A fixed point index approach to Krasnosel’skiĭ-Precup fixed point theorem in cones and applications

Nonlinear Analysis 226 (2023) 113138 Contents lists available at ScienceDirect Nonlinear Analysis www.elsevier.com/locate/na A fixed point index approach to Krasnosel’ski˘ı-Precup fixed point theorem in cones and applications Jorge Rodríguez–López CITMAga & Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain a r t i c l e i n f o Article history: Received 21 June 2022 Accepted 2 September 2022 Communicated by Tobias Weth MSC: 47H10 47H11 45G15 34B18 35J92 Keywords: Coexistence fixed point fixed point index positive solution Hammerstein systems p-Laplacian system radial solution abstract We present an alternative approach to the vector version of Krasnosel’ski˘ı compression–expansion fixed point theorem due to Precup, which is based on the fixed point index. It allows us to obtain new general versions of this fixed point theorem and also multiplicity results. We emphasize that all of them are coexistence fixed point theorems for operator systems, that means that every component of the fixed points obtained is non-trivial. Finally, these coexistence fixed point theorems are applied to obtain results concerning the existence of positive solutions for systems of Hammerstein integral equations and radially symmetric solutions of (p1, p2)-Laplacian systems. ©2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction Krasnosel’ski˘ı compression–expansion fixed point theorem is between the main tools of Nonlinear Analysis for proving the existence of non-trivial solutions of different types of boundary value problems. Basically, assuming cone-compression or cone-expansion conditions on the boundary of an annulus, it ensures the existence of fixed points of compact operators defined in cones of normed linear spaces. In the case of systems, the localization of the fixed points obtained by means of Krasnosel’ski˘ı theorem is not given independently in each component, so there is not guarantee that all the components of the fixed point are non-trivial, as already pointed out in [3,25,26]. This fact motivated Precup to establish the vector version of Krasnosel’ski˘ı fixed point theorem [25,26] (see Theorem 2.3 below), which provides a componentwise localization of the fixed points. Thus, it gives sufficient conditions for the existence of a coexistence fixed E-mail address: jorgerodriguez.lop[email protected]. https://doi.org/10.1016/j.na.2022.113138 0362-546X/©2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 point as coined by Lan [20], that is, a fixed point with all the components different from zero. As far as we know, there are only a few papers in the literature which deal with theoretical results concerning coexistence fixed points of compact maps, see [3,15,16,20,25,26,29]. These results have direct applications in population models when studying the coexistence of several competing species. In this context, the reader may find some useful comments in the paper by Dancer [4]. Moreover, under the assumptions of the vector version of Krasnosel’ski˘ı fixed point theorem, each component of the compact map may have a different behavior, namely, compression or expansion (see Remark 2.1 below). To the best of our knowledge, fixed point theorems for expansive–compressive maps are not common in the literature, we refer the interested reader to the results due to Mawhin [23] in this direction. In this paper, we will refer to the mentioned vector version of Krasnosel’ski˘ı compression–expansion fixed point theorem due to Precup as Krasnosel’ski˘ı-Precup fixed point theorem in cones. It is well-known that the classical Krasnosel’ski˘ı fixed