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On positive solutions of a nonlinear fourth order boundary value problem via a fixed point theorem in ordered sets. (English) Zbl 1225.47132

The authors consider the boundary value problem
\[ \begin{cases} u^{(4)}(t)=f(t,u),\;t\in(0,1),\\ u(0)=u(1)=0=u''(0)=u''(1), \end{cases} \]
where \(f\) is continuous, non-decreasing with respect to the second argument and satisfies the growth conditions
\[ f(t,y)-f(t,x)\leq\alpha\sqrt{\ln[(y-x)^2+1]}, \] where \(0<\alpha\leq\sqrt{\frac{80640}{17}}\). The problem is written as a fixed point problem via a suitable Green’s function. Using a recent fixed point theorem in ordered sets [J. Harjani and K. Sadarangani, “Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations”, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3–4, A, 1188–1197 (2010; Zbl 1220.54025)], the authors prove existence and uniqueness of a solution.

MSC:

47N20 Applications of operator theory to differential and integral equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
54H25 Fixed-point and coincidence theorems (topological aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces

Citations:

Zbl 1220.54025
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