On the existence of a countable set of positive solutions for a nonlocal boundary-value problem with vector-valued response. (English) Zbl 1086.34024
The aim of the author is to provide sufficient conditions for the existence of a countable set of positive solutions for a differential equation of the form
\[
{{\partial}\over{\partial{t}}}(H_z[t,x'(t)])=-V_x[t,x'(t)]\text{ a.e. in }[0, T]
\]
satisfying the nonlocal boundary condition
\[ H_z[t,x'(T)]=\int_{t_0}^TH_z[s,x'(s)]dg(s). \] The results of the paper are obtained by using the idea of Fenchel conjugate to describe a duality of the problem, according to which a solution of the problem is a minimizer of a suitable integral functional.
satisfying the nonlocal boundary condition
\[ H_z[t,x'(T)]=\int_{t_0}^TH_z[s,x'(s)]dg(s). \] The results of the paper are obtained by using the idea of Fenchel conjugate to describe a duality of the problem, according to which a solution of the problem is a minimizer of a suitable integral functional.
Reviewer: George Karakostas (Ioannina)
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 47J30 | Variational methods involving nonlinear operators |
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