point theorem can be proved via fixed point index for compact maps, so our aim here is to present an alternative approach to Krasnosel’ski˘ı-Precup fixed point theorem based on this tool. It has its own interest since (a) it provides a way to extend Krasnosel’ski˘ı-Precup fixed point theorem to operators defined in more general domains; (b) the computation of the fixed point index allows to obtain easily new multiplicity results; (c) the proof can be replicated for other classes of maps for which a fixed point index theory is available (as, for instance, upper semicontinuous multivalued maps). Note that the fixed point index for compact maps was also the main tool employed in order to prove the coexistence fixed point theorems in [15,16,20]. In addition, we prove that the finite-dimensional version of Krasnosel’ski˘ı-Precup fixed point theorem is equivalent to Poincar´e–Miranda zeros theorem. This fact gives a connection between it and classical results. In the last section, we apply the theoretical results obtained in Section 2to two different problems: systems of Hammerstein integral equations and radially symmetric solutions of Dirichlet problems for (p1, p2)-Laplacian systems. Note that Krasnosel’ski˘ı-Precup fixed point theorem has been already employed by several authors in order to study the existence, localization and multiplicity of positive solutions for different types of systems of boundary value problems, see for instance [8,27,31–33]. Our intention is to emphasize the applicability of the new fixed point theorems established here and so we present a multiplicity result for a system of Hammerstein type equations, which complements previous results in the literature, see [2,9,17,20,29] and the references therein. Moreover, concerning radial solutions of (p1, p2)-Laplacian systems, our sufficient conditions provide not only the existence of positive solutions, but also a novel localization of them, cf. [24,33]. It is worth mentioning that existence of (not necessarily radial) solutions of (p1, p2)-Laplacian systems with both nonzero components was already studied in [16]. 2. Krasnosel’ski˘ ı-Precup fixed point theorem in cones First, we recall Krasnosel’ski˘ı compression–expansion fixed point theorem in cones [18] (see also [1,12]). In the sequel, we need the following notions. A closed convex subset Kof a normed linear space (X, ∥·∥) is a cone if λ u ∈Kfor every u∈Kand for all λ≥0, and K∩(−K) = {0}. A cone Kinduces the partial order in Xgiven by u⪯vif and only if v−u∈K. Moreover, we shall say that u≺vif v−u∈K\{0}. The following notations will be useful: for given r, R ∈R+:= [0,∞), 0 < r < R, we define Kr,R := {u∈K:r < ∥u∥< R}and Kr,R := {u∈K:r≤ ∥u∥ ≤ R}. 2 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 Theorem 2.1 (Krasnosel’ski˘ı).Let (X, ∥·∥)be a normed linear space, Ka cone in Xand r, R ∈R+, 0< r < R. Consider a compact map T:Kr,R →Ksatisfying one of the following conditions: (a)T(u)⊀uif ∥u∥=rand T(u)⊁uif ∥u∥=R; (b)T(u)⊁uif ∥u∥=rand T(u)⊀uif ∥u∥=R. Then Thas at least a fixed point u∈Kwith r≤ ∥u∥ ≤ R. Condition (a) in Krasnosel’ski˘ı theorem is usually called a compression type condition, whereas authors often refer to condition (b) as the cone-expansion condition. Similarly, in case (a) we will say that the operator Tis compressive, while in case (b)Tis called an expansive operator. It is well-known that conditions (a) and (b) can be weakened as homotopy type conditions. In this way, we have the homotopy version of Krasnosel’ski˘ı theorem or Krasnosel’ski˘ı-Benjamin theorem, see for instance [1]. Theorem 2.2 (Krasnosel’ski˘ı-Benjamin).Let (X, ∥·∥)be a normed linear space, Ka cone in Xand r, R ∈R+,0< r < R. Assume