Abstract:
Based on the theory of fractional calculus, the contraction mapping principle, Krasnoselskii fixed point theorem and the inequality technique, a class of Caputo fractional-order BAM neural networks with delays in the leakage terms is investigated in this paper. Some new sufficient conditions are established to guarantee the existence and uniqueness of the nontrivial solution. Moreover, uniform stability of such networks is proposed in fixed time intervals. Finally, an illustrative example is also given to demonstrate the effectiveness of the obtained results.

Abstract:
With the help of the fixed point index theorem in cones, we get an existence theorem concerning the existence of positive solution for a second-order three-point eigenvalue problem ,？？ , where is a parameter. An illustrative example is given to demonstrate the effectiveness of the obtained result. 1. Introduction Motivated by the work of Bitsadze and Samarskii [1] and Ilyin and Moiseev [2], much attention has been paid to the study of certain nonlocal boundary value problems (BVPs) in recent years. In the last twenty years, many mathematician, have considered the existence of positive solutions of nonlinear three-point boundary value problems; see, for example, Graef et al. [3] Webb [4], Gupta and Trofimchuk [5], Infante [6], Ehrke [7], Ma [8], Feng [9], He and Ge [10], Bai and Fang [11], and Guo [12]. Recently, by applying the Avery-Henderson [13] double fixed point theorem, Henderson [14] studied the existence of two positive solutions of the three-point boundary value problem for the second-order differential equation where and is continuous. In this paper, motivated and inspired by the above work and Wong [15], we apply a fixed point index theorem in cones to investigate the existence of positive solutions for nonlinear three-point eigenvalue problems where and . We need the following well-known lemma. See [16] for a proof and further discussion of the fixed point index . Lemma 1.1. Assume that is a Banach space, and is a cone in . Let . Furthermore, assume that is a completely continuous map, and for . Then, one has the following conclusions: if for , then ; if for , then . 2. Main Results In the following, we will denote by the space of all continuous functions . This is a Banach space when it is furnished with usual sup-norm . By [14], the Green's function for the three-point boundary-value problem is given by From the Green's function , we have that a function is a solution of the boundary value problem (1.2) if and only if it satisfies Lemma 2.1. Suppose that is defined as above. Then we have the following results: . Proof. It is easy to see that (1) holds. To show that (2) holds, we distinguish four cases.(i)If , then (ii)If and , then (iii)If and , then (iv)Finally, if , then Remark 2.2. If and , then and , respectively. Define Obviously, is a cone in the Banach space . Define an operator as follows: It is easy to know that fixed points of are solutions of the BVP (1.2). Now, we can state and prove our main results. Lemma 2.3. is completely continuous. Proof. For any , by Lemma 2.1 (1), we have , for each . It follows from Lemma 2.1 that

Abstract:
This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows: (/)((1/2)0？1(0())？(1/2)？1(()))

Abstract:
We study the existence of positive solutions for a boundary value problem of fractional-order functional differential equations. Several new existence results are obtained.

Abstract:
This article concerns the existence of solutions to the nonlinear fractional boundary-value problem $$displaylines{ frac{d}{dt} Big({}_0 D_t^{alpha-1}({}_0^c D_t^{alpha} u(t)) -{}_t D_T^{alpha-1}({}_t^c D_T^{alpha} u(t))Big) +lambda f(u(t)) = 0, quadhbox{a.e. } t in [0, T], cr u(0) = u(T) = 0, }$$ where $alpha in (1/2, 1]$, and $lambda$ is a positive real parameter. The approach is based on a local minimum theorem established by Bonanno.

Abstract:
We are concerned with the nonlinear fourth-order three-point boundary value problem , , , , . By using Krasnoselskii's fixed point theorem in a cone, we get some existence results of positive solutions.

Abstract:
We consider a second-order ordinary differential equation with antiperiodic boundary conditions and impulses. By using Schaefer's fixed-point theorem, some existence results are obtained.

Abstract:
We consider a second-order ordinary differential equation with antiperiodic boundary conditions and impulses. By using Schaefer's fixed-point theorem, some existence results are obtained.

Abstract:
In this paper, a Hopfield neural network with neutral time-varying delays is investigated by using the continuation theorem of Mawhin's coincidence degree theory and some analysis technique. Without assuming the continuous differentiability of time-varying delays, sufficient conditions for the existence of the periodic solutions are given. The result of this paper is new and extends previous known result.