Research Article | Open Access
Nontrivial Solutions for a Class of Fractional Differential Equations with Integral Boundary Conditions and a Parameter in a Banach Space with Lattice
Existence of nontrivial solutions for the following fractional differential equation with integral boundary conditions , , , is investigated by using results for the computation of topological degree under the lattice structure, where , , , is the standard Riemann-Liouville derivative. is allowed to be singular at and .
Fractional differential equations have been of great interest for many researchers recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various fields of sciences and engineering such as control, porous media, electromagnetic, and other fields. For an extensive collection of such results, we refer the readers to the monographs by Samko et al. , Podlubny , and Kilbas et al. . Recently, there are some papers dealing with the existence of solutions (or positive solutions) for nonlinear fractional differential equation by means of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, adomian decomposition method, lower and upper solution method, etc.); see [4–16].
As is well known, the first eigenvalue is a character of great significance for the linear operator. For some integer order differential equations, many authors have investigated the existence of positive and nontrivial solutions concerning the first eigenvalue corresponding to the relevant linear operators when the nonlinearities are sublinear, see [17–22] for reference. On the other hand, papers [23–26] obtained similar results to the sublinear case. The main discussion is based on the concepts of dual space, dual cone, and a constructed cone on them.
Recently, Xu et al.  and Bai  obtained the existence results of positive solutions for some fractional differential equations under the conditions with respect to the first eigenvalue corresponding to the relevant linear operators.
Motivated by the above papers, by using results for the computation of topological degree under the lattice structure, we investigate the existence of nontrivial solutions for the following nonlinear fractional differential equations with integral boundary conditions: where , , , is the standard Riemann-Liouville derivative. In this paper, it is not required that nonlinearity , for all . To the author's knowledge, few papers are available in the literature to study the existence of solutions for fractional differential equations with integral boundary conditions under the lattice structure. The method used in this paper is different from those in previous works.
This paper is organized as follows. In Section 2 corresponding Green's function for BVP (1.1) is derived and its properties are also discussed. The main results and their proof are presented in Section 3.
2. Background Materials and Green's Function
Let be a Banach space with a cone . Then becomes an ordered Banach space under the partial ordering which is induced by . is said to be normal if there exists a positive constant such that implies . is called solid if it contains interior points, that is, . is called total if . If is solid, then is total. For the concepts and the properties about the cone we refer to [31, 32].
We call a lattice under the partial ordering if and exist for arbitrary . For , let , , and are called the positive part and the negative part of , respectively, and obviously . Take , then . One can refer to  for the definition and the properties of the lattice. Let , as denoted in [29, 30]. Then , and .
Let be a bounded linear operator. is said to be positive if . In this case, is an increase operator, namely, for , implies . Let be a positively completely continuous operator, a spectral radius of , the conjugated operator of , the conjugated cone of . Since is a total cone, according to the famous Krein-Rutman theorem (see ), we infer that if , then there exist and such that For a given constant , set then is also a cone in .
Definition 2.1 (see ). Let and a nonlinear operator. is said to be quasiadditive on lattice, if there exists such that
Definition 2.3 (see ). The Riemann-Liouville fractional integral of order of a function is given by provided the right-hand side is pointwise defined on .
Definition 2.4 (see ). The Riemann-Liouville fractional derivative of order of a continuous function is given by where , denotes the integer part of the number , provided that the right-hand side is pointwise defined on .
Lemma 2.5 (see ). Let be a normal solid cone in and completely continuous and quasiadditive on lattice. Suppose that the following conditions are satisfied:(i)there exist a positive bounded linear operator , and , such that (ii)there exist a positive bounded linear operator and , such that (iii) , , where is the spectral radius of . Then there exists such that for , the topological degree .
Lemma 2.6 (see ). Let be a normal cone of , and a completely continuous operator. Suppose that there exist positive bounded linear operator and , such that If , then there exists such that for the topological degree .
Lemma 2.7 (see ). Let be a solid cone in and a completely continuous operator with , where is quasiadditive on lattice, and is a positive bounded linear operator satisfying condition. Suppose that (i)there exist and such that (ii)there exist and such that Then there exists such that for the topological degree .
Lemma 2.8 (see ). Let be a bounded open set which contains . Suppose that is a completely continuous operator which has no fixed point on . If (i)there exists a positive bounded linear operator such that (ii), then the topological degree .
Lemma 2.9 (see ). Let . If one assumes , then the fractional differential equation has , , , as unique solution, where is the smallest integer greater than or equal to.
Lemma 2.10 (see ). Assume that with a fractional derivative of order that belongs to .
Then for some , , where is the smallest integer greater than or equal to .
In the following, we present Green's function of the fractional differential equation boundary value problem.
Lemma 2.11. Given , the problem where , , , is equivalent to where Here, , is called the Green function of BVP (2.14). Obviously, is continuous on .
Proof. We may apply Lemma 2.10 to reduce (2.14) to an equivalent integral equation for some . Consequently, the general solution of (2.14) is By , we get that . On the other hand, boundary condition combining with yields Therefore, the unique solution of the problem (2.14) is For , one has For , one has The proof is complete.
Lemma 2.12. The function defined by (2.16) satisfies(a1), for all ;(a2), for all ;(a3), for all ;(a4), and is not decreasing on ;(a5), for all ,
Proof. For , , For , For , For , , From above, (a1), (a2), (a3), (a5) are complete. Clearly, (a4) is true. The proof is complete.
3. Main Results and Proof
Let , , . Obviously, is a normal solid cone with normal constant 1 in Banach space , and is a lattice under the partial ordering which is deduced by .
Throughout this paper, we always assume that (H1) is continuous; (H2) is continuous and not identical zero on any closed subinterval of with .
Remark 3.1. In the assumption , it is not required that , .
Define operators and as follows:
Lemma 3.3. Suppose that (H2) holds, then the spectral radius and has a positive eigenfunction corresponding to the first eigenvalue .
By standard argument, we have the following.
Lemma 3.4. Suppose that (H1) and (H2) hold, then is completely continuous.
Proof. From Lemma 3.4, we know that is completely continuous. By (3.4), there exist , such that This implies where . Set Obviously, . Let , where is defined as (3.1). It is clear that is a positive bounded linear operator and It follows from (3.3), (3.6) that It follows from Lemma 2.5 that there exists big enough such that which means that has at least one solution.
Theorem 3.6. Suppose that (H1) and (H2) hold. In addition, Then BVP (1.1) has at least one solution.
Proof. Let , and let and be defined by (3.1) and (3.2), respectively. Obviously, by remark 3.1 in [27, 28], is continuous and quasiadditive on lattice, and . By Lemma 3.4, we know that is completely continuous.
It follows from (3.14) and (3.15) that there exist constants and such that Therefore, there exists a constant such that From (3.18), one can see that (2.9) and (2.10) hold for and .
Next, we are in position to show that satisfies condition. Let By Lemma 3.3, we know that , and there exits , such that By Lemma 2.12, we have Therefore, for , we get by Lemma 2.12, (3.20) and (3.21) that