that T:Kr,R →Kis a compact map and there exists h∈K\{0}such that one of the following conditions is satisfied: (a)T(u) + µ h =uif ∥u∥=rand µ > 0, and T(u)=λ u if ∥u∥=Rand λ > 1; (b)T(u)=λ u if ∥u∥=rand λ > 1, and T(u) + µ h =uif ∥u∥=Rand µ > 0. Then Thas at least a fixed point u∈Kwith r≤ ∥u∥ ≤ R. In [25,26], Precup proposed a compression–expansion type fixed point theorem for systems of operators. The main novelty is that compression–expansion conditions are given in a component-wise manner, in what was called the vector version of Krasnosel’ski˘ı fixed point theorem. Let us recall this result. Consider two cones K1and K2of a normed linear space X, and so K:= K1×K2is a cone of X2=X×X. For r, R ∈R2 +,r= (r1, r2), R= (R1, R2), with 0 < ri< Ri(i= 1,2), we denote (Ki)ri,Ri:= {u∈Ki:ri≤ ∥u∥ ≤ Ri}(i= 1,2), Kr,R := {u= (u1, u2)∈K:ri≤ ∥ui∥ ≤ Rifor i= 1,2}. Clearly, Kr,R = (K1)r1,R1×(K2)r2,R2. The aim of the vector version of Krasnosel’ski˘ı theorem is to obtain a solution u= (u1, u2) to the operator system {u1=T1(u1, u2), u2=T2(u1, u2), located in the set Kr,R, that is, u= (u1, u2)∈Kand ri≤ ∥ui∥ ≤ Ri,i= 1,2. Theorem 2.3 (Krasnosel’ski˘ı-Precup).Let (X, ∥·∥)be a normed linear space, K1and K2two cones in X and r, R ∈R2 +,r= (r1, r2),R= (R1, R2), with 0< ri< Ri(i= 1,2). Assume that T= (T1, T2) : Kr,R →Kis a compact map and for each i∈ {1,2}there exists hi∈Ki\{0} such that one of the following conditions is satisfied in Kr,R: (a)Ti(u) + µ hi=uiif ∥ui∥=riand µ > 0, and Ti(u)=λ uiif ∥ui∥=Riand λ > 1; (b)Ti(u)=λ uiif ∥ui∥=riand λ > 1, and Ti(u) + µ hi=uiif ∥ui∥=Riand µ > 0. Then Thas at least a fixed point u= (u1, u2)∈Kwith ri≤ ∥ui∥ ≤ Ri(i= 1,2). 3 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 Remark 2.1. As already pointed out in [25,26], the operator Tmay exhibit a different behavior (compression or expansion) in each component. More exactly, the following options are possible: (i) both operators T1and T2are compressive; (ii) both operators T1and T2are expansive; (iii) one of the operators T1or T2is compressive, while the other one is expansive. Let us recall briefly the main ideas of the proof of Theorem 2.3 given in [25,26], which is essentially divided into two cases: (1) both operators T1and T2are compressive; (2) one of the operators is expansive. The first case relies on Schauder fixed point theorem. In the second case, the fixed point problem is reduced to an equivalent one in which both operators satisfy the compression type condition and so the existence of a fixed point is ensured by the former case. Note that the same proof due to Precup remains valid for the following n-dimensional vector version of Krasnosel’ski˘ı fixed point theorem for an operator T= (T1, T2, . . . , Tn) defined in Xn. Theorem 2.4. Let (X, ∥·∥)be a normed linear space, K1, . . . , Kncones in X,K:= K1×···×Kn,r, R ∈Rn +, r= (r1, . . . , rn),R= (R1, . . . , Rn), with 0< ri< Ri(i= 1, . . . , n), and Kr,R := {u= (u1, . . . , un)∈K: ri≤ ∥ui∥ ≤ Rifor i= 1, . . . , n}. Assume that T= (T1, . . . , Tn) : Kr,R →Kis a compact map and for each i∈ {1, . . . , n}there exists hi∈Ki\{0}such that one of the following conditions is satisfied in Kr,R: (a)Ti(u) + µ hi=uiif ∥ui∥=riand µ > 0, and Ti(u)=λ uiif ∥ui∥=Riand λ > 1; (b)Ti(u)=λ uiif ∥ui∥=riand λ > 1, and Ti(u) + µ hi=uiif ∥ui∥=Riand µ > 0. Then Thas at least a fixed point u= (u1, . . . , un)∈Kwith ri≤ ∥ui∥ ≤ Ri(i= 1, . . . , n). We highlight that our proof here is completely different to that due to Precup, since it is based on fixed point index theory independently of the possibility (i)–(iii) in Remark 2.1. In particular, our approach does not require to turn the expansive operators into compressive ones. 2.1. Fixed