A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions

Abstract

In this article, we focus on the existence of positive solutions and establish a corresponding iterative scheme for a nonlinear fourth-order equation with indefinite weight and Neumann boundary conditions y(4)(x)+(k1+k2)y″(x)+k1k2y(x)=λh(x)f(y(x)),x∈[0,1],y′(0)=y′(1)=y‴(0)=y‴(1)=0,\left\{\begin{array}{l}{y}^{\left(4)}\left(x)+\left({k}_{1}+{k}_{2}){y}^{^{\prime\prime} }\left(x)+{k}_{1}{k}_{2}y\left(x)=\lambda h\left(x)f(y\left(x)),\hspace{1em}x\in \left[0,1],\\ y^{\prime} \left(0)=y^{\prime} \left(1)={y}^{\prime\prime\prime }\left(0)={y}^{\prime\prime\prime }\left(1)=0,\\ \end{array}\right. where k1{k}_{1} and k2{k}_{2} are constants, λ>0\lambda \gt 0 is a parameter, h(x)∈L1(0,1)h\left(x)\in {L}^{1}\left(0,1) may change sign, and f∈C([0,1]×R+,R)f\in C\left(\left[0,1]\times {{\mathbb{R}}}^{+},{\mathbb{R}}), R+≔[0,∞){{\mathbb{R}}}^{+}:= \left[0,\infty ). We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we establish some new results of the existence of positive solutions to this problem if the nonlinearity ff is monotone on R+{{\mathbb{R}}}^{+}. The technique for dealing with this article relies on a monotone iteration technique and Schauder’s fixed point theorem. Finally, an example is presented to illustrate the application of our main results.

Keywords

• euler-bernoulli beam equations
• positive solutions
• indefinite weight
• monotone iteration technique
• 34b15
• 34b18
• 34b27
• 34c23
• 34g20