.1) c 2019 The Royal Society of Edinburgh 1 use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 2C. Lois–Prados and R. Rodr´ıguez–L´opez Figure 1. Motivation to work with more general sets. or T(x)−x/∈C, ∀x∈C,||x|| ⩽r, x−T(x)/∈C, ∀x∈C,||x|| ⩾R, (1.2) then Thas a nontrivial fixed point in {x∈C:r⩽||x|| ⩽R}. Next, we mention the different directions in which these classical results have been generalized, some of these extensions can be found in [7]. The first direction of extension is to relax conditions (1.1) and (1.2), in such a way that it is only required that x−T(x)/∈C, ∀x∈C,||x|| =r, ∀ε>0,T(x)−(1 + ε)x/∈C, ∀x∈C,||x|| =R, (1.3) or ∀ε>0,T(x)−(1 + ε)x/∈C, ∀x∈C,||x|| =r, x−T(x)/∈C, ∀x∈C,||x|| =R. (1.4) The second direction of extension is to seek more general regions, in which the theorem provides the fixed point. As an example, a result due to G¨uo and Lakshmikhantan [5] states that, if Ω1,Ω 2are bounded open sets in the Banach space X such that 0 ∈Ω1⊂Ω1⊂Ω2and T:C∩(Ω2\Ω1)−→ Cis a continuous and compact mapping, then Thas a fixed point in the region C∩(Ω2\Ω1), where Ωiis the closure of Ωi,fori=1,2. We can find another example in [1], where the region is generalized using general functionals instead of the norm. Recently, Webb [11] provides new results for nonlinear integral operators by using fixed point index and modifying the underlying sets. A third way to extend the mentioned classical results consists in considering another type of mappings instead of compact ones, the maps that we consider will be referred to as k-set contractions. Many of the generalizations of the classical results due to Krasnosel’skii have been proved using topological degree theory, such as those appearing in [1,5]. It is important to mention that this theory is not the approach used here. Now, we can explain properly the motivation of our work. Assume that T:D⊂ X−→ Xhas two fixed points with the same norm. We cannot prove their existence by using the mentioned fixed point results due to Krasnosel’skii. However, use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 3 if we prove similar results working with more general sets, it could be possible to distinguish these two fixed points. Figure 1illustrates an example of sets that allow us to separate two fixed points with the same norm. 2. Preliminaries For the sake of completeness, we provide the following definitions, results and notations, which are useful to our procedure. We refer to [8,10] for some basic monographs. Notation 2.1. Let ε>0and da distance in X, we denote Bd(x, ε)={y∈ X:d(x, y)<ε}and Bd(x, ε)={y∈X:d(x, y)⩽ε}. If there is no possibility of confusion, we fix the notation B(x, ε)≡Bd(x, ε)and B(x, ε)≡Bd(x, ε). Notation 2.2. Let Xbe a set, A, B subsets of Xand λ∈R, we establish the notation: •If Xis a topological space, Adenotes the closure of Ain X. •If Xis a real vector space, we define A+B={a+b:a∈A, b ∈B}and λA = {λa :a∈A}. Definition 2.3. Let Xbe a real vector space and Aa subset of X. The convex hull of Ais the set co(A):=n  i=1 λixi:n∈N, n  i=1 λi=1,x i∈A, λi∈[0,1],∀i∈{1,...,n}. Furthermore, if Xis a topological space, we denote the closure of co(A)byco(A). Definition 2.4. Let (X,d) be a metric space, K=∅a subset of Xand x∈X. The distance between xand Kis defined by dK(x)≡d(x, K):= inf y∈Kd(x, y). When K=∅, we define d(x, ∅):=+∞. Remark 2.5. If A⊂Xand A=∅, it is satisfied that d(x, A)<+∞, for all x∈X. Definition 2.6. Let X,Y be metric spaces and D⊂X. The mapping T:D⊂ X−→ Yis compact if, for all A⊂Dbounded, T(A) is a compact set. The following concept allows us to relax the compactness hypothesis assumed in the mentioned classical results due to Krasnosel’skii. Some of its principal properties can be found in [9]. