Abstract
This research concerns the existence and location of solutions for coupled system of differential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented.
Highlights
Between 1938 and 1941, the English mathematician George James Lidstone (1870–1952) published notes on interpolation, where it was shown that poly-Copyright c 2021 The Author(s)
Nomial interpolation is the solution of boundary value problem given by u(2m)(t) = 0, t ∈ [a, b], u(j)(a) = Aj, u(j)(b) = Bj, j = 0, 2, ..., 2m − 2
The BVP (1.1) can be generalized and coupled into the following problem u(2m)(t) =f (t, u(t), u (t), ..., u(2m−1)(t)) t ∈ [0, 1], u(j)(0) =Aj, u(j)(1) = Bj, j = 0, 2, ..., 2m − 2, which appears in the literature as Lidstone boundary value problems
Summary
Between 1938 and 1941, the English mathematician George James Lidstone (1870–1952) published notes on interpolation, where it was shown that poly-. In [8], de Sousa and Minhos used Lidstone boundary conditions in a coupled system composed by two and fourth order differential equations, to model the bending of the main beam in suspension bridges. Lidstone boundary value problems can be found on [18], where Minhos et al prove an existence and location result for the fourth order fully nonlinear equation u(iv) = f (t, u, u , u , u ), 0 < t < 1 with the Lidstone boundary conditions u(0) = u(1) = u (0) = u (1) = 0, where f : [0, 1] × R4 → R is a continuous function satisfying a Nagumo type condition. The authors ellaborate on how different definitions of lower and upper solutions can generalize existence and location results for boundary value problems with Lidstone boundary data.
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