Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance

Qu and Liu Boundary Value Problems 2013, 2013:127 http://www.boundaryvalueproblems.com/content/2013/1/127 RESEARCH Open Access Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance Haidong Qu1* and Xuan Liu2 * Correspondence: 1 Department of Mathematics, Hanshan Normal University, Chaozhou, Guangdong 521041, China Full list of author information is available at the end of the article Abstract We consider the fractional differential equation q D0+ u(t) = f (t, u(t)), 0 < t < 1, satisfying the boundary conditions p p–1 p–n+1 D0+ u(t)|t=0 = D0+ u(t)|t=0 = · · · = D0+ u(t)|t=0 = 0, u(1) = m–2  αi u(ξi ), i=1 q where D0+ is the Riemann-Liouville fractional order derivative. The parameters in the multi-point boundary conditions are such that the corresponding differential operator is a Fredholm map of index zero. As a result, the minimal and maximal nonnegative solutions for the problem are obtained by using a fixed point theorem of increasing operators. MSC: 26A33; 34A08 Keywords: fractional order; coincidence degree; at resonance 1 Introduction Let us consider the fractional differential equation   q D+ u(t) = f t, u(t) ,  < t < , (.) with the boundary conditions (BCs) ⎧ ⎨Dp u(t)| = Dp– u(t)| = · · · = Dp–n+ u(t)| = , t= t= t= + + + ⎩u() = m– αi u(ξi ), i= (.)  q– where n ≥ , max{q – , } ≤ p < q – , n < q ≤ n + , m– = , αi > ,  < ξ < i= αi ξi ξ < · · · < ξm– < , m ≥ . We assume that f : [, ] × [, ∞) → [, ∞) is continuous. A boundary value problem at resonance for ordinary or fractional differential equations has been studied by several authors, including the most recent works [–] and the references therein. In the most papers mentioned above, the coincidence degree theory was © 2013 Qu and Liu; 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. Qu and Liu Boundary Value Problems 2013, 2013:127 http://www.boundaryvalueproblems.com/content/2013/1/127 Page 2 of 10 applied to establish existence theorems. But in [], Wang obtained the minimal and maximal nonnegative solutions for a second-order m-point boundary value problem at resonance by using a new fixed point theorem of increasing operators, and in this paper we use this method of Wang to establish the existence theorem of equations (.) and (.). For the convenience of the reader, we briefly recall some notations. Let X, Z be real Banach spaces, L : dom(L) ⊂ X → Z be a Fredholm map of index zero and P : X → X, Q : Z → Z be continuous projectors such that Im(P) = Ker(L), Ker(Q) = Im(L) and X = Ker(L) ⊕ Ker(P), Z = Im(L) ⊕ Im(Q). It follows that L|Ker(P)∩dom(L) : Ker(P) ∩ dom(L) → Im(L) is invertible. We denote the inverse of the map by KP : Im(L) → Ker(P) ∩ dom(L). Since dim Im(Q) = dim Ker(L), there exists an isomorphism J : Im(Q) → Ker(L). Let Ω be an open bounded subset of X. The map N : X → Z will be called L-compact on Ω if QN(Ω) and KP (I – Q)(Ω) are compact. We take H = L + J – P, then H : dom(L) ⊂ X → Z is a linear bijection with bounded inverse and (JQ + KP (I – Q))(L + J – P) = (L + J – P)(JQ + KP (I – Q)) = I. We know from [] that K = H(K ∩ dom(L)) is a cone in Z. Theorem . [] N(u) + J – P(u) = H(ũ), where ũ = P(u) + JQN(u) + KP (I – Q)N(u) and ũ is uniquely