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 4C. Lois–Prados and R. Rodr´ıguez–L´opez Definition 2.7. Let (X, d) be a metric space and A⊂Xa bounded set. The measure of noncompactness of Ais the nonnegative real number α(A):=infε>0:A⊂ n  i=1 Ai,diam(Ai)⩽ε, ∀i∈{1,...,n}, where diam(Ai):=sup{d(x, y):x, y ∈Ai}, for all i∈{1,...,n}. Proposition 2.8. Let A, B be subsets of a metric space (X,d). The measure of noncompactness satisfies the following properties: (i) A⊆B⇒α(A)⩽α(B). (ii) α(A∪B) = max{α(A),α(B)}. (iii) α(A)=α(A). Moreover, if (X,·)is a Banach space, then: (iv) α(A+B)⩽α(A)+α(B). (v) α(λA)=|λ|α(A),∀λ∈R. (vi) α(co(A)) = α(A). (vii) Ais a compact set if and only if α(A)=0. The measure of noncompactness can be considered as a tool to determine how much a particular set differs from being compact. In this way, we will be able to define a concept close to compact mapping, this one will be known as k-set contraction. So, we give a formal definition of this kind of mappings and some results about them. This information can be found in [9]. Definition 2.9. Let X,Y be metric spaces and D⊂X. Assume that the mapping T:D⊂X−→ Yis continuous. We say that Tis a k-set contraction if there exists a constant k⩾0 such that α(T(A)) ⩽kα(A),for all bounded A⊂D. Remark 2.10. If Tis a k-set contraction, it is implicitly required that T(A)is bounded when A⊂Dis bounded since it is a necessary condition to calculate the measure of noncompactness of T(A). Example 2.11. Continuous and compact mappings are in correspondence with 0-set contractions when Xis a Banach space. Proposition 2.12. Let (Xi,d i)be metric spaces for i∈{1,2,3},and(X,·)a Banach space. The following properties are satisfied: (i) If T1:X1−→ X2,T2:X2−→ X3are, respectively, k1,k 2-set contractions, then T2◦T1:X1−→ X3is a k1k2-set contraction. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 5 (ii) If S1:X1−→ X,S2:X1−→ Xare, respectively, k1,k 2-set contractions, then S1+S2:X1−→ Xis a k1+k2-set contraction. Proposition 2.13. Let D, ˆ Dbe closed subsets of a metric space (X, d). Assume that T:D−→ X,ˆ T:ˆ D−→ Xare k-set contractions and T|D∩ˆ D=ˆ T|D∩ˆ D.Ifwe define another mapping by ˜ T:D∪ˆ D−→ X x−→ ˜ T(x):=T(x),x∈D, ˆ T(x),x∈ˆ D, then ˜ Tis a k-set contraction. Corollary 2.14. Let D, ˆ Dbe closed subsets of a metric space (X, d). Suppose that T:D−→ Xis a k-set contraction and ˆ T:ˆ D−→ Xis a ˆ k-set contraction such that T|D∩ˆ D=ˆ T|D∩ˆ D. Define ˜ T:D∪ˆ D−→ X x−→ ˜ T(x):=T(x),x∈D, ˆ T(x),x∈ˆ D, then ˜ Tis a ˜ k-set contraction with ˜ k= max{k, ˆ k}. Proposition 2.15. Let (X, ·)be a Banach space, T:D⊂X−→ Xak-set contraction and λ:D−→ R+∪{0}a continuous function such that supx∈Dλ(x)=l< ∞. Define ˆ T:D⊂X−→ X x−→ ˆ T(x):=λ(x)T(x), then ˆ Tis a kl-set contraction. 3. Main fixed point results In this section, we include some known results proved by Potter [9] that generalize one of the fixed-point results due to Krasnosel’skii, in two of the mentioned directions. Besides, we show our contribution proving a result that generalizes the one proved by Potter in other of the directions mentioned working with more general sets which do not need to be convex. The following result is basic in the proof of the fixed point theorem due to Potter. This can be found in [4]. Proposition 3.1 (Fixed point theorem for k-set contractions). Let (X, ·)be a Banach space and B⊂Xa closed, convex and bounded set. Assume that T:B−→ Bis a k-set contraction with k<1, then there exists x∈BafixedpointofT. