$\varphi$-压缩不动点理论在Lidstone边界条件下弹性梁方程中的应用

Application of $\varphi$-Compressed Fixed Point Theory to Elastic Beam Equations with Lidstone Boundary Conditions

摘要: 运用几乎$\varphi$ -压缩不动点理论讨论了带Lidstone边界条件的弹性梁方程$\{\begin{array}{l}{y}^{\left(4\right)}\left(x\right)+\left(k_1+k_2\right)y^{″}\left(x\right)+k_1{k}_{2}y\left(x\right)=f\left(x,y\left(x\right)\right),x\in \left[0,1\right],\\ y\left(0\right)=y\left(1\right)=y^{″}\left(0\right)=y^{″}\left(1\right)=0\end{array}$ 非平凡解的存在唯一性，其中$f:\left[0,1\right]\times \left[0,+\infty \right)\to \left[0,+\infty \right)$ 为连续函数，$k_1$ 和$k_2$ 均为常数。

Abstract: The existence and uniqueness of nontrivial solution to the elastic beam equation$\{\begin{array}{l}{y}^{\left(4\right)}\left(x\right)+\left(k_1+k_2\right)y^{″}\left(x\right)+k_1{k}_{2}y\left(x\right)=f\left(x,y\left(x\right)\right),x\in \left[0,1\right],\\ y\left(0\right)=y\left(1\right)=y^{″}\left(0\right)=y^{″}\left(1\right)=0\end{array}$ with Lidstone boundary conditions were obtained using almost $\varphi$ -compressed fixed point theory, where $f:\left[0,1\right]\times \left[0,+\infty \right)\to \left[0,+\infty \right)$ is a continuous function, $k_1$ and $k_2$ are constants.