We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $$a\left(t\right)x^{\text{'}\text{'}}+b\left(t\right)g(x,x^{\text{'}})+c\left(t\right)h\left(x(t-r)\right)m\left(x^{\text{'}}\right)=p(t,x,x^{\text{'}})$$ and $${\left(a\left(t\right)x^{\text{'}}\right)}^{\text{'}}+b\left(t\right)g(x,x^{\text{'}})+c\left(t\right)h\left(x(t-r)\right)m\left(x^{\text{'}}\right)=p(t,x,x^{\text{'}}),$$ where $a$, $b$, $c$, $g$, $h$, $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski functional to establish our results....

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.

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.

We study the existence of positive solutions to the fourth-order two-point boundary value problem $$\left\{\begin{array}{cc}{u}^{\text{'}\text{'}\text{'}\text{'}}\left(t\right)+f(t,u\left(t\right))=0,\hfill & 0<t<1,\hfill \\ u^{\text{'}}\left(0\right)=u^{\text{'}}\left(1\right)=u^{\text{'}\text{'}}\left(0\right)=0,\hfill & u\left(0\right)=\alpha \left[u\right],\hfill \end{array}\right.$$ where $\alpha \left[u\right]={\int }_0^{1}u\left(t\right)\mathrm{d}A\left(t\right)$ is a Riemann-Stieltjes integral with $A\ge 0$ being a nondecreasing function of bounded variation and $f\in 𝒞([0,1]\times {ℝ}_{+},{ℝ}_{+})$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

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 ${\int }_a^{\infty }1/\left(t\omega \left(t\right)\right)dt<\infty$.

Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\text{sgn}x\ge 0$ $for|x|>R$, si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

This paper concerns the following system of nonlinear third-order boundary value problem: $$u_i^{\text{'}\text{'}\text{'}}\left(t\right)+f_i(t,u_1\left(t\right),\cdots ,u_n\left(t\right),u_1^{\text{'}}\left(t\right),\cdots ,u_n^{\text{'}}\left(t\right))=0,0<t<1,i\in \{1,\cdots ,n\}$$ with the following multi-point and integral boundary conditions: $$\left\{\begin{array}{c}{u}_i\left(0\right)=0u_i^{\text{'}}\left(0\right)=0u_i^{\text{'}}\left(1\right)=\sum _{j=1}^p{\beta }_{j,i}{u}_i^{\text{'}}\left({\eta }_{j,i}\right)+{\int }_0^1{h}_i(u_1\left(s\right),\cdots ,u_n\left(s\right))ds\hfill \end{array}\right.$$ where ${\beta }_{j,i}>0$, $0<{\eta }_{1,i}<\cdots <{\eta }_{p,i}<\frac{1}{2}$, $f_i:[0,1]\times {ℝ}^n\times {ℝ}^n\to ℝ$ and $h_i:[0,1]\times {ℝ}^n\to ℝ$ are continuous functions for all $i\in \{1,\cdots ,n\}$ and $j\in \{1,\cdots ,p\}$. Using Guo-Krasnosel’skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.

For a double complex $(A,d^{\text{'}},d^{\text{'}\text{'}})$, we show that if it satisfies the $d^{\text{'}}{d}^{\text{'}\text{'}}$-lemma and the spectral sequence $\left\{E_r^{p,q}\right\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d^{\text{'}}{d}^{\text{'}\text{'}}$-lemma.

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...

Let $p_1,p_2,\cdots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k\ge 5$, then $$(p_{k+1}-1)!\mid \left(\frac{1}{2}(p_{k+1}-1)\right)!p_k!,$$ which improves a previous result of the second author.

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...

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^{\text{'}}$ (the magnetization) of two variables $x^{\text{'}}$: $E_{\epsilon }\left(m^{\text{'}}\right)=\epsilon \int {|{\nabla }^{\text{'}}·m^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla }^{\text{'}}{|}^{-1/2}{\nabla }^{\text{'}}·m^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be $S^1$-valued maps $m^{\text{'}}$ of vanishing distributional divergence ${\nabla }^{\text{'}}·m^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel...

Viene dimostrata l'esistenza di soluzioni del problema di Darboux per l'equazione iperbolica $z_{xy}^{\mathrm{\prime \prime }}=f(x,y,z,Z_x^{\prime },z_y)$ sul planiquarto $x\ge 0$, $y\ge 0$. Qui, $f$ è una funzione continua, con valori in uno spazio Banach che soddisfano alcune condizioni di regolarità espresse in termini della misura di non-compattezza $\alpha$.

We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_q\left(n\right)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${\left(\alpha s_q\left(P\left(n\right)\right)\right)}_{n\in \mathbb{N}}$ is uniformly distributed modulo 1.