In the current investigation, the notions of ‐module biprojectivity and ‐module biflatness for a Banach algebra , are introduced, where is a closed ideal of . Moreover, the relationships between these new concepts with the notions of (module) biflatness and biprojectivity of Banach algebras are found. Furthermore, these concepts are connected to module amenability and super amenability. The ‐module biprojectivity of a module projective tensor product of Banach algebras is studied as well. Eventually, for the validity of our abstract results, some applications and typical examples in the category of inverse semigroup algebras are presented.
The present paper develops a new two‐step iterative method, referred to as the P ∗ iteration, for the approximation of fixed points of weak contractions mappings in Banach spaces. The theoretical analysis establishes the strong convergence of the proposed scheme to the unique fixed point, along with its almost I ‐stability. Further, a comparative study with well‐known iterative methods, including the Picard, Mann, S , Varat, and F ∗ algorithms demonstrates that the P ∗ iteration converges at a faster rate. To validate the theoretical findings, numerical examples are provided, along with graphical and numerical illustrations implemented in MATLAB.
We establish the Mikhlin‐type Fourier multiplier theorems on variable exponent Lebesgue–Bochner spaces using the Fourier multiplier theorems on weighted Lebesgue–Bochner spaces and the transform results of bounded operators from weighted spaces to variable exponent spaces. Using the multiplier theorems, we discuss the L p (·) ‐well‐posedness of the first‐order and second‐order differential equations.
In this paper, we introduce a new class of multiple Stancu‐type Bernstein operators defined via Beta function based integral modification. We study their approximation properties and establish uniform convergence by means of Korovkin′s theorem under suitable conditions on the parameter α η . Rates of convergence are obtained in terms of the modulus of continuity and Peetre′s κ ‐functional, and a Voronovskaja‐type asymptotic formula is derived. Furthermore, we obtain representations of the operator in terms of special functions, which, combined with its moment properties, lead to finite summation formulas for the generalized hypergeometric function 3 F 2 and coefficient identities associated with the confluent hypergeometric function 1 F 1 . Moreover, the operators unify several classical constructions: They reduce to the multiple Bernstein operators as α ⟶0 + , and yield a multiple Szász‐type operator under an appropriate scaling, recovering a result of (Varma, 2013). Numerical examples and graphical simulations are provided to illustrate the approximation behavior, confirm the theoretical findings, and highlight the crucial role of the parameter α η in controlling the approximation behavior.
The present work investigates perturbation determinants associated with dissipative operators derived from fractal Sturm–Liouville expressions within the Weyl limit‐circle framework. By employing Livšic′s theorem, we rigorously establish the completeness of the root vector system for these dissipative Sturm–Liouville operators.
