Existence and multiplicity of solutions of Stieltjes differential equations

This is a joint work with Alberto Cabada and F. Adrián F. Tojo (Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela). We investigate a first order non-linear boundary value problem $$u_{g}^{\prime}(t)+b(t) u(t)=f(t, u(t)), \quad t \in[0, T]$$ with linear boundary conditions $$u(0)=u(T)+k\,B(u),$$ where $k$ is a constant and $B$ is a linear functional. Note that boundary conditions extend the periodic case. We use techniques from Stieltjes calculus and fixed point index theory to show the existence and multiplicity of solution.