# Some surjectivity theorems with applications

## H. K. Pathak and S. N. Mishra

Address:

School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur 492010, India

Corresponding author S. N. Mishra Department of Mathematics, Walter Sisulu University, Mthatha 5117, South Africa

E-mail:

hkpathak05@gmail.com

smishra@wsu.ac.za

Abstract: In this paper a new class of mappings, known as locally $\lambda $-strongly $\phi $-accretive mappings, where $\lambda $ and $\phi $ have special meanings, is introduced. This class of mappings constitutes a generalization of the well-known monotone mappings, accretive mappings and strongly $\phi $-accretive mappings. Subsequently, the above notion is used to extend the results of Park and Park, Browder and Ray to locally $\lambda $-strongly $\phi $-accretive mappings by using Caristi-Kirk fixed point theorem. In the sequel, we introduce the notion of generalized directional contractor and prove a surjectivity theorem which is used to solve certain functional equations in Banach spaces.

AMSclassification: primary 47H15; secondary 47H05.

Keywords: strongly \phi -accretive, locally strongly \phi -accretive, locally \lambda -strongly \phi -accretive, fixed point theorem.

DOI: 10.5817/AM2013-1-17