Krasnosel'skii type compression-expansion fixed point theorem for set contractions and star convex sets

Krasnosel’skii type compression-expansion fixed point theorem for set contractions and star convex sets Cristina Lois-Prados, Radu Precup and Rosana Rodr´ıguez-L´opez Abstract. In this paper, we give or improve compression-expansion results for set contractions in conical domains determined by balls or star convex sets. In the compression case, we use Potter’s idea of proof, while the expansion case is reduced to the compression one by means of a change of variable. Finally, to illustrate the theory, we give an application to the initial value problem for a system of implicit first order differential equations. Mathematics Subject Classification (2010). Primary 47H10, 34A09. Keywords. compression-expansion fixed point theorem, set contraction, star convex set, implicit differential system. 1. Introduction Krasnosel’skii compression-expansion fixed point theorem is a powerful tool to prove the existence of positive solutions to several classes of boundary value problems and also to obtain multiple solutions (see, for example [4], [5], [8], [11], [15], [16] and [17]). Krasnosel’skii proved his theorem directly, using only arguments of fixed point theory, particularly Schauder’s fixed point theorem (see [6] and [7]). However, it is well-known that this result appears as a consequence of the topological degree theory (see [3]). Nevertheless, for applications, it is more convenient to use Krasnosel’skii theorem, because it offers directly the conditions - of compression or expansion - that have to be verified. Also for the theory, a direct approach without using degree arguments could be useful when trying to extend the results from compact mappings to more general ones. Such a possibility is shown in Chapter 10 of [10], where some compression-expansion results are established for mappings from a smaller family for which the topological degree has not been developed. 2 Cristina Lois-Prados, Radu Precup and Rosana Rodr´ıguez-L´opez The direct approach owed to Krasnosel’skii was also followed by Potter [12], who extended the compression result from compact mappings to set contractions. Notice that both Krasnosel’skii and Potter obtained a solution localized in annular conical sets defined using the norm. An expansion type result for set contractions was proved by C´ac and Gatica [1], but under a more restrictive condition on the lower boundary of the annular set. In this paper, we prove the expansive type result for set contractions without restrictive conditions on the lower boundary, using the idea of reducing the expansive case to the compressing one, as shown in [14] for compact operators. We shall do even more, proving the results on star convex conical sets. The motivation for working with star convex sets is given in the Introduction of [9]. It comes from the necessity to distinguish between two solutions in case that they have the same norm. The difficulty when working with k-set contractions (k∈[0,1)) comes from the fact that the geometric transformations which are used in the proofs can change uncontrollably the constant k. As regards the use of star convex sets, they introduce much more complicated geometric transformations connected to their retro-activity property, which have to be put in accordance with the constant k, as Lemma 3.1 shows. The results are presented as follows. First, in Section 2, we give some preliminary