Abstract
Abstract This article is concerned with the existence of nontrivial solutions for a non-positive fourth-order two-point boundary value problem (BVP) and the existence of positive solutions for a semipositive fourth-order two-point BVP. In mechanics, the problem describes the deflection of an elastic beam rigidly fixed at both ends. The method to show our main results is the topological degree and fixed point theory of nonlinear operator on lattice. Mathematics Subject Classification 2010: 34B18; 34B16; 34B15.
Highlights
The purpose of this article is to investigate the existence of nontrivial solutions and positive solutions of the following nonlinear fourth-order two-point boundary value problemu(4)(t) = λf (t, u(t)), 0 ≤ t ≤ 1, u(0) = u(1) = u (0) = u (1) = 0, (P)where l is a positive parameter, f : [0,1] × R1 ® R1 is continuous
Yao [11] considered the existence of positive solutions of semipositive elastic beam equations by constructing control functions and a special cone and using fixed point theorem of cone expansion-compression type
By the topological degree and fixed point theory of superlinear operator on lattice, we obtain the existence of nontrivial solutions for the non-positive boundary value problem (BVP) (P) and the existence of positive solutions for the semipositive BVP (P)
Summary
The purpose of this article is to investigate the existence of nontrivial solutions and positive solutions of the following nonlinear fourth-order two-point boundary value problem (for short, BVP). Yao [11] considered the existence of positive solutions of semipositive elastic beam equations by constructing control functions and a special cone and using fixed point theorem of cone expansion-compression type. Let P be a generating cone and B a u0bounded completely continuous linear operator. Let B : E ® E be a positive completely continuous linear operator, r(B) a spectral radius of B, B* the conjugated operator of B, and P* the conjugated cone of P. Let P be a solid cone, A : E ® E be a completely continuous operator satisfying A = BF, where F is quasi-additive on lattice, B is a positive bounded operator satisfying H condition. If (i) there exists a positive bounded linear operator B such that |Ax| ≤ B|x|, for all x Î ∂Ω;.
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