Abstract
In this paper, we consider the existence of positive solutions for the fully fourth-order boundary value problemu4t=ft,ut,u′t,u″t,u‴t, 0≤t≤1,u0=u1=u″0=u″1=0, wheref:0,1×0,+∞×−∞,+∞×−∞,0×−∞,+∞⟶0,+∞is continuous. This equation can simulate the deformation of an elastic beam simply supported at both ends in a balanced state. By using the fixed-point index theory and the cone theory, we discuss the existence of positive solutions of the fully fourth-order boundary value problem. We transform the fourth-order differential equation into a second-order differential equation by order reduction method. And then, we examine the spectral radius of linear operators and the equivalent norm on continuous space. After that, we obtain the existence of positive solutions of such BVP.
Highlights
We study the existence of positive solutions for the fully fourth-order boundary value problem:
Where f: [0, 1] × [0, +∞] × (−∞, +∞) × (−∞, 0) × (−∞, +∞) ⟶ [0, +∞] is continuous. is boundary value problem can simulate the deformation of an elastic beam, whose one end is fixed and the other end is free in a balanced state
Each derivative of u(t) has its physical meaning: u′(t) is the slope, u′′(t) is the bending moment stiffness, u′′′(t) is the shear force stiffness, and u(4)(t) is the load density stiffness. e nonlinear fourthorder differential equation boundary value problem can simulate the deformation of an elastic beam under external force, and different boundary value conditions can show its force under different conditions
Summary
We study the existence of positive solutions for the fully fourth-order boundary value problem: Because of its importance in mechanics, many scholars have done a lot of research on the existence of solutions for fourth-order ordinary differential equations using various nonlinear methods [1,2,3,4,5,6,7,8,9,10,11]. If f(t, u) is superlinear or sublinear growth on u, the authors in [1] used the fixed-point theorem on the cone to obtain the existence of the positive solution of equation (2).
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