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In this paper, we apply the Cădariu–Radu method derived from the Diaz–Margolis theorem to investigate existence, uniqueness approximation of Ξ-Hilfer fractional differential equations, and Hypergeometric stability for both finite and infinite domains. An example is given to illustrate the main result for a fractional system. [ABSTRACT FROM AUTHOR]

Let A be a unital prime *-algebra with characteristic not 2 and with a projection e ≠ 0 , 1 . Suppose that d : A → Q s (A) is a map satisfying d (x y * z + z y * x 2) = d (x) y * z + x d (y) * z + x y * d (z) + d (z) y * x + z d (y) * x + z y * d (x) 2 for all x , y , z ∈ A , where Q s (A) denotes the symmetric Martindale algebra of quotients of A. It is shown that d is an additive triple derivation. Moreover, there exist a ∈ Q s (A) with a * = − a and an additive *-derivation δ : A → Q s (A) such that d (x) = a x + δ (x) for all x ∈ A . The analogous results for prime locally matrix *-algebras, prime *-algebras with nonzero socle, factor von Neumann algebras and standard operator *-algebras on Hilbert spaces are also described. [ABSTRACT FROM AUTHOR]

Our main goal in this article is to investigate the Hyers–Ulam–Rassias stability (HURS) for a type of integral equation called Volterra integral equation with delay (VIE-D). First, by considering special functions such as the Wright function (WR), Mittag–Leffler function (ML), Gauss hypergeometric function (GH), H -Fox function (H -F), and also by introducing the aggregation function, we select the best control function by performing numerical calculations to investigate the stability of the desired equation. In the following, using the selected optimal function, i.e., the minimum function, we prove the existence of a unique solution and the HURS of the VI-D equation in the matrix-valued fuzzy space (MVFS) with two different intervals. At the end of each section, we provide a numerical example of the obtained results. [ABSTRACT FROM AUTHOR]

Let S(0,1) be the *-algebra of all classes of Lebesgue measurable functions on the unit interval (0,1) and let be a complete symmetric Δ-normed *-subalgebra of S(0,1), in which simple functions are dense, e.g., L∞(0,1), Llog(0,1), S(0,1) and the Arens algebra Lω(0,1) equipped with their natural Δ-norms. We show that there exists no non-trivial derivation commuting with all dyadic translations of the unit interval. Let be a type II (or I∞) von Neumann algebra, be an arbitrary abelian von Neumann subalgebra of , let be the algebra of all measurable operators affiliated with. We show that there exists no non-trivial derivation which admits an extension to a derivation on. In particular, we answer an untreated question in [8]. [ABSTRACT FROM AUTHOR]

We apply stochastic controllers to stable the pseudo stochastic bi-homomorphisms and stochastic bi-derivations in MB-algebra and unital MC- ⋄ -algebras and we obtain some approximation by stochastic bi-homomorphisms and stochastic bi-derivations and we compute the optimum error of the estimate. [ABSTRACT FROM AUTHOR]

Let A{\mathcal{A}} be a ∗\ast -algebra, ℳ{\mathcal{ {\mathcal M} }} be a ∗\ast -A{\mathcal{A}}-bimodule, and δ\delta be a linear mapping from A{\mathcal{A}} into ℳ{\mathcal{ {\mathcal M} }}. δ\delta is called a ∗\ast -derivation if δ(AB)=Aδ(B)+δ(A)B\delta \left(AB)=A\delta \left(B)+\delta \left(A)B and δ(A∗)=δ(A)∗\delta \left({A}^{\ast })=\delta {\left(A)}^{\ast } for each A,BA,B in A{\mathcal{A}}. Let GG be an element in A{\mathcal{A}}, δ\delta is called a ∗\ast -antiderivable mapping at GG if AB∗=G⇒δ(G)=B∗δ(A)+δ(B)∗AA{B}^{\ast }=G\Rightarrow \delta \left(G)={B}^{\ast }\delta \left(A)+\delta {\left(B)}^{\ast }A for each A,BA,B in A{\mathcal{A}}. We prove that if A{\mathcal{A}} is a C∗{C}^{\ast }-algebra, ℳ{\mathcal{ {\mathcal M} }} is a Banach ∗\ast -A{\mathcal{A}}-bimodule and GG in A{\mathcal{A}} is a separating point of ℳ{\mathcal{ {\mathcal M} }} with AG=GAAG=GA for every AA in A{\mathcal{A}}, then every ∗\ast -antiderivable mapping at GG from A{\mathcal{A}} into ℳ{\mathcal{ {\mathcal M} }} is a ∗\ast -derivation. We also prove that if A{\mathcal{A}} is a zero product determined Banach ∗\ast -algebra with a bounded approximate identity, ℳ{\mathcal{ {\mathcal M} }} is an essential Banach ∗\ast -A{\mathcal{A}}-bimodule and δ\delta is a continuous ∗\ast -antiderivable mapping at the point zero from A{\mathcal{A}} into ℳ{\mathcal{ {\mathcal M} }}, then there exists a ∗\ast -Jordan derivation Δ\Delta from A{\mathcal{A}} into ℳ♯♯{{\mathcal{ {\mathcal M} }}}^{\sharp \sharp } and an element ξ\xi in ℳ♯♯{{\mathcal{ {\mathcal M} }}}^{\sharp \sharp } such that δ(A)=Δ(A)+Aξ\delta \left(A)=\Delta \left(A)+A\xi for every AA in A{\mathcal{A}}. Finally, we show that if A{\mathcal{A}} is a von Neumann algebra and δ\delta is a ∗\ast -antiderivable mapping (not necessary continuous) at the point zero from A{\mathcal{A}} into itself, then there exists a ∗\ast -derivation Δ\Delta from A{\mathcal{A}} into itself such that δ(A)=Δ(A)+Aδ(I)\delta \left(A)=\Delta \left(A)+A\delta \left(I) for every AA in A{\mathcal{A}}.