point index computation First, let us recall some of the useful properties of the fixed point index for compact maps. For more details, we refer the reader to [1,5,12] (see also [14]). Proposition 2.1. Let Pbe a cone of a normed linear space, U⊂Pbe a bounded relatively open set and T:U→Pbe a compact map such that Thas no fixed points on the boundary of U(denoted by ∂ U). Then the fixed point index of Tin Pover U,iP(T, U), has the following properties: 1. (Additivity) Let Ube the disjoint union of two open sets U1and U2. If 0∈ (I−T)(U\(U1∪U2)), then iP(T, U) = iP(T, U1) + iP(T, U2). 2. (Existence) If iP(T, U)= 0, then there exists u∈Usuch that u=Tu. 3. (Homotopy invariance) If H:U×[0,1] →Pis a compact homotopy and 0∈ (I−H)(∂ U ×[0,1]), then iP(H(·,0), U) = iP(H(·,1), U). 4. (Normalization) If Tis a constant map with T(u) = u0for every u∈U, then iP(T, U) = {1,if u0∈U, 0,if u0∈ U. Moreover, we have the following conditions concerning the computation of the fixed point index. 4 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 Proposition 2.2. Let Ube a bounded relatively open subset of a cone Psuch that 0∈Uand T:U→P be a compact map. (a) If T(u)=λ u for all u∈∂ U and all λ≥1, then iP(T, U)=1. (b) If there exists h∈P\{0}such that T(u) + λ h =ufor every λ≥0and all u∈∂ U, then iP(T, U)=0. In the sequel, let (X, ∥·∥X) and (Y, ∥·∥Y) be normed linear spaces, K1⊂X,K2⊂Ytwo cones and K:= K1×K2the corresponding cone of X×Y. When no confusion may occur, both norms ∥·∥Xand ∥·∥Y will be simply denoted by ∥·∥. Now, we present a technical result which will be crucial in the computation of the fixed point index in the main results of this section. It was already proven in [29, Lemma 2.3], but we include here the proof again for the reader’s convenience. Lemma 2.1. Let Uand Vbe bounded relatively open subsets of K1and K2, respectively, such that 0∈U. Assume that T:U×V→K,T= (T1, T2), is a compact map and there exists h∈K2\{0}such that T1(u, v)=λ u for u∈∂K1U, v ∈Vand λ≥1; (2.1) T2(u, v) + µ h =vfor u∈U, v ∈∂K2Vand µ≥0.(2.2) Then iK(T, U ×V)=0. Proof. Consider the homotopy H:U×V×[0,1] →Kgiven by H((u, v), t)=(t T1(u, v), T2(u, v) + (1 −t)µ0h), with µ0>0 big enough such that v=T2(0, v)+µ0hfor all v∈V. Note that the existence of such a positive number µ0is guaranteed since Vis bounded and Tis compact. Assumptions (2.1) and (2.2) guarantee that the homotopy function Hhas no fixed points on ∂K(U×V). Therefore, by the homotopy invariance property of the fixed point index, we have that iK(H(·,0), U ×V) = iK(H(·,1), U ×V) = iK(T, U ×V).(2.3) On the other hand, for t= 0, the map H((u, v),0) = (0, T2(u, v)+µ0h) has no fixed points in U×V. Indeed, if (u, v)∈U×Vis such a fixed point, then u= 0 and v=T2(0, v)+µ0h, a contradiction with the hypothesis about µ0. Hence, iK(H(·,0), U ×V) = 0 and so the conclusion follows from (2.3).□ Remark 2.2. Obviously, the roles that play T1and T2in the statement of Lemma 2.1, given by assumptions (2.1) and (2.2), are interchangeable. For r, R ∈R2 +, 0 < ri< Ri(i= 1,2), fixed, our aim is to compute the fixed point index of a compact operator T= (T1, T2) : Kr,R →Kin the relatively open set Kr,R := {u= (u1, u2)∈K:ri<∥ui∥< Rifor i= 1,2} under the conditions of Krasnosel’ski˘ı-Precup fixed point theorem. Obviously, we need to assume also that Thas no fixed points on the boundary of Kr,R in order to have the fixed point index well-defined over this set. In the sequel, we will also use the following notations: (Ki)ri={u∈Ki:∥u∥< ri}, 5 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 (Ki)ri={u∈Ki:∥u∥ ≤ ri}(i= 1,2), Kr={u= (u1, u2)∈K:∥ui∥< rifor i= 1,2}, Kr={u= (u1, u2)∈K:∥ui∥ ≤ rifor i= 1,2}. To compute the fixed point index over