‎In the present‎ ‎paper we introduce the notion Bochner integral for measurable sections and study some properties such integrals‎. ‎Given necessary and successfully condition for integrability of a‎ ‎measurable section‎. ‎Dominated convergence theorem and analogue of‎ ‎Hille's theorem are proved‎.

‎Given a completely rational conformal net $\mathcal{A}$ on $S^1$‎, ‎its fusion‎ ‎ring acts faithfully on the K-group $K_0(\mathfrak{K}_{\mathcal{A}})$ of a certain‎ ‎universal $C^*$-algebra $\mathfrak{K}_{\mathcal{A}}$ associated to $\mathcal{A}$‎, ‎as shown in a‎ ‎previous paper‎. ‎We prove here that this action can actually be‎ ‎identified with a Kasparov product‎, ‎thus paving the way for a‎ ‎fruitful interplay between conformal field theory and KK-theory‎.

‎We offer a global bound for an abstract Jensen's functional‎. ‎Particularly‎, ‎the results from Simić [Rocky Mount‎. ‎J‎. ‎Math.‎, ‎41‎ ‎(2011)‎, ‎no‎. ‎6‎, ‎2021-2031] are reobtained‎. ‎Applications to integral‎ ‎inequalities and the theory of means are pointed out‎.

‎In this paper, we study eigenvalue problem for a class of nonlinear fractional differential equations $$D^\alpha_{0^+}u(t)=\lambda f(u(t)),\quad 0 \lt t \lt 1,$$ $$u(0)=u(1)=u'(0)=u'(1)=0,$$ where $3 \lt \alpha\leq4$ is a real number, $D^\alpha_{0^+}$ is the Riemann-Liouville fractional derivative, $\lambda$ is a positive parameter and $f:(0,+\infty)\to(0,+\infty)$ is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results.

‎A Banach space operator $T$ satisfies property($Bgw$)‎, ‎a variant‎ ‎property$(gw)$‎, ‎if the complement in the approximate point spectrum‎ ‎$\sigma_{a}(T)$ of the semi-B-essential approximate point spectrum‎ ‎$\sigma_{SBF^{-}_{+}}(T)$ coincides with set of isolated eigenvalues‎ ‎of $T$ of finite multiplicity $E^{0}(T)$‎. ‎We also introduce‎ ‎properties $(Bb)$‎, ‎and property $(Bgb)$ in connection with Weyl type‎ ‎theorems‎, ‎which are analogous‎, ‎respectively‎, ‎to generalized‎ ‎Browder's theorem and property($gb$)‎. ‎We obtain relation among‎ ‎these new properties‎.

‎An asymmetric Fuglede-Putnam's Theorem for $w$-hyponormal operators and class $\mathcal{Y}$ operators is proved‎, ‎as a consequence of this result‎, ‎we obtain that the range of the generalized derivation induced by the above classes of‎ ‎operators is orthogonal to its kernel‎.

‎We show that a unital involutive Banach algebra‎, ‎with identity of‎ ‎norm one and continuous involution‎, ‎is a $C^*$-algebra‎, ‎with the‎ ‎given involution and norm‎, ‎if every continuous linear functional‎ ‎attaining its norm at the identity is positive‎.

Positive definiteness‎, ‎reproducing kernel Hilbert spaces‎, ‎integral operators and Mercer's‎ ‎theorem in its various formats are common topics in many branches of‎ ‎mathematics. In this paper we review and upgrade upon some recent‎ ‎results that always involve at least two of them and indicate a few‎ ‎directions in which additional research could be carried out‎.

‎In this article we present a generalization of the concept of‎ ‎function of bounded variation‎, ‎in the sense of Riesz‎, ‎for functions‎ ‎defined on a rectangle in $\mathbb{R}^{2}$‎, ‎which take values in a‎ ‎Banach space‎. ‎As applications‎, ‎we obtain generalizations of some‎ ‎results due to Chistyakov and a counterpart of the classical‎ ‎Riesz's Lemma‎.

