Electron. J. Diff. Equ., Vol. 2012 (2012), No. 13, pp. 1-10.

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

Rajib Haloi, Dhirendra Bahuguna, Dwijendra N. Pandey

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.

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Rajib Haloi
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Pin 208016, India
Tel. +91-512-2597053, Fax +91-512-2597500
email: rajib.haloi@gmail.com
Dhirendra Bahuguna
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Pin 208016, India
email: dhiren@iitk.ac.in
Dwijendra N. Pandey
Department of Mathematics
Indian Institute of Technology Roorkee
Pin 247667, India
email: dwij.iitk@gmail.com

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