A Halpern-Type Iteration Method for Bregman Nonspreading Mapping and Monotone Operators in Reflexive Banach Spaces

In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mappings in a reflexive Banach space. Using the Bregman distance function, we study the composition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotone operator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and a Bregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping. We further state and prove a strong convergence result using the iterative algorithm introduced. This result extends many works on finding a common solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to a reflexive Banach space.

On the generalized asymptotically nonspreading mappings in convex metric spaces

<p>In this article, we propose a new class of nonlinear mappings, namely, generalized asymptotically nonspreading mapping, and prove the existence of fixed points for such mapping in convex metric spaces. Furthermore, we also obtain the demiclosed principle and a delta-convergence theorem of Mann iteration for generalized asymptotically nonspreading mappings in CAT(0) spaces.</p>

On Mann’s Method with Viscosity for Nonexpansive and Nonspreading Mappings in Hilbert Spaces

In the setting of Hilbert spaces, inspired by Iemoto and Takahashi (2009), we study a Mann’s method with viscosity to approximate strongly (common) fixed points of a nonexpansive mapping and a nonspreading mapping. A crucial tool in our results is the nonspreading-average type mapping.

On Strong Convergence of Halpern’s Method Using Averaged Type Mappings

Under suitable hypotheses on control coefficients, we study Halpern’s method to approximate strongly common fixed points of a nonexpansive mapping and of a nonspreading mapping or a fixed point of one of them. A crucial tool in our results is the regularization with the averaged type mappings.