On the lower and upper solution method for higher order functional boundary value problems

Abstract

The authors consider the nth-order differential equation ?(?(u(n?1)(x)))?= f(x, u(x), ..., u(n?1)(x)), for 2?(0, 1), where ?: R? R is an increasing homeomorphism such that ?(0) = 0, n?2, I:= [0,1], and f : I ?Rn ? R is a L1-Carath?odory function, together with the boundary conditions gi(u, u?, ..., u(n?2), u(i)(1)) = 0, i = 0, ..., n? 3, gn?2 (u, u?, ..., u(n?2), u(n?2)(0), u(n?1)(0)) = 0, gn?1 (u, u?, ..., u(n?2), u(n?2)(1), u(n?1)(1)) = 0, where gi : (C(I))n?1?R ? R, i = 0, ..., n?3, and gn?2, gn?1 : (C(I))n?1?R2 ? R are continuous functions satisfying certain monotonicity assumptions. The main result establishes sufficient conditions for the existence of solutions and some location sets for the solution and its derivatives up to order (n?1). Moreover, it is shown how the monotone properties of the nonlinearity and the boundary functions depend on n and upon the relation between lower and upper solutions and their derivatives.

Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

“…• the function a(t, x), defined by (27), verifies (H 2 ) ; • the constant functions α(t) ≡ −1 and β(t) ≡ 1 are lower and upper solutions of Problem (26), respectively. • f (t, x, y) verifies (8) for ρ ∈ [1.09, 5.91] and satisfies a Nagumo-type condition for −1 ≤ x ≤ 1 with:…”

“…Motivated by these works, we prove, in this paper, the existence of heteroclinic solutions for (1) assuming a Nagumo-type condition on the real line and without asymptotic assumptions on the nonlinearities φ and f . The method follows arguments suggested in [3][4][5], applying the technique of [3] to a more general function a, with an adequate functional problem and to classical and singular φ-Laplacian equations. The most common application for φ is the so-called p-Laplacian, i.e., φ(y) = |y| p−2 p, p > 1, and even in this particular case, verifying (4), the new assumption on φ.Moreover, this type of equation includes, for example, the mean curvature operator.…”

First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model

“…Indeed, the functional part can deal with global boundary assumptions, such as minimum or maximum arguments, infinite multi-point data, and integral conditions on the several unknown functions. Functional problems, along with their features, can be seen in [14][15][16][17][18][19] and the references therein.…”

Existence results for functional first‐order coupled systems and applications

“…Indeed, the functional part can deal with global boundary assumptions, such as with minimum or maximum arguments, infinite multipoint data, integral conditions, … , on the several unknown functions. More details on functional problems can be seen in the previous studies [7][8][9][10][11][12] and the references therein.…”