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 6C. Lois–Prados and R. Rodr´ıguez–L´opez Now, we fix the notation about the subsets of a Banach space in which the main results will locate the fixed points. Definition 3.2. Let (X,·) be a Banach space. A subset Cof Xis a cone if •Cis closed; •for all x, y ∈C,a, b ∈R+, it is satisfied that ax +by ∈C; •x∈C,−x∈Cif and only if x=0. Example 3.3. Let us consider the Banach space (R2,||·||), where || · || :R2−→ [0,+∞),(x, y)−→ | | (x, y)|| := x2+y2. The set C:= {(x, y)∈R2:x, y ⩾0}is a cone in R2. We will make representations with this cone to illustrate the restrictions imposed in the statements of the main fixed point results. Example 3.4. Let us consider the Banach space (C([0,1],R),|| · ||), where the elements in C([0,1],R) are continuous functions on the interval [0,1] with values in R and || · || :C([0,1],R)−→ [0,+∞),x−→ | | x|| := sup t∈[0,1] |x(t)|. The set C:= {x∈C([0,1],R):x⩾0}is a cone in C([0,1],R). Notation 3.5. Let (X, ·)be a Banach space and C⊂Xacone.Forr, R ∈R, with 0<r<R,let •Fr,R ={x∈C:r⩽x⩽R}; •Br={x∈C:x⩽r}; •Sr={x∈C:x=r}. We outline the mentioned result due to Potter. The following lemma is required to prove this fixed point theorem. It allows us to extend the domain of some k-set contractions, with k<1, preserving this property of the mapping. Lemma 3.6. Assume that T:Sr−→ Cis a k-set contraction. Let us consider ˜ T:Br−→ C, x −→ ˜ T(x):=x rTr xx,x=0, 0,x=0, then ˜ Tis a ˜ k-set contraction, with ˜ k>k,˜ kas near kas we please. The conditions of the classical results due to Krasnosel’skii, which have been formulated in the Introduction (1.1), can be replaced by the ones in the following definition. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 7 Definition 3.7. Let (X,|| · ||)beaBanachspaceandCa cone. A mapping T: Fr,R −→ Cis said to be a compression of the cone Cif •x−Tx /∈Cfor all x∈Cwith x=r; •for all ε>0andx∈Cwith x=R,Tx−(1 + ε)x/∈C. Theorem 3.8. Let (X, || · ||)be a Banach space, CaconeinXand 0<r<R real numbers. Suppose that T:Fr,R −→ Cis a k-set contraction with k<1and a compression of the cone C.ThenThas at least one fixed point in Fr,R ⊂C. Now, we prove a more general result working with star convex sets that are not necessarily convex. Definition 3.9. Let (X,·) be a Banach space, E⊂Xand x0∈Esuch that λx0+(1−λ)x∈E, for all λ∈[0,1] and x∈E. If x0=0,Eis said to be an x0-star convex set. If x0= 0, it is called a star convex set. In our main results, we will consider the following hypothesis. Condition 3.10. Let (X, ·)be a Banach space, CaconeinXand Ea star convex set (which trivially satisfies that E∩C=∅). We assume the following essential conditions for the star convex set E: •Eis bounded, closed and has nonempty interior. •If Fis the boundary of Ein X, then 0/∈F. •There exists a continuous mapping ∂:E\{0}−→F, x −→ ∂(x), such that (see figure 2): ∂(x)=∂(λx),∀x∈E, ∀λ∈(0,1], ∂(x)=x, ∀x∈F. If the boundary Fsatisfies the appropriate conditions, there is a unique procedure valid to define the mapping ∂, related to the replication of its values through rays travelling to 0. The last property in condition 3.10 is too strong since the interesting properties arise in C∩E\{0}. Remark 3.11. Suppose that (X, ·) is a Banach space, CaconeinXand Ea star convex set satisfying condition 3.10.AsEis a closed and star convex set such use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 8C. Lois–Prados and R. Rodr´ıguez–L´opez Figure 2. An example of mappings ∂and ∂C. that 0 ∈˚ E, it is possible to extend the domain of ∂|C∩(E\{0})to C\{0}by ∂C:C\{0}−→F x−→ ∂C(x):=∂(x),x∈E\{0}, ∂d(0,F ) ||x|| x,x∈C\(C∩˚ E). Besides, ∂Cis a continuous function. Figure 2illustrates the behaviour of ∂Cin a particular case. Proposition 3.12. Let (X, ·)be a Banach space, CaconeinXand Ea star convex set satisfying condition 3.10.Then,forallx∈C∩(E\{0}), there exists a unique number βx∈R+such that βxx∈C∩F. Proof. The existence of such a number is clear since Cis a cone and Eis closed, bounded, with nonempty interior and a star convex set. Suppose that there exist β1 x,β2 x∈R+such that β1 x=β2 xand β1 xx,β2 xx∈C∩F. Assume that β2 x>β 1 x, then 0<((β1 x)/(β2 x)) <1 and, therefore, ∂(β1 xx)=∂β1 x β2 x β2 xx=∂(β2 xx)=β2 xx∈F. Besides, since β1 xx∈F,∂(β1 xx)=β1 xx∈F. As a consequence, β1 xx=β2 xx. Taking the norm, we get β1 xx=β2 xx, and, since x= 0, then β1 x=β2 x. We conclude that the element βxin the statement is unique and the proof is finished.  