In this paper, considering fuzzy measure theory and matrix-valued fuzzy norm spaces, we study a differential system of non-autonomous cellular neural networks with mixed delays (NA-CNN-MD). Our main goal is to investigate the existence of a unique solution for the presented system of NA-CNN-MD in the matrix-valued fuzzy spaces and using the fuzzy measure. After investigating the existence of a unique solution of NA-CNNs-MD which are ϱ -pseudo almost periodically, we prove the Mittag–Leffler stability and Mittag-Leffler attractiveness for ϱ -pseudo almost periodic functions. Finally, two examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We introduce a new class of fuzzy control functions and define the concept of matrix-valued fuzzy Mittag–Leffler–Hypergeometric–Wright stability. Then, we apply Radu–Mihet method derived from an alternative fixed point theorem to investigate existence, uniqueness and matrix-valued fuzzy Mittag–Leffler–Hypergeometric–Wright stability of a a class of P -Hilfer fractional differential equations in matrix-valued fuzzy Banach space. Next, we show the main results for unbounded domains. An example is given to illustrate the Mittag–Leffler–Hypergeometric–Wright stability for a fractional system. [ABSTRACT FROM AUTHOR]

Applying the Radu-Miheţ method derived from an alternative FPT, we stabilize the tri-additive Υ-random operator inequality by providing a class of stochastic matrix control functions in matrix MB-algebras. In addition, we get an estimation for permuting tri-homomorphisms and permuting tri-derivations in unital matrix MC-⋄-algebras and matrix MB-algebras. [ABSTRACT FROM AUTHOR]

Let A be a von Neumann algebra, a linear map δ : A → A is said to be derivable at Ω ∈ A , if δ (A B) = δ (A) B + A δ (B) for every A , B ∈ A with A B = Ω. In this paper, we show that a norm-continuous linear map δ : A → A is derivable at an arbitrary but fixed operator Ω ∈ A if and only if there exists a linear derivation τ : A → A such that δ (A) = τ (A) + δ (I) A for all A ∈ A , where δ (I) is in the centre of A and δ (I) Ω = 0. When A is a von Neumann algebra with no summands of type I 1 or a properly infinite von Neumann algebra, we obtain the same result by weakening the linearity and norm-continuity assumption of δ into additivity. [ABSTRACT FROM AUTHOR]

2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra M 2 n (A) ( n ≥ 2 ) is a derivation, it is shown that any 2-local derivation on real AW ∗ -algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property. [ABSTRACT FROM AUTHOR]

Let T : A → X be a bounded linear operator, where A is a C ∗ -algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent: (a) T is anti-derivable at zero (i.e., a b = 0 in A implies T (b) a + b T (a) = 0 ); (b) There exist an anti-derivation d : A → X ∗ ∗ and an element ξ ∈ X ∗ ∗ satisfying ξ a = a ξ , ξ [ a , b ] = 0 , T (a b) = b T (a) + T (b) a - b ξ a , and T (a) = d (a) + ξ a , for all a , b ∈ A . We also prove a similar equivalence when X is replaced with A ∗ ∗ . This provides a complete characterization of those bounded linear maps from A into X or into A ∗ ∗ which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are ∗ -anti-derivable at zero. [ABSTRACT FROM AUTHOR]

