Existence and multiplicity of solutions for fourth-order boundary value problems with parameters

By Xi-Lan Liu, Wan-Tong Li. 2007

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/01fd709be2d5d1b765080d5a6faf30a9c24c1c8a

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

Positive fixed points and fourth‐order equations (2009, methodology): “2): lower and upper solutions [5, 9, 10, 19], monotone iterative technique [2, 11, 13], Krasnoselskii fixed point theorem [6], fixed point index [3, 16, 18, 24], Leray–Schauder degree [12, 14], variational methods [8, 21] and bifurcation theory [17, 20].”
Infinitely many solutions for a fourth-order elastic beam equation (2011, background): “Owing to the importance of fourth-order two-point boundary value problems in describing a large class of elastic deflection, there is a wide literature that deals with multiplicity results for such a problem (see, for instance, [4,5, 8 ], and the references therein).”