‎In this paper we will give a new characterization of the inner product space which use the trigonometry‎. ‎We conclude that a normed space $\left( X,\left\Vert \cdot \right\Vert \right)$ is an inner product space if and only if there exists $\alpha\in\mathbb{R}\backslash \pi\mathbb{Q}$ so that‎ ‎$$\left\Vert x\cos\alpha‎ + ‎y\sin\alpha\right\Vert^{2}+\left\Vert y\cos\alpha‎ - ‎x\sin\alpha\right\Vert^{2}=\left\Vert x\right\Vert^{2}+\left\Vert y\right\Vert^{2},$$‎ ‎for any $x,y\in X$‎.

‎We suggest a concept of generalized angles in arbitrary real normed vector spaces‎. ‎We give for each real number a definition of an `angle' by means of the shape of the unit ball‎. ‎They all yield the well known Euclidean angle in the special case of real inner product spaces‎. ‎With these different‎ ‎angles we achieve a classification of normed spaces‎, ‎and we obtain a characterization of inner product‎ ‎spaces‎. ‎Moreover we consider this construction also for a generalization of normed spaces‎, ‎i.e‎. ‎for spaces which may have a non-convex unit ball‎.

‎In this paper‎, ‎among others‎, ‎we prove the following results:\\ $(1)$‎ ‎Let $(X,d)$ be a complete cone metric space partially ordered by‎ ‎$\sqsubseteq$ and $q$ be a c-distance on $X$‎. ‎Suppose $F‎ : ‎X \times‎ ‎X \to X$ and $g‎ : ‎X \to X$ be two continuous and commuting functions‎ ‎with $F(X \times X)\subseteq g(X)$.\ Let $F$ satisfy mixed‎ ‎g-monotone property and $q(F(x‎, ‎y)‎, ‎F(u‎, ‎v)) \preceq \frac{k}{2}‎ ‎(q(gx‎, ‎gu)+q(gy,gv))$ for some $k \in [0‎, ‎1)$ and all $x‎, ‎y‎, ‎u‎, ‎v‎ ‎\in X$ with $(gx \sqsubseteq gu)$ and $(gy \sqsupseteq gv)$ or $(gx‎ ‎\sqsupseteq gu)$ and $(gy \sqsubseteq gv)$.\ If there exist $x_0‎, ‎y_0 \in X$ satisfying $gx_0 \sqsubseteq F(x_0‎, ‎y_0)$ and $F(y_0‎, ‎x_0) \sqsubseteq gy_0$‎, ‎then there exist $x^*‎, ‎y^*\in X$ such that‎ ‎$F(x^*‎, ‎y^*) = gx^*$ and $F(y^*‎, ‎x^*) = gy^*$‎, ‎that is‎, ‎$F$ and $g$‎ ‎have a coupled coincidence point $(x^*‎, ‎y^*)$.\ $(2)$ If‎, ‎in $(1)$‎, ‎we replace completeness of $(X,d)$ by completeness of $(g(X),d)$ and‎ ‎commutativity‎, ‎continuity of mappings $F$ and $g$ by the condition‎: ‎$(i)$ for any nondecreasing sequence $\{x_n\}$ in $X$ converging to‎ ‎$x$ we have $x_n \sqsubseteq x$ for all $n$.\ $(ii)$ for any‎ ‎nonincreasing sequence $\{y_n\}$ in $Y$ converging to $y$ we have $y‎ ‎\sqsubseteq y_n$ for all $n$‎, ‎then $F$ and $g$ have a coupled‎ ‎coincidence point $(x^*,y^*)$‎.

‎In this paper we establish some general multiple Opial-type‎ ‎inequalities involving the Canavati fractional derivatives‎. ‎In some‎ ‎cases the best possible constants are discussed‎.

‎We characterize boundedness and compactness‎ ‎of weighted composition operators from the space of Cauchy integral transforms to logarithmic weighted-type spaces‎. ‎We also manage to compute‎ ‎norm of weighted composition operators‎ ‎acting between these spaces‎.

‎This survey is devoted to the theory of Abram Vilgelmovich Shtraus‎ ‎and his disciples and followers‎. ‎This theory studies the so-called‎ ‎generalized resolvents of symmetric and isometric operators in a‎ ‎Hilbert space and provides alternative formulas to the well-known‎ ‎Krein-type formulas‎.