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Symmetric positive solutions to a fourth-order nonlinear differential equation with nonlocal boundary conditions. (Chinese) Zbl 1140.34334

Summary: The author considers the nonlocal boundary value problem for a nonlinear fourth order ordinary differential equation of the form

u (4) (t)=g(t)f(t,u(t)),0<t<1,u(0)=u(1)= 0 1 a(s)u(s)ds,u '' (0)=u '' (1)= 0 1 b(s)u '' (s)ds

where a,bL 1 [0,1],g:(0,1)[0,) is continuous, symmetric on (0,1) and may be singular at t=0 and t=1·f:[0,1]×[0,)[0,) is continuous and f(·,x) is symmetric on [0,1] for all x[0,). Under some suitable growth conditions, the author shows the existence and multiplicity of symmetric positive solutions of the above problem by applying Krasnoselskii‚Äôs fixed point theorem in a cone.

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47H10Fixed point theorems for nonlinear operators on topological linear spaces
34B16Singular nonlinear boundary value problems for ODE