Existence of solutions for a cantilever beam problem
This paper is concerned with the existence and multiplicity of positive solutions to the fourth-order dynamic equation on a measure chain \(\mathbb{X}\) \[ (I(x) W^{\triangle \nabla}(x))^{\triangle \nabla} = f(x, W(x)), \quad x \in (a,b), \] satisfying the boundary conditions \[ W(a)= W^{\triangle}(a) = 0, \quad I(b)W^{\triangle \nabla}(b) = \left(I(x)W^{\triangle \nabla}(x)\right)^{\triangle}| _{x = b} = 0, \] where \(f: [a,b] \times[0,\infty) \to [0,\infty)\) is continuous and \(I: [\rho(a),\sigma(b)] \to (0,\infty)\) is left-dense continuous. Existence and multiplicity of positive solutions are based on analysis of the Green function and follow from cone-theoretic fixed point theorems for ordered Banach spaces due to Krasnosel'skiĭ, \textit{R. Avery} and \textit{J. Henderson} [Commun. Appl. Nonlinear Anal. 8, No. 1, 27--36 (2001; Zbl 1014.47025)], and \textit{R. Avery}, [Math. Sci. Res. Hot-Line 3, No. 7, 9--14 (1998; Zbl 0965.47038)].
scientific article; zbMATH DE number 5077880