determined. From the above theorem, the author [] obtained that the assertions (i) P(u) + JQN(u) + KP (I – Q)N(u) : K ∩ dom(L) → K ∩ dom(L) and (ii) N(u) + J – P(u) : K ∩ dom(L) → K are equivalent. We also need the following definition and theorem. Definition . [] Let K be a normal cone in a Banach space X, u ≤ v , and u , v ∈ K ∩ dom(L) are said to be coupled lower and upper solutions of the equation Lx = Nx if ⎧ ⎨Lu ≤ Nu ,   ⎩Lv ≥ Nv . Theorem . [] Let L : dom(L) ⊂ X → Z be a Fredholm operator of index zero, K be a normal cone in a Banach space X, u , v ∈ K ∩ dom(L), u ≤ v , and N : [u , v ] → Z be L-compact and continuous. Suppose that the following conditions are satisfied: (C ) u and v are coupled lower and upper solutions of the equation Lx = Nx; (C ) N + J – P : K ∩ dom(L) → K is an increasing operator. Then the equation Lx = Nx has a minimal solution u∗ and a maximal solution v∗ in [u , v ]. Moreover, u∗ = lim un , n→∞ v∗ = lim vn , n→∞ where  –   N + J – P un– , un = L + J – P  –   N + J – P vn– , vn = L + J – P n = , , , . . . and u ≤ u ≤ u ≤ · · · ≤ un ≤ · · · ≤ vn ≤ · · · ≤ v ≤ v ≤ v . Qu and Liu Boundary Value Problems 2013, 2013:127 http://www.boundaryvalueproblems.com/content/2013/1/127 Page 3 of 10 2 Preliminaries In this section, we present some necessary basic knowledge and definitions about fractional calculus theory. q Definition . (see Equation .. in []) The R-L fractional integral I+ u of order q ∈ R (q > ) is defined by q I+ u(t) := t  Γ (q) u(τ ) dτ (t – τ )–q  (t > ). Here Γ (q) is the gamma function. q Definition . (see Equation .. in []) The R-L fractional derivative D+ u of order q ∈ R (q > ) is defined by d dt q D+ u(t) = = n n–q I+ u(t)  d Γ (n – q) dt n t u(τ ) dτ (t – τ )q–n+    n = [q] + , t >  , where [q] means the integral part of q. Lemma . [] If q , q > , q > , then, for u(t) ∈ Lp (, ), the relations q q +q q I+ I+ u(t) = I+  u(t) and q q D+ I+ u(t) = u(t) hold a.e. on [, ]. q Lemma . (see []) Let q > , n = [q] + , D+ u(t) ∈ L (, ), then we have the equality q q I+ D+ u(t) = u(t) + n  Ci t q–i , i= where Ci ∈ R (i = , , . . . , n) are some constants. q Lemma . (see Corollary . in []) Let q >  and n = [q] + , the equation D+ u(t) =  is  valid if and only if u(t) = ni= Ci t q–i , where Ci ∈ R (i = , , . . . , n) are arbitrary constants. Let X = Z = C[, ] with the norm u = supt∈[,] |u(t)|, then X and Z are Banach spaces. Let K = {u ∈ X : u(t) ≥ , t ∈ [, ]}. It follows from Theorem .. in [] that K is a normal cone. q Let dom(L) = {u(t) ∈ X | D+ u(t) ∈ Z, u(t) satisfies BCs (.)}. We define the operators L : dom(L) → Z by q (Lu)(t) = D+ u(t) (.) Qu and Liu Boundary Value Problems 2013, 2013:127 http://www.boundaryvalueproblems.com/content/2013/1/127 Page 4 of 10 and N : K → Z by   (Nu)(t) = f t, u(t) , then BVPs (.) and (.) can be written as Lu = Nu, u ∈ K ∩ dom(L). Lemma . If the operator L is defined in (.), then (i) Ker(L) = {c · t q– | c ∈ R},   q– s (ii) Im(L) = {y ∈ Z |  ( – s)q– m– i= αi ξi ξi s y(τ ) dτ ds = } =: L. Proof (i) It can be seen from Lemma . and BCs (.) that Ker(L) = {c · t q– | c ∈ R}. q (ii) If y ∈ Im(L), then there exists a function u ∈ dom(L) such that y(t) = D+ u(t), by Lemma ., we have q I+ y(t) = u(t) + c t q– + · · · (...truncated)