LAUSR

In this study, as the domain of four-dimensional backward difference matrix in the space $$\mathcal {L}_u(t)$$ L u ( t ) , we introduce the complete paranormed space $$\mathcal {BV}(t)$$ BV ( t ) of bounded variation double sequences and examine some properties of that space. Also, we determine the $$\gamma $$ γ -dual and $$\beta (\vartheta )$$ β ( ϑ ) -dual of the space $$\mathcal {BV}(t)$$ BV ( t ) . Finally, we characterize the classes $$(\mathcal {BV}(t):\mathcal {M}_{u})$$ ( BV ( t ) : M u ) , $$(\mathcal {BV}(t):\mathcal {C}_{\vartheta })$$ ( BV ( t ) : C ϑ ) and $$(\mathcal {L}_u(t):\mu )$$ ( L u ( t ) : μ ) with $$\mu \in \{\mathcal {BS},\mathcal {CS}_{\vartheta },\mathcal {M}_{u}(\Delta ),\mathcal {C}_{\vartheta }(\Delta )\}$$ μ ∈ { BS , CS ϑ , M u ( Δ ) , C ϑ ( Δ ) } , where $$\mathcal {M}_{u}(\Delta )$$ M u ( Δ ) and $$\mathcal {C}_{\vartheta }(\Delta )$$ C ϑ ( Δ ) denote the spaces of all double sequences whose $$\Delta $$ Δ -transforms are in the spaces $$\mathcal {M}_{u}$$ M u and $$\mathcal {C}_{\vartheta }$$ C ϑ , respectively.