Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, $E{*}_{\omega *}$ be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put $G\left(x\right):={\int }_{\Omega }g(s,x\left(s\right))d\mu \left(s\right)$. Consider the integral functional G defined on some non-$L^p$-type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued...

Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into ${ℝ}^d$. The paper deals with Y-weak cluster points ϕ̅ of the sequence $\varphi (·,z_j\left(·\right))$ in X, where $z_j:\Omega \to {ℝ}^m$ is measurable for j ∈ ℕ and $\varphi :\Omega \times {ℝ}^m\to {ℝ}^d$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_{\varphi }$, the integral $I(\varphi ,{\nu }_x):={\int }_{{ℝ}^m}\varphi (x,\lambda )d{\nu }_x\left(\lambda \right)$ exists for $x\in \Omega \setminus A_{\varphi }$ and $\varphi ̅\left(x\right)=I(\varphi ,{\nu }_x)$ on $\Omega \setminus A_{\varphi }$, where $\nu ={{\nu }_x}_{x\in \Omega }$ is a measurable-dependent family of Radon probability measures on ${ℝ}^m$.

In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions y = y”” - (A² + B²)y” + A²B²y ∈ F(t,y), (*) with the initial conditions y(0) = y’(0) = y”(0) = y”’(0) = 0, (**) where the matrices $A,B\in {ℝ}^{d\times d}$ are commutative and the multifunction $F:[0,1]\times {ℝ}^d⇝cl\left({ℝ}^d\right)$ is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that $F:[0,1]\times {ℝ}^d⇝cl\left({ℝ}^d\right)ismeasurableintandintegrablybounded.Let$y₀ ∈ W4,1$beanarbitraryfunctionsatisfying(**)andsuchthat$ $d_H(y₀\left(t\right),F(t,y₀\left(t\right)))\le p₀\left(t\right)$ a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*)...

The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: $in{f}_{y\in Y}su{p}_{x\in X}f(x,y)\le su{p}_{x\in X}in{f}_{y\in Y}g(x,y)$. We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $su{p}_{x\in X}f(x,y)\le su{p}_{x\in X}g(x,y)$, ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some...

We give some equivalent conditions (independent from the Young functions) for inclusions between some classes of $X^{\Phi }$ spaces, where $\Phi$ is a Young function and $X$ is a quasi-Banach function space on a $\sigma$-finite measure space $(\Omega ,𝒜,\mu )$.

Let $H\subseteq Z\subseteq 2^{\omega }$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{\omega }$ to Z give as preimages of H all Σₙ¹ subsets of $2^{\omega }$, then so do continuous injections.

Let G be a locally compact group, let (φ,ψ) be a complementary pair of Young functions, and let $L^{\phi }\left(G\right)$ and $L^{\psi }\left(G\right)$ be the corresponding Orlicz spaces. Under some conditions on φ, we will show that for a Banach $L^{\phi }\left(G\right)$-submodule X of $L^{\psi }\left(G\right)$, the multiplier space $Ho{m}_{L^{\phi }\left(G\right)}(L^{\phi }\left(G\right),X*)$ is a dual Banach space with predual $L^{\phi }\left(G\right)\bullet X:=\overline{span}ux:u\in L^{\phi }\left(G\right),x\in X$, where the closure is taken in the dual space of $Ho{m}_{L^{\phi }\left(G\right)}(L^{\phi }\left(G\right),X*)$. We also prove that if $\phi$ is a Δ₂-regular N-function, then $C{v}_{\phi }\left(G\right)$, the space of convolutors of $M^{\phi }\left(G\right)$, is identified with the dual of a Banach algebra of functions on G...

The lower semicontinuity of functionals of the type ${\int }_{\Omega }f(x,u,v,\nabla u)dx$ with respect to the $\left(W^{1,1}\times L^p\right)$-weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1,1}\times L^p$ for the lower semicontinuous envelope is also provided.

Let $L:=-\Delta +V$ be a Schrödinger operator on ${ℝ}^n$ with $n\ge 3$ and $V\ge 0$ satisfying ${\Delta }^{-1}V\in L^{\infty }\left({ℝ}^n\right)$. Assume that $\varphi :{ℝ}^n\times [0,\infty )\to [0,\infty )$ is a function such that $\varphi (x,·)$ is an Orlicz function, $\varphi (·,t)\in {𝔸}_{\infty }\left({ℝ}^n\right)$ (the class of uniformly Muckenhoupt weights). Let $w$ be an $L$-harmonic function on ${ℝ}^n$ with $0<C_1\le w\le C_2$, where $C_1$ and $C_2$ are positive constants. In this article, the author proves that the mapping $H_{\varphi ,L}\left({ℝ}^n\right)\ni f\mapsto wf\in H_{\varphi }\left({ℝ}^n\right)$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi ,L}\left({ℝ}^n\right)$, to the Musielak-Orlicz-Hardy space $H_{\varphi }\left({ℝ}^n\right)$ under some assumptions on $\varphi$. As applications, the author further...

Let $M(X,𝒜)$ ($M^{*}(X,𝒜)$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X,𝒜)$, let $M_K(X,𝒜)$ be the family of all $f\in M(X,𝒜)$ such that $\mathrm{coz}\left(f\right)$ is compact, and let $M_{\infty }(X,𝒜)$ be all $f\in M(X,𝒜)$ that $\{x\in X:|f\left(x\right)|\ge \frac{1}{n}\}$ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X,𝒜)$, we show that $M^{*}(X,𝒜)$ is a realcompact subring of $M(X,𝒜)$, and also $M(X,𝒜)$ is a realcompact if and only if $(X,𝒜)$ is a compact measurable space. For every nonzero real Riesz map $\varphi :M(X,𝒜)\to ℝ$, we prove that there is an element $x_0\in X$ such that $\varphi \left(f\right)=f\left(x_0\right)$ for every $f\in M(X,𝒜)$ if $(X,𝒜)$ is a compact measurable space....

In this review article we present an overview on some a priori estimates in $L^p$, $p>1$, recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^p$-bounds, $p>2$, for...

For the dynamics $x^{\text{'}\text{'}}=-{\nabla }_{x}V\left(x\right)$, an equilibrium point $\underline{x}$ are always unstable when on a neighborhood of $\underline{x}$ the non constant $V$ satisfies $P(x,\partial )V=0$ for a real second order elliptic $P$. The proof uses a result of Kozlov [6].

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...