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.

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.

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 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.

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....

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.

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.

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...

Improving on a theorem of Heath-Brown, we show that if X is sufficiently large then a positive proportion of the values n³ + 2: n ∈ (X,2X] have a prime factor larger than $X^{1+{10}^{-52}}$.

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 consider elementary operators $x\mapsto {\sum }_{j=1}^n{a}_{j}x{b}_j$, acting on a unital Banach algebra, where $a_j$ and $b_j$ are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families $a_j$ and $b_j$, i.e. $a_j=a_j^{\text{'}}+i{a}_j^{\text{'}\text{'}}$ ($b_j=b_j^{\text{'}}+i{b}_j^{\text{'}\text{'}}$), where all $a_j^{\text{'}}$ and $a_j^{\text{'}\text{'}}$ ($b_j^{\text{'}}$ and $b_j^{\text{'}\text{'}}$) commute. The main tool is an L¹ estimate of the Fourier transform of a certain class...

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.

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.

We determine the duals of the homogeneous matrix-weighted Besov spaces ${Ḃ}_p^{\alpha q}\left(W\right)$ and ${ḃ}_p^{\alpha q}\left(W\right)$ which were previously defined in [5]. If W is a matrix $A_p$ weight, then the dual of ${Ḃ}_p^{\alpha q}\left(W\right)$ can be identified with ${Ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(W^{-p^{\text{'}}/p}\right)$ and, similarly, $\left[{ḃ}_p^{\alpha q}\left(W\right)\right]*\approx {ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(W^{-p^{\text{'}}/p}\right)$. Moreover, for certain W which may not be in the $A_p$ class, the duals of ${Ḃ}_p^{\alpha q}\left(W\right)$ and ${ḃ}_p^{\alpha q}\left(W\right)$ are determined and expressed in terms of the Besov spaces ${Ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(A_Q^{-1}\right)$ and ${ḃ}_{p^{\text{'}}}^{-\alpha q^{\text{'}}}\left(A_Q^{-1}\right)$, which we define in terms of reducing operators ${A_Q}_Q$ associated with W. We also develop the basic theory of these reducing operator Besov spaces....