Lemma 3.13. Let (X, ·)be a Banach space, CaconeinXand Ea star convex set satisfying condition 3.10.Then β:C∩(E\{0})−→ [1,+∞) x−→ β(x):=βx, use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 9 where βxis the unique βx∈R+such that βxx∈F∩C, satisfies the following properties: (a) βis a continuous function. (b) lim x→0β(x)=+∞. Proof. First of all, we prove that the image of βis a subset of [1,+∞). Let x∈ C∩(E\{0}), then it is satisfied that βxx=∂(x), where βx=∂(x) x⩾1. Secondly, we prove the properties (a) and (b). Indeed: (a) The function d0:X−→ [0,∞),x−→ d0(x):=d(0,x) is continuous because of the properties of the distance. By hypothesis, ∂: E\{0}−→Fis continuous too. Since βcanbeexpressedas β:C∩(E\{0})−→ [1,+∞) x−→ β(x)=d(0,∂(x)) d(0,x)=(d0◦∂)(x) d0(x), then βis a continuous function. (b) For all M∈R+,welookforδ∈R+such that β(x)>M, for all x∈C∩(E\{0}) with ||x|| <δ. If M∈(0,1), using that β∈[1,+∞), then β(x)>M is trivially satisfied for all x∈C∩(E\{0}). If M⩾1, let 0 <δ=((d(0,F))/(M)) ⩽d(0,F)<+∞. We prove that, if x∈C∩(E\{0}) with ||x|| <δ, then β(x)>M: β(x)=d(0,∂(x)) d(0,x)⩾d(0,F) d(0,x)=d(0,F) ||x|| >d(0,F) δ=M. The proof is concluded.  Remark 3.14. Assume that Eis a star convex set satisfying condition 3.10. Since Eis a bounded and closed set and Fis its boundary, then there exists L∈R+such that d(0,∂(x)) = ∂(x)⩽Lfor all x∈E\{0}. It is enough to take L=sup{d(0,x):x∈F}. We have stated sufficient conditions on the sets. The next step will be to reformulate definition 3.7 dealing with these more general sets. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 16 C. Lois–Prados and R. Rodr´ıguez–L´opez •T|BRis a k-set contraction with k<1. To prove it, we define another helpful auxiliary mappings: T1:BR∩E1−→ BRis given by T1(x):=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ δh, x =0; 1 β1(x)T(β1(x)x)+δh, x ∈BR∩Eδ 1\{0}; 1 β1(x)T(β1(x)x)+d(x, F1)h, x ∈BR∩E1\˚ Eδ 1. T2:BR\(BR∩˚ E1)−→ BRis given by T2(x):=⎧ ⎨ ⎩ T(x),x∈BR∩(E2\˚ E1), T(∂C 2(x)),x∈BR\(BR∩˚ E2). It is possible to express T1as the sum of two mappings T1 1and T2 1.The mapping T1 1:BR∩E1−→ BR x−→ T1 1(x):=δh, x ∈BR∩Eδ 1, d(x, F1)h, x ∈BR∩(E1\˚ Eδ 1), is a 0-set contraction. In fact, let A⊂BR∩E1, then Ais bounded and T1 1(A)= T1 1(A∩(BR∩Eδ 1)) ∪T1 1(A∩(BR∩(E1\˚ Eδ 1))). As T1 1(A∩(BR∩(E1\Eδ 1))) ⊂co{{0}∪{δh}}, by using properties (i), (ii), (iii), (vi) and (vii) of proposition 2.8,wecan conclude α(T1 1(A)) ⩽max{α({δh}),α(co{{0}∪{δh}})}=0. Furthermore, the mapping T2 1:BR∩E1−→ BR x−→ T2 1(x):=⎧ ⎨ ⎩ 0,x=0, 1 β1(x)T(β1(x)x),x=0, is a k2 1-set contraction with k2 1<1, because it is the restriction to BR∩E1of ˜ Tin lemma 3.16 with E=E1and F=F1. Using (ii) of proposition 2.12, then T1=T1 1+T2 1is a k1=0+k2 1-set contraction with k1<1. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 17 Besides, T2is a k2-set contraction with k2=k, because it can be written as the composition T◦S, where S:BR\(BR∩˚ E1)−→ BR∩(E2\˚ E1) is given by S(x):=⎧ ⎨ ⎩ x, x ∈BR∩(E2\˚ E1), ∂C 2(x)=β2d(0,F 2) ||x|| xd(0,F 2) ||x|| x, x ∈BR\(BR∩˚ E2), is a 1-set contraction. Therefore, hypothesis (i) of proposition 2.12 is satisfied and, as a consequence, T2is a k-set contraction. We now prove that Sis a 1-set contraction. For this, let us consider λ:BR\(BR∩˚ E1)−→ R+given by λ(x):=⎧ ⎨ ⎩ 1,x∈BR∩(E2\˚ E1), β2d(0,F 2) ||x|| xd(0,F 2) ||x|| ,x∈BR\(BR∩˚ E2), which