Positive solutions of second order semipositone singular three-point boundary value problems.
Let h ∈ L¹[0,1] ∩ C(0,1) be nonnegative and f(t,u,v) + h(t) ≥ 0. We study the existence and multiplicity of positive solutions for the nonlinear fourth-order two-point boundary value problem , 0 < t < 1, u(0) = u’(0) = u’(1) =u”’(1) =0, where the nonlinear term f(t,u,v) may be singular at t=0 and t=1. By constructing a suitable cone and integrating certain height functions of f(t,u,v) on some bounded sets, several new results are obtained. In mechanics, the problem models the deflection of...
This paper studies the existence of multiple positive solutions to a nonlinear fourth-order two-point boundary value problem, where the nonlinear term may be singular with respect to both time and space variables. In order to estimate the growth of the nonlinear term, we introduce new control functions. By applying the Hammerstein integral equation and the Guo-Krasnosel'skii fixed point theorem of cone expansion-compression type, several local existence theorems are proved.
Existence theorems of positive solutions to a class of singular second order boundary value problems of the form y'' + f(x,y,y') = 0, 0 < x < 1, are established. It is not required that the function (x,y,z) → f(x,y,z) be nonincreasing in y and/or z, as is generally assumed. However, when (x,y,z) → f(x,y,z) is nonincreasing in y and z, some of O'Regan's results [J. Differential Equations 84 (1990), 228-251] are improved. The proofs of the main theorems are based on a fixed point theorem for...
We study a nonlocal boundary value problem for the equation x''(t) + f(t,x(t),x'(t)) = 0, t ∈ [0,1]. By applying fixed point theorems on appropriate cones, we prove that this boundary value problem admits positive solutions with slope in a given annulus. It is remarkable that we do not assume f≥0. Here the sign of the function f may change.
We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.