Abstract
We study the existence of solutions for a periodic fourth order problem. We prove an associated uniform antimaxi- mum principle and develop a method of upper and lower solutions in reversed order. Furthermore, by the quasilinearization method we construct an iterative sequence that converges quadratically to a solution.
Highlights
In this work, we study the fourth order problem (1.1)u(4) + g(t, u, u ) = 0 under periodic conditions (1.2)u(j)(0) = u(j)(T ) for j = 0, 1, 2, 3.In the last years there has been an increasing interest on higher order problems, both because of their intrinsic mathematical interest and their applications to different problems in Mathematical Physics.In order to study the existence of solutions of problem (1.1)-(1.2), we shall apply method of upper and lower solutions and the so-called quasilinearization technique.The method of upper and lower solutions is one of the most extensively used tools in nonlinear analysis, both for ODE’s and PDE’s problems
We study the fourth order problem u(4) + g(t, u, u ) = 0 under periodic conditions u(j)(0) = u(j)(T ) for j = 0, 1, 2, 3
It is worth to mention that this method has been applied mostly to second order equations, since it relies on the maximum principle associated to the problem
Summary
In order to study the existence of solutions of problem (1.1)-(1.2), we shall apply method of upper and lower solutions and the so-called quasilinearization technique. Fourth order periodic problems - Antimaximum principle - Quasilinearization method - Upper and lower solutions. We develop the method of upper and lower solutions for (1.1)-(1.2) in the following “reversed order” cases:. It has been applied to different nonlinear problems in the presence of an ordered couple of a lower and an upper solution.
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