A linear mapping ϕ from an algebra A into its bimodule ℳ is called a centralizable mapping at G ∈ A if ϕ(AB) = ϕ(A)B = Aϕ(B) for each A and B in A with AB = G. In this paper, we prove that if ℳ is a von Neumann algebra without direct summands of type I1 and type II, A is a *-subalgebra with ℳ ⊆ A ⊆ LS (ℳ ) and G is a fixed element in A , then every continuous (with respect to the local measure topology t(ℳ )) centralizable mapping at G from A into ℳ is a centralizer. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and solve the following additive (ρ1,ρ2)-functional inequalities ‖ f (x + y + z) − f (x) − f (y) − f (z) ‖ ≤ ‖ ρ 1 (f (x + z) − f (x) − f (z)) ‖ + ‖ ρ 2 (f (y + z) − f (y) − f (z)) ‖ , where ρ1 and ρ2 are fixed nonzero complex numbers with ∣ρ1∣ + ∣ρ2∣ < 2. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the above additive (ρ1,ρ2)-functional inequality in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of hom-derivations in C*-ternary algebras. [ABSTRACT FROM AUTHOR]

Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of biderivations and bihomomorphisms in Banach algebras and unital C ∗ -algebras, associated with the bi-additive functional inequality: 1 ‖ f (x + y , z + w) + f (x + y , z - w) + f (x - y , z + w) + f (x - y , z - w) - 4 f (x , z) ‖ ≤ s 2 f x + y , z - w + 2 f x - y , z + w - 4 f (x , z) + 4 f (y , w) , where s is a fixed nonzero complex number with | s | < 1 . [ABSTRACT FROM AUTHOR]

By a stochastic controller, we make stable the pseudo stochastic Lie bracket (derivation, derivation) in complex MB-algebras. Next, we get an approximation by a stochastic Lie bracket (derivation, derivation) and calculate the maximum error of the estimate. [ABSTRACT FROM AUTHOR]

Let A be a C ⋆ -algebra, and let δ : A → A ∗ ∗ be a continuous linear map. We assume that δ acts like a derivation or like an anti-derivation at orthogonal elements for several types of orthogonality conditions such as a b = 0 , a b ⋆ = 0 , a ⋆ b = 0 , a b = b a = 0 and a b ⋆ = b ⋆ a = 0 . In each case, we characterize the precise form of δ . [ABSTRACT FROM AUTHOR]

Letbe a generalized matrix ring, whereandare 2-torsion free. We prove that ifis an additive mapping such thatwheneverthen, whereis a Jordan derivation andis a multiplier. As its applications, we prove that the similar conclusion remains valid on full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras, CDCSL algebras and von Neumann algebras. [ABSTRACT FROM PUBLISHER]

In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation. [ABSTRACT FROM AUTHOR]

We prove that every weak-local triple derivation on a JB ⁎ -triple E (i.e. a linear map T : E → E such that for each ϕ ∈ E ⁎ and each a ∈ E , there exists a triple derivation δ a , ϕ : E → E , depending on ϕ and a , such that ϕ T ( a ) = ϕ δ a , ϕ ( a ) ) is a (continuous) triple derivation. We also prove that conditions ( h 1 ) { a , T ( b ) , c } = 0 for every a , b , c in E with a , c ⊥ b ; ( h 2 ) P 2 ( e ) T ( a ) = − Q ( e ) T ( a ) for every norm-one element a in E , and every tripotent e in E ⁎ ⁎ such that e ≤ s ( a ) in E 2 ⁎ ⁎ ( e ) , where s ( a ) is the support tripotent of a in E ⁎ ⁎ , are necessary and sufficient to show that a linear map T on a JB ⁎ -triple E is a triple derivation. [ABSTRACT FROM AUTHOR]

The paper is devoted to the description of 2-local derivations on von Neumann algebras. Earlier it was proved that every 2-local derivation on a semi-finite von Neumann algebra is a derivation. In this paper, using the analogue of Gleason Theorem for signed measures, we extend this result to type $$III$$ von Neumann algebras. This implies that on arbitrary von Neumann algebra each 2-local derivation is a derivation. [ABSTRACT FROM AUTHOR]