notions and results about α−Lipschitz mappings (set contractions, in particular) and star convex sets, and we recall Potter’s compression result for balls. In Section 3, we first state and prove a technical lemma improving the corresponding one in [9], which allows us to extend the compression fixed point theorem in star convex sets given in [9], to general set contractions. Next, in Subsection 3.2, we improve the expansion result of C´ac and Gatica by simplifying the condition on the lower boundary of the conical annular set. We conclude this section by proving the expansion fixed point theorem for set contractions and star convex sets. Finally, in Section 4, as an application, we consider the initial value problem for a system of implicit first order differential equations, which leads to a fixed point equation with a non-compact operator. Adding a Lipschitz condition over the derivative-dependent term, this operator becomes a set contraction. 2. Preliminaries We start by briefly giving some notations, definitions and preliminary results, for details, see [9], [12]. If Xis a real vector space and Aa subset of X, we denote by co(A) the convex hull of A, that is co(A) := (n X i=1 λixi:n∈N, xi∈A, λi∈[0,1] for i= 1,···, n, n X i=1 λi= 1) Krasnosel’skii fixed point theorem for set contractions 3 and, in case that Xis a topological vector space, by co(A) the closure of co(A). The following concept allows us to relax the compactness hypothesis assumed in the mentioned classical results owed to Krasnosel’skii. Some of its principal properties can be found in [12], [9]. Let (X, d) be a metric space and A⊂Xbe a bounded set. The Kuratowski measure of noncompactness of Ais the non negative real number α(A) := inf{ε > 0 : there exist finitely many sets of diameter at most ε which cover A}. The measure of noncompactness can be considered as a tool to determine how much a particular set differs from being compact. In this way, it is possible to define a concept close to compact mapping, known as set contraction. Let X, Y be metric spaces and D⊂X. Assume that the mapping T:D⊂X−→ Yis continuous. We say that Tis α-Lipschitz (see [2]) if there exists a constant k≥0 such that α(T(A)) ≤kα(A),for all bounded A⊂D. When kis important to be mentioned, we say that Tis a k-α-Lipschitz mapping, or that Tis an α-Lipschitz mapping with constant k. In case that k∈[0,1), we say that Tis a set contraction or a k-set contraction. Note that any continuous and compact mapping is a 0-set contraction. Recall the following properties. Proposition 2.1. Let (Xi, di)be metric spaces for i= 1,2,3, and (X, k·k)be a Banach space. (i)If T1:X1→X2is a k1-α-Lipschitz mapping and T2:X2→X3is a k2-α-Lipschitz mapping, then T2◦T1:X1→X3is a k1k2-α-Lipschitz mapping. (ii)If T:D⊂X→Xis a k-α-Lipschitz mapping and λ:D→R+is a continuous function such that supx∈Dλ(x) = l < ∞,then the mapping b T:D⊂X→X, b T(x) = λ(x)T(x) is kl-α-Lipschitz. Now, we give some definitions and results about star convex sets, which are needed in what follows. For details, see [9]. For a fixed x0∈X, we say that a set E⊂Xis x0-star convex if λx0+ (1 −λ)x∈E, for all λ∈[0,1] and x∈E. In case that x0= 0, E is simply called a star convex set. Clearly, if Eis x0-star convex, then x0∈E. We have the following result about the projection mapping on the boundary of an x0-star convex set. 4 Cristina Lois-Prados, Radu Precup and Rosana Rodr´ıguez-L´opez Theorem 2.2. Let (X, k·k)be a Banach space, x0∈Xand E⊂Xbe a