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We consider the existence of countably many positive solutions for nonlinear nth-order three-point boundary value problem u n t a t f u t, 1 for some p ≥ 1 and has countably many singularities in 0, 1/2 . The associated Green's function for the nth-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity f which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.

Summary This article focuses on the consensus problem of leader‐following fractional‐order multi‐agent systems (MASs) with general linear and Lipschitz nonlinear dynamics. First, the distributed adaptive protocols for linear and nonlinear fractional‐order MASs are constructed, respectively. We allow the control coupling gains to be time varying for each agent. Moreover, the adaptive modification schemes for the control gain are designed, which renders smaller control gains and thus requires smaller amplitude on the control input without sacrificing consensus convergence. Second, based on fractional‐order Lyapunov stability theorem and Barbalat's lemma, two novel sufficient conditions in terms of linear matrix inequalities are provided to ensure that the leader‐following consensus can be obtained in the case for any undirected connected communication graph. Furthermore, we show that the proposed algorithm also works for consensus of agents with intrinsic Lipschitz nonlinear dynamics. As a result, the proposed framework requires no global information and thus can be implemented in a fully distributed manner. Finally, the numerical simulations are given to demonstrate the effectiveness of obtained the theoretical results.

This paper investigates the existence and nonexistence of positive solutions for a class of fourth-order nonlinear differential equation with integral boundary conditions. The associated Green's function for the fourth-order boundary value problems is first given, and the arguments are based on Krasnoselskii's fixed point theorem for operators on a cone.

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In this paper, we present a stabilization method on the non‐linear fractional‐order uncertain systems. Firstly, a sufficient condition for the robust asymptotic stabilization of the non‐linear fractional‐order uncertain system is presented based on direct Lyapunov approach. Secondly, utilising the matrix's singular value decomposition (SVD) method, the systematic robust stabilization design algorithm is then proposed. Finally, two numerical examples are provided to illustrate the efficiency and advantage of the proposed algorithm.