Kr,R, we need to extend the definition of Tto the set KR. A key ingredient in our purpose is the definition of a retraction from KRinto Kr,R. To do so, we use that (Ki)ri,Ri is a retract of (Ki)Ri(i= 1,2), see [7, Example 3]. Indeed, we have the retraction ρi: (Ki)Ri→(Ki)ri,Ri defined as ρi(ui) = ⎧ ⎪ ⎨ ⎪ ⎩ ri ui+ (ri−∥ui∥)2hi   ui+ (ri−∥ui∥)2hi   ,if ∥ui∥< ri, ui,if ri≤ ∥ui∥ ≤ Ri, (2.4) where hi∈Ki\ {0}is fixed. Note that ρiis well-defined:   ui+ (ri−∥ui∥)2hi  = 0 for all ui∈(Ki)ri. Otherwise, −ui= (ri−∥ui∥)2hi∈Ki, what together with ui∈Kiimplies ui= 0, from the definition of cone. Taking ui= 0, we have  r2 ihi >0 since ri>0 and hi∈Ki\ {0}. Moreover, it is clear that ρiis continuous and ρi(ui) = uifor all ui∈(Ki)ri,Ri. Now we are in a position to compute the fixed point index over the set Kr,R. First, we study the case in which both operators are compressive. Theorem 2.5. Assume that T= (T1, T2) : Kr,R →Kis a compact map and for each i∈ {1,2}there exists hi∈Ki\{0}such that the following conditions are satisfied in Kr,R: (i) Ti(u) + µ hi=uiif ∥ui∥=riand µ≥0; (ii) Ti(u)=λ uiif ∥ui∥=Riand λ≥1. Then iK(T, Kr,R)=1. Proof. Consider the retraction ρ:KR→Kr,R defined as ρ(u1, u2)=(ρ1(u1), ρ2(u2)), where the functions ρiare given by (2.4),i= 1,2. Now, define the auxiliary map N= (N1, N2) : KR→Kas follows N(u) := (T◦ρ)(u).(2.5) Clearly, Nis a compact operator, N(u) = T(u) for every u∈Kr,R and for each i∈ {1,2}the following conditions hold in KR: (i∗)Ni(u) + µ hi=uiif ∥ui∥=riand µ≥0; (ii∗)Ni(u)=λ uiif ∥ui∥=Riand λ≥1. Note that (i∗) implies that N(u) + µ h =ufor all u∈∂Krand µ≥0 (where h= (h1, h2)), so Proposition 2.2 yields iK(N, Kr)=0. Similarly, condition (ii∗) in conjunction with Proposition 2.2 guarantee that iK(N, KR)=1. Moreover, by Lemma 2.1, iK(N, (K1)R1×(K2)r2) = iK(N, (K1)r1×(K2)R2)=0. 6 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 By the additivity property of the fixed point index, iK(N, (K1)r1,R1×(K2)r2) = iK(N, (K1)R1×(K2)r2)−iK(N, Kr)=0, so we obtain iK(N, Kr,R) = iK(N, KR)−iK(N, (K1)r1,R1×(K2)r2)−iK(N, (K1)r1×(K2)R2) = 1. Finally, since T=Non the set Kr,R, we have iK(T, Kr,R) = iK(N, Kr,R) = 1. □ Next, let us consider the case in which we have compression for one operator and expansion for the other one. Theorem 2.6. Assume that T= (T1, T2) : Kr,R →Kis a compact map and for each i∈ {1,2}there exists hi∈Ki\{0}such that the following conditions are satisfied in Kr,R: (i) T1(u) + µ h1=u1if ∥u1∥=r1and µ≥0, and T1(u)=λ u1if ∥u1∥=R1and λ≥1; (ii) T2(u) + µ h2=u2if ∥u2∥=R2and µ≥0, and T2(u)=λ u2if ∥u2∥=r2and λ≥1. Then iK(T, Kr,R) = −1. Proof. Consider the map N:KR→Kdefined as in (2.5).ByProposition 2.2, iK(N, (K1)R1×(K2)r2)=1, iK(N, (K1)r1×(K2)R2)=0. Moreover, Lemma 2.1 yields iK(N, Kr) = iK(N, KR)=0. Hence, it follows from the additivity property of the index that iK(N, (K1)r1,R1×(K2)r2) = iK(N, (K1)R1×(K2)r2)−iK(N, Kr)=1, and so iK(N, Kr,R) = iK(N, KR)−iK(N, (K1)r1×(K2)R2)−iK(N, (K1)r1,R1×(K2)r2) = −1. Then, iK(T, Kr,R) = −1. □ Remark 2.3. The statement of Theorem 2.6 corresponds to the case in which T1is compressive and T2is expansive. Clearly, the same conclusion can be obtained if T1is expansive and T2is compressive. Finally, we deal with the case in which both T1and T2are expansive. Theorem 2.7. Assume that T= (T1, T2) : Kr,R →Kis a compact map and for each i∈ {1,2}there exists hi∈Ki\{0}such that the following conditions are satisfied in Kr,R: (i) Ti(u) + µ hi=uiif ∥ui∥=Riand µ≥0; (ii) Ti(u)=λ uiif ∥ui∥=riand λ≥1. Then iK(T, Kr,R)=1. 