We study the existence and multiplicity of positive solutions of the nonlinear equation u''(x) + λh(x)f(u(x)) = 0 subject to nonlinear boundary conditions. The method of upper and lower solutions and degree theory arguments are used.

We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ $-d/dt(1/2_0{D}_t^{-\sigma }\left(u^{\text{'}}\left(t\right)\right)+1/2_t{D}_T^{-\sigma }\left(u^{\text{'}}\left(t\right)\right))-\lambda \beta \left(t\right)f\left(u\left(t\right)\right)-\mu \gamma \left(t\right)g\left(u\left(t\right)\right)=0$, a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where ${}_0{D}_t^{-\sigma }$ and ${}_t{D}_T^{-\sigma }$ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].

We apply the method of quasilinearization to multipoint boundary value problems for ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.

The existence of synchronization is an important issue in complex dynamical networks. In this paper, we study the synchronization of impulsive coupled oscillator networks with the aid of rotating periodic solutions of impulsive system. The type of synchronization is closely related to the rotating matrix, which gives an insight for finding various types of synchronization in a united way. We transform the synchronization of impulsive coupled oscillators into the existence of rotating periodic solutions...

In this paper, two multi-valued versions of the well-known hybrid fixed point theorem of Dhage [6] in Banach algebras are proved. As an application, an existence theorem for a certain differential inclusion in Banach algebras is also proved under the mixed Lipschitz and compactness type conditions.