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We prove some relations between the discrete Gehring classes 풢q and discrete Muckenhoupt classes 풜p. Specifically, by using some known Hardy-type and Carleman-type inequalities, we study the relationship between 풢1, 풜∞, and 풜1 for nonincreasing and nondecreasing weights. Finally, we establish some general results by introducing the notions of 풢φ classes defined for a nonnegative convex function φ. [ABSTRACT FROM AUTHOR]

In this paper, we will prove some fundamental properties of the power mean operator M p g (t) = (1 ϒ (t) ∫ 0 t λ (s) g p (s) d s) 1 / p , for t ∈ I ⊆ R + , of order p and establish some lower and upper bounds of the compositions of operators of different powers, where g, λ are a nonnegative real valued functions defined on I and ϒ (t) = ∫ 0 t λ (s) d s . Next, we will study the structure of the generalized class U p q (B) of weights that satisfy the reverse Hölder inequality M q u ≤ B M p u , for some p < q , p. q ≠ 0 , and B > 1 is a constant. For applications, we will prove some self-improving properties of weights in the class U p q (B) and derive the self improving properties of the weighted Muckenhoupt and Gehring classes. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new dynamic inequalities of Opial type involving higher-order derivatives of two functions, with two different weights on time scales. From these inequalities, we will derive some special cases and give an improvement of some versions of recent results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new class of dynamic inequalities of Hardy's type which generalize and improve some recent results given in the literature. More precisely, we prove some new Hardy-type inequalities involving many functions on time scales. Some new discrete inequalities are deduced in seeking applications. [ABSTRACT FROM AUTHOR]

In this paper, first we prove some new refinements of discrete weighted inequalities with negative powers on finite intervals. Next, by employing these inequalities, we prove that the self-improving property (backward propagation property) of the weighted discrete Muckenhoupt classes holds. The main results give exact values of the limit exponents as well as the new constants of the new classes. As an application, we establish the self-improving property (forward propagation property) of the discrete Gehring class. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the self-improving property of the weighted Gehring class G λ p with a weight λ holds in the non-homogeneous spaces. The results give sharp bounds of exponents and will be used to obtain the self-improving property of the Muckenhoupt class A q . By using the rearrangement (nonincreasing rearrangement) of the functions and applying the Jensen inequality, we show that the results cover the cases of non-monotonic functions. For applications, we prove a higher integrability theorem and report that the solutions of partial differential equations can be solved in an extended space by using the self-improving property. Our approach in this paper is different from the ones used before and is based on proving some new inequalities of Hardy type designed for this purpose. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new Opial-type dynamic inequalities on time scales. Our results are obtained in frame of convexity property and by using the chain rule and Jensen and Hölder inequalities. For illustration purpose, we obtain some particular Opial-type inequalities reported in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when T = R and T = N , respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new. [ABSTRACT FROM AUTHOR]

We first prove some new weighted refinements of inequalities of Hardy's type with negative powers. Next, we prove that any A λ 1 Muckenhoupt class with a weight λ belongs to some weighted Gehring class G λ p for p > 1 . We also prove that the self-improving property of the weighted Muckenhoupt class A λ q holds. The main results give exact values of the limit exponents and the constants of the new classes. The self-improving property of the weighted Muckenhoupt class will then be applied to prove the self-improving property of the Gehring class with a sharp value on the exponents. [ABSTRACT FROM AUTHOR]

In this paper, we establish some sufficient conditions which ensure that the solutions of the third order delay difference equation with a negative middle term Δ (a n Δ (Δ w n) α) − p n (Δ w n + 1) α − q n h (w n − l) = 0 , n ≥ n 0 , are oscillatory. Moreover, we study the asymptotic behavior of the nonoscillatory solutions. Two illustrative examples are included for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new characterizations of weighted functions of dynamic inequalities containing a Hardy operator on time scales. These inequalities contain the characterization of Ariňo and Muckenhoupt when T = R , whereas they contain the characterizations of Bennett–Erdmann and Gao when T = N . [ABSTRACT FROM AUTHOR]

