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In this paper, we consider a class of Kirchhoff-type equations with singularity and nonlinearity in a bounded domain. Using the approximation method, a unique positive solution is obtained.

Cooperative relationship in social network can be described effectively by one-mode RDP model (‘RDP’ represents the initials of Ramasco, Dorogovtsev and Pastor-Satorras, who propose the model.), where n > 2 participants including m ⩾ 1 new participants and n - m > 1 ‘old’ participants collaborate on one project together at each time step. In this paper, the synchronization in this evolution network model is studied systematically. We discover that the synchronization capacity of one-mode RDP model is independent of the network size N, can be reduced by slightly increasing m when m is small, and also be increased by increasing n - m . By calculating the specific expression of average path length (APL) of one-mode RDP model, we analyze the correlation between APL and synchronizability of this model. In some cases, there is no correlation between the APL and the synchronizability of one-mode RDP model. But when N and m are fixed, as n - m increases, the synchronizability of this model is negatively correlated with APL. Furthermore, we explore the robustness and fragility of synchronization in one-mode RDP model. The synchronizability of this model against random fault is not correlated with m and N, and the synchronizability to resist intentional attack is independent of m and N. The synchronizability of this model against random fault and intentional attack can be improved by increasing n - m . Our study fills in the gaps of previous studies that did not consider the synchronization and average path length of one-mode RDP model. Our research results also contain the previous relevant research results and expand them.

In this paper, using the theory of fixed point index, we prove some results on the multiplicity of positive solutions for a class of nonlocal elliptic problem.

In this paper, using the fixed-point index method, we are concerned with the existence of sign-changing solutions of the following problem (0.1) { − ( a + b ∫ Ω | u ( x ) | γ d x ) Δ u = f ( u ) , x in Ω , u ( x ) = 0 , x on ∂ Ω , where a > 0 , b > 0 and Ω ⊆ R N is a bounded open set with smooth boundary and γ ≥ 1 .

In this paper, we establish the existence of an unbounded solution to a second order boundary value problem on the half line. Also the existence of multiple unbounded positive solutions is discussed using the theory of fixed point index.

The existence of at least two positive solutions is presented for the singular second-order boundary value problems {1p(t)(p(t)x^'(t))^'+@F(t)f(t,x(t),p(t)x^'(t))=0,00p(t)x^'(t)=0,x(1)=0 using the fixed point index, where @!"0^11p(r)dr=+~ and f may be singular at x=0 and px^'=0. The solutions we construct may be unbounded.

This paper presents a lower and upper solution technique for singular second-order boundary value problems on the half-line and discusses sufficient and necessary conditions for the existence of positive solutions. Also multiplicity is discussed using the fixed point index.

This paper presents some results on the existence of multiple unbounded solutions for the boundary value problems for second order differential equations on half line and on the uniqueness of unbounded solution.

By constructing a special cone and using fixed point index theory in cone, this paper investigates the existence of multiple solutions of singular boundary value problems for differential systems.

This paper presents some results on the existence and uniqueness of solutions for the boundary value problems on the half-line with impulses and infinite delay. Moreover, an existence theorem of multiple solutions is obtained also. The problems may be singular at the boundary.

This paper improves Tychonov fixed point theorem and discusses the existence of solutions of nonlinear Fredholm integral equations on [0. +∞) in Banach spaces with Frechet space theory.