Approximating Monotone Positive Solutions of a Nonlinear Fourth-Order Boundary Value Problem via Sum Operator Method

In this article, the authors investigate the existence and uniqueness as well as approximations of monotone positive solutions for a nonlinear fourth-order boundary value problem of the form \( u^{(4)}(t)=f(t,u(t)),\ 0< t<1; u(0)=u'(0)= u'(1)=0, u^{(3)}(1)+g(u(1))=0,\) where \(f\in C([0,1]\times [0,+\infty ),[0,+\infty )),\ g\in C([0,+\infty ),[0,+\infty )).\) It is shown that the above boundary value problem has a unique monotone positive solution and the sequence of successive approximations converges to the monotone positive solution under some proper conditions. These results are based upon two fixed point theorems of a sum operator in partial ordering Banach space. Finally, two examples are also given to illustrate the main abstract results.