Search

Let Λ be a direct set, { A α } α ∈ Λ be a family of Banach algebras with bounded approximate identity (or unital) and I be a set. We consider the Banach algebra ℓ ∞ (ℓ A β (I , A α)). We show that this algebra has a bounded approximate identity (or is unital) if and only if I is finite. We also characterize the left multipliers of these algebras and investigate their amenability of them. Moreover, we characterize the character spaces (Gelfand spaces) of these algebras in a special case. [ABSTRACT FROM AUTHOR]

In this paper, we define the Jordan notions of contractibility and biprojectivity of Banach algebras and comparing to the classical notions of contractibility and biprojectivity. Also, we introduce the notion of Jordan diagonal and find relations between these Jordan notions. As a main result, we show that the Banach algebra is unital and Jordan contractible if and only if it has a Jordan diagonal if and only if is unital and Jordan biprojective, whenever ⊗ ̂ is a Banach (, op ) -bimodule. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and thoroughly examine new generalized  ψ -conformable fractional integral and derivative operators associated with the auxiliary function  ψ (t) . We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit  ψ -conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications. [ABSTRACT FROM AUTHOR]

In this paper, we apply the well-known results concerning derivations and generalized derivations of commutative Banach algebras and of prime rings to certain Banach algebras that are neither commutative Banach algebras nor prime rings. For example, we investigate the truth of SingerWermer conjecture and Posner’s second theorem for this class of Banach algebras. [ABSTRACT FROM AUTHOR]

This paper considers a coupled hybrid thermostat system driven by the Ã-Caputo fractional derivative in a Banach algebra. We employ a version of Darbo’s fixed-point theorem combined with the measure of noncompactness (MNC) technique to establish certain existence results. Additionally, an example is provided to illustrate our findings. [ABSTRACT FROM AUTHOR]

Let A and B be complex unital Banach algebras, and let φ , ψ : A → B be surjective mappings. If A is semisimple with an essential socle and φ and ψ together preserve the invertibility of linear pencils in both directions, that is, for any x , y ∈ A and λ ∈ C , λ x + y is invertible in A if and only if λ φ (x) + ψ (y) is invertible in B , then we show that there exists an invertible element u in B and a Jordan isomorphism J : A → B such that φ (x) = ψ (x) = u J (x) for all x ∈ A. [ABSTRACT FROM AUTHOR]

This paper is an attempt to give an axiomatic approach to the investigation of various kinds of generalizations of Drazin invertibility in Banach algebras. We shall say that an element a of a Banach algebra $ \mathcal {A} $ A is generalized Drazin invertible relative to a regularity $ \mathcal {R} $ R if there is $ b\in \mathcal {A} $ b ∈ A such that $ ab=ba,\ bab=b $ ab = ba , bab = b and $ \sigma _{\mathcal {R}}(a-aba)\subset \{0\} $ σ R (a − aba) ⊂ { 0 }. The concept of Koliha-Drazin invertible elements, as well as some generalizations of this concept are described via the concept of generalized Drazin invertible elements relative to a regularity $ \mathcal {R} $ R which satisfies two properties: (D1) if $ a, b\in \mathcal {R} $ a , b ∈ R , p is an idempotent commuting with a and b, then $ ap+b(1-p)\in \mathcal {R} $ ap + b (1 − p) ∈ R ; (D2) if $ a\in \mathcal {R} $ a ∈ R , then a is almost invertible. If a regularity $ \mathcal {R} $ R satisfies the properties (D1) and (D2), we prove that $ a\in \mathcal {A} $ a ∈ A is generalized Drazin invertible relative to $ \mathcal {R} $ R if and only if 0 is not an accumulation point of $ \sigma _{\mathcal {R}}(a) $ σ R (a). In particular we define and characterize generalized Drazin-T-Riesz invertible elements relative to an arbitrary (not necessarily bounded) Banach algebra homomorphism T and so extend the concept of generalized Drazin-Riesz invertible operators introduced in [Živković-Zlatanović SČ, Cvetković MD. Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators. Linear Multilinear A. 2017;65(6):1171–1193]. Also we consider generalized Drazin invertibles relative to $ \mathcal {R} $ R in the case when $ \mathcal {R} $ R is the set of Drazin invertibles, as well as when $ \mathcal {R} $ R is the set of Koliha-Drazin invertibles. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as C-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as $ C^* $-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions.

