Using the Caputo derivative of order q ∈ (α − 1, α), we examine boundary value problems for fractional integro-differential equations in Banach spaces. Using an a priori estimate technique, the Holder's inequality, a suitable singular Gronwall's inequality, and the fixed point theorem are utilised to prove the existence and uniqueness of solutions.

In this paper, we introduce the statistical summability by the weighted mean method in a two-normed space and establish necessary and sufficient Tauberian conditions under which statistical convergence of a sequence (x n ) follows from its statistical summability by weighted mean method in a two-normed space. In particular, our Tauberian conditions are satisfied if (x n ) is statistically slowly oscillating in a two-normed space. The main theorem of this paper is the statistical extension of a Tauberian theorem in C ¸anak et al. [Acta Comment. Univ. Tartu. Math. 24, 49-57 (2020)].

In this paper, we study the existence of entropy solutions for some generalized elliptic p(u)-Laplacian problem when p(u) is a local quantity. We get the results by assuming the right-hand side function f to be an integrable function, and by using the regularization approach combined with the theory of Sobolev spaces with variable exponents.

Let M be an R-module.If every essential submodule of M has a g-supplement in M, then M is called an essential g-supplemented (or briefly, eg-supplemented) module.If every essential submodule has ample g-supplements in M, then M is called an amply essential gsupplemented (or briefly, amply eg-supplemented) module.In this work, some properties of these modules are investigated.It is proved that every factor module and every homomorphic image of an amply eg-supplemented module are amply eg-supplemented.Let M be a projective and eg-supplemented module.Then every finitely Mgenerated R-module is amply egsupplemented.

The investigation of Ma and Minda classes of functions associated with q-calculus has been on increase in recent times. Newly, limac ¸on functions, the classes of starlike and convex limac ¸on functions were initiated and investigated in the literature. As a result, the present article is aimed to present the q-analogue of the limac ¸on functions and utilize it to establish the classes of starlike and convex limac ¸on functions that are correlated with quantum calculus. To this end, the geometric characterization of these functions is examined. Moreover, radius, subordination and some other related results with these novel classes are verified. Overall, some consequences of our investigations are also illustrated.

This paper is a continuation of a recent work on a new norm, christened the (α, β)norm, on the space of bounded linear operators on a Hilbert space. We obtain some upper bounds for the said norm of n × n operator matrices. As an application of the present study, we estimate bounds for the numerical radius and the usual operator norm of n × n operator matrices, which generalize the existing ones.

In this paper we introduce the concepts of Wijsman ρ-statistical convergence of order α and Wijsman strongly ρ-convergence of order α. In addition, some inclusion theorems are also presented.

We characterize the α-large submodules of α-modules in terms of certain sequences of ordinals, and give some of their interesting properties.Also, we deal with α-large submodules of the closure of an unbounded direct sum of uniserial modules, and the α-large submodules of the smallest α-pure fully invariant submodules of the closure containing a given element.

In this paper, we prove the stability of Hadamard homomorphisms and Hadamard derivations in Banach algebras.This is applied to investigate Hadamard isomorphisms between Banach algebras.

examine a generalized convexity type inequality, called e-convexity.