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Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications. (English) Zbl 1205.34019

Let wL p (0,1) and λ be a parameter. The authors considered the second-order differential equation

λu '' (t)-w(t)f(t,u(t))=0

subject to one of the following sets of conditions:

a. x(0)=x(1)= 0 1 g(t)x(t)dt;

b. x(0)= 0 1 g(t)x(t)dt, x(1)=0;

c. x(0)=0, x(1)= 0 1 g(t)x(t)dt.

By applying Hölder’s inequality and fixed point theory in cones, they prove the existence and multiplicity of positive solutions of the above problems.

MSC:
34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
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