bounded closed x0-star convex set such that its boundary Fdoes not contain x0,and for all x∈E\ {x0},there is a unique λx>0with λxx+ (1 −λx)x0∈F. (2.1) Then there exists a continuous mapping ∂:E\ {x0} → Fsuch that ∂(x) = ∂(λx + (1 −λ)x0),for all x∈E\ {x0}, λ ∈(0,1], ∂(x) = x, for all x∈F. (2.2) Proof. As x0∈E\F, there exists γ > 0 such that B:= {x∈X:kx−x0k ≤ γ} ⊂ E\F. Let Sbe the boundary of B, i.e., S:= {x∈X:kx−x0k=γ}. We define the mapping ∂as the composition η◦η0,where η0is the radial projection η0:E\ {x0} → S, η0(x) = γ kx−x0k(x−x0), and η:S→F, η (x) = λxx+ (1 −λx)x0. From (2.1), the mapping ηis well-defined. Also, it is easy to see that condition (2.2) is satisfied. Clearly, η0is continuous. It remains to prove the continuity of η. To this aim, it suffices to prove the continuity of the function λ:S−→ R+, λ (x) = λx. For that purpose, let {yn}n∈Nbe any sequence in Sconverging to some y∈S. Since Eis bounded and λ(S)⊂[0,+∞), we can assert that there exists m∈R+such that λ(S)⊂[0, m]. Then the sequence {λ(yn)}n∈Nis included in the compact interval [0, m], so any of its limit points is finite. Let lbe any limit point of {λ(yn)}n∈N.From η(yn) = λ(yn)yn+ (1 −λ(yn))x0∈F, we find that ly + (1 −l)x0∈F. This, in view of (2.1), gives l=λ(y).Hence λ(yn)→λ(y) as n→+∞. Therefore, λis continuous as wished.  In the following, we shall consider only star convex sets Ewhich satisfy the following condition. Condition 1.Eis bounded, closed, 0 /∈Fwhere Fis the boundary of Ein X, and for every x∈E\ {0}there exists a unique λx>0 with λxx∈F. Let us now consider a cone Cin Xand its intersection with a star convex set E. Clearly 0 ∈C∩Eand we can make the following remark. Krasnosel’skii fixed point theorem for set contractions 5 Figure 1. An example of mappings ∂and ∂C. Remark 2.3.If Eis a star convex set satisfying Condition 1, then the restriction of the mapping ∂to C∩(E\ {0}) can be continuously extended to C\ {0},as follows ∂C:C\ {0} → F, ∂C(x) = (∂(x), x ∈(C∩E)\ {0}, ∂d(0,F ) kxkx, x ∈C\E. Figure 1 illustrates the behavior of ∂Cin a particular case. Proposition 2.4 ([9]). Let Ebe a star convex set satisfying Condition 1. Then, for every x∈(C∩E)\ {0}, there exists a unique number βx∈[1,+∞)such that βxx∈C∩F. Moreover, the mapping β: (C∩E)\ {0} → [1,+∞), β (x) = βx=d(0, ∂(x)) d(0, x) is continuous and β(x)→+∞as x→0. Remark 2.5.By using ∂C, the map βcan be continuously extended to C\{0} as follows: βC(x) = d(0, ∂C(x))/d(0, x), for x∈C\{0}. For two star convex sets Ei,i= 1,2, satisfying Condition 1, we let: •Fibe the boundary of Ei,˚ Eibe the interior of Ei. •∂C i:C\ {0} −→ Fibe the continuous mapping associated whit Ei according to Remark 2.3. •βi: (C∩Ei)\{0} −→ [1,+∞) be the continuous mapping associated to Eias shown in Proposition 2.4, and βC ibe the corresponding continuous extension to C\ {0}. Assume next that E1⊂E2, F1∩F2=∅,and let T:C∩E2˚ \E1→C be a continuous mapping. We say that Tis a compression of the set C∩E2\˚ E1(see Figure 2 (a)) if: 6 Cristina Lois-Prados, Radu Precup and Rosana Rodr´ıguez-L´opez (C1)x−T(x)/∈C, for all x∈C∩F1, (C2)T(x)−(1 + ε)x /∈C, for all ε > 0 and x∈C∩F2. We say that T:C∩E2\˚ E1→Cis an expansion of the set C∩E2\˚ E1 (see Figure 2 (b)) if (E1)T(x)−(1 + ε)x /∈C, for all ε > 0 and x∈C∩F1, (E2)x−T(x)/∈C, for all x∈C∩F2. (a) (b) Figure 2. Figures (a) and (b) represent, respectively, the boundary conditions for a compression and an expansion of the set C∩E2\˚ E1, where C={(x, y)∈R2:x, y ≥0}, and E1,E2are star convex sets. We