Unique Results for a New Fourth-Order Boundary Value Problem

Abstract:

      In this paper, we investigate the existence and uniqueness of solutions for a new fourth-order differential equation boundary value problem: $$\left\{ \begin{array}{l} u^{(4)}(t)=f(t,u(t))-b,\ 0< t<1,\\ u(0)=u'(0)= u'(1)= u^{(3)}(1)=0, \ \end{array}\right. $$ where $f\in C([0,1]\times(-\infty,+\infty),(-\infty,+\infty)),\ b\geq 0$ is a constant. The novelty of this paper is that the boundary value problem is a new type and the method is a new fixed point theorem of $\varphi$-$(h,e)$-concave operators.