Existence of concave symmetric positive solutions for a three-point boundary value problems

Abstract

In this paper, we investigate the existence of triple concave symmetric positive solutions for the nonlinear boundary value problems with integral boundary conditions. The proof is based upon the Avery and Peterson fixed point theorem. An example which supports our theoretical result is also indicated.

The lower and upper solution method for three-point boundary value problems with integral boundary conditions on a half-line

“…From the following three parts we shall conclude that T : X → X is completely continuous. (1) T : X → X is well defined: For each u ∈ X, in view of (10), (12) and 16, we have…”

“…Suppose that f : [0, +∞) × R 2 → R is continuous function satisfying the Nagumo's condition with respect to the pair of functions α 1 , β 2 . If (12) and (13) hold, then (1) has at least three solutions u 1 , u 2 , u 3 ∈ X ∩ C 2 (0, +∞) satisfying…”

“…Multi-point boundary value problems for second-order differential equations in a finite interval and on an infinite interval included the large amount of priori work and many excellent results are obtained by using Avery-Peterson fixed point theorem, shooting method, lower and upper solution method, Leray-Schauder continuation theorem and so on, see for instance [1][2][3][4][5][6][7][8][9][10][11][12][13]15]. Meanwhile, BVPs with integral boundary conditions for ordinary differential equations have been extensively examined by many authors, for example see [11][12][13][14][15][16]. But, there is a little work related to boundary value problems with integral boundary conditions on an infinite interval.…”

“…Moreover, we have α (x) + q(x) f (x, α(x), α (x)) = 2(1 + x) e x arctan(x + 1) > 0, x ∈ (0, +∞), ) + q(x) f (x, β(x), β (x)) = 2(1 + x) e x arctan(−x − 1) < 0, x ∈ (0, +∞),Thus α, β are lower and upper solutions of (30), respectively. Furthermore, α, β ∈ X, α(x) ≤ β(x), x ∈ [0, +∞).Clearly, f is continuous, moreover, f satisfies the Naguma's condition with respect to α(x) = −x − 1 and β(x) = x + 1; that is, when 0 ≤x < +∞, −x − 1 ≤ u ≤ x + 1 and v ∈ R, it holds | f (x, u, v)| ≤ φ(x)h(|v|),whereφ(x) = (1 + x) and h(v) + x) γ q(x)φ(x) = sup x∈[0,+∞) 65 < ∞,that is,(12) and(13) are satisfied. Therefore, from Theorem 3.1, the boundary problem (30) has at least one solution u such thatα(x) = −x − 1 ≤ u(x) ≤ x + 1 = β(x), |u (x)| < N for all x ∈ [0, +∞),where N > e 108π e 2 (r 2 + 1) − 1, r ≥ 1 with γ = 2.…”

Variational principle for a three-point boundary value problem

“…For example, a differential equation describing a truss bridge requires multiple boundary conditions. An unsuitable boundary condition might make the problem ill-posed, though the solution does exist, and this is the reason that existence of solution was widely studied for three-point boundary problems [1,10]. It becomes an important issue in multiple point problems to incorporate a suitable boundary condition into the governing equations.…”