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...

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...

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

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.

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.

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 → ∞.

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 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....

Consider the third order nonlinear dynamic equation $x^{\Delta \Delta \Delta }\left(t\right)+p\left(t\right)f\left(x\right)=g\left(t\right)$, (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation $\Delta ³x\left(n\right)+n^{\alpha }{\left|x\right|}^{\gamma }sgn\left(n\right)=(-1)ⁿn^c$, where α ≥ -1, γ > 0, c > 3, is oscillatory.

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 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.

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 $u_1^{\text{'}}\left(t\right)\sim \lambda {\mathrm{e}}^{\lambda t}$ ($t\to \infty$).

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.

We study the existence of solutions of the system $$\left\{\begin{array}{c}{\left({\phi }_1\left(u_1^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}=f_1(t,u_1\left(t\right),u_2\left(t\right),u_1^{\text{'}}\left(t\right),u_2^{\text{'}}\left(t\right)),\text{a.e.}t\in [0,T],{\left({\phi }_2\left(u_2^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}=f_2(t,u_1\left(t\right),u_2\left(t\right),u_1^{\text{'}}\left(t\right),u_2^{\text{'}}\left(t\right)),\text{a.e.}t\in [0,T],\hfill \end{array}\right.$$ submitted to nonlinear coupled boundary conditions on $[0,T]$ where ${\phi }_1,{\phi }_2:(-a,a)\to ℝ$, with $0<a<+\infty$, are two increasing homeomorphisms such that ${\phi }_1\left(0\right)={\phi }_2\left(0\right)=0$, and $f_i:[0,T]\times {ℝ}^4\to ℝ$, $i\in \{1,2\}$ are two $L^1$-Carathéodory functions. Using some new conditions and Schauder fixed point Theorem, we obtain solvability result.

Let $K_{s,t}$ be the complete bipartite graph with partite sets $\{x_1,...,x_s\}$ and $\{y_1,...,y_t\}$. A split bipartite-graph on $(s+s^{\text{'}})+(t+t^{\text{'}})$ vertices, denoted by ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$, is the graph obtained from $K_{s,t}$ by adding $s^{\text{'}}+t^{\text{'}}$ new vertices $x_{s+1},...,x_{s+s^{\text{'}}}$, $y_{t+1},...,y_{t+t^{\text{'}}}$ such that each of $x_{s+1},...,x_{s+s^{\text{'}}}$ is adjacent to each of $y_1,...,y_t$ and each of $y_{t+1},...,y_{t+t^{\text{'}}}$ is adjacent to each of $x_1,...,x_s$. Let $A$ and $B$ be nonincreasing lists of nonnegative integers, having lengths $m$ and $n$, respectively. The pair $(A;B)$ is potentially ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$-bigraphic if there is a simple bipartite graph containing ${\mathrm{SB}}_{s+s^{\text{'}},t+t^{\text{'}}}$ (with $s+s^{\text{'}}$ vertices $x_1,...,x_{s+s^{\text{'}}}$ in the part of size $m$...

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...