is a continuous function and satisfies sup{λ(x):x∈BR∩(BR\˚ E1)}⩽1. Hence, by using proposition 2.15, we conclude that Sis a 1-set contraction since the identity also fulfills this property. Applying corollary 2.14 to T1and T2, we get that T|BRis a k-set contraction with k= max{k1,k}<1. Therefore, the hypotheses of proposition 3.1 are satisfied and T|BRhas at least one fixed point x.Weonlyhavetoprovethatx∈C∩(E2\˚ E1). To that purpose, we consider four cases: Case 1: Suppose that x=0. So T(0) = 0, then δh= 0 and this is not possible since δ>0,h>0. Case 2: Assume that x∈BR∩(Eδ 1\{0}). Consequently, T(x) = ((1)/(β1(x)))T(β1(x)x)+δh =x,so h⩽1 δx+1 δβ1(x)T(β1(x)x) and it is a contradiction with the selection of hin the definition of the mapping T:C−→ C. Case 3: Let x∈BR∩(E1\Eδ 1). Since xis a fixed point, then T(x) = ((1)/(β1(x)))T(β1(x)x)+ d(x, F1)h=x,so x−1 β1(x)T(β1(x)x)=d(x, F1)h∈C, due to d(x, F1)⩾0. Moreover, β1(x)∈[1,+∞), thus β1(x)x−T(β1(x)x)∈C, where β1(x)x∈F1∩C, which contradicts the hypothesis (C1) for Ta compression of the cone C. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 18 C. Lois–Prados and R. Rodr´ıguez–L´opez Case 4: Suppose that x∈BR\(BR∩E2). Let us define yx=((d(0,F 2))/(||x||))x, then T(x)=T(∂C 2(x)) = T(∂2(yx)) =T(β2(yx)yx)=Td(0,∂ 2(yx)) yx d(0,F 2) xx. As yx=((d(0,F 2))/(x))x=d(0,F 2), then T(x)=T(((d(0,∂ 2 (yx)))/(x))x). Take ε=((x)/(d(0,∂ 2(yx)))) −1, then ε>0 since ((x)/(d(0,∂ 2(yx)))) >1. We can express xas (1 + ε)((d(0,∂ 2(yx)))/ (x))x, then T(x)=Td(0,∂ 2(yx)) xx=(1+ε)d(0,∂ 2(yx)) xx. Due to ((d(0,∂ 2(yx)))/(x))x∈C∩F2and T(((d(0,∂ 2(yx)))/(x))x) −(1 + ε)((d(0,∂ 2(yx)))/(x))x=0∈C, we obtain a contradiction with the hypothesis (C2) for Ta compression of the cone C. Finally, the fixed point of Tbelongs to C∩(E2\˚ E1), since Tand Tcoincide on this set, then we conclude that Thas a fixed point in the mentioned set.  Potter, in [9], asserts that, using similar techniques as the ones in the proof of theorem 3.8, it is possible to generalize the fixed point result due to Krasnosel’skii by considering hypothesis (1.2) instead of (1.1), but the proof is not given. On the other hand, C´ac and Gatica prove it in [3] following the steps of Krasnosel’skii to prove the mentioned classical result with hypothesis (1.2). In the future, we will consider the case of expansive mappings working with sets which satisfy condition 3.10. 3.1. Admissible sets defined by functionals The above-mentioned fixed point theorem due to Potter provides the existence of fixed points in certain subsets of a Banach space (X, || · ||). These subsets are determined by the norm || · ||, for instance, given r∈R,r>0andCaconeinX, we define Br:= C∩{x∈X:||x|| ⩽r},S r:= C∩{x∈X:||x|| =r}. This result guarantees, for some r, R ∈R,0<r<R, and some particular mapping T, that there exists at least a fixed point of Tin BR\˚ Br=C∩{x∈X:r⩽||x|| ⩽ R}. We have just proved a generalization of this result working with some particular star convex sets. Now, our aim is to determine conditions over a functional ϕ: X−→ [0,+∞) such that, for r∈R,r>0, the sets Er:= {x∈X:ϕ(x)⩽r},F r:= {x∈X:ϕ(x)=r} use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 19 satisfy the condition 3.10, and derive some consequences as corollary of theorem 3.19. Remark 3.20. If ϕ=||·||, then Br=C∩Erand Sr=C∩Fr. This allows to consider functionals with weaker properties in comparison with the norm. Theorem 3.21. Let (X, ·)be a Banach space, r∈R,r>0and ϕ:X−→ [0,+∞)satisfying: (F1) If x∈Xis such that ϕ(x)⩽r, then ϕ(λx)⩽rfor all λ∈[0,1]. (F2) ϕis a continuous functional. (F3) For all x∈Er,ϕ(x)=0⇔x=0. (F4) ϕ(λx)=λϕ(x)for all λ∈(0,+∞),x∈Er. (F5) There exists m∈R,m>0such that mx⩽ϕ(x)for all x∈Xwith x> ϕ(x),or lim inf x→+∞ϕ(x)>r. Under these assumptions, Erand