Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form $$\left(\right|y^{\text{'}\text{'}}{|}^{\alpha }\mathrm{sgn}{y}^{\text{'}\text{'}}{)}^{\text{'}\text{'}}+q\left(t\right){\left|y\right|}^{\alpha }\mathrm{sgn}y=0,t\ge a>0,A$$ where $\alpha >0$ is a constant and $q\left(t\right)$ is positive continuous function on $[a,\infty )$, are given in terms of an increasing continuously differentiable function $\omega \left(t\right)$ from $[a,\infty )$ to $(0,\infty )$ which satisfies ${\int }_a^{\infty }1/\left(t\omega \left(t\right)\right)dt<\infty$.

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...

The second order linear differential equation $${\left(p\left(x\right)y^{\text{'}}\right)}^{\text{'}}+q\left(x\right)y=0,x\in (0,x_0]$$ is considered, where $p$, $q\in C^1(0,x_0]$, $p\left(x\right)>0$, $q\left(x\right)>0$ for $x\in (0,x_0]$. Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near $x=0$ without the Hartman–Wintner condition.

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.

We show in two dimensions that if $Kf={\int }_{ℝ₊²}k(x,y)f\left(y\right)dy$, $k(x,y)=\left(e^{i{x}^a·y^b}\right){/\left(\right|x-y|}^{\eta })$, p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), $v_p\left(y\right)=y^{(p/p^{\text{'}})(1̅-b/a)}$, then ${\left|\right|Kf\left|\right|}_p{\le C\left|\right|f\left|\right|}_{p,v_p}$ if η + α₁ + α₂ < 2, ${\alpha }_j=1-b_j/a_j$, j = 1,2. Our methods apply in all dimensions and also for more general kernels.

The half-linear differential equation $$\left(\right|u^{\text{'}}{|}^{\alpha }\mathrm{sgn}{u}^{\text{'}}{)}^{\text{'}}=\alpha ({\lambda }^{\alpha +1}+b\left(t\right)){\left|u\right|}^{\alpha }\mathrm{sgn}u,t\ge t_0,$$ is considered, where $\alpha$ and $\lambda$ are positive constants and $b\left(t\right)$ is a real-valued continuous function on $[t_0,\infty )$. It is proved that, under a mild integral smallness condition of $b\left(t\right)$ which is weaker than the absolutely integrable condition of $b\left(t\right)$, the above equation has a nonoscillatory solution $u_0\left(t\right)$ such that $u_0\left(t\right)\sim {\mathrm{e}}^{-\lambda t}$ and $u_0^{\text{'}}\left(t\right)\sim -\lambda {\mathrm{e}}^{-\lambda t}$ ($t\to \infty$), and a nonoscillatory solution $u_1\left(t\right)$ such that $u_1\left(t\right)\sim {\mathrm{e}}^{\lambda t}$ and $u_1^{\text{'}}\left(t\right)\sim \lambda {\mathrm{e}}^{\lambda t}$ ($t\to \infty$).

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.

For the equation $$y^{\left(n\right)}+{\left|y\right|}^k\mathrm{sgn}y=0,k>1,n=3,4,$$ existence of oscillatory solutions $$y={(x^{*}-x)}^{-\alpha }h(log(x^{*}-x)),\alpha =\frac{n}{k-1},x<x^{*},$$ is proved, where $x^{*}$ is an arbitrary point and $h$ is a periodic non-constant function on $ℝ$. The result on existence of such solutions with a positive periodic non-constant function $h$ on $ℝ$ is formulated for the equation $$y^{\left(n\right)}={\left|y\right|}^k\mathrm{sgn}y,k>1,n=12,13,14.$$

In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation $${(y\left(t\right)-\sum _{i=1}^k{p}_i\left(t\right)y\left(r_i\left(t\right)\right))}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ oscillates or tends to zero as $t\to \infty$, where, $n\ge 1$ is any positive integer, $p_i$, $r_i\in C^{\left(n\right)}([0,\infty ),ℝ)$  and $p_i$ are bounded for each $i=1,2,\cdots ,k$. Further, $f\in C\left(\right[0,\infty ),ℝ)$, $g$, $h$, $v$, $u\in C\left(\right[0,\infty ),[0,\infty \left)\right)$, $G$ and $H\in C(ℝ,ℝ)$. The functional delays $r_i\left(t\right)\le t$, $g\left(t\right)\le t$ and $h\left(t\right)\le t$ and all of them approach $\infty$ as $t\to \infty$. The results hold when $u\equiv 0$ and $f\left(t\right)\equiv 0$. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature. ...

Let d be a positive integer and μ a generalized Cantor measure satisfying $\mu ={\sum }_{j=1}^m{a}_j\mu \circ S_j^{-1}$, where $0<a_j<1$, ${\sum }_{j=1}^m{a}_j=1$, $S_j=\rho R+b_j$ with 0 < ρ < 1 and R an orthogonal transformation of ${ℝ}^d$. Then ⎧1 < p ≤ 2 ⇒ ⎨$su{p}_{r>0}{r}^{d(1/{\alpha }^{\text{'}}-1/p^{\text{'}})}({\int }_{J_x^r}{\left|\mu ̂\left(y\right)\right|}^{p^{\text{'}}}{dy)}^{1/p^{\text{'}}}\le D₁{\rho }^{-d/{\alpha }^{\text{'}}}$, $x\in {ℝ}^d$, ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’$,$where $J_x^r={\prod }_{i=1}^d(x_i-r/2,x_i+r/2)$, α’ is defined by ${\rho }^{d/{\alpha }^{\text{'}}}={\left({\sum }_{j=1}^m{a}_j^p\right)}^{1/p}$ and the constants D₁ and D₂ depend only on d and p.

We extend a result of the second author [27, Theorem 1.1] to dimensions $d\ge 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<2(d+1)/d-1$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "$ϵ$ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] $L^2$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive...

Let $r(G_1,G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V\left(S(n_1,n_2)\right)=\{v_0,v_1,...,v_{n_1},w_0$, $w_1,...,w_{n_2}\}$ and $E\left(S(n_1,n_2)\right)=\{v_0{v}_1,...,v_0{v}_{n_1},v_0{w}_0,w_0{w}_1,...,w_0{w}_{n_2}\}$. We determine $r(K_{1,m-1},$ $S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{\text{'}\text{'}}=(V,E_2)$ and $T_n^{\text{'}\text{'}\text{'}}=(V,E_3)$, where $V=\{v_0,v_1,...,v_{n-1}\}$, $E_2=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3}$, $v_1{v}_{n-2},v_2{v}_{n-1}\}$ and $E_3=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3},$ $v_2{v}_{n-2},v_3{v}_{n-1}\}$. We also obtain explicit formulas for $r(K_{1,m-1},T_n)$, $r(T_m^{\text{'}},T_n)$ $(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n^{\text{'}},T_n)$ and $r(P_n,T_n)$, where $T_n\in \{T_n^{\text{'}\text{'}},T_n^{\text{'}\text{'}\text{'}},T_n^3\}$, $P_n$ is the path on $n$ vertices and $T_n^{\text{'}}$ is the unique tree with $n$ vertices and maximal degree $n-2$.