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Zbl 1195.34037
Ma, Ruyun; Xu, Ling
Existence of positive solutions of a nonlinear fourth-order boundary value problem. (English)
[J] Appl. Math. Lett. 23, No. 5, 537-543 (2010). ISSN 0893-9659

Summary: We study the existence of positive solutions of fourth-order boundary value problem $$u^{(4)}(t) = f(t, u(t), u^{\prime \prime }(t)), t \in (0, 1).$$ $$u(0) = u(1) = u^{\prime \prime }(0) = u^{\prime \prime }(1) = 0,$$ where $f: [0, 1] \times [0, \infty ) \times (-\infty , 0] \to [0, \infty )$ is continuous. The proof of our main result is based upon the Krein-Rutman theorem and the global bifurcation techniques.

MSC 2000:: *34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations
47J15 Abstract bifurcation theory

Keywords: Krein-Rutman theorem; fourth-order ordinary differential equations; elastic beam; bifurcation; positive solutions; eigenvalue

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