In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if f : X × Y × Z −→ W is a bounded tri-linear mapping and h : W −→ S is a bounded linear mapping, then f is regular if and only if hof is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint D^∗∗∗∗ of a tri-derivation D is again tri-derivation.

‎Let $X$ be a real normed  space, then  $C(subseteq X)$  is  functionally  convex  (briefly, $F$-convex), if  $T(C)subseteq Bbb R $ is  convex for all bounded linear transformations $Tin B(X,R)$; and $K(subseteq X)$  is  functionally   closed (briefly, $F$-closed), if  $T(K)subseteq Bbb R $ is  closed  for all bounded linear transformations $Tin B(X,R)$. We improve the    Krein-Milman theorem  ...

for any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. the condition is that every minimal ideal of the ring consist entirely of bounded elements. it is not too stringent, and is satisfied, for instance, by rings of conti...

‎We determine the‎ ‎form of polynomially bounded solutions to the Loewner differential ‎equation that is satisfied by univalent subordination chains of the‎ ‎form $f(z,t)=e^{int_0^t A(tau){rm d}tau}z+cdots$‎, ‎where‎ ‎$A:[0,infty]rightarrow L(mathbb{C}^n,mathbb{C}^n)$ is a locally‎ ‎Lebesgue integrable mapping and satisfying the condition‎ ‎$$sup_{sgeq0}int_0^inftyleft|expleft{int_s^t‎ ‎[A(tau)...

In this paper, we are concerned with the following fractional Schrödinger-Poisson system:    (−∆s)u + V (x)u + φu = m(x)|u|q−2|u|+ f(x,u), x ∈ Ω, (−∆t)φ = u2, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where s,t ∈ (0,1], 2t + 4s > 3, 1 < q < 2 and Ω is a bounded smooth domain of R3, and f(x,u) is linearly bounded in u at infinity. Under some assumptions on m, V and f we obtain the existence of non-trivial so...

For any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. The condition is that every minimal ideal of the ring consist entirely of bounded elements. It is not too stringent, and is satisfied, for instance, by rings of ...

‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...

‎we determine the‎ ‎form of polynomially bounded solutions to the loewner differential ‎equation that is satisfied by univalent subordination chains of the‎ ‎form $f(z,t)=e^{int_0^t a(tau){rm d}tau}z+cdots$‎, ‎where‎ ‎$a:[0,infty]rightarrow l(mathbb{c}^n,mathbb{c}^n)$ is a locally‎ ‎lebesgue integrable mapping and satisfying the condition‎ ‎$$sup_{sgeq0}int_0^inftyleft|expleft{int_s^t‎ ‎[a(tau)...