A note on the theorem of Leray–Schauder

Using elemental methods of Topology, theory of degree and theory of metric continua, we prove a new version of the theorem of Leray–Schauder. It provides the existence of arc-connected set of solutions. This result may have a lot of applications in a large variety of problems. Although the assumption of the theorem is not easy to verify in practice, this theorem could be an important tool to prove not only the existence of set of solutions but also the existence of a set of solution which is homeomorphic to the interval [ 0 , 1 ] ⊂ R .