Frsatisfy condition 3.10. Proof. We proceed step by step, that is, we prove each one of the properties required to Erand Frby using the appropriate hypotheses of ϕ. Indeed: •(F1) is equivalent to Erbeing a star convex set. •(F2) implies that Eris closed and ϕ−1([0,r)) is open in X. Indeed, we can write Er=ϕ−1([0,r]). As [0,r] is a closed subset of ([0,+∞),|·|), where |·|is the absolute value for real numbers, then Eris closed since it is the preimage of a closed set by a continuous function. Besides, we can assert that ϕ−1([0,r)) is open in Xsince [0,r) is an open subset of [0,+∞). •(F3) implies that 0 /∈Fr, because ϕ(x)=r>0 for all x∈Fr. •By hypothesis (F4), we prove two conditions over the sets Erand Fr.First, we show that Fris the boundary of Er. We have just proved that Eris closed and Er\Fr=ϕ−1([0,r)) is open, so the boundary of Eris a subset of Fr.Let us consider y∈Frarbitrarily, we want to prove that yis a boundary point of Er.Asy∈Fr⊂Er, we must prove that, for all ε>0, B(y,ε)∩(X\Er)=∅. We take z=(1+((ε)/(2y)))y∈B(y,ε), then it is satisfied ϕ(z)=1+ ε 2yϕ(y), and, as (1 + ((ε)/(2y))) >1, we can conclude that ϕ(z)>ϕ(y)=r. Therefore, z∈X\Erand yis a boundary point of Er. Secondly, there exists a mapping ∂:Er\{0}−→Fr,x−→ ∂(x):=((r)/(ϕ(x)))x, which satisfies the desired conditions for mapping ∂. The function ∂is clearly welldefined and continuous, by using (F2) and (F3). Let x∈Er\{0},ϕ(∂(x)) = ϕ(((r)/(ϕ(x)))x)=r, then ∂(x)∈Fr. Moreover, if x∈Fr, then ϕ(x)=rand, use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 20 C. Lois–Prados and R. Rodr´ıguez–L´opez Figure 5. Graph of function ϕ, which is upper semicontinuous. therefore, ∂(x)=x. Furthermore, by (F4), ∂(λx)=∂(x) for all x∈Erand λ∈(0,1]. •Finally, (F5) implies that Eris a bounded set.  Remark 3.22. We know that the norm is a continuous function, so we think if it is possible to relax this condition required to the functional. Is it enough to work with an upper or lower semicontinuous function ϕ? Next, we prove that this assumption is not enough. Assume that ϕis upper semicontinuous. We justify that Er\Fris open and Er is not necessarily closed: •Let r∈R,r>0, then ϕ−1([0,r)) is open. We prove it by contradiction, so suppose that ϕ−1([0,r)) is not open, then there exists y∈ϕ−1([0,r)) such that yis not an interior point. Therefore, for all δ∈R,δ>0, it holds that B(y,δ)ϕ−1([0,r)).(3.5) As ϕ(y)<r, we can take ε>0 such that ϕ(y)+ε<r. Besides, by using that ϕis upper semicontinuous, there exists δy εsuch that, for all x∈B(y,δy ε), 0⩽ϕ(x)<ϕ(y)+ε<r. As a consequence, B(y,δy ε)⊂ϕ−1([0,r)) and it contradicts (3.5). Therefore, we conclude that ϕ−1([0,r)) is an open set. •However, ϕ−1([0,r]) is not always closed. As an example, we work with the function ϕ:[0,+∞)−→ [0,+∞) x−→ ϕ(x)=x+1, where · denotes the floor function (see figure 5). use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 21 First of all, we prove that ϕis an upper semicontinuous function. Let ε∈R, ε>0andy∈[0,+∞) be arbitrarily fixed. We take δy=((d(y,ϕ(y)))/(2)) >0, so ϕ(B[0,+∞)(y,δy)) ⊂[0,ϕ(y+δy)) ⊂[0,ϕ(y)) ⊂[0,ϕ(y)+ε). Then, ϕis upper semicontinuous. Secondly, for each r∈R,r>0, it is satisfied that: If r∈(0,1), then ϕ−1([0,r]) = ∅.Ifr∈[1,+∞), then ϕ−1([0,r]) = [0,r), which is an open set of ([0,+∞),|·|). Assume that ϕis lower semicontinuous. We justify that Eris closed and Er\Fr is not necessarily open. ◦Let r∈R,r>0, then ϕ−1([0,r]) is closed. As ϕ−1([0,r]) = X\ϕ−1((r, +∞)), we prove that ϕ−1((r, +∞)) is an open subset of [0,+∞) by contradiction. Let us assume that ϕ−1((r, +∞)) is not open, so there exists at least a point y∈ϕ−1(r, +∞) such that for each δ∈R,δ>0, it is fulfilled B(y,δ)ϕ−1((r, +∞)).