Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 13, pp. 1-10.

Title: Existence and uniqueness of solutions for quasi-linear 
       differential equations with deviating arguments

Authors: Rajib Haloi (Indian Institute of Technology Kanpur, India)
         Dhirendra Bahuguna (Indian Institute of Technology Kanpur, India)
	 Dwijendra N. Pandey (Indian Institute of Technology Roorkee, India)

Abstract:
 We prove the  existence and uniqueness of a local
 solution to a quasi-linear differential equation of parabolic type
 with deviated argument in an arbitrary Banach space. The results are
 obtained by applying the Sobolevskii-Tanabe theory of parabolic
 equations, fractional powers of operators, and the Banach fixed point
 theorem. We include an example that illustrates the
 theory.
 
Submitted August 15, 2011. Published January 17, 2012.
Math Subject Classifications: 34G20, 34K30, 35K90, 47N20.
Key Words: Analytic semigroup; parabolic equation;
           deviated argument; Banach fixed point theorem.