Sufficient conditions are obtained so that every solution of ${[y\left(t\right)-p\left(t\right)y(t-\tau )]}^{\left(n\right)}+Q\left(t\right)G\left(y(t-\sigma )\right)=f\left(t\right)$ where n ≥ 2, p,f ∈ C([0,∞),ℝ), Q ∈ C([0,∞),[0,∞)), G ∈ C(ℝ,ℝ), τ > 0 and σ ≥ 0, oscillates or tends to zero as $t\to \infty$. Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed that ${\int }_0^{\infty }Q\left(t\right)dt=\infty$. Through examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $${\left(r\left(t\right){\left(z^{\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+\sum _{i=1}^m{q}_i\left(t\right)x^{{\alpha }_i}\left({\sigma }_i\left(t\right)\right)=0,t\ge t_0,$$ where $z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)$. Under the assumption ${\int }^{\infty }{\left(r\left(\eta \right)\right)}^{-1/\gamma }\mathrm{d}\eta =\infty$, we consider two cases when $\gamma >{\alpha }_i$ and $\gamma <{\alpha }_i$. Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

A necessary and sufficient condition is given on the basis of a rare maximal function $M_l$ such that $M_{l}f\in L¹\left([0,1]\right)$ implies f ∈ L log L([0,1]).

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ${\left(a\left(t\right){\phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right){\phi }_p\left(x^{\text{'}}\right)+c\left(t\right){\phi }_p\left(x\right)=0$, where ${\phi }_p\left(x\right)={\left|x\right|}^{p-1}\mathrm{sgn}x$ for $x\in ℝ$ and $p>1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial...

This paper deals with the oscillation problems on the nonlinear differential equation $\left(a\left(t\right)\right|x^{\text{'}}{|}^{p\left(t\right)-2}{x}^{\text{'}}{)}^{\text{'}}+b\left(t\right){\left|x\right|}^{\lambda -2}x=0$ involving $p\left(t\right)$-Laplacian. Sufficient conditions are given under which all proper solutions are oscillatory. In addition, we give a-priori estimates for nonoscillatory solutions and propose an open problem.

The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $$\Delta x\left(n\right)+\sum _{k=-p}^q{a}_k\left(n\right)x(n+k)=0,n>n_0,$$ where $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ is the difference operator and $\left\{a_k\left(n\right)\right\}$ are sequences of real numbers for $k=-p,...,q$, and $p>0$, $q\ge 0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

Let $M_S$ denote the strong maximal operator. Let $M_x$ and $M_y$ denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that ${\int }_Q{M}_y{M}_{x}h<\infty$ but ${\int }_Q{M}_x{M}_{y}h=\infty$. It is shown that if f is a function supported on Q such that ${\int }_Q{M}_y{M}_{x}f<\infty$ but ${\int }_Q{M}_x{M}_{y}f=\infty$, then there exists a set A of finite measure in ℝ² such that ${\int }_A{M}_{S}f=\infty$.

We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to $L^{1,\infty }$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^p$ regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^p$...

Recently there has been an increasing interest in studying $p\left(t\right)$-Laplacian equations, an example of which is given in the following form $$\left(\right|u^{\text{'}}{\left(t\right)|}^{p\left(t\right)-2}{u}^{\text{'}}{\left(t\right))}^{\text{'}}+c\left(t\right){\left|u\left(t\right)\right|}^{q\left(t\right)-2}u\left(t\right)=0,t>0.$$ In particular, the first study of sufficient conditions for oscillatory solution of $p\left(t\right)$-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations...

We consider the maximal function $\left|\right|\left(S^{a}f\right)\left[x\right]{\left|\right|}_{L^{\infty }[-1,1]}$ where $\left(S^{a}f\right){\left(t\right)}^{\wedge }\left(\xi \right)=e^{{it\left|\xi \right|}^a}f̂\left(\xi \right)$ and 0 < a < 1. We prove the global estimate $\left|\right|S^{a}f{\left|\right|}_{L²(ℝ,L^{\infty }[-1,1])}{\le C\left|\right|f\left|\right|}_{H^s\left(ℝ\right)}$, s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$, there exists a $p_0\in (n/(n+1),1)$ such that for certain classes of distributions, the $L^p\left(𝒳\right)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p\in (p_0,\infty ]$. This result yields a radial maximal function characterization for Hardy spaces on $𝒳$. ...

In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form $$\Delta \left[\begin{array}{c}x\left(n\right)+p\left(n\right)x(n-m)\\ y\left(n\right)+p\left(n\right)y(n-m)\end{array}\right]=\left[\begin{array}{cc}a\left(n\right)& b\left(n\right)\\ c\left(n\right)& d\left(n\right)\end{array}\right]\left[\begin{array}{c}x(n-\alpha )\\ y(n-\beta )\end{array}\right]$$ are established, where $m>0$, $\alpha \ge 0$, $\beta \ge 0$ are integers and $a\left(n\right)$, $b\left(n\right)$, $c\left(n\right)$, $d\left(n\right)$, $p\left(n\right)$ are sequences of real numbers.

This paper is concerned with the delay partial difference equation (1) $A_{m+1,n}+A_{m,n+1}-A_{m,n}+{\Sigma }_{i=1}^u{p}_i{A}_{m-k_i,n-l_i}=0$ where $p_i$ are real numbers, $k_i$ and $l_i$ are nonnegative integers, u is a positive integer. Sufficient and necessary conditions for all solutions of (1) to be oscillatory are obtained.

Consider the first-order linear delay (advanced) differential equation$$x^{\text{'}}\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0(x^{\text{'}}\left(t\right)-q\left(t\right)x\left(\sigma \left(t\right)\right)=0),t\ge t_0,$$ where $p$ $\left(q\right)$ is a continuous function of nonnegative real numbers and the argument $\tau \left(t\right)$ $\left(\sigma \right(t\left)\right)$ is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions$$\underset{t\to \infty }{lim\; sup}{\int }_{\tau \left(t\right)}^{t}p\left(s\right)\mathrm{d}s>1\left(\underset{t\to \infty }{lim\; sup}{\int }_t^{\sigma \left(t\right)}q\left(s\right)\mathrm{d}s>1\right)$$ and $$\underset{t\to \infty }{lim\; inf}{\int }_{\tau \left(t\right)}^{t}p\left(s\right)\mathrm{d}s>\frac{1}{\mathrm{e}}\left(\underset{t\to \infty }{lim\; inf}{\int }_t^{\sigma \left(t\right)}q\left(s\right)\mathrm{d}s>\frac{1}{\mathrm{e}}\right)$$ are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known...

We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p,q}$, 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p,q,r}$ such that for any $\varphi \in L^{p,q}$, ${\left|\right|ℳ\varphi \left|\right|}_{r,\infty }\le C_{p,q,r}{\left|\right|\varphi \left|\right|}_{p,q}$.

Let Y be a Banach space and let $S\subset L_p$ be a subspace of an $L_p$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S\left(Y\right)\subset L_p\left(Y\right)$. We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{-tB}$ is a positive contraction...

Sufficient oscillation conditions involving $lim\; sup$ and $lim\; inf$ for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.