In this paper, we establish some conditions on nonnegative rd-continuous weight functions u x and υ x which ensure that a reverse dynamic inequality of the form ∫ a ∞ f p (x) υ x Δ x 1 p ≤ C ∫ a ∞ u x ∫ a σ x K σ x , σ y f (y) Δ y q Δ x 1 q , holds when q ≤ p < 0 and 0 < q ≤ p < 1. Corresponding dual results are also obtained. In particular, we prove some reverse dynamic weighted Hardy-type inequalities with kernels on time scales which as special cases contain some generalizations of integral and discrete inequalities due to Beesack and Heinig. [ABSTRACT FROM AUTHOR]

In this paper, we prove some properties of weighted Cesàro and Copson sequences spaces by establishing some factorization theorems. The results lead to two-sided norm discrete inequalities with best possible constants and also give conditions for the boundedness of the generalized discrete weighted Hardy and Copson operators. [ABSTRACT FROM AUTHOR]

In this paper, we establish some factorization theorems for weighted Cesàro and Copson spaces, obtain two sided norm dynamic inequalities, and give conditions for the boundedness of the Hardy and Copson dynamic operators on the weighted space L λ p (T) . We obtain, as special cases, the classical integral inequalities on R and the discrete inequalities on N . [ABSTRACT FROM AUTHOR]

In this paper, we employ some algebraic equations due to Hardy and Littlewood to establish some conditions on weights in dynamic inequalities of Hardy and Copson type. For illustrations, we derive some dynamic inequalities of Wirtinger, Copson and Hardy types and formulate the classical integral and discrete inequalities with sharp constants as particular cases. The results improve some results obtained in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we prove some factorization theorems of Cesàro and Copson spaces on an arbitrary time scale , which offer enhancements of dynamic Copson's and Hardy's inequalities. Our results enhance, among others, the best-known forms of dynamic Hardy's inequality. The main results will be proved by employing the time scales Hölder's inequality and the derived time scales power rule of integration. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with oscillation of the nonlinear difference equation Δ(cn[Δ(dnΔxn)]γ) +qnf(xg(n))=0, n≥n0, where γ > 0 is the quotient of odd positive integers, cn, dn and qn are positive sequences of real numbers, g(n) is a sequence of nonnegative integers and f ϵ C(R,R) such that u f (u) > 0 for u ≠ 0. We establish some new sufficient conditions for oscillation by employing the Riccati substitution and the analysis of the associated Riccati difference inequality. Our results extend and improve some previously obtained ones. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

Some new oscillation criteria for nonlinear delay difference equation with damping are given. Our results partially solve the open problem posed in [ Math. Bohemica, 125 (2000), 421– 430]. Also, we will establish some new oscillation criteria for special cases of (∗), which improve some of the well–known results in the literature. [ABSTRACT FROM AUTHOR]

We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ + p(t)xγ(t) = 0 and (r(t)(xΔ)γ)Δ + p(t)xγδ(t) = 0, on a time scale 핋, where γ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when 핋 = ℝ involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Došlý and Řehák and some results of Řehák for half-linear equations on time scales. Several examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper studies the nonlinear delay impulsive respiratory dynamics model. The model describes the sudden changes of the concentration of CO2 in the blood of the mammal. It is proved that the model has a unique positive periodic solution. Some sufficient conditions for oscillation of all positive solutions about the positive periodic solution are established and also some sufficient conditions for the global attractivity of the periodic solution are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the discrete Muckenhoupt class A p (C) and the discrete Gehring class G q (K) . In particular, we prove that the self-improving property of the Muckenhoupt class holds, i.e., we prove that if u ∈ A p (C) then there exists q < p such that u ∈ A q (C 1) . Next, we prove that the power rule also holds, i.e., we prove that if u ∈ A p then u q ∈ A p for some q > 1 . The relation between the Muckenhoupt class A 1 (C) and the Gehring class is also discussed. For illustrations, we give exact values of the norms of Muckenhoupt and Gehring classes for power-low sequences. The results are proved by some algebraic inequalities and some new inequalities designed and proved for this purpose. [ABSTRACT FROM AUTHOR]

In this paper, we give an affirmative answer to the following question: Is the solvability of some nonlinear dynamic equations on a time scale T not only sufficient but in a certain sense also necessary for the validity of some dynamic Hardy-type inequalities with two different weights? In fact, this answer will give a new characterization of the weights in a weighted Hardy-type inequality on time scales. The results contain the results when T = R , T = N , and when T = q N 0 as special cases. Some applications are given for illustrations. [ABSTRACT FROM AUTHOR]