Abstract
The existence and uniqueness of positive solutions are obtained for singular fourth-order four-point boundary value problem with p-Laplace operator $[\varphi_{p}(u''(t))]''=f(t,u(t))$ , $0< t<1$ , $u(0)=0$ , $u(1)=au(\xi)$ , $u''(0)=0$ , $u''(1)=bu''(\eta)$ , where $f(t,u)$ is singular at $t=0,1$ and $u=0$ . A fixed point theorem for mappings that are decreasing with respect to a cone in a Banach space plays a key role in the proof.
Highlights
In this paper, we investigate the existence and uniqueness of positive solutions for the singular fourth-order differential equation involving the p-Laplace operator φp u (t) = f t, u(t), t ∈ (, ), ( . )with the four-point boundary conditions u( ) =, u( ) = au(ξ ), u ( ) =, u ( ) = bu (η), It is well known that the bending of elastic beam can be described by some fourth-order boundary value problems
The existence and uniqueness of positive solutions are obtained for singular fourth-order four-point boundary value problem with p-Laplace operator [φp(u (t))] = f (t, u(t)), 0 < t < 1, u(0) = 0, u(1) = au(ξ ), u (0) = 0, u (1) = bu (η), where f (t, u) is singular at t = 0, 1 and u = 0
1 Introduction In this paper, we investigate the existence and uniqueness of positive solutions for the singular fourth-order differential equation involving the p-Laplace operator φp u (t) = f t, u(t), t ∈ (, ), ( . )
Summary
The existence and uniqueness of positive solutions are obtained for singular fourth-order four-point boundary value problem with p-Laplace operator [φp(u (t))] = f (t, u(t)), 0 < t < 1, u(0) = 0, u(1) = au(ξ ), u (0) = 0, u (1) = bu (η), where f (t, u) is singular at t = 0, 1 and u = 0. In the case ≤ a < , ≤ b < , using the lower and upper solution method and the Schauder fixed-point theorem, Zhang and Liu [ ] proved that the SBVP
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