We study the L 2 -type contraction property of large perturbations around shock waves scalar viscous conservation laws with strictly convex fluxes in one space dimension. The holds up to a shift, and it is measured by weighted relative entropy, for which we choose an appropriate entropy associated flux. In particular, handle shocks small amplitude. This result improves recent article [19] autho...

In this paper we establish new forward and backward versions of Ekeland’s variational principle for the class of strictly-decreasing forward(resp. backward-) lower-semicontinuou functionals in pseudo-quasimetric spaces. We do not require that the space under consideration either is complete or enjoys the limit uniqueness property due to the fact that the collections of forward and backward limi...

In this paper, we prove a Stolarsky type inequality for pseudo-integrals. More precisely, we show that: ∫ sup [0,1] f(x 1 a+b )dx ≥ ( ∫ sup [0,1] f(x 1 a )dx ) ̄ ( ∫ sup [0,1] f(x 1 b )dx ) , where a, b > 0, f : [0, 1] → [0, 1] is a continuous and strictly decreasing function ( strictly increasing function ) and μ is the sup-measure the same as Theorem 2.4. Also ̄ is represented by an increasing...

The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, ihen there exî sts an .v,, such thai A ".\/\\A "x\\ converges to x,j for all .v > 0, There are many classical proofs of this theorem, all depending on a connection between positivily of a matrix and properties of ils eigenvalues. A more modern proof, due to Garrett Birkhoff. is based on t...

The purpose of this paper is to present the definition of [Formula: see text]-contractive mapping and to discuss the relation of [Formula: see text]-contractive mappings and [Formula: see text]-contractive mappings. Furthermore, the generalized [Formula: see text]-contraction mapping principle has been proved without the uniqueness condition. Meanwhile, the generalized [Formula: see text]-contr...

and Applied Analysis 3 Definition 1.10 see 5 . Let X,G be a G-metric space and f, g : X → X be given mappings. We say that f is a generalized weakly G-contraction mapping of type B with respect to g if for all x, y, z ∈ X, the following inequality holds: ψ ( G ( fx, fy, fz )) ≤ ψ ( G ( gx, gx, fy ) G ( gy, gy, fz ) G ( gz, gz, fx ) 3 ) − φGgx, gx, fy, Ggy, gy, fz, Ggz, gz, fx, 1.2