- Open Access
Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure
© Wu et al; licensee Springer. 2012
- Received: 26 November 2011
- Accepted: 30 May 2012
- Published: 30 May 2012
In this article, we consider the existence of positive solutions of the (n - 1, 1) conjugate-type nonlocal fractional differential equation
where α ≥ 2, is the standard Riemann-Liouville derivative, is a linear functional given by the Stieltjes integral, A is a function of bounded variation, and dA may be a changing-sign measure, namely the value of the linear functional is not assumed to be positive for all positive x. By constructing upper and lower solutions, some sufficient conditions for the existence of positive solutions to the problem are established utilizing Schauder's fixed point theorem in the case in which the nonlinearities f(t, x) are allowed to have the singularities at t = 0 and (or) 1 and also at x = 0.
AMS (MOS) Subject Classification: 34B15; 34B25.
- upper and lower solutions
- fractional differential equation
- Schauder's fixed point theorem
- positive solution.
where α ≥ 2, is the standard Riemann-Liouville derivative, f: (0, 1) × (0, +∞) → [0, +∞) is continuous and f may be singular at x = 0 and t = 0, 1.
In the BVP (1.1), denotes the Riemann-Stieltijes integral, where A is a function of bounded variation, that is dA can be a signed measure. In this work we do not suppose that for all x ≥ 0, and hence the BVP (1.1) has a wider range of applications as positive or negative values of the linear functional are viable in some cases. But in most previous works of nonlocal boundary value problems, some kind of positivity on the functional is often required in order to obtain positive solutions of the problem, and in particular the study of m-point boundary value problems with the boundary condition is often required (see [1–4]). So from this point of view, the BVP (1.1) not only includes the multi-point boundary value problem and the integral boundary value problem (A(s) = s or dA(s) = h(s)ds) as special cases, but also generalizes the multi-point boundary value problem and integral boundary value problem for more general cases.
The nonlocal BVPs have been studied extensively. The methods used therein mainly depend on the fixed-point theorems, the degree theory, the upper and lower solution techniques, and the monotone iterations. Particularly, when α is an integer and , where 0 < η < 1 and 0 < μηn-1< 1, the authors of [5–8] established the existence and multiplicity of positive solutions for the n th-order three-point BVP (1.1) by applying the fixed-point theorems on cones. If , where 0 < η1< η2< · · · < η n -2 < 1, μ i > 0 with , the n th-order m-point BVP (1.1) has been studied in [1–3]. A more general equation with f depending on derivatives and the boundary conditions with two nonlocal terms was studied by Zhang , where f can be singular at t = 0 and/or t = 1 and be allowed to change sign. In addition, the nonlocal integral boundary value problems also represent a very interesting and important class of problems arising in physical, biological and chemical processes and have attracted the attention of Khan , Gallardo , Karakostas and Tsamatos , Ahmad et al. , Feng et al.  and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer the readers to Corduneanu  and Agarwal and O'Regan .
where a may be singular at t = 0, 1, f may be singular at x = 0 but has no singularity at t = 0, 1. The existence of positive solutions of the BVP (1.2) is obtained by means of the fixed point index theory in cones. In , in order to overcome the singularity of f at x = 0, the authors adopted the condition below :
where . This condition was also adopted by Wang et al.  to study the existence and multiplicity of positive solutions for more general fractional order BVP (1.1) by using fixed point theorem when f(t, x) has singularity at x = 0. But we notice that the condition (H) used in [19, 20] is a mixed condition involving integrability condition and supremum and limit condition, and is quite difficult to verify. Thus, in this article, by finding a simple integrability condition, we establish the existence of positive solutions for the BVP (1.1) when the nonlinearity f may be singular at both t = 0, 1 and x = 0 by utilizing different techniques [19, 20], through establishing a maximal principle and constructing upper and lower solutions, instead of using a fixed point theorem on cone, some sufficient conditions for the existence of positive solutions are established via Schauder's fixed point theorem.
For the convenience of the reader, we present here some definitions in fractional calculus which are to be used in the later sections.
provided that the right-hand side is pointwise defined on (0, +∞).
where n = [α] + 1, [α] denotes the integer part of the number α, provided that the right-hand side is pointwisely defined on (0, +∞).
where c i ∈ ℝ (i = 1, 2,..., n), and n is the smallest integer greater than or equal to α.
holds for f ∈ L1(a, b).
Lemma 2.2 (see). The: function G(t, s) has the following properties:
Lemma 2.3 Let 0 ≤ C < 1 andfor s ∈ [0, 1], then the Green function defined by (2.6) satisfies:
(1) H(t, s) > 0, \forall t, s ∈ (0, 1).
Proof. The conclusion is obvious from Lemma 2.3, we omit the proof.
We make the following assumptions throughout the rest of this article:
(B0) A is a function of bounded variation, and for s ∈ [0,1], .
(B1) f ∈ C((0, 1) × (0, ∞), [0, +∞)), and f(t, x) is decreasing in x.
which imply that they are well defined.
where c and d are defined in Lemma 2.3.
Then T is well defined and T (P) ⊂ P.
It follows from (3.3) and (3.4) that T is well defined and T (P) ⊂ P:
Notice that (3.5) implies that satisfy the boundary conditions in the BVP (1.1). Thus by (3.8) and (3.9), are the lower and upper solutions of the BVP (1.1), respectively, and ψ(t), ϕ(t) ∈ P.
Obviously, a fixed point of the operator T0 is a solution of the BVP (3.11).
So T0 is bounded. It is easy to see T0: E → E is continuous from the continuity of H and (B2).
That is to say, T0(Ω) is equicontinuous.
In the following, we divide the proof into two cases.
This implies that T0(Ω) is equicontinuous.
From the Arzela-Ascoli theorem, we get that T0: E → E is completely continuous. Thus, by using the Schauder fixed point theorem, T0 has at least one fixed point w such that w = T0w.
Consequently, F (t, w(t)) = f(t, w(t)), t ∈ [0, 1]. Then w(t) is a positive solution of the BVP (1.1).
Corollary 3.1. Suppose the following conditions hold:
(C1) f ∈ C((0, 1) × [0, ∞), [0, +∞)), and f(t, x) is decreasing in x.
Thus the rest of proof is similar to those of Theorem 3.1.
Thus , and (B0) holds.
which implies that (B2) holds.
i.e., (B3) also holds.
Since x(t) is a unknown function, it is difficult for us to verify whether it is equal to 0. But we see that the integrability condition (B2)-(B3) are easier to be checked than (H) by a simple calculation.
Remark 3.2. In Example 3.1, the boundary condition reveals that positive or negative values of the linear functional, in the condition , are viable in some cases. This implies that we here removed the nonnegative requirements of μ i used in most of the literature, for example [1–4] and other related literature on multi-point boundary-value problems.
where A is a function of bounded variation, and , where .
then f(t, x): [0, 1] × [0, ∞) → [0, +∞) is continuous and decreasing in x, and for any t ∈ [0, 1], and
The authors thank the referees' for helpful comments and suggestions, which led to improvement of the article. The authors were supported financially by the National Natural Science Foundation of China (11071141, 11126231) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017).
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