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Existence and uniqueness of solutions for a fourth-order boundary value problem. (English) Zbl 1169.34308

Summary: We study the fourth-order two-point boundary value problem
\[ \begin{aligned} & x''''(t)-f(t,x(t),x'(t),x'',x'''(t))=0,\quad t\in(0,1),\\ & x(0)=x'(1)=0,\quad ax''(0)-bx'''(0)=0,\quad cx''(1)+dx'''(1)=0.\end{aligned} \]
By means of lower and upper solution method, growth conditions on the nonlinear term \(f\) which guarantee the existence of solutions for the above boundary value problem are given. In particular, we obtain the uniqueness of the solution by imposing a monotonicity condition of the term \(f\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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