Kızmaz introduced the concept of difference sequence spaces, studied their topological properties and inclusion relations, and computed their duals. In this paper, we generalize difference sequence spaces by introducing difference net spaces over directed sets. We define the norm on these spaces and study their properties like completeness, separability, and inclusion relations. We characterize the dual spaces and establish the isometric isomorphisms $ \bigl(\ell^\infty(\Delta, D)\bigr)^* \cong \frac{\mathbb{K}\times ba(D)} {\left(T(\ell^\infty(\Delta, D))\right)^\perp} $, $ c(\Delta, D)^* \cong \frac{ \mathbb{K}\times (\ell^1(D)\oplus\mathbb{K}) }{ R^\perp } $ and $ (c_0(\Delta, D))^* \cong \frac{\mathbb{K}\times\ell^1(D)} {\left(T(c_0(\Delta, D))\right)^\perp} $. Lastly, we establish an application to fixed-point theory, demonstrating that iterative nets generated by contraction mappings constitute natural elements of the space $ C_{0}(\Delta, D) $.
Citation: Aadil Hussain Dar, Lateef Ahmad Wani, Mudasir Younis, Saiful R. Mondal. Topological properties and duals of difference net spaces over directed sets with fixed-point applications[J]. AIMS Mathematics, 2026, 11(9): 29610-29635. doi: 10.3934/math.20261175
Kızmaz introduced the concept of difference sequence spaces, studied their topological properties and inclusion relations, and computed their duals. In this paper, we generalize difference sequence spaces by introducing difference net spaces over directed sets. We define the norm on these spaces and study their properties like completeness, separability, and inclusion relations. We characterize the dual spaces and establish the isometric isomorphisms $ \bigl(\ell^\infty(\Delta, D)\bigr)^* \cong \frac{\mathbb{K}\times ba(D)} {\left(T(\ell^\infty(\Delta, D))\right)^\perp} $, $ c(\Delta, D)^* \cong \frac{ \mathbb{K}\times (\ell^1(D)\oplus\mathbb{K}) }{ R^\perp } $ and $ (c_0(\Delta, D))^* \cong \frac{\mathbb{K}\times\ell^1(D)} {\left(T(c_0(\Delta, D))\right)^\perp} $. Lastly, we establish an application to fixed-point theory, demonstrating that iterative nets generated by contraction mappings constitute natural elements of the space $ C_{0}(\Delta, D) $.
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Aadil Hussain Dar, Lateef Ahmad Wani, Mudasir Younis, Saiful R. Mondal. Topological properties and duals of difference net spaces over directed sets with fixed-point applications[J]. AIMS Mathematics, 2026, 11(9): 29610-29635. doi: 10.3934/math.20261175
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