conclude this section by recalling Potter’s compression result for balls. Given two real numbers r, R with 0 < r < R, denote Cr,R ={x∈C:r≤ kxk ≤ R}, Cr={x∈C:kxk ≤ r}, Sr={x∈C:kxk=r}. Theorem 2.6 ([12]). If T:Cr,R →Cis a k-set contraction and a compression of the set Cr,R,i.e., x−T(x)/∈C, for all x∈Sr, T(x)−(1 + ε)x /∈C, for all ε > 0and x∈SR, then Thas at least one fixed point in Cr,R. 3. Main results In this section, we first give an extension of the main compression result from [9]. This extension allows kto be any number with 0 ≤k < 1, compared to the more restrictive assumption 0 ≤k < d(0, F)/L, where L:= sup{d(0, x) : x∈F}, considered in [9]. Next, we improve the expansion type result given Krasnosel’skii fixed point theorem for set contractions 7 in [1]. This is done by reducing the expansion case to the compression one. Finally, the expansion result is extended to star convex sets. 3.1. Compression fixed point theorem for set contractions and star convex sets In order to accomplish our first aim, we need to improve the technical Lemma 3.16 in [9]. Lemma 3.1. Assume that (X, k · k)is a Banach space, Ca cone in Xand Ea set satisfying Condition 1. Let T:C∩F−→ Cbe α−Lipschitz with constant kand consider the mapping ˜ T:C∩E−→ C x7−→ ˜ T(x) := (1 β(x)T(β(x)x), x 6= 0, 0, x = 0. Then ˜ Tis α−Lipschitz with constant ˜ kfor every ˜ k > k as close to kas we wish. Proof. We only need to adapt the part of the proof where we show the existence of ˜ kas close to kas we wish such that α˜ T(A)≤˜ kα(A),for every A⊂C∩E, α(A)6= 0.(3.1) Thus, let A⊂C∩Ebe such that α(A)6= 0 and k > 0 (if this is not true, we can consider ˆ k > 0 as close to kas we wish). As k, α(A)>0, there exists d > 0 such that k α(A)/2> d > 0. We fix dsatisfying these conditions. To accomplish the aim of proving that (3.1) is satisfied, we proceed as follows. Step 1: Cover any set Aby a finite number of subsets with the property that the restrictions of ˜ Tto each of them is α−Lipschitz for some suitable constant. To this aim, we start by using the continuity of ˜ Tproved in [9], that ensures the existence of δd>0 such that ˜ T((C∩E)∩B(0, δd)) ⊂B(0, d). Then, we have ˜ T(A∩B(0, δd)) ⊂B(0, d).(3.2) On the other hand, for each natural number n≥1, take εn=δd/n and, for all natural number m≥1, define An m:= {x∈A:β(x)∈[1 + (m−1)εn,1 + mεn]}. For any natural number n≥1, there exists Nn∈N,Nn>1, such that, if β(x)>1 + Nnεn, then x∈A∩B(0, δd). Therefore, A⊂(A∩B(0, δd)) ∪ Nn [ m=1 An m!.(3.3) 8 Cristina Lois-Prados, Radu Precup and Rosana Rodr´ıguez-L´opez Step 2: Fix n∈N,n > 1, arbitrarily and, for each m∈ {1, . . . , Nn}, we show that ˜ T|An mis α−Lipschitz with constant k(1+mεn)/(1+(m−1)εn). For each m∈ {1, . . . , Nn}, we have 0 /∈An m, so ˜ T|An m:An m−→ C x7−→ ˜ T|An m(x) = 1 β(x)T(β(x)x). Let m∈ {1, . . . , Nn}and define the following auxiliary mappings: 1 β|An m :An m−→ C, x 7−→ 1 β|An m (x) := 1 β(x), Sn m:An m−→ C, x 7−→ Sn m(x) := β(x)x. Since βis continuous and its image is a subset of [1,∞), we deduce that 1/β|An mis a continuous function. Besides, for all x∈An m, it is satisfied that β(x)∈[1 + (m−1) εn,1 + m εn]⇔1 β(x)∈1 1 + m εn ,1 1+(m−1) εn, hence sup 1 β(x):x∈An m≤1 1+(m−1) εn . As Tis a k-set contraction, in view of Proposition 2.1, it remains to prove that Sn mis α−Lipschitz with constant (1 + mεn). Let B⊂An m, then B is bounded and, using the definition of An m, we have Sn m(B) = nβ(x)x:x∈Bo ⊂n[λ(1 + (m−1)εn) + (1 −λ)(1 + mεn)]x:λ∈[0,1], x ∈Bo = co[1 + (m−1)εn]B∪[1 + mεn]B. Now, by using the properties of the measure of noncompactness, we obtain α(Sn m(B)) ≤max nα[1 + (m−1)εn]B, α[1 + mεn]Bo= (1 + mεn)α(B). Also, as βis continuous, then Sn mis continuous. Therefore Sn mis α−Lipschitz with constant (1 + mεn). As a consequence, by using the properties of α−Lipschitz mapping in Proposition 2.1, one has that ˜ T|An mis α−Lipschitz with constant k(1 + mεn)/(1 + (m−1)εn). Step 3: Taking into account the two previous steps, we can conclude that ˜ T is α−Lipschitz with constant ˜ kas close to kas we wish. It follows from (3.2), Krasnosel’skii fixed point theorem for set contractions 9 (3.3), and the properties of the measure of noncompactness, that α˜ T(A)=α ˜ T(A∩B(0, δd)) ∪ Nn [ m=1 ˜ T(An m)! =α˜ T(A∩B(0, δd)) ∪˜ T(An 1)∪. . . ∪˜ T(An Nn) = max nα˜ T(A∩B(0, δd)), α ˜ T(An 1), . . . , α ˜ T(An Nn)o ≤max nα(B(0, d)) , α ˜ T(An 1), . . . , α ˜ T(An Nn)o ≤max 2d, 1 + εn 1kα(An 1),..., 1 + Nnεn 1+(Nn−1)εn kα(An Nn) ≤max 2d, 1 + εn 1kα(A),..., 1 + Nnεn 1+(Nn−1)εn kα(A) ≤(1 + εn)kα(A), where the last inequality holds since 2d < kα(A) and 1 + mεn 1+(m−1)εn ≤1 + εn 1for all m∈ {1, . . . , Nn}. Since εn→0 as n→ ∞, one has that, if nis large enough, the number ˜ k:= (1 + εn)kis as close to kas we wish.  Now, we can state the extension of the compression result from [9], for any constant 0 ≤k < 1. Theorem 3.2. Let (X, k·k)be a Banach space, Cbe a cone in X, and E1,E2 be star convex sets fulfilling Condition 1. Assume that T:C∩E2\˚ E1−→ C is a k-set contraction and a compression of the set C∩E2\˚ E1. Then T has at least one fixed point in C∩E2\˚ E1. Proof. The proof follows the one of the corresponding theorem in [9], using this time the improved Lemma 3.1.  3.2. Expansion fixed point theorem for set contractions and balls Our second aim is to improve the expansion result given in [1]. We shall do this by simplifying the condition on the lower boundary of the conical annular set. Theorem 3.3. Let (X, k·k)be a Banach space, Cbe a cone in X,r, R ∈R, 0< r < R, and T:Cr,R −→ Cbe a k−set contraction verifying the following properties: Tx −(1 + ε)x /∈C, for all ε > 0and all x∈C, kxk=r; x−Tx /∈C, for all x∈C, kxk=R. Then, Thas a fixed point in Cr,R. 16 Cristina Lois-Prados, Radu Precup and Rosana Rodr´ıguez-L´opez and assume that the following conditions are satisfied: (H4,C) For each i∈ {1, . . . , n}, fi+gi> ri,(4.5) fi+gi≤Ri ai+bi .(4.6) Under these conditions, E1⊂˚ E2,and (C1), (C2) hold, that is, the operator Tis a compression of the set C∩(E2\˚ E1). First, using the inequalities (ai+bi)ri< Ri,we have E1⊂˚ E2.Indeed, if y∈E1,then, for every i∈ {1, . . . , n}, one has ϕi(yi) = aimin t∈[0,1] |yi(t)|+bikyik∞≤(ai+bi)kyik∞≤(ai+bi)ri< Ri, which gives y∈˚ E2. Next, we show that (4.5) guarantees that condition (C1) is fulfilled. Indeed, if we assume the contrary, then there exists y∈Cwith kyjk∞≤rj, for all j∈ {1, . . . , n}, and kykk∞=rkfor some k∈ {1, . . . , n}, such that y(t)≥T(y)(t),for every t∈[0,1]. Let t0∈[0,1] be such that yk(t0) = kykk∞=rk.From the previous inequality, by using the definition of T, we obtain rk=yk(t0)≥Tk(y)(t0) = fkt0,Zt0 0 y(s)ds+gk(t0, y(t0)) ≥fk+gk, which contradicts (4.5). Hence, (C1) holds. Finally, we prove that (C2) is also satisfied by using (4.6). If we assume the contrary, then there exist ε > 0 and y∈Cwith ϕj(yj)≤Rjfor all j∈ {1, . . . , n}, and ϕk(yk) = Rkfor some k∈ {1, . . . , n}, such that T(y)(t)≥(1 + ε)y(t),for any t∈[0,1]. Let t0∈[0,1] be such that yk(t0) = kykk∞.Then, using the last inequality, the expression of T, and (4.4), we obtain fk+gk≥fkt0,Zt0 0 y(s)ds+gk(t0, y(t0)) > yk(t0)≥Rk ak+bk , which contradicts (4.6). Hence, (C2) holds. Therefore, since all the assumptions of Theorem 3.2 are fulfilled, we have the following existence and localization result. Theorem 4.1. Under conditions (H1)−(H4,C), problem (4.1) has a nonnegative and increasing solution x∈ C1([0,1],Rn)such that rk≤ kx0 kk∞,for at least one k∈ {1, . . . , n},and ϕi(x0 i)≤Ri,for all i∈ {1, . . . , n}.