Unique Results for a New Fourth-Order Boundary Value Problem
Received:January 17, 2018  Revised:July 17, 2018
Key Word: fourth-order boundary value problem   unique solution   $\varphi$-$(h,e)$-concave operator   existence and uniqueness
Fund ProjectL:Supported by the Shanxi Scholarship Council of China (Grant No.2016-009) and the Natural Science Foundation of Shanxi Province (Grant No.201701D121004).
 Author Name Affiliation Shunyong LI School of Mathematical Sciences, Shanxi University, Shanxi 030006, P. R. China
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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.