Abstract
We consider the nonlinear n-th order boundary value problem Lu=u(n)=f(t,u(t),u′(t),…,u(n−1)(t))=Nu given arbitrary bounded linear functional conditions Bi(u)=0, i=1,…,n and develop a method that allows us to study all such resonance problems of order one, as well as implementing a more general constructive method for deriving existence criteria in the framework of the coincidence degree method of Mawhin. We demonstrate applicability of the formalism by giving an example for n=4.
Highlights
We provide a method for these of the form of (1) of resonance one in which we assume very little about the boundary conditions (2), in order to illuminate this process of constructing projectors
We know that the proposed κ0 is a solution to the problem K κ0 = − Ĝ ( g) by a simple calculation and noting that − B4 ( G ( g)) = − 21 (3B1 ( G ( g)) + B2 ( G ( g)) + B3 ( G ( g)))
We considered the nonlinear n-th order boundary value problem at resonance subject to abstract linear functional conditions and developed a method that allows us to study all resonance scenarios of order one
Summary
The above shows a similar trend: they offer a particular type of bounded linear boundary conditions Using said conditions they obtain projectors P and Q and conclude that solutions exist because of coincidence degree theory due to Mawhin. We provide a method for these of the form of (1) of resonance one in which we assume very little about the boundary conditions (2), in order to illuminate this process of constructing projectors. This is a further extension of [7] in which they apply a similar methodology to solve all similar problems to that of (1) but n = 2.
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