(4.7) In particular, if we assume that n= 1, and the monotonicity of fand g, more exactly: (H5) For each t∈[0,1],the functions f(t, ·) and g(t, ·) are increasing in R+, Krasnosel’skii fixed point theorem for set contractions 17 then conditions (4.5) and (4.6) in (H4,C), with r1, R1, a1, b1simply denoted by r, R, a, b, turn into f(t, 0) + g(t, r)> r, for every t∈[0,1],(4.8) ft, R b+gt, R b≤R a+b,for every t∈[0,1].(4.9) Let us present two examples. The first one, which in fact is a solvable equation, is given to test the compression conditions. Example 1.Let f(s) = λs and g(s) = αs +β(s∈R), where λ, α ≥0, β > 0 and λ+α < b a+b. If r < β 1−αand R1 a+b−λ+α b≥β, (4.10) then the assumptions of Theorem 4.1 are fulfilled. For example, condition (4.10) is satisfied for a=b= 1, r = 1, R = 6, β= 1 and λ=α= 1/6.In this case, the exact solution of the problem is x(t)=6et 5−1, for t∈[0,1], and x0(t) = 6 5et 5, for t∈[0,1], then kx0k∞=6 5e1 5and ϕ(x0) = 6 51 + e1 5, and it is easy to see that conditions (4.7) hold. The second example deals with equations that can not be explicitly solved. Example 2.Consider the equation x0(t) = λx(t) + αx0(t) + β+γsin(x0(t)), t ∈[0,1].(4.11) In this case, f(s) = λs and g(s) = αs +β+γsin s(s∈R), where we assume that α, β, γ and λare non-negative. Now, we explain how to fulfill the conditions of Theorem 4.1. Clearly, condition (H1) holds. Next, if α < 1−γ, then |g0(s)|=|α+γcos s| ≤ α+γ < 1. Therefore, (H2) is satisfied for k=α+γ < 1. To guarantee condition (H3), we need g(R+)⊂R+, which takes place if β≥γ. Furthermore, condition (H5) is fulfilled if gis increasing in R+, and this happens if α≥γ. This condition, together with α < 1−γ, gives γ≤α < 1−γ. Then, obviously, γhas to satisfy 0 ≤γ < 1 2. Finally, we have to check conditions (4.8) and (4.9). For the first, we need r > 0 such that g(r)> r, that is, αr +β+γsin r > r. This clearly happens if αr +β−γ > r, or, equivalently, r < β−γ 1−α, which requires β > γ since rhas to be positive. Condition (4.9) reads as λR b+αR b+β+γsin R b≤R a+b.(4.12) 18 Cristina Lois-Prados, Radu Precup and Rosana Rodr´ıguez-L´opez We show that there exists Rlarge enough that satisfies this inequality. Indeed, if we divide by R b, we obtain λ+α+βb R+γsinR b R b ≤b a+b. The limit of the left hand side, when Rtends to ∞, being λ+αguarantees the existence of Rprovided that λ+α < b a+bor, equivalently, λ < b a+b−α. In view of λ≥0, it requires that α < b a+b. Therefore, the conditions of Theorem 4.1 are fulfilled if the non-negative parameters α,β,γand λsatisfy: γ≤α < 1−γ, λ+α < b a+b, γ < min 1 2,b a+b, β. Under these conditions, for every r < β−γ 1−α, there exists a solution x∈ C1([0,1],R) of equation (4.11) with x(0) = 0 that is non-negative, increasing and with kx0k∞≥r. If, in addition, a number Ris chosen such that inequality (4.12) holds, then the solution xalso satisfies amin t∈[0,1] x0(t) + bmax t∈[0,1] x0(t)≤R. Notice that, if γ= 0, Example 2 reduces to Example 1 and also our sufficient