Nonlocal semilinear integro-differential inclusions via vectorial measures of noncompactness

Abstract

In this note, we show the existence of mild solutions to a nonlocal semilinear integro-differential problem via a new -valued measure of noncompactness involving a Volterra integral operator. Our existence result extends in a broad sense some recent theorems for integro-differential equations. Then we provide a compactness result on the mild solution set and an application to a reaction–diffusion problem involving a distributed time delay.

Keywords:

• Integro-differential semilinear inclusion
• nonlocal condition
• mild solution
• vectorial measure of noncompactness
• reaction–diffusion equation

AMS Subject Classifications:

• 34G25
• 34B10
• 47H08
• 35K57

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second two authors are members of INdAM and GNAMPA. The paper was partially supported by the project GNAMPA 2014 “Metodi topologici: sviluppi ed applicazioni a problemi differenziali non lineari” and by the project “Fondi Ricerca di Base 2015” (Department of Mathematics and Computer Science, University of Perugia).