7 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 Proof. Consider again the map N:KR→Kdefined as in (2.5). Now, by Proposition 2.2, iK(N, KR)=0, iK(N, Kr)=1. Moreover, according with Lemma 2.1, iK(N, (K1)r1×(K2)R2) = iK(N, (K1)R1×(K2)r2)=0. As a consequence of the additivity property of the index, we deduce that iK(N, (K1)r1×(K2)r2,R2) = iK(N, (K1)r1×(K2)R2)−iK(N, Kr) = −1. Therefore, iK(N, Kr,R) = iK(N, KR)−iK(N, (K1)r1×(K2)r2,R2)−iK(N, (K1)R1×(K2)r2)=1. In conclusion, iK(T, Kr,R) = 1, as wished. □ Remark 2.4. The computation of the fixed point index given by Theorems 2.5–2.7 is independent of that obtained in [29, Theorem 2.8]. Indeed, in [29], the assumptions on Tand, in particular, the homotopy conditions (i)–(ii) were imposed in the whole set KRinstead of its subset Kr,R, as here. As a straightforward consequence of the computation of the fixed point index provided by Theorems 2.5– 2.7, we have an alternative version of Theorem 2.3. Theorem 2.8. Assume that T= (T1, T2) : Kr,R →Kis a compact map and for each i∈ {1,2}there exists hi∈Ki\{0}such that one of the following conditions is satisfied in Kr,R: (a)Ti(u) + µ hi=uiif ∥ui∥=riand µ≥0, and Ti(u)=λ uiif ∥ui∥=Riand λ≥1; (b)Ti(u)=λ uiif ∥ui∥=riand λ≥1, and Ti(u) + µ hi=uiif ∥ui∥=Riand µ≥0. Then Thas at least a fixed point u= (u1, u2)∈Kwith ri<∥ui∥< Ri(i= 1,2). Proof. By Theorems 2.5–2.7, we have that iK(T, Kr,R) = ±1, and so the existence property of the fixed point index ensures that Thas at least a fixed point in Kr,R.□ 2.2. Other versions of Krasnosel’ski˘ı -Precup fixed point theorem: different domains After the previous computation of the fixed point index of Tover the set Kr,R, we can think of proving similar results for operators Tdefined in other regions different from Kr,R. In this way, we increase the range of applicability of the original Krasnosel’ski˘ı-Precup fixed point theorem. For each i∈ {1,2}, let φi:Ki→R+be a continuous concave functional on Ki, that is, φiis a continuous function and φi(λ u + (1 −λ)v)≥λ φi(u) + (1 −λ)φi(v),for all u, v ∈Ki, λ ∈[0,1]. Then, for r, R ∈R2 +, 0 < ri< Ri(i= 1,2), fixed, consider the sets Kφ r,R := {u= (u1, u2)∈K:ri< φi(ui) and ∥ui∥< Rifor i= 1,2}, Kφ r,R := {u= (u1, u2)∈K:ri≤φi(ui) and ∥ui∥ ≤ Rifor i= 1,2}. 8 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 This type of sets has been already considered by Leggett and Williams [21] in the context of their celebrated fixed point theorem. Note that Kφ r,R is a closed convex set. Hence, by Dugundji extension theorem (see [6, Theorem 4.1] or [12]), the subset Kφ r,R is a retract of K. With this in mind, one can easily establish alternative versions of Theorems 2.5–2.7. We sum up them in the following result. Theorem 2.9. Assume that there exist continuous concave functionals φi:Ki→R+such that φi(u)≤ ∥u∥ for all u∈Ki(i= 1,2), the set Kφ r,R is nonempty, T= (T1, T2) : Kφ r,R →Kis a compact map and for each i∈ {1,2}there exists hi∈Ki\{0}such that one of the following conditions is satisfied in Kφ r,R: (i) Ti(u) + µ hi=uiif φi(ui) = riand µ≥0, and Ti(u)=λ uiif ∥ui∥=Riand λ≥1; (ii) Ti(u)=λ uiif φi(ui) = riand λ≥1, and Ti(u) + µ hi=uiif ∥ui∥=Riand µ≥0. Then iK(T, K φ r,R)=(−1)k, where k= 0 if both T1and T2are compressive, k= 1 if one of the operators T1or T2is compressive and the other one is expansive and k= 2 if both operators are expansive. Proof. Consider a retraction ρ:K→Kφ r,R and define the map N:K→Kas follows: N(u) := (T◦ρ)(u). We introduce the following useful notation: (Ki)φi ri={u∈Ki:φi(u)< ri}and (Ki)φi ri={u∈Ki:φi(u)≤ri}. Clearly, ∂(Ki)φi ri⊂ {u∈Ki:φi(u) = ri}(i= 1,2) and, moreover, Kφ r,R =((K1)R1\(K1)φ1 r1)×((K2)R2\(K2)φ2 r2), so now the proof follows as those of Theorems 2.5–2.7, replacing ∥·∥ with φi(·) where needed. □ Remark 2.5. The previous fixed point index computation remains true for an operator Tdefined in a much more general domain of type (U1\V1)×(U2\V2), where