AN EXISTENCE RESULT FOR SYSTEMS OF SECOND-ORDER BOUNDARY VALUE PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS

作者：Goodrich Christopher S^*

来源：Dynamic Systems and Applications, 2014, 23(4): 601-618.

In this note we consider the system %26lt;br%26gt;x %26apos;%26apos;(t) = -lambda(1)f(t,x(t), y(t)), t is an element of (0,1) %26lt;br%26gt;y %26apos;%26apos;(t) = -lambda(2g)(t, x(t), y(t)), t is an element of (0,1) %26lt;br%26gt;x(0) = phi(x), y(0) = psi(y) %26lt;br%26gt;x(1) = 0 = y(1), %26lt;br%26gt;and demonstrate that under suitable conditions on the functions f, g: [0,1] x R x R -%26gt; [0, +infinity) and the functionals phi, psi : C([0,1]) -%26gt; R this problem admits at least one positive solution. Since the functionals phi and psi can be nonlinear, the boundary condition at t = 0 can be quite general. Our approach is based on supposing that each of x bar right arrow phi(x) and y bar right arrow psi(y) behaves, in some sense, like a linear functional as parallel to(x, y)parallel to -%26gt; +infinity, and to this end we utilize the concept of the Frechet derivative at +infinity in our existence proof. We conclude by providing an explicit example of and discussion regarding our existence theorem.

• 出版日期2014-12