Existence and multiplicity results for systems of first-order differential equations via the method of solution-regions
This excellent paper is devoted to prove the existence and multiplicity of solutions of the following system of first order differential equations \[ u'(t)=f(t,u(t))\text{ for a.e }t \in [0,T], \] coupled to either initial, \(u(0)=r\), or periodic conditions, \(u(0)=u(T)\). Here \(f:[0,T] \times {\mathbb R}^N \to {\mathbb R}^N\) is a Carathéodory function. The main novelty of this paper is the introduction of the new concepts of solution-regions and strict solution-regions. These new methods generalize the previous ones of ordered lower and upper solutions and solution-tube. In the paper, it is shown that this method covers more general regions and, as a consequence, it is possible to deduce existence and multiplicity results under weaker assumptions on the function \(f\) that defines the equation. The results are deduced from fixed point and index theory. Several examples are shown along the paper that point out the applicability of the obtained results.
system of first order differential equation
initial value and periodic boundary value problem
scientific article; zbMATH DE number 6921845
scientific article