Abstract
We establish convergence in the modular sense of an iteration scheme associated with a pair of mappings on a nonlinear domain in modular function spaces. In particular, we prove that such a scheme converges to a common fixed point of the mappings. Our results are generalization of known similar results in the non-modular setting. In particular, we avoid smoothness of the norm in the case of Banach spaces and that of the triangle inequality of the distance in metric spaces. MSC:47H09, 46B20, 47H10, 47E10.
Highlights
Introduction and basic definitionsThe earliest attempts to generalize the classical function spaces Lp of Lebesgue type were made in the early s by Orlicz and Birnbaum in connection with orthogonal expansions
Where φ : [, ∞] → [, ∞] was assumed to be a convex function increasing to infinity, i.e. the function which to some extent behaves similar to power functions φ(t) = tp
We present a useful tool for applications whenever there is a need to introduce a function space by means of functionals which have some reasonable properties but are far from being norms or F-norms
Summary
Introduction and basic definitionsThe earliest attempts to generalize the classical function spaces Lp of Lebesgue type were made in the early s by Orlicz and Birnbaum in connection with orthogonal expansions. They considered the function spaces defined as follows: Lφ = f : R → R; ∃λ > : φ λ f (x) dx < ∞ , R Let us assume that there exists an increasing sequence of sets Kn ∈ P such that = Kn. By E we denote the linear space of all simple functions with support from P. [ – ] Let ρ be a convex function modular.
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