Let n ≥ 2 be a fixed integer and A be a C ∗ -algebra. A permuting n-linear map G : A n → A is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D : A n → A such that G ς 1 , ς 2 , ... , ς i ς i ′ , ... , ς n = G ς 1 , ς 2 , ... , ς i , ... , ς n ς i ′ + ς i D (ς 1 , ς 2 , ... , ς i ′ , ... , ς n) holds ∀ ς i , ς i ′ ∈ A . In this paper, we investigate the structure of C ∗ -algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity. [ABSTRACT FROM AUTHOR]

Let A be a unital Banach algebra with unit e, M be a Banach A-bimodule, and w ∈ A . In this paper, we characterize those continuous linear maps δ : A → M that satisfy one of the following conditions: δ (a b) = δ (a) b + a δ (b) , 2 δ (w) = δ (a) b + a δ (b) , δ (a b) = δ (a) b + a δ (b) - a δ (e) b , for any a , b ∈ A with a b = b a = w , where w is either a separating point with w ∈ Z (A) or an idempotent. [ABSTRACT FROM AUTHOR]

Let 풜 {\mathcal{A}} be a complex unital Banach algebra and let R ⊆ 풜 {R\subseteq\mathcal{A}} be a non-empty set. This paper defines the property such that R is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity R with (CID) property, R D {R^{D}} is constructed as an extension of R to axiomatically study the accumulation of σ R ⁢ ( a ) {\sigma_{R}(a)} for any a ∈ 풜 {a\in\mathcal{A}} . At last, several illustrative examples on Banach algebra and operator algebra are provided. [ABSTRACT FROM AUTHOR]

We give analogues of Lévy and Wiener theorems for some Banach algebras or p-normed algebras that are subalgebras of algebra of absolutely convergent Vilenkin series. [ABSTRACT FROM AUTHOR]

We present new results for the generalized Drazin inverse of 2 × 2 anti-triangular matrices with commuting entries over a Banach algebra. As an application, the g-Drazin invertibility of block-operator matrices is obtained under new wider conditions. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the converse problems of some well-known results related to the generalized Drazin (g-Drazin for short) inverse in Banach algebras. Let A be a Banach algebra and a , b ∈ A . First, we give the relationship between the Drazin (g-Drazin, group) invertibility of a, b and that of the sum a + b under certain conditions. Then, for a given polynomial f (x) ∈ C [ x ] , the g-Drazin invertibility of f(a), f (a d) , f(ab), f (1 - a b) and f (a + b) are investigated. [ABSTRACT FROM AUTHOR]

In the present manuscript our intention is to prove the following: Suppose that R is a semiprime ring and n is a positive integer. If d , F : R → R are two additive mappings satisfying the identity F (r 2 n) = F (r n) r n + r n d (r n) , ∀ r ∈ R , then d will be a derivation and F will be a generalized derivation associated with d on R. Alongside, it is shown that F 1 is generalized left derivation with associated left derivation d 1 on R if they satisfy the algebraic identity F 1 (r 2 n) = r n F 1 (r n) + r n d 1 (r n) , ∀ r ∈ R . These theorems have been validated by providing instances demonstrating that they are not inconsequential. Moreover, we furnish an application in context with Banach algebra. [ABSTRACT FROM AUTHOR]

Let A be a complex and unital Banach algebra, and let E denote the collection of idempotents of A. An old paper of J. Zemánek’s, which was really the starting point of studies of idempotents in general Banach algebras, exhibits a multitude of results concerning the set E. More specifically, it was shown that the connected component of E which contains p ∈ E, say Ep, is precisely the set of elements of the form wpw−1 where w runs through the principal component of the invertible group of A. Following Zemánek’s work, a considerable number of papers concerning the form of paths connecting the members of Ep have seen the light of day. In the present paper, we elaborate on the second part of Zemánek’s article; by utilizing modern techniques and results, together with a surprisingly general connection between the spectra of products of idempotents and linear combinations of idempotents, we show that the global algebraic conditions which characterize central idempotents of A can be replaced by significantly weaker local spectral conditions. [ABSTRACT FROM AUTHOR]

For a general complex unital Banach algebraA, the spectrumalways commutes : for all a, b ∈ A, σ(ab) \ {0} = σ(ba) \ {0}. In this paper, we prove that the above commutative property is not true if we replace the usual spectrum by condition spectrum. Further we study the similar question for more general spec- trum called Ransford spectrum. [ABSTRACT FROM AUTHOR]

