Monotone positive solutions for a fourth order equation with nonlinear boundary conditions

Constant sign solutions of two-point fourth order problems

“…The obtained results are in the framework of the classical monotone iterative techniques [1,2,4,6,7,14]. …”

“…In this paper we describe the set of the real parameters M for which the fourth order boundary value problem L M u(t) ≡ u (4) (t) + M u(t) = σ (t), a.e. t ∈ [0, 1]; u(1) = u(0) = u (0) = u (0) = 0, (1) has nonpositive solutions, for any nonnegative L 1 -function σ .…”

“…If operator L M is disconjugate in I then the Green's function g M , related to Problem(1), satisfies that g M (t, s) < 0, on (0, 1) × (0, 1).…”

Monotone Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions

“…Up to now, a little work has been done for fourth-order BVPs with nonlinear boundary conditions. It is worth mentioning that, in 2009, Alves et al 16 studied some fourth-order BVPs with nonlinear boundary conditions, which models an elastic beam whose left end is fixed and right end is attached to a bearing device or sliding clamped. Their main tool was monotone iterative method.…”

Existence of nontrivial solutions for a nonlinear fourth-order boundary value problem via iterative method

“…Among these papers, the existence and multiplicity of solutions or positive solutions for fourth-order boundary value problems were discussed widely by using various methods. Some of the main tools are the lower and upper solution method (see [5,6,8,12]), monotone iterative technique (see [1,2,8,11]), Krasnoselskii fixed point theorem (see [7,17]), fixed point index (see [4,18,23]), Leray-Schauder degree (see [10,12]), bifurcation theory (see [14,19,21]), the critical point theory (see [24]), the shooting method (see [3]) and fixed point theorems on cones (see [13,16,15,22,26,25]). In this paper, we use monotone iterative technique and lower and upper solutions to get the existence of nonzero solutions for the problem (1.1).…”