for each i∈ {1,2}, one has 0 ∈Vi⊂Vi⊂Ui,Ui and Viare bounded and relatively open sets in Kiand Ui\Viis a retract of Ui. Observe that, in particular, Ui\Viis a retract of Uiprovided that ∂ Viis a retract of Vi, what allows Uito be an arbitrary bounded open set large enough. It can be immediately deduced that this is the case for Vi= (Ki)ri. Indeed, for a fixed hi∈Ki\{0}, one can define the retraction ρi:Vi→∂ Vias ρi(ui) = ri ui+ (ri−∥ui∥)2hi   ui+ (ri−∥ui∥)2hi   . Note that in this case the set Uineeds not be the intersection of a ball with the cone Ki, which enlarges the applicability of Theorems 2.5–2.7. In this context, conditions (i) and (ii) above can be written in the following way: (i) Ti(u) + µ hi=uiif ui∈∂ Viand µ≥0, and Ti(u)=λ uiif ui∈∂ Uiand λ≥1; (ii) Ti(u)=λ uiif ui∈∂ Viand λ≥1, and Ti(u) + µ hi=uiif ui∈∂ Uiand µ≥0; and they must be satisfied over the set (U1\V1)×(U2\V2). Obviously, from Theorem 2.9 it follows immediately a new fixed point theorem in the line of Theorem 2.3. 9 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 64 <∥v1∥∞<522,2<∥v2∥∞<512, 1 4≤ ∥w1∥∞≤64,2<∥w2∥∞<512. We emphasize that the second component of all the three solutions is situated in the same region, so the multiplicity is obtained due to we are able to localize their first component in distinct sets. 3.2. Radial solutions of (p1, p2)-Laplacian systems In this section, we consider the existence of positive radial solutions for the (p1, p2)-Laplacian system −∆p1u=f1(u, v) in B, −∆p2v=f2(u, v) in B, u=0=von ∂B, (3.13) where ∆pu= div (∥∇u∥p−2∇u),Bis the unit open ball in Rncentered at origin, p1, p2> n ≥2 and f1, f2:R2 +→R+are continuous and nondecreasing functions (that is, if (u1, v1),(u2, v2)∈R2 +with u1≤u2 and v1≤v2, then fi(u1, v1)≤fi(u2, v2) for i= 1,2). Setting, as usual, r=∥x∥,u(x) = u1(r) and v(x) = u2(r), the Dirichlet system (3.13) is reduced to the following system of ordinary differential equations with mixed boundary conditions −[rn−1ϕp1(u′ 1)]′=rn−1f1(u1, u2) in (0,1), −[rn−1ϕp2(u′ 2)]′=rn−1f2(u1, u2) in (0,1), u′ 1(0) = u1(1) = 0 = u′ 2(0) = u2(1), (3.14) where ϕp(t) := |t|p−2tis the p-Laplacian homeomorphism. We will look for positive solutions of (3.14), that is, radially symmetric solutions of (3.13). A Harnack type inequality has been established in [28] for problem −[rn−1ϕp(v′)]′=rn−1h(r, v) in (0,1), v′(0) = v(1) = 0, in terms of the energetic norm. By using H¨older inequality, one can derive a Harnack type inequality in terms of the usual max-norm, see [13]. The result can be summarized as follows. Lemma 3.1. Let p > n. Every function v∈C1[0,1] with rn−1ϕp(v′)∈C1[0,1] and [rn−1ϕp(v′)]′≤0on [0,1] satisfies that v′≤0on [0,1]. If, in addition, −r1−n[rn−1ϕp(v′)]′is nonincreasing on (0,1], then v(r)≥p−n p−1(1 −r)rn p−1∥v∥∞, r ∈[0,1] . Let us consider the following cones in the space of continuous functions C(I), with I:= [0,1], Ki={v∈ C(I) : v≥0 on I, v is nonincreasing on Iand min r∈[a,b]v(r)≥ci∥v∥∞}(i= 1,2), where [a, b]⊂(0,1) and ci:= pi−n pi−1(1 −b)an pi−1,i= 1,2. As before, we define the cone K:= K1×K2in the product space. We will look for positive solutions of problem (3.14) as fixed points of the operator T= (T1, T2) : K→K defined as Ti(u1, u2)(r) = ∫1 r ϕ−1 pi(1 sn−1∫s 0 τn−1fi(u1(τ), u2(τ)) dτ)ds, (i= 1,2).