Let and be Banach algebras, let be a Banach algebra epimorphism from to , and let be a nonzero character on . As is known if is -contractible (amenable) then is -contractible (amenable). We prove that the converse is true under some conditions. As an important application, we study the -contractibility and amenability of the convolution algebra of trace class operators , where is a locally compact quantum group, and is a nonzero character on . [ABSTRACT FROM AUTHOR]

In this paper, we study and compare the types of diagonals for Lau product Banach algebras × φ ℬ and diagonals of two Banach algebras and ℬ , where φ is a homomorphism from ℬ to . As a consequence, we obtain the (approximate) amenability and contractibility × φ ℬ in terms of (approximate) amenability and contractibility of and ℬ. Moreover, we introduce the notion of pseudo-virtual diagonal for a Banach algebra and present some characterization results for Lau product Banach algebras. [ABSTRACT FROM AUTHOR]

Let A be a unital Banach algebra and M be a unital A -bimodule. We show that additive mappings δ , τ : A → M satisfy the identity A − 1 δ (A) + δ (A) A − 1 + A τ (A − 1) + τ (A − 1) A = 0 for all invertible element A in A if and only if there exists a Jordan derivation D such that δ (A) = D (A) + A δ (I) and τ (A) = D (A) − δ (I) A for all A ∈ A . We also characterize additive mappings δ , τ : A → M satisfying the identity δ (A) B + A τ (B) = M for all A , B ∈ A with AB = N, where N is an invertible element in A and M is a fixed element in M. Similar conclusions are obtained for additive mappings from a matrix ring M n (D) into its unital bimodule satisfying any of the above identities, where D is a division ring and char D ≠ 2 . [ABSTRACT FROM AUTHOR]

We introduce a new generalized inverse, called generalized core inverse in a Banach *-algebra. This new inverse is an extension of weak core inverse defined for complex square matrix and bounded linear operators over Hilbert spaces. We present various characterizations of this new inverse. The relationship between the generalized core inverse and other generalized inverses is investigated. Finally, we consider the necessary and sufficient conditions under which generalized core inverse and generalized core-EP inverse coincide with each other in a Banach *-algebra. [ABSTRACT FROM AUTHOR]

In this paper, using fixed point method, we investigate the stability and also the superstability of cubic bi-derivations on Banach algebras. [ABSTRACT FROM AUTHOR]

Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic identity $ 2\mathcal{H}(a^{p+q}) = \mathcal{H}(a^p) a^q+ a^p \mathcal{D}(a^q)+\mathcal{H}(a^q) a^p+ a^q \mathcal{D}(a^p) $ for all $ a\in \mathcal{A} $, then $ \mathcal{H} $ is a generalized derivation with $ \mathcal{D} $ as an associated derivation on $ \mathcal{A} $. In addition to that, it is also proved in this article that $ \mathcal{H}_1 $ is a generalized left derivation associated with a left derivation $ \delta $ on $ \mathcal{A} $ if they fulfilled the algebraic identity $ 2\mathcal{H}_1(a^{p+q}) = a^p \mathcal{H}_1(a^q)+ a^q \delta(a^p)+a^q \mathcal{H}_1(a^p)+ a^p \delta(a^q) $ for all $ a \in \mathcal{A} $. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.

We show that l¹(ℕʌ) is ᵠ-amenable for each multiplicative linear functional ᵠ : l¹(ℕʌ) → C. This is a counterexample to the final corollary of Jaberi and Mahmoodi ['On ᵠ-amenability of dual Banach algebras', Bull. Aust. Math. Soc. 105 (2022), 303-313] and shows that the final theorem in that paper is not valid. [ABSTRACT FROM AUTHOR]

Let A be a (p + q)!-torsion free semiprime ring. We proved that if H, D : A → A are two additive mappings fulfilling the algebraic identity 2H(ap+q) = H(ap)aq + apD(aq) + H(aq)ap + aqD(ap) for all a e A, then H is a generalized derivation with D as an associated derivation on A. In addition to that, it is also proved in this article that H1 is a generalized left derivation associated with a left derivation δ on A if they fulfilled the algebraic identity 2H1(ap+q) = apH1(aq) + aqδ(ap) + aqH1(ap) + apδ(aq) for all a ∈ A. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented. [ABSTRACT FROM AUTHOR]

