An order-type existence theorem and applications to periodic problems

Chu and Wang Boundary Value Problems 2013, 2013:37 http://www.boundaryvalueproblems.com/content/2013/1/37 RESEARCH Open Access An order-type existence theorem and applications to periodic problems Jifeng Chu1* and Feng Wang1,2 * Correspondence: Department of Mathematics, Hohai University, Nanjing, 210098, China Full list of author information is available at the end of the article 1 Abstract Based on the fixed point index and partial order method, one new order-type existence theorem concerning cone expansion and compression is established. As applications, we present sufficient existence conditions for the first- and second-order periodic problems. MSC: 34B15 Keywords: fixed point index; order-type existence theorem; cone expansion and compression; positive solutions; periodic boundary value problems 1 Introduction and preliminaries Let X, Y be real Banach spaces. Consider a linear mapping L : dom L ⊂ X → Y and a nonlinear operator N : X → Y . Here we assume that L is a Fredholm operator of index zero, that is, Im L is closed and dim Ker L = codim Im L < ∞. Then the solvability of the operator equation Lx = Nx has been studied by many researchers in the literature; see [–] and the references therein. In [], Cremins established a fixed point index for A-proper semilinear operators defined on cones which includes and improves the results in [, , ]. Using the fixed point index and the concept of a quasi-normal cone introduced in [], Cremins established a norm-type existence theorem concerning cone expansion and compression in [], which generalizes some corresponding results contained in []. In this paper, we will use the properties of the fixed point index in [] and partial order to present a new order-type existence theorem concerning cone expansion and compression which extends the corresponding results in []. We recall that a partial order in X induced by a cone K ⊂ X is defined by x≤y ⇐⇒ y – x ∈ K. As applications, we study the first- and second-order periodic boundary problems and obtain new existence results. During the last few decades, periodic boundary value problems have been studied by many researchers in the literature; see, for example, [–] and the references therein. Our new results improve those contained in [, ]. © 2013 Chu and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Chu and Wang Boundary Value Problems 2013, 2013:37 http://www.boundaryvalueproblems.com/content/2013/1/37 Next we recall some notations and results which will be needed in this paper. Let X and Y be Banach spaces, D be a linear subspace of X, {Xn } ⊂ D and {Yn } ⊂ Y be the sequences of oriented finite dimensional subspaces such that Qn y → y in Y for every y and dist(x, Xn ) →  for every x ∈ D, where Qn : Y → Yn and Pn : X → Xn are sequences of continuous linear projections. The projection scheme  = {Xn , Yn , Pn , Qn } is then said to be admissible for maps from D ⊂ X to Y . A map T : D ⊂ X → Y is called approximationproper (abbreviated A-proper) at a point y ∈ Y with respect to an admissible scheme  if Tn ≡ Qn T|D∩Xn is continuous for each n ∈ N and whenever {xnj : xnj ∈ D ∩ Xnj } is bounded with Tnj xnj → y, then there exists a subsequence {xnjk } such that xnjk → x ∈ D and Tx = y. T is simply called A-proper if it is A-proper at all points of Y . L : dom L ⊂ X → Y is a Fredholm operator of index zero if Im L is closed and dim Ker L = codim Im L < ∞. As a  consequence of this property, X and Y may be expressed as direct sums; X = X X , Y = Y ⊕ Y with continuous linear projections P : X → Ker L = X and Q : Y → Y . The restriction of L to dom L ∩ X , denoted L , is a bijection onto Im L = Y with continuous inverse L–  : Y → dom L ∩ X . Since X and Y have the same finite dimension, there exists a continuous bijection J : Y → X . Let H = L + J – P, then H : dom L ⊂ X → Y is a linear bijection with bounded inverse. Let K be a cone in a Banach space X. Then K = H(K ∩ dom L) is a cone in Y . In [], Petryshyn has shown that an admissible scheme L can be constructed such that L is A-proper with respect to L . The following properties of the fixed point index indK and two lemmas can be found in []. Proposition . Let  ⊂ X be open and bounded and ∂K = ∂ ∩ K . Assume that Qn K ⊂ K , P + JQN + L–  (I – Q)N maps K to K , and Lx = Nx on ∂K . (P ) (Existence property) If indK ([L, N], ) = {}, then there exists x ∈ K such that Lx = Nx. (P ) (Normality) If x ∈ K , then indK ([L, –J – P + ŷ ], ) = {}, where ŷ = Hx and ŷ (y) = y for every y ∈ HK . (P ) (Additivity) If Lx = Nx for x ∈ K \( ∪  ), where  and  are disjoint relatively open subsets of K , then       indK [L, N],  ⊆ indK [L, N],  + indK [L, N],  with equality if either of indices on the right is a singleton. (P ) (Homotopy invariance) If L – N(λ, x) is an A-proper homotopy on K for λ ∈ [, ] and (N(λ, x) + J – P)H – : K → K and θ ∈/ (L – N(λ, x))(dom L ∩ ∂K ) for λ ∈ [, ], then indK ([L, N(λ, x)], ) = indK (Tλ , U) is independent of λ ∈ [, ], where Tλ = (N(λ, x) + J – P)H – . Lemma . If L : dom L → Y is Fredholm of index zero,  is an open bounded set and K ∩ dom L = ∅, θ ∈  ⊂ X. Let L – λN be A-proper for λ ∈ [, ]. Assume that N is bounded – and P + JQN + L–  (I – Q)N maps K to K . If Lx = μNx – ( – μ)J Px on ∂K for μ ∈ [, ], then   indK [L, N],  = {}. Lemma . If L : dom L → Y is Fredholm of index zero,  is an open bounded set and K ∩ dom L = ∅. Let L – λN be A-proper for λ ∈ [, ]. Assume that N is bounded and Page 2 of 12 Chu and Wang Boundary Value Problems 2013, 2013:37 http://www.boundaryvalueproblems.com/content/2013/1/37 Page 3 of 12 P + JQN + L–  (I – Q)N maps K to K . If there exists e ∈ K \{θ } such that Lx – Nx = μe, for every x ∈ ∂K and all μ ≥ , then   indK [L, N],  = {}. 2 An abstract result We will establish an abstract existence theorem concerning cone expansion and compression of order type, which reads as follows. Theorem . If L : dom L → Y is Fredholm of index zero, let L – λN be A-proper for λ ∈ [, ]. Assume that N is bounded and P + JQN + L–  (I – Q)N maps K to K . Suppose further that  and  are two bounded open sets in X such that θ ∈  ⊂  ⊂  ,  ∩ K ∩ dom L = ∅ and  ∩ K ∩ dom L = ∅. If one of the following two conditions is satisfied: – (C ) (P + JQN)x + L–  (I – Q)Nx  x for all x ∈ ∂ ∩ K and (P + JQN)x + L (I – Q)Nx  x for all x ∈ ∂ ∩ K ; – (C ) (P + JQN)x + L–  (I – Q)Nx  x for all x ∈ ∂ ∩ K and (P + JQN)x + L (I – Q)Nx  x for all x ∈ ∂ ∩ K . Then there exists x ∈ ( \ ) ∩ K such that Lx = Nx. Proof We assume that (C ) is satisfied. First we show that Lx = μNx – ( – μ)J – Px, (...truncated)