Global Structure of Positive Solutions for Some Second-Order Multipoint Boundary Value Problems

Abstract

We investigate in this paper the following second-order multipoint boundary value problem:( ). Under some conditions, we obtain global structure of positive solution set of this boundary value problem and the behavior of positive solutions with respect to parameter by using global bifurcation method. We also obtain the infinite interval of parameter about the existence of positive solution.

Existence of multiple solutions for nonlinear multi-point boundary value problems

“…Since then, many authors have considered the existence of nontrivial solutions for nonlinear multi-point boundary value problems and obtained many great results. We can refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein. For example, in [4], Xu has studied the following multi-point boundary value problem:…”

“…By means of fixed point theorems with lattice structure, the author has obtained the existence results of negative and sign-changing solution for BVP (1.4) for superlinear case. Inspired by [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], we consider boundary value problem (1.1) in this paper. By the existing fixed point theorems due to Sang et al [19], we obtain the existence results of multiple nontrivial solutions for BVP (1.1) for asymptotically linear case, including two positive solutions, one sign-changing, and two negative solutions.…”

Solvability of integral boundary value problems at resonance in $R^{n}$

“…Let A = (a ij ) n×n be a square matrix of order n. α : [0, 1] → R is a bounded variation function, and 1 0 u(t) dα(t) = ( under the following assumptions: (H1) B is a diagonalization matrix, and det(I -B) = 0; (H2) 1 0 t(1t) dα(t) = 0; (H3) f : [0, 1] × R 2n → R n satisfies the Carathéodory conditions. If the condition (H 1 ) is considered, the associated linear problem -u (t) = 0, u(0) = 0, u(1) = A 1 that, when n = 1, the existence theory of integral boundary value problems for ordinary differential equations or fractional differential equations has been well studied; we refer the reader to [4,10,17,20,21,24,25,[29][30][31][32][33][34][35]37] for some recent results at non-resonance and to [2,3,16,18,22,23,27,36] for results at resonance. When n ≥ 2 and A is not a diagonal matrix, IBVP (1.1) becomes a system of ordinary differential equations with coupled boundary conditions.…”

Existence of Nontrivial Solutions for Some Second-Order Multipoint Boundary Value Problems

“…By using fixed point index and Leray-Schauder degree methods, the author showed existence of multiple sign-changing solutions for the boundary value problem (3). In [14], the authors have considered the following multipoint boundary value problem:…”