We establish some Lie–Trotter formulae for unital complex Jordan–Banach algebras, showing that for any elements a, b, c in a unital complex Jordan–Banach algebra A the identities lim n → ∞ e a n ∘ e b n n = e a + b , lim n → ∞ U e a n e b n n = e 2 a + b , and lim n → ∞ U e a n , e c n e b n n = e a + b + c hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in A . These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals f : A → C satisfying f (U x (y)) = U f (x) f (y) , for all x , y ∈ A . We prove that for any such a functional f, there exists a unique continuous (Jordan-) multiplicative linear functional ψ : A → C such that f (x) = ψ (x) , for every x in the connected component of the set of all invertible elements of A containing the unit element. If we additionally assume that A is a JB ∗ -algebra and f is continuous, then f is a linear multiplicative functional on A . The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Schulz, and Touré. [ABSTRACT FROM AUTHOR]

We illustrate the interplay between multiplicative structure and ordered structure on Banach spaces of real-valued functions via various versions of the Stone–Weierstrass Approximation Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications. [ABSTRACT FROM AUTHOR]

Let U be an associative unital Banach algebra endowed with the Lie product [ x , y ] = x y - y x ( x , y ∈ U ) and create a Lie algebra. In this article, we are going to study the additive maps on U that act at idempotent-products such as centralizers on the Lie structure of U . More precisely, we consider the subsequent condition on an additive map φ on a unital Banach algebra U with a non-trivial idempotent p: x , y ∈ U , x y = p ⟹ φ ([ x , y ]) = [ φ (x) , y ] = [ x , φ (y) ] , and we show under certain conditions that φ (x) = c x + μ (x) for all x ∈ U , where c ∈ Z (U) , μ : U → Z (U) ( Z (U) is the center of U ) is an additive map in which μ ([ x , y ]) = 0 for any x , y ∈ U with x y = p . The obtained results will be used for some Banach algebras, especially, for von Neumann algebras. [ABSTRACT FROM AUTHOR]

In this paper we try to approximate any properties of quasi-block Toeplitz matrix (QBT), by means of a finite number of parameters. A quasi-block Toeplitz (QBT) matrix is a semi-infinite block matrix of the kind F = T(F) + E where... These matrices are a Banach algebra with the standard arithmetic operations. We try to analysis some structures and computational properties for arithmetic operations of QBT matrices with some MATLAB commands. [ABSTRACT FROM AUTHOR]

The focus of this paper is to extend some stability results of the analysis of perturbed operators on the Banach space to the case of Fredholm perturbation theory in a Banach algebra relative to a bounded homomorphism. Our approach is to define some subsets in a Banach algebra to enrich the theory of perturbed left-right Browder and left-right Weyl elements in a Banach algebra relative to a bounded homomorphism. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to obtain some results on the image of reverse derivations on Banach algebras. For instance, we prove that every reverse derivation on a semiprime Banach algebra maps the algebra into its radical. Some other related results are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

In this paper, we present new presentations of group inverses for the sum of two group invertible elements in a Banach algebra. We then apply these results to block complex matrices. The group invertibility of certain block complex matrices is thereby obtained.

The goal of this study is to bring out the following conclusion. Let $ R $ be a noncommutative prime ring with $ 2(m+n)! $ torsion freeness and let $ m $ and $ n $ be fixed, non-negative integers and $ d, g $ be Jordan derivations on $ R $. If $ x^{m+n}d(x)+x^mg(x)x^n\in Z(R) $ or $ d(x)x^{m+n}+x^mg(x)x^n\in Z(R) $ or $ x^{n}d(x)x^{m}+x^mg(x)x^n\in Z(R) $ then $ d = g = 0 $ follows for every $ x\in R $.

Abstract In this paper, we introduce the notion of graphical cone metric spaces over Banach algebra and prove some fixed point results for a particular type of contractive mappings defined on such spaces. These results extend and generalize several results from metric, graphical metric, and cone metric spaces. Some examples that demonstrate the results proved herein are provided. An application of our results to the existence of solution of a pair of initial value problems is provided.

Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value problem associated with φ -Caputo fractional derivatives of variable order. The primary objectives are to establish the existence and uniqueness of solutions, as well as explore their stability through the Ulam-Hyers concept. To achieve these goals, Banach's and Krasnoselskii's fixed point theorems are employed as powerful mathematical tools. Additionally, we provide numerical examples to illustrate results and enhance comprehension of theoretical findings. This comprehensive analysis significantly advances our understanding of variable-order fractional differential equations, providing a strong foundation for future research. Future directions include exploring more complex boundary value problems, studying the effects of varying fractional differentiation orders, extending the analysis to systems of equations, and applying these findings to real-world scenarios, all of which promise to deepen our understanding of Caputo fractional differential equations with variable order, driving progress in both theoretical and applied mathematics. [ABSTRACT FROM AUTHOR]

Introduction A Banach algebra A is called contractible if for any Banach A-bimodule E, every continuous derivation from A into E is inner. Contractibility is among of the various notions of amenability which have been defined after a very bold paper of Johnson on cohomology of Banach algebras. It is well-known that full matrix algebras and their finite direct sum are contractible. Also, every finite-dimensional contractible Banach algebra is of this form. One of the oldest unconfirmed conjectures in amenability says that every contractible Banach algebra is finite dimensional. The special case of this conjecture which is still unconfirmed says that for any Banach space X, if the Banach algebra B(X) of all bounded linear operators on X, is contractible then X is finite- dimensional. Results and discussion It is well-known that a Banach algebra A is contractible if and only if it is unital and has a diagonal, that is a member M in the Banach algebra A⊗π A such that satisfies in ∆(M) = 1 and (a ⊗ 1)M = M(1 ⊗ a) for every a in A. Here, denotes the completed projective tensor product of Banach spaces, and ∆: A ⊗π A → Ais the unique bounded linear operator defined by (a ⊗ b) ⟼ ab. For a Banach space X of finite dimension m, B(X) is isomorphic to the matrix algebra Mm. It is well-known that Mm has a unique diagonal of the form m¹ ∑ikδij where δij's denote the standard basis of Mm -1 The aim of this short note is to prove a property (Theorem 6) for the diagonal of any contractible B (X) when X is infinite-dimensional. For the proof we use the famous estimate of Kadec and Snobar on the norms of projection operators on finite-dimensional subspaces. We hope that this property of diagonal and the technics we have used here, helps to solve the mentioned conjecture on finite-dimensionality of X. Consider the following operator: Ψ: Β(Χ) ⊗π Β(X) → В(Х⊗πX), Ψ(T ⊗ S)(x ⊗ y) = T(x) ⊗ T(y). Then Ψ is a Banach algebra homomorphism with norm equals to 1. We denote the image of M€ Β(Χ) ⊗π Β(X) under Ψ by MOP. The main result of this note is Theorem 6: Theorem 6. Suppose that there is an infinite-dimensional Banach space X such that B(X) is contractible. Then for every diagonal M of B(X) we have MOP = 0. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and uniqueness of implicit solutions in a coupled symmetry system of hybrid fractional order differential equations, along with hybrid integral boundary conditions in Banach Algebras. The methodology centers on a hybrid fixed-point theorem that involves mixed Lipschitz and Carathéodory conditions, serving to establish the existence of solutions. Moreover, it derives sufficient conditions for solution uniqueness and establishes the Hyers–Ulam types of solution stability. This study contributes valuable insights into complex hybrid fractional order systems and their practical implications. [ABSTRACT FROM AUTHOR]

The multiplicative version of the Gleason–Kahane–Żelazko theorem for -algebras given by Brits et al. in [4] is extended to maps from -algebras to commutative semisimple Banach algebras. In particular, it is proved that if a multiplicative map from a -algebra to a commutative semisimple Banach algebra is continuous on the set of all noninvertible elements of and for any , then is a linear map. The multiplicative variation of the Kowalski–Słodkowski theorem given by Touré et al. in [14] is also generalized. Specifically, if is a continuous map from a -algebra to a commutative semisimple Banach algebra satisfying the conditions and for all , then generates a linear multiplicative map on which coincides with on the principal component of the invertible group of . If is a Banach algebra such that each element of has totally disconnected spectrum, then the map itself is linear and multiplicative on . It is shown that a similar statement is valid for a map with semisimple domain under a stricter spectral condition. Examples which demonstrate that some hypothesis in the results cannot be discarded. [ABSTRACT FROM AUTHOR]