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

Feng, Hanying; Ji, Dehong; Ge, Weigao

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 10, 3561-3566 (2009).

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

Keywords:

fourth-order boundary value problem; lower and upper solutions; Nagumo condition; integral-differential equation; existence and uniqueness

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