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In this paper we show that each composition operator is isolated in comp(Lp) (1 ≤ p ≤ ∞) under the norm topology. We also prove that while each composition operator is non-isolated in comp(≤p) (1 ≤ p < ∞) under the strong operator topology, every composition operator is isolated in comp(L∞) under the strong operator topology. [ABSTRACT FROM AUTHOR]

The following model p(t)u′(t)−p(0)β−∫0t((gu)(s)+∑i=1m(ξi(s)x(νi(s))−ψi(s)x(μi(s))))dQ(s)=−F(t)of the Stieltjes string on the segment [0,l]with a nonlocal initial value condition u(0)=αis considered. D-stability conditions of the unique solution of the mentioned problem are established. Also, example is presented.

We study Beck-like coloring of measurable functions on a measure space O taking values in a measurable semigroup ?. To any measure space O and any measurable semigroup ?, we assign a graph (called a zero-divisor graph) whose vertices are labeled by the classes of measurable functions defined on O and having values in ?, with two vertices f and g adjacent if f - g = 0 a.e. We show that, if O is atomic, then not only the Beck's conjecture holds but also the domination number coincides to the clique number and chromatic number as well. We also determine some other graph properties of such a graph. [ABSTRACT FROM AUTHOR]

Being motivated by Dini's Theorem, the uniform convergence of a sequence of functions is characterized by that of another decreasing sequence of functions. This result allows us to describe the uniform convergence of a sequence of measurable functions on a closed interval in terms of that of a decreasing sequence of Lebesgue integrable functions. For a sequence of continuous functions on a compact set, its uniform convergence is further characterized by the pointwise convergence of another decreasing sequence of continuous functions. It also turns out that the uniform convergence of a sequence of Riemann integrable functions on a closed interval can be determined by that of another decreasing sequence of Riemann integrable functions. As applications, characterizations for a uniformly convergent series of functions have been presented and one fixed point theorem has been established. [ABSTRACT FROM AUTHOR]

In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function F. Also, we establish some new double-weighted inequalities, which are particular cases of the main result and are represented by corollaries. These inequalities are further refinement of all other inequalities associated with increasing, positive, monotone and measurable function existing in literature. The existing inequalities associated with increasing, positive, monotone and measurable function are also restored by applying specific conditions as given in Remarks. Many other types of fractional integral inequalities can be obtained by applying certain conditions on F and Ψ given in the literature.

Let A be a zero measure dense Gδ subset of I=[0,1], with the set of measurable self-maps of I. There exists a residual set R⊂M such that for each f in R, the range of f is contained in A, and the function f is one-to-one. Moreover, there exists h:I→I, a Baire-2 function, such that f(x)=h(x) a.e., and for any x∈I, the trajectory τ(x,h) is ∞-adic, so that the ω-limit set ω(x,h) is a Cantor set. Since the range of f is contained in A, it follows that for any x in I, there exists y in A such that the trajectory τ(f(x),f)=τ(y,f)⊂A. Speaking loosely, the dynamical structures of f are completely determined by its behavior on the set A. [ABSTRACT FROM AUTHOR]

In this paper we introduce the metric function on L0(Ω, F, μ) be the space of measurable functions defined on measure space (Ω, F, μ) .We prove the converge in this metric implies converge in measure ,but the converse not true only if μ is finite and prove L0(Ω, F, μ) with respect metric functions defined by d(f,g)=ʃΩ |f-g|/1+|f-g| dμ for all f,g ∈L0(Ω, F, μ) is complete metric space and a topological linear space and F-space . [ABSTRACT FROM AUTHOR]

In this work we introduce and study a class of linear operators related to a finite measure space, for which its L 2 -Hilbert space is separable. These linear operators represent a generalization of pseudo-differential operators on S 1 , where S 1 is the unit circle with centre at the origin. We call these operators, S-operators or generalized pseudo-differential operators on S, where (S , B , m) is the corresponding finite measure space for which the L 2 -space is separable. We give some L 2 -boundedness and compactness results, and we also study the Hilbert–Schmidt property in connection with this class of linear operators. In the end we give some examples of finite measure spaces and corresponding S-operators for which we can apply our results. [ABSTRACT FROM AUTHOR]

An operator T is said to be k-quasi class A*n operator if T*k(|Tn+1|2/n+1 - |T*|2) Tk ≥ 0, for some positive integers n and k. In this paper, we prove that the k-quasi class A* n operators have Bishop;s property (ß). Then, we give a necessary and suffcient condition for T⊗S to be a k-quasi class A*n operator, whenever T and S are both non-zero operators. Moreover, it is shown that the Riesz idempotent for a non-zero isolated point λ0 of a k-quasi class A*n operator T say Ri, is self-adjoint and ran(Ri) = ker(T -λ0) = ker(T -λ0)*. Finally, as an application in the last section, a necessary and suffcient condition is given in such a way that the weighted conditional type operators on L2(Σ), defined by Tw,u(f) := wE(uf), belong to k-quasi- A*n class. [ABSTRACT FROM AUTHOR]

We study (semi)measurable functions on σ -frames, extending the theory of real-valued functions from frames to σ -frames. The new objects of study are the σ -frame homomorphisms ▪ from the usual frame of reals into the congruence lattice of a σ -frame L , and its subclass of measurable functions L (R) → L. The desired extension faces two obstacles: (1) in general, σ -frames have no uncountable joins and are not pseudocomplemented; (2) σ -sublocales, that is, the subobjects in the dual category of the category of σ -frames and σ -frame homomorphisms, cannot be described as concrete subsets of the σ -frame L , unlike their counterpart in the category of locales, forcing us to work in the congruence lattice of L. Nevertheless, it is shown that, despite (1), the familiar method for generating real functions on frames via scales can be extended to arbitrary σ -frames. This is achieved by the new notions of σ -scale and finite σ -scale. It is also shown that, despite (2), the familiar insertion, extension and separation results for real-valued functions in several classes of frames (normal, extremally disconnected, G -perfect, F -perfect, perfectly normal) can be proved without involving uncountable joins and pseudocomplements, thus allowing their extension to measurable and semimeasurable functions. [ABSTRACT FROM AUTHOR]