Abstract
In this paper we study the existence and multiplicity of the solutions for the fourth-order boundary value problem (BVP) u ( 4 ) ( t ) + η u ″ ( t ) − ζ u ( t ) = λ f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f : [ 0 , 1 ] × R → R is continuous, ζ , η ∈ R and λ ∈ R + are parameters. By means of the idea of the decomposition of operators shown by Chen [W.Y. Chen, A decomposition problem for operators, Xuebao of Dongbei Renmin University 1 (1957) 95–98], see also [M. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gostehizdat, Moscow, 1956], and the critical point theory, we obtain that if the pair ( η , ζ ) is on the curve ζ = − η 2 / 4 satisfying η < 2 π 2 , then the above BVP has at least one, two, three, and infinitely many solutions for λ being in different interval, respectively.
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