On composition operators of Fibonacci matrix and applications of Hausdorff measure of noncompactness

The aim of the paper is introduced the composition of the two infinite matrices $\Lambda=(\lambda_{nk})$ and $\widehat{F}=\left( f_{nk} \right).$ Further, we determine the $\alpha$-, $\beta$-, $\gamma$-duals of new spaces and also construct the basis for the space $\ell_{p}^{\lambda}(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$Finally we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators by applying the Hausdorff measure of noncompactness, where $X\in\{\ell_{\infty}^{\lambda}(\widehat{F}),\ell_{p}^{\lambda}(\widehat{F})\}$ and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\},$ and $1\leq p<\infty.$

Some Compact Operators on the Hahn Space

We establish the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space $h_{d}$, where $d$ is an unbounded monotone increasing sequence of positive real numbers, into the spaces $[c_{0}]$, $[c]$ and $[c_{\infty}]$ of sequences that are strongly convergent to zero, strongly convergent and strongly bounded. Furthermore, we prove estimates for the Hausdorff measure of noncompactness of bounded linear operators from $h_{d}$ into $[c]$, and identities for the Hausdorff measure of noncompactness of bounded linear operators from $h_{d}$ to $[c_{0}]$, and use these results to characterise the classes of compact operators from $h_{d}$ to $[c]$ and $[c_{0}]$.

Impulsive Fractional Differential Inclusions and Almost Periodic Waves

In the present paper, the concept of almost periodic waves is introduced to discontinuous impulsive fractional inclusions involving Caputo fractional derivative. New results on the existence and uniqueness are established by using the theory of operator semigroups, Hausdorff measure of noncompactness, fixed point theorems and fractional calculus techniques. Applications to a class of fractional-order impulsive gene regulatory network (GRN) models are proposed to illustrate the results.

Banach spaces of absolutely k-summable series

In this paper, we present a general Banach space of absolutely k-summable series using a triangle matrix operator and prove that this is a BK-space isometrically isomorphic to the space ℓ k {\ell_{k}} . We also establish the α - {\alpha-} , β - {\beta-} , γ-duals and base of the new space. Finally, we qualify some matrix and compact operators on the new space making use of the Hausdorff measure of noncompactness. Our results include, as particular cases, a number of well-known results.

Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space

A new sequence space related to the space $\ell _{p}$ ℓ p , $1\leq p<\infty $ 1 ≤ p < ∞ (the space of all absolutely p-summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.

Hausdorff measure of noncompactness of matrix mappings on Cesàro spaces

In this study we establish some identities or estimates for operator norms and the Hausdorff measure of noncompactness of certain operators on spaces |C_{α}|_{k}, which have more recently been introduced in [14]. Further, by applying the Hausdorff measure of noncompactness, we establish the necessary and sufficient conditions for such operators to be compact and so the some well known results are generalized.

Existence of Solutions of Infinite Systems of Nonlinear Functional Integral Equations of N-Variables in C(I × I ×⋯ × I,m(ϕ))

The aim of this paper is to investigate the solvability of infinite systems of nonlinear functional integral equations of [Formula: see text]-variables in [Formula: see text] by using the Hausdorff measure of noncompactness with the help of Meir–Keeler condensing operators. We also provide an illustrative example in support of our existence theorems.

On difference sequence spaces of fractional-order involving Padovan numbers

In this paper, we introduce Padovan difference sequence spaces of fractional-order [Formula: see text] [Formula: see text] [Formula: see text] by the composition of the fractional-order difference operator [Formula: see text] and the Padovan matrix [Formula: see text] defined by [Formula: see text] and [Formula: see text] respectively, where the sequence [Formula: see text] is the Padovan sequence. We give some topological properties, Schauder basis and [Formula: see text]-, [Formula: see text]- and [Formula: see text]-duals of the newly defined spaces. We characterize certain matrix classes related to the [Formula: see text] space. Finally, we characterize certain classes of compact operators on [Formula: see text] using Hausdorff measure of noncompactness.

Family of mappings with an equicontractive-type condition

In a real Banach space X and a complete metric space M, we consider a compact mapping C defined on a closed and bounded subset A of X with values in M and the operator $$T:A\times C(A) \rightarrow X$$ T : A × C ( A ) → X . Using a new type of equicontractive condition for a certain family of mappings and $$\beta $$ β -condensing operators defined by the Hausdorff measure of noncompactness we prove that the operator $$x\mapsto T(x,C(x))$$ x ↦ T ( x , C ( x ) ) has a fixed point. The obtained results are applied to the initial value problem.

Generalised regularity and compactness of matrix operators

A well-known result by Cohen and Dunford ([2], 1937) characterises the class of all regular compact linear operators. It follows that a regular matrix transformation cannot be compact. This means that if c denotes the set of all complex sequences of complex numbers, then an infinite matrix that maps c into c and preserves the limits cannot be compact. We obtained this result in a different way applying the theory of BK spaces from functional analysis and summability, and using the Hausdorff measure of noncompactness. Furthermore, we present the extension of this result to matrix transformations between the spaces c and the spaces of strongly summable sequences by the Cesaro method of order 1, and of strongly convergent sequences. We present new unified proofs for our main results.