Search

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 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 will prove a discrete Rubio De Francia extrapolation theorem in the theory of discrete Ap- Muckenhoupt weights for which the discrete Hardy-Littlewood maximal operator is bounded on lwp (Z+). The results will be proved by employing the self-improving property of the discrete Ap- Muckenhoupt weights and the Marcinkiewicz Interpolation Theorem. [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]

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, by using the Riccati technique which reduces the higher order dynamic equations to a Riccati dynamic inequality, we will establish some new sufficient conditions for oscillation of the second order nonlinear neutral dynamic equation (r(t)((x(t) + p(t)x(τ(t)))Δ)γ)Δ + q(t)xα(δ(t)) + v(t)xβ(η(t)) = 0; on time scales where γ, α, β, are quotient of odd positive integers. [ABSTRACT FROM AUTHOR]

In this paper, we will establish some new sufficient condition for oscillation of solutions of a certain class of first-order neutral delay difference equations of the form Δ (xn -- pnxn-1) + qnxn-τγ = 0, where γ is a quotient of odd positive integers. We will consider the sublinear and super linear cases. The results will be obtained by using the oscillation theorems of second order delay difference equations. [ABSTRACT FROM AUTHOR]

In this paper, we will prove some fundamental properties of the discrete power mean operator M p u n = 1 / n ∑ k = 1 n u p k 1 / p , for n ∈ I ⊆ ℤ + , of order p , where u is a nonnegative discrete weight defined on I ⊆ ℤ + the set of the nonnegative integers. We also establish some lower and upper bounds of the composition of different operators with different powers. Next, we will study the structure of the generalized discrete class B p q B of weights that satisfy the reverse Hölder inequality M q u ≤ B M p u , for positive real numbers p , q , and B such that 0 < p < q and B > 1. For applications, we will prove some self-improving properties of weights from B p q B and derive the self improving properties of the discrete Gehring weights as a special case. The paper ends by a conjecture with an illustrative sharp example. [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 prove some new characterizations of weights in some Hardy-type inequalities on time scales. The results as special cases contain the results due to Beesack and Heinig, Leindler and Bloom and Kerman. Some new integral and discrete inequalities related to Copson's, Flett's, Bliss's and Bennett's will be formulated. The main results will be proved by using new generalizations of Opial's type inequalities, Hölder's inequality, Minkowski's inequality and the chain rule on time scales. [ABSTRACT FROM AUTHOR]

In this paper, we extend and improve some dynamic inequalities by using the sneak-out principle with di¤erent exponents on time scales. The main results can be used to formulate the corresponding discrete inequalities of Bennett and G-Erdmann type. [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 establish some interval oscillation criteria for impulsive conformable fractional partial delay differential equations with a forced term. The main results will be obtained by employing Riccati technique. Our results extend and improve some results reported in the literature for the classical differential equations without impulses. An example is provided to illustrate the relevance of the new theorems. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the solvability of dynamic equations of second order is sufficient for the validity of some Hardy and Opial type inequalities with two different weights on time scales. In particular, the results give new characterizations of two different weights in inequalities containing Hardy and Opial operators. The main contribution in this paper is the characterizations of weights in discrete inequalities that will be formulated from our results as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new dynamic inequalities involving C- monotonic functions on time scales. The main results will be proved by employing Hölder's inequality, integration by parts, and a chain rule on time scales. As a special case when T=R, our results contain the continuous inequalities proved by Heinig, Maligranda, Pečarić, Perić, and Persson and when T=N, the results to the best of the authors' knowledge are essentially new. [ABSTRACT FROM AUTHOR]

In this paper, we study the gaps between zeros of nontrivial solutions of fourth order differential equations subject to some boundary conditions, and establish some sufficient conditions for (2, 2)-disconjugacy. The results, which cover some boundary value problems in bending of beams, will be proved by making use of a generalization of Hardy's inequality and some Opial and Wirtinger type inequalities. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new dynamic inequalities on time scales using Hölder's inequality and Keller's chain rule on time scales. These inequalities, as special cases when the time scale [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]

