EXISTENCE OF OPTIMAL POSITIVE SOLUTIONS FOR NONLINEAR EULER–BERNOULLI BEAM EQUATION WHOSE ENDS ARE SLIDING CLAMPED

Abstract

We consider the nonlinear boundary value problem

$$\{\begin{array}{c}{y}^{(4)}(x)+(k_1+k_2)y^{\prime \prime }(x)+k_1{k}_{2}y(x)=\lambda f(x,y(x)),x\in [0,1],\hfill \\ y^{\prime }(0)=y^{\prime }(1)=y^{\prime \prime \prime }(0)=y^{\prime \prime \prime }(1)=0,\hfill \end{array}$$

where $k_1$ and $k_2$ are constants, $\lambda$ is a parameter. Based on this, by using the fixed-point index theory in cones and the upper and lower solution method, the criteria of the existence, multiplicity and nonexistence of positive solutions are established in terms of different values of $\lambda$.