We investigate the existence of positive solutions for a nonlinear second-order differential system subject to some m-point boundary conditions. The nonexistence of positive solutions is also studied.
We study the boundary value problem $-{\mathrm div}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in ${\mathbb{R}} ^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x)=\max \lbrace p_1(x),p_2(x)\rbrace $ for any $x\in \overline{\Omega }$ or $m(x)<q(x)< \frac{N\cdot m(x)}{(N-m(x))}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda $ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}} _2$-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.
We say that a real normed lattice is quasi-Baire if the intersection of each sequence of monotonic open dense sets is dense. An example of a Baire-convex space, due to M. Valdivia, which is not quasi-Baire is given. We obtain that $E$ is a quasi-Baire space iff $(E, T({\mathcal U}),T({\mathcal U}^{-1}))$, is a pairwise Baire bitopological space, where $\mathcal U$, is a quasi-uniformity that determines, in $L$. Nachbin’s sense, the topological ordered space $E$.
The actual construction of the Szász-Mirakyan operators and its various modifications require estimations of infinite series which in a certain sense restrict their usefulness from the computational point of view. Thus the question arises whether the Szász-Mirakyan operators and their generalizations cannot be replaced by a finite sum. In connection with this question we propose a new family of linear positive operators.
We assign to each positive integer $n$ a digraph whose set of vertices is $H=\lbrace 0,1,\dots ,n-1\rbrace $ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^2\equiv b\hspace{4.44443pt}(\@mod \; n)$. We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.
Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see \cite{Combs_Andrews_1998,Jayaram2008} and \cite{Baczynski_Jayaram_2008}). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see \cite{Baczynski_Jayaram_2009})
f(min(x+y,a))=min(f(x)+f(y),b),
where a,b>0 and f:[0,a]→[0,b] is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation
f(m1(x+y))=m2(f(x)+f(y)),
where m1,m2 are functions defined on some intervals of \R satisfying additional assumptions. We analyze the cases when m2 is injective and when m2 is not injective.
Let $C$ be a closed convex subset of a complete convex metric space $X$. In this paper a class of selfmappings on $C$, which satisfy the nonexpansive type condition $(2)$ below, is introduced and investigated. The main result is that such mappings have a unique fixed point.
A theorem of Burnside asserts that a finite group G is p-nilpotent if for some prime p a Sylow p-subgroup of G lies in the center of its normalizer. In this paper, let G be a finite group and p the smallest prime divisor of |G|, the order of G. Let P \in Syl_{p} (G). As a generalization of Burnside’s theorem, it is shown that if every non-cyclic p-subgroup of G is self-normalizing or normal in G then G is solvable. In particular, if P \not\cong \left\langle {a,b;{a^{{p^{n - 1}}}} = 1,{b^2} = 1,{b^{ - 1}}ab = {a^{1 + {p^{n - 2}}}}} \right\rangle, where n\geq 3 for p > 2 and n\geq 4 for p = 2, then G is p-nilpotent or p-closed., Jiangtao Shi., and Obsahuje seznam literatury
We consider the functional equation f(xf(x)) = ϕ(f(x)) where ϕ: J → J is a given increasing homeomorphism of an open interval J ⊂ (0, ∞) and f : (0, ∞) → J is an unknown continuous function. In a previous paper we proved that no continuous solution can cross the line y = p where p is a fixed point of ϕ, with a possible exception for p = 1. The range of any non-constant continuous solution is an interval whose end-points are fixed by ϕ and which contains in its interior no fixed point except for 1. We also gave a characterization of the class of continuous monotone solutions and proved a sufficient condition for any continuous function to be monotone. In the present paper we give a characterization of the equations (or equivalently, of the functions ϕ) which have all continuous solutions monotone. In particular, all continuous solutions are monotone if either (i) 1 is an end-point of J and J contains no fixed point of ϕ, or (ii) 1 ∈ J and J contains no fixed points different from 1.
In this paper we deal with a homogeneity condition for an $MV$-algebra concerning a generalized cardinal property. As an application, we consider the homogeneity with respect to $\alpha $-completeness, where $\alpha $ runs over the class of all infinite cardinals.