Solvability for a Fully Elastic Beam Equation with Left-End Fixed and Right-End Simply Supported

Abstract

The aim of the present paper is to consider a fully elastic beam equation with left-end fixed and right-end simply supported, i.e., $\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right),u^{\prime }\left(t\right),u^{″}\left(t\right),u^{‴}\left(t\right)\right),t\in \left[\mathrm{0,1}\right]\hfill \\ u\left(0\right)=u^{\prime }\left(0\right)=u\left(1\right)=u^{″}\left(1\right)=0\hfill \end{array}\right.$ , where $f:\left[\mathrm{0,1}\right]\times {ℝ}^4⟶ℝ$ is a continuous function. By applying Leray–Schauder fixed point theorem of the completely continuous operator, the existence and uniqueness of solutions are obtained under the conditions that the nonlinear function satisfies the linear growth and superlinear growth. For the case of superlinear growth, a Nagumo-type condition is introduced to limit that $f\left(t,x_0,x_1,x_2,x_3\right)$ is quadratical growth on $x_3$ at most.