Narrow Results

In this paper, we introduce Padovan difference sequence spaces of fractional-order $\alpha ,$${\ell }_p({\tilde{P}}^{(\alpha )})$$(1\le p\le \infty )$ by the composition of the fractional-order difference operator ${\mathrm{\Delta }}^{(\alpha )}$ and the Padovan matrix $\tilde{P}=({\tilde{p}}_{nk})$ defined by ${\mathrm{\Delta }}_k^{(\alpha )}(x)={\sum }_{i=0}^{\infty }{(-1)}^i\frac{\mathrm{\Gamma }(\alpha +1)}{i!\mathrm{\Gamma }(\alpha -i+1)}{x}_{k-i}$ and

respectively, where the sequence $({\tilde{p}}_k)$ is the Padovan sequence. We give some topological properties, Schauder basis and $\alpha$-, $\beta$- and $\gamma$-duals of the newly defined spaces. We characterize certain matrix classes related to the ${\ell }_p({\tilde{P}}^{(\alpha )})$ space. Finally, we characterize certain classes of compact operators on ${\ell }_p({\tilde{P}}^{(\alpha )})$ using Hausdorff measure of noncompactness.

The aim of this paper is to investigate the solvability of infinite systems of nonlinear functional integral equations of $N$-variables in $C(I\times I\times \cdots \times I,m(\varphi ))$ 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.

Roopaei [J. Inequal. Appl.2020 (2020) 120] introduced the sequence space ${{𝒞}}_p^{\alpha }$ which is a matrix domain of Copson matrix in the space ${\ell }_p$$(1\le p<\infty ).$ In this paper, we characterize the matrix classes $({{𝒞}}_p^{\alpha },{\ell }_{\infty })$, $({{𝒞}}_p^{\alpha },c)$ and $({{𝒞}}_p^{\alpha },c_0)$$(1\le p<\infty )$. Moreover, we characterize compact operators associated with matrices belonging to these classes via the techniques of measures of noncompactness.

In this work, we study the existence of mild solutions for some nondensely partial functional integro-differential equations with state-dependent delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an integrated resolvant operators. The nonlinear part satisfies Carathéodory’s conditions. Our approach is based on the generalized Darbo Fixed Point Theorem on Banach spaces and the integrated resolvent operators theory. For illustration, we propose a reaction–diffusion equation with state-dependent delay. The results obtained in this paper are a generalization of some existing results in this area.