On positive solutions of some nonlinear fourth-order beam equations

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

“…In the last decade or so, several papers have been devoted to the boundary value problems (P1) and (P2), for example, see [5], [7]- [9], [11], [15], [21]. However, in the literature, the problems (P1) and (P2) have not received as much attention as the fourth-order boundary value problems with boundary condition u(0) = u(1) = u ′′ (0) = u ′′ (1) = 0, which were considered, for example, in [4], [6], [12], [14], [16], [19]. This paper focuses on the positive eigenvalue intervals for which there exist one or two positive solutions.…”

Unique solution for a fourth-order boundary value problem

“…In this paper, we shall study the unique solution for the fourth order boundary value problem Recent years, there have been many papers to study the existence of solutions for some fourth order boundary value problems, see [1][2][3][4][5]. In [1,2], for the problems of type (1), the authors all obtained the existence results of positive solutions when f is either superlinear or sublinear in u by employing the cone expansion or compression fixed point theorem.…”

“…In [1,2], for the problems of type (1), the authors all obtained the existence results of positive solutions when f is either superlinear or sublinear in u by employing the cone expansion or compression fixed point theorem. In [3][4][5], the authors applied critical point theory to consider the existence of solutions for a class of fourth order boundary value problems and obtained some excellent results.…”

Global structure of positive solutions for superlinear 2mth-boundary value problems

“…In the past twenty years, the existence of solutions, especially the existence of positive solutions, of (1.1), (1.2) and its general cases, has been extensively studied by using the Leray-Schauder degree and the fixed point theorem in cones, see Agarwal [1], Agarwal and Wong [2], Aftabizadeh [3], Yang [4], Del Pino and Manásevich [5], Ma and Wang [6], Ma, Zhang and Fu [7], Bai and Wang [8], Bai and Ge [9], Yao [10], Y. Li [11], F. Li et al [12] and references therein. Also, the global structure of positive solution set (and nodal solutions set) are investigated by several authors, see for example, the interesting contributions [13]- [15] by Bari and Rynne. Very recently Ma [16]- [18] studied the global bifurcation phenomena of nodal solutions of (1.1), (1.2) when m = 2 and f 0 ∈ (0, ∞), where f 0 = lim…”