conditions on parameters α, β and λrequired in Example 2 become those in Example 1. To conclude this paper, we claim that a similar approach with expansion type conditions does not work, that is, Theorem 3.4 does not apply for the initial value problem (4.1). Indeed, if we take E1={y∈ C([0,1],R) : ϕ(y)≤r}, E2={y∈ C([0,1],R) : kyk∞≤R}, where r, R are positive numbers with r < bR, and we proceed similarly to the compression case, we arrive to the following sufficient conditions of expansion: (H4,E) max t∈[0,1], y∈[0,r b]f(t, y) + max t∈[0,1], y∈[r a+b,r b]g(t, y)≤r a+b,(4.13) min t∈[0,1], y∈[0,R]f(t, y) + min t∈[0,1] g(t, R)> R. (4.14) Krasnosel’skii fixed point theorem for set contractions 19 These conditions ensure that E1⊂˚ E2and (E1),(E2) are fulfilled but, unfortunately, they are not compatible with hypothesis (H2), and thus Theorem 3.4 can not be applied. We shall prove this incompatibility in the autonomous case, that is, when fand gdo not depend on t, and conditions (4.13), (4.14) are max y∈[0,r b]f(y) + max y∈[r a+b,r b]g(y)≤r a+b, min y∈[0,R]f(y) + g(R)> R. Subtracting the two inequalities yields g(R)−max y∈[r a+b,r b]g(y)> R −r a+b+ max y∈[0,r b]f(y)−min y∈[0,R]f(y).(4.15) From r < bR, we have [0, r/b]⊂[0, R],whence min y∈[0,R]f(y)≤min y∈[0,r b]f(y)≤max y∈[0,r b]f(y). Hence, the right-hand side in (4.15) is greater than or equal to R−r/ (a+b) and, so, g(R)−max y∈[r a+b,r b]g(y)> R −r a+b.(4.16) On the other hand, if ˆy∈[r/ (a+b), r/b] is such that g(ˆy) = max y∈[r/(a+b), r/b]g(y), then, also using (H2), we find g(R)−max y∈[r a+b,r b]g(y) = g(R)−g(ˆy)≤k(R−ˆy) ≤kR−r a+b< R −r a+b. This together with (4.16) clearly yields a contradiction, proving our claim. Nevertheless, expansion conditions are possible for many other problems involving compact operators, as shown by lots of papers in the literature. Acknowledgements The authors thank the anonymous referee for suggesting to give the application more generally to systems than to a single equation. Also for giving us the idea to apply the results to boundary value problems for PDEs, a theme for possible future research. Cristina Lois-Prados and Rosana Rodr´ıguez-L´opez acknowledge the support of the research grant MTM2016-75140-P (AEI/FEDER, UE). The research of Cristina Lois-Prados has been partially supported by grant ED481A- 2018/080 from Xunta de Galicia. The authors also thank Daniel Cao Labora for his idea of the proof of Theorem 2.2. 20 Cristina Lois-Prados, Radu Precup and Rosana Rodr´ıguez-L´opez References [1] N. P. C´ac, J. A. Gatica, Fixed point theorems for mappings in ordered Banach spaces, J. Math. Anal. Appl. 71 (1979) 547–557. [2] K. Deimling, Nonlinear Functional Analysis, Springer, New York, 1985. [3] J. Dugundji, A. Granas, Fixed Point Theory, Springer, New York, 2013. [4] L. H. Erbe, H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994) 743–748. [5] D. Guo, V. Lakshmikantham, X. 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Cristina Lois-Prados Instituto de Matem´aticas, Facultade de Matem´aticas, Campus Vida, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain e-mail: [email protected] Krasnosel’skii fixed point theorem for set contractions 21 Radu Precup Department of Mathematics, Babe¸s–Bolyai University, 400084 Cluj-Napoca, Romania e-mail: [email protected] Rosana Rodr´ıguez-L´opez Instituto de Matem´aticas, Facultade de Matem´aticas, Campus Vida, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain e-mail: [email protected]