(3.6) Now, as ϕ(y)>r, we can take ε>0 such that ϕ(y)−ε>r. Besides, by using that ϕis lower semicontinuous, there exists δy ε>0 such that, for all x∈B(y,δy ε), r<ϕ(y)−ε<ϕ(x)<+∞. As a consequence, B(y,δy ε)⊂ ϕ−1((r, +∞)), which contradicts (3.6). ◦Nevertheless, ϕ−1([0,r)) is not always open. For example, we consider the function ϕ:[0,+∞)−→ [0,+∞) x−→ ϕ(x):=x,x/∈N; x−1,x∈N; where · is the floor function (see figure 6). First, we show that ϕis lower semicontinuous. Let y=0andε∈R,ε>0, since ϕ([0,δ)) ⊂[0,+∞)=[ϕ(0),+∞)⊂(ϕ(0) −ε, +∞),∀δ∈R,δ>0, ϕis clearly lower semicontinuous at 0. Let ε∈R,ε>0, and y∈(0,+∞)be arbitrarily fixed. We take δy=((d(y,ϕ(y)))/(2)) >0 and it is possible to prove that ϕ(B[0,+∞)(y,δy)) ⊂(ϕ(y)−ε, +∞). As a consequence, we can assert that ϕis lower semicontinuous. Next, let r∈R,r>0, it is satisfied that: If r∈N, then ϕ−1([0,r)) = [0,r], which is a closed subset of [0,+∞). If r/∈N, then ϕ−1([0,r)) = [0,r+ 1], which is a closed subset of [0,+∞). To sum up, it is not enough to work with an upper or lower semicontinuous function. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 22 C. Lois–Prados and R. Rodr´ıguez–L´opez Figure 6. Graph of function ϕ, which is lower semicontinuous. Now, we rewrite theorem 3.19 by using functionals. Theorem 3.23. Let (X, ·)be a Banach space, CaconeinXand r, R ∈ R,with0<r<R. Assume that ϕ, ψ :X−→ [0,+∞)are functionals satisfying the hypotheses (F1) −(F5) of theorem 3.21,forr, R respectively, and ψ(x)< (R/r)ϕ(x)for all x∈Xsuch that ϕ(x)⩽r. We consider E1:= {x∈X:ϕ(x)⩽ r}and E2:= {x∈X:ψ(x)⩽R}.ThenE1and E2fulfill condition 3.10,0∈E1⊂ E2and F1∩F2=∅. Moreover, suppose that T:C∩(E2\˚ E1)−→ Cis a k-set contraction, with k<((d(0,F 1))/(L1)) and a compression of the cone Caccording to definition 3.15.ThenThas at least one fixed point in C∩(E2\˚ E1). Proof. First of all, by theorem 3.21, since ϕ, ψ satisfy the hypotheses (F1) −(F5), then Ei,Fifulfill the condition 3.10,fori=1,2. Secondly, we check that 0 ∈E1⊂E2and F1∩F2=∅in order to deal with the concept of a compression of the cone C. Since ψ(x)<(R/r)ϕ(x) for all x∈E1: •Let y∈E1,ψ(y)<(R/r)ϕ(y)⩽R, then y∈E2. •Let y∈F1,ψ(y)<(R/r)ϕ(y)=R, then y/∈F2. Therefore, the hypotheses of theorem 3.19 are satisfied and, as a consequence, T hasatleastafixedpointinC∩(E2\˚ E1).  Remark 3.24. The hypothesis ψ(x)<(R/r)ϕ(x), for all x∈Xsuch that ϕ(x)⩽r, of theorem 3.23, can be replaced by ψ(x)⩽R rϕ(x),for all x∈Xsuch that ϕ(x)<r, ψ(x)<R rϕ(x),for all x∈Xsuch that ϕ(x)=r. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 23 4. Application In this last section, we consider an ordinary differential equation subject to boundary conditions and apply theorems 3.19 and 3.21 to study the existence of solutions localized in a region determined by two star convex sets defined by functionals. The same problem was considered by Avery, Henderson and O’Regan in [2], where the authors applied other different fixed point theorems. To proceed, let us consider the second-order nonlinear boundary value problem x(t)+f(x(t)) = 0,t∈[0,1], x(0) = 0 = x(1),(4.1) where f:R−→ Ris a continuous function such that f(z)⩾0 for all z⩾0. We seek some solution x∈C2([0,1],R)to(4.1). We prove that there exists at least one solution that satisfies some properties as concavity and symmetry with respect to t=1/2. First of all, we obtain a mapping Twhose fixed points correspond to the solutions of the boundary value problem. This mapping is given as follows: T:C([0,1],R)−→ C ([0,1],R) x−→ T(x):=Tx:[0,1] −→ R t−→ [Tx](t):=1 0 G(t, s)f(x(s))ds, where Gis the function: G:[0,1] ×[0,1] −→ [0,1] (t, s)−→ G(t, s):=t(1 −s),0⩽t⩽s⩽1, s(1 −t),0⩽s⩽t⩽1. Now, we apply theorem 3.19 to the mapping T. We consider the Banach space (C([0,1],R),|| · ||), where || · || is the norm defined in example 3.4. However, we take aconeinC([0,1],R) different from the one in this example, as follows: C:= x∈C([0,1],R):x⩾0,xis concave and symmetric with respect to 1 2. Besides, we require that the function fwhich defines the second-order differential equation in (4.1) satisfies the following conditions. Condition 4.1. There exist real numbers 0<r<R and 0<α<((R)/(r)) −1 such that: (H1)f(z)>8r, for all z∈[0,r]. (H2)f(z)⩽((8R)/((1 + α))), for all z∈[0,R/α]. