On some discrete boundary value problem with parameters
fourth-order discrete boundary value problem
The authors study (with using the critical point theory) the existence and multiplicity of solutions of the discrete boundary value problem NEWLINE\[NEWLINE\begin{gathered}\Delta^4 u(k-2)+\eta \Delta^2 u(k-1)-\xi u(k)=\lambda f(k,u(k)),\quad k\in[a+1,b+1],\\ u(a)=\Delta^2 u(a-1)=0,\qquad u(b+2)=\Delta^2 u(b+1)=0,\end{gathered}NEWLINE\]NEWLINE where \(f:[a+1,b+1]\times\mathbb{R}\to\mathbb{R}\) is continuous and \(\eta,\xi,\lambda\) are real parameters such that NEWLINE\[NEWLINE\begin{gathered} \eta<8\sin^2 \frac{\pi}{2(b-a+2)}, \quad \eta^2+4\xi\geq0,\\ \xi+4\eta\sin^2\frac{\pi}{2(b-a+2)}<16\sin^4\frac{\pi}{2(b-a+2)},\quad \lambda>0.\end{gathered}NEWLINE\]NEWLINE The main results are obtained for the boundary value problem with superlinear (see Section 3) and sublinear (see Section 4) growth conditions imposed on the nonlinear terms. Finally, some improvements of results from \textit{Y. Yang} and \textit{J. Zhang} [Appl. Math. Comput. 211, No. 2, 293--302 (2009; Zbl 1169.39009)] are discussed in Section 5.
scientific article; zbMATH DE number 6081866
scientific article