ABSTRACT
It is well know that many physically significant problems in different areas of research can be transformed into an equation of the form Au = 0, (0.0.1) where A is a nonlinear monotone operator from a real Banach space E into its dual E∗ . For instance, in optimization, if f : E −→ R ∪ {+∞} is a convex, Gˆateaux differentiable function and x ∗ is a minimizer of f, then f 0 (x ∗ ) = 0. This gives a criterion for obtaining a minimizer of f explicitly. However, most of the operators that are involved in several significant optimization problems are not differentiable. For instance, the absolute value function x 7→ |x| has a minimizer, which, in fact, is 0. But, the absolute value function is not differentiable at 0. So, in a case where the operator under consideration is not differentiable, it becomes difficult to know a minimizer even when it exists. Thus, the above characterization only works for differentiable operators. A generalization of differentiability called subdifferentiability allows us to recover the above result for non differentiable maps.
Bello, A (2021). Algorithms For Approximation Of Solutions Of Equations Involving Nonlinear Monotone-Type And Multi-Valued Mappings. Afribary. Retrieved from https://afribary.com/works/algorithms-for-approximation-of-solutions-of-equations-involving-nonlinear-monotone-type-and-multi-valued-mappings
Bello, Abdulmalik "Algorithms For Approximation Of Solutions Of Equations Involving Nonlinear Monotone-Type And Multi-Valued Mappings" Afribary. Afribary, 13 Apr. 2021, https://afribary.com/works/algorithms-for-approximation-of-solutions-of-equations-involving-nonlinear-monotone-type-and-multi-valued-mappings. Accessed 16 Nov. 2024.
Bello, Abdulmalik . "Algorithms For Approximation Of Solutions Of Equations Involving Nonlinear Monotone-Type And Multi-Valued Mappings". Afribary, Afribary, 13 Apr. 2021. Web. 16 Nov. 2024. < https://afribary.com/works/algorithms-for-approximation-of-solutions-of-equations-involving-nonlinear-monotone-type-and-multi-valued-mappings >.
Bello, Abdulmalik . "Algorithms For Approximation Of Solutions Of Equations Involving Nonlinear Monotone-Type And Multi-Valued Mappings" Afribary (2021). Accessed November 16, 2024. https://afribary.com/works/algorithms-for-approximation-of-solutions-of-equations-involving-nonlinear-monotone-type-and-multi-valued-mappings