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of 24 C. Lois–Prados and R. Rodr´ıguez–L´opez Given real numbers 0 <r<Rand 0 <α<((R)/(r)) −1 such that the hypotheses (H1) and (H2) of condition 4.1 are fulfilled, and 0 ⩽δ<1/2 fixed, we consider the following sets E1:= x∈C([0,1],R):||x|| = max t∈[0,1] |x(t)|⩽r, E2:= x∈C([0,1],R) : min t∈[1/2−δ,1/2+δ]|x(t)|+αx⩽R. We assert that E1and E2are star convex sets fulfilling condition 3.10.E1is clearly under the required hypotheses. On the other hand, E2is a set defined by the functional ϕ:C([0,1],R)−→ [0,+∞) x−→ ϕ(x) := min t∈[1/2−δ,1/2+δ]|x(t)|+αx. This functional satisfies the hypotheses required in theorem 3.21. Therefore, E2 is a star convex set fulfilling condition 3.10. Besides, the following properties are satisfied: Since α<R/r−1, then E1⊂E2and F1∩F2=∅. If x∈C∩F1, then x= maxt∈[0,1] |x(t)|=x(1/2) = r. If x∈C∩F2, then mint∈[1/2−δ,1/2+δ]|x(t)|+αx=x(1/2−δ)+αx(1/2) = R. The next result proves the existence of a nonzero, concave and symmetric with respect to 1/2 solution to the boundary value problem (4.1). Theorem 4.2. For all continuous function f:R−→ Rsuch that f(z)⩾0for all z⩾0and fulfilling condition 4.1, the boundary value problem (4.1)has at least a solution in C∩(E2\˚ E1). Proof. We prove the result by checking the hypotheses of theorem 3.19. First, (C([0,1],R),|| · ||) is a Banach space, CaconeinC([0,1],R)andE1,E2 star convex sets satisfying condition 3.10. Secondly, it is possible to prove that T:C([0,1],R)−→ C ([0,1],R) is a continuous and compact mapping. Then, we have to prove the following properties: (i) T(C∩(E2\˚ E1)) ⊂C. (ii) x−Tx /∈C, for all x∈C∩F1. (iii) For all ε>0andx∈C∩F2,Tx−(1 + ε)x/∈C. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of A generalization of Krasnosel’skii fixed point theorem 25 We begin by proving (i). Let x∈C∩(E2\˚ E1), we show that T(x)∈C.Tothat purpose, we check the following items: •For each t∈[0,1], since G(t, s)∈[0,1] for all (t, s)∈[0,1] ×[0,1] and f(x(s)) ⩾ 0 for all s∈[0,1], then [Tx](t)=1 0 G(t, s)f(x(s))ds⩾0. •By construction of T(x), for all t∈(0,1), it is satisfied that [Tx](t)= −f(x(t)) ⩽0, so Tx is a concave function. •Tx is symmetric with respect to 1/2, because [Tx](t)=[Tx](1 −t),for all t∈[0,1]. In fact, for all t∈[0,1], it holds that [Tx](t)=1 0 G(t, s)f(x(s))ds=t 0 s(1 −t)f(x(s))ds+1 t t(1 −s)f(x(s))ds. Making the change of variable s=1−u, we obtain [Tx](t)=1 1−t (1 −u)(1 −t)f(x(1 −u))du+1−t 0 tuf(x(1 −u))du. Since x∈C,xis symmetric with respect to 1/2, then [Tx](t)=1−t 0 u[1 −(1 −t)]f(x(u))du+1 1−t (1 −t)(1 −u)f(x(u))du =1 0 G(1 −t, u)f(x(u))du=[Tx](1 −t). Therefore, property (i) has been proved. In fact, we have just proved Tx∈Cfor all x∈C. By hypothesis (H1), we prove that condition (ii) is fulfilled. In fact, let x∈ C∩F1,asx⩾0and||x|| =r, then 0 ⩽x(s)⩽rfor all s∈[0,1]. Let us consider t=1/2, then x1 2−[Tx]1 2=r−1 0 G1 2,s f(x(s))ds. Now, by using (H1), we obtain that f(x(s)) >8rfor all s∈[0,1], then x1 2−[Tx]1 2<r−8r1 0 G1 2,s ds=0, and, therefore, we conclude that x−Tx /∈C, because x−Tx⩾0 is not satisfied. As x∈C∩F1was arbitrarily fixed, then (ii) has been proved. use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/prm.2018.119 Downloaded from https://www.cambridge.org/core. Universidade de Santiago de Compostela, on 07 Jun 2019 at 09:02:34, subject to the Cambridge Core terms of