(3.15) The operator Tis well-defined, that is, it maps the cone Kinto itself. Indeed, take u= (u1, u2)∈K and let us show that vi:= Ti(u)∈Ki, for i= 1,2. Clearly, vi∈ C(I) and vi≥0 on I, since fiis 16 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 continuous and nonnegative. Moreover, [rn−1ϕpi(v′ i)]′≤0 on I, so viis nonincreasing on I, as a consequence of Lemma 3.1. On the other hand, u1and u2are nonincreasing and fiis nondecreasing, which implies that the map r↦→ fi(u1(r), u2(r)) is nonincreasing on Iand then so is −r1−n[rn−1ϕpi(v′ i)]′=fi(u1(r), u2(r)). Hence, again by Lemma 3.1, one has that minr∈[a,b]vi(r)≥ci∥vi∥∞. In conclusion, vi∈Ki, as desired. It is a routine to check that Tis completely continuous. Now, let us define a continuous concave functional on Ki,φi:Ki→R+, as follows φi(v) = min r∈[a,b]v(r), i = 1,2. We intend to apply Theorem 2.10 in order to obtain sufficient conditions for the existence of fixed points of Tlocated in a set of the form Kφ r,R. Note that this set can also be written as Kφ r,R = (U1\V1)×(U2\V2), where Vi={u∈Ki: min r∈[a,b]u(r)< ri}, Ui={u∈Ki:∥u∥∞< Ri}(i= 1,2). The bounded open sets Viwere introduced by Lan in [19] and later employed by several authors, see [14] and the references therein. Theorem 3.3. Assume that there exist αi, βi>0with βi/ci< αi,i= 1,2, such that fi(β1, β2)>βpi−1 i (b−a)an−1(1 −b)pi−1, fi(α1, α2)< αpi−1 i(i= 1,2).(3.16) Then the system (3.14) has at least one positive solution (u1, u2)∈Ksuch that βi< φi(ui)and ∥ui∥∞< αi, i = 1,2. Proof. Consider the operator T= (T1, T2) : Kφ r,R →Kdefined as in (3.15), with ri=βiand Ri=αi, i= 1,2. Let us check that it fulfills the assumptions of Theorem 2.10. First, fix i∈ {1,2}and take u= (u1, u2)∈Kφ r,R with φi(ui) = ri. Then (u1(r), u2(r)) ≥(r1, r2) for all r∈[a, b] and thus, by the monotonicity assumption on fi, we have that fi(u1(r), u2(r)) ≥fi(r1, r2) for all r∈[a, b]. Hence, for r∈[a, b], Ti(u)(r)≥∫1 b ϕ−1 pi(1 sn−1∫s 0 τn−1fi(u1(τ), u2(τ)) dτ)ds ≥∫1 b ϕ−1 pi(1 sn−1∫b a τn−1fi(u1(τ), u2(τ)) dτ)ds ≥∫1 b ϕ−1 pi(1 sn−1∫b a τn−1fi(r1, r2)dτ)ds ≥(1 −b)ϕ−1 pi((b−a)an−1fi(r1, r2))> ri, where the last inequality follows from (3.16). This clearly implies that Ti(u) + µ1 1 1=uiif u∈Kφ r,R with φi(ui) = riand µ≥0. Suppose now that u∈Kφ r,R with ∥ui∥=Rifor some i∈ {1,2}. Then fi(u1(r), u2(r)) ≤fi(R1, R2) for every r∈Iand so, by (3.16), we have ∥Ti(u)∥∞≤∫1 0 ϕ−1 pi(1 sn−1∫s 0 τn−1fi(u1(τ), u2(τ)) dτ)ds ≤ϕ−1 pi(fi(R1, R2)) < R1. Therefore, alternative (a) in Theorem 2.10 holds and hence we reach the thesis. □ 17 J. Rodríguez–López Nonlinear Analysis 226 (2023) 113138 Remark 3.2. Assumption (3.16) is guaranteed by the following asymptotic conditions: for every i∈ {1,2}, lim ui→0 fi(u1, u2) upi−1 i = +∞and lim ui→∞ fi(u1, u2) upi−1 i = 0 uniformly with respect to uj,j=i. In this case, it is said that both functions f1and f2are superlinear at 0 and sublinear at infinity with respect to ϕp1and ϕp2, respectively. Remark 3.3. Under the assumptions of Theorem 3.3, both operators T1and T2are compressive. Notice that the behaviors compressive–expansive and expansive–expansive are also possible: 1. (Compressive–expansive) Assume that there exist αi, βi>0, i= 1,2, with β1/c1< α1and α2< β2, such that f1(β1, c2α2)>βp1−1 1 (b−a)an−1(1 −b)p1−1, f1(α1, β2/c2)< αp1−1 1, f2(β1, β2)>βp2−1 2 (b−a)an−1(1 −b)p2−1, f2(α1, α2)< αp2−1 2. Then the operator T= (T1, T2) defined as in (3.15) has at least one fixed point in (U1\V1)×(U2\V2), where V1={u∈K1:φ1(u)< β1}, U1={u∈K1:∥u∥∞< α1}, V2={u∈K2:∥u∥∞< α2}, U2={u∈K2:φ2(u)< β2}. In this case, the operator T1is compressive and T2is expansive on (U1\V1)×(U2\V2). 2. (Expansive–expansive) Assume that there exist αi, βi>0, with αi< βi,i= 1,2, such that f1(β1, c2α2)>βp1−1 1 (b−a)an−1(1 −b)p1−1, f1(α1, β2/c2)< αp1−1 1, f2(c1α1, β2)>βp2−1 2 (b−a)an−1(1 −b)p2−1, f2(β1/c1, α2)< αp2−1 2. Then the operator T= (T1, T2) has at least one fixed point in (U1\V1)×(U2\V2), where Vi={u∈Ki:∥u∥∞< αi}, Ui={u∈Ki:φi(u)< βi}(i= 1,2). Note that both operators T1and T2are of expansive type on (U1\V1)×(U2\V2). 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