Our aim in this paper is to establish some explicit bounds of the unknown function in a certain class of nonlinear dynamic inequalities in two independent variables on time scales which are unbounded above. These on the one hand generalize and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of partial dynamic equations on time scales. Some examples are considered to demonstrate the applications of the results. [ABSTRACT FROM AUTHOR]

We establish a necessary and sufficient condition for global stability of the nonlinear discrete red blood cells survival model and demonstrate that local asymptotic stability implies global stability. Oscillation and solution bounds are investigated. We also show that, for different values of the parameters, the solution exhibits some time-varying dynamics, that is, if the system is moved in a direction away from stability by increasing the parameters, then it undergoes a series of bifurcations that leads to increasingly long periodic cycles and finally to deterministic chaos. We also study the chaotic behavior of the model with a constant positive perturbation and prove that, for large enough values of one of the parameters, the perturbed system is again stable. [ABSTRACT FROM AUTHOR]

We will prove some dynamic inequalities of Opial type on time scales which not only extend some results in the literature but also improve some of them. Some discrete inequalities are derived from the main results as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we develop a general approach to the existence of periodic solutions for the difference equation[image omitted] with variable delays and apply it to find necessary and sufficient conditions for the existence of a positive periodic solution [image omitted] for the host macroparasite model. We present some sufficient conditions for the global attractivity of [image omitted] . For the constant coefficients nondelay case, we obtain necessary and sufficient attractivity conditions using recent results of Liz. [ABSTRACT FROM AUTHOR]

We consider the second part of the Cushing-Henson conjecture (the cycle's average is less than the average of carrying capacities; the first part of the conjecture deals with the existence and global stability of periodic cycles) for a periodic delay difference equation [image omitted] Sufficient conditions on f and hi are obtained, when the second part of the conjecture is valid. We demonstrate the sharpness of these conditions by presenting several counterexamples. In addition, sufficient global attractivity conditions are deduced for the Pielou equation. [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]

In this paper, we consider the discrete non-linear delay population dynamics model [ABSTRACT FROM PUBLISHER]

In this paper, we discuss the oscillatory behavior of the second-order forced nonlinear dynamic equation (a(t)xΔ(t))+p(t)f(xο)=r(t), on a time scale T when a(t) > 0. We establish some sufficient conditions which ensure that every solution oscillates or satisfies lim inft→∞ |x(t)| = 0. Our oscillation results when r(t) = 0 improve the oscillation results for dynamic equations on time scales that has been established by Erbe and Peterson [Proc. Amer. Math. Soc 132 (2004), 735-744], Bohner, Erbe and Peterson [J. Math. Anal. Appl. 301 (2005), 491-507] since our results do not require ∫∞t0 q(t)Δt > 0 and ∫±∞± t0 du/f(u) < ∞. Also, as a special case when T = R, and r(t) = 0 our results improve some oscillation results for differential equations. Some examples are given 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]

New oscillation criteria for the second order nonlinear neutral delay differential equation [y(t)=p(t)y(t-ϒ)]-+q(t) f(y(g(t))) = 0, t ≥ t0 are given. The relevance of our theorems becomes clear due to a carefully selected example. [ABSTRACT FROM AUTHOR]

By means of the Riccati transformation techniques, we will establish some new oscillation criteria for hyperbolic nonlinear neutral delay difference equations. Our results can be considered as the discrete analogues of the Kamenev-type and Philos-type oscillation criteria. [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]

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 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]

In this paper, we will establish some new interval oscillation criteria for the forced second-order nonlinear dynamic equation[image omitted] on a time scale  where γ ≥ 1. As a special case when  =  our results not only include the oscillation results for second-order differential equations established by Wong (Oscillation criteria for a forced second-order linear differential equation, J. Math. Anal. Appl. 231 (1999), pp. 233-240) and Nasr (Sufficient condition for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. math. Soc. 126 (1998), pp. 123-125) but also improve these results. When  = ,  = h or  = qN, i.e. for difference equations, generalized difference equations or q-difference equations our results are essentially new and also can be applied on different types of time scales, as illustrated in several examples. [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]

In this paper, we prove some new reverse dynamic inequalities of Renaud- and Bennett-type on time scales. The results are established using the time scales Fubini theorem, the reverse Hölder inequality and a time scales chain rule. [ABSTRACT FROM AUTHOR]