Wendroff type inequalities on time scales via Picard operators. (English) Zbl 1262.35006

András, Szilárd; Mészáros, Alpár

Math. Inequal. Appl. 16, No. 1, 159-174 (2013).

Summary: Recently, R. A. C. Ferreira and D. F. M. Torres [Nonlinear Dyn. Syst. Theory 9, No. 2, 161–169 (2009; Zbl 1180.26020)] proved some linear and nonlinear Wendroff type inequalities on time scales. It is well known that the Wendroff inequality is not the best possible upper estimate for the solutions of the corresponding integral inequality. The aim of our paper is to improve the known Wendroff type inequalities on time scales and to give a different proof for the existing inequalities. The method we use is based on a variant of the abstract comparison Gronwall lemma and on the theory of Picard operators.

MSC:

35A23	Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
26D10	Inequalities involving derivatives and differential and integral operators
47H10	Fixed-point theorems
45G10	Other nonlinear integral equations
35R45	Partial differential inequalities and systems of partial differential inequalities

Keywords:

Picard operators; abstract comparison Gronwall lemma