We investigate a weakly coupled infinite system of nonlinear strong parabolic differential–functional inequalities of the following form (1) ∂tz (t, x) < f (t, x, z(t, x), ∂xz (t, x), ∂ xxz (t, x), z), i ∈ S, in an arbitrary domain D. The right-hand sides f i of these inequalites are functionals of an unknown function z and Volterra functionals only will be regarded in this paper. We give a fun...

The integral inequalities occupy privileged position in the theory of differential and integral equations. In the recent years nonlinear integral inequalities have received considerable attention because of their important applications to a variety of problems in diverse fields of nonlinear differential and integral equations. In 1919, Gronwall [4] introduced the famous Gronwall inequality in t...

Various singular integral inequalities play an important role in the development of the theory of differential equations, functional differential equations, and integral equations. For example, Henry 1 proposed a linear integral inequality with singular kernel to investigate some qualitative properties for a parabolic differential equation, and Sano and Kunimatsu 2 gave a modified version of He...

The theory of time scales was introduced by Hilger 1 in 1988 in order to contain both difference and differential calculus in a consistent way. Recently, many authors have extended some fundamental integral inequalities used in the theory of differential and integral equations on time scales. For example, we refer the reader to the papers 2–12 and the references cited there in. In this paper, w...

Friedrichsand Poincaré-type inequalities are important and widely used in the area of partial differential equations and numerical analysis. Most of their proofs appearing in references are the argument of reduction to absurdity. In this paper, we give direct proofs of Friedrichs-type inequalities in H 1(Ω) and Poincaré-type inequalities in some subspaces of W1,p(Ω). The dependencies of the ine...

Differential forms are interesting and important generalizations of real functions and distributions. Many interesting results and applications of differential forms have recently been found in some fields, such as tensor analysis, potential theory, partial differential equations and quasiregular mappings, see [B], [C], [D1], [HKM], [I], [IL] and [IM]. In many cases, we need to know the integra...

OBJECTIVE Smoking usually develops in adolescence and is patterned by socioeconomic position (SEP). We examined whether early adolescent smoking development and associations with SEP have changed over time in a population with well-developed tobacco control policies. We additionally investigated the relative importance of socioeconomic inequalities at different stages of smoking development. ...

This paper deals with some nonlinear integral inequalities that involve the maximum of the unknown scalar function of one variable. The considered inequalities are generalizations of the classical integral inequality of Gronwall–Bellman. The importance of these integral inequalities is due to their wide applications in qualitative investigations of differential equations with “maxima”, and it i...

Gronwall-Bellman inequalities and their various linear and nonlinear generalizations play very important roles in the discussion of existence, uniqueness, continuation, boundedness, and stability properties of solutions of differential equations and difference equations. The literature on such inequalities and their applications is vast. For example, see 1–12 for continuous cases, and 13–20 for...