The Extended Cone b-Metric-like Spaces over Banach Algebra and Some Applications

In this paper, we introduce the structure of extended cone b-metric-like spaces over Banach algebra as a generalization of cone b-metric-like spaces over Banach algebra. In this generalized space we define the notion of generalized Lipschitz mappings in the setup of extended cone b-metric-like spaces over Banach algebra and investigated some fixed point results. We also provide examples to illustrate the results presented herein. Finally, as an application of our main result, we examine the existence and uniqueness of solution for a Fredholm integral equation.

General Higher-Order Lipschitz Mappings

In this paper, we introduce a new class of mappings and investigate their fixed point property. In the first direction, we prove a fixed point theorem for general higher-order contraction mappings in a given metric space and finally prove an approximate fixed point property for general higher-order nonexpansive mappings in a Banach space.

Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1 n ∑ i = 0 n − 1 T i ( x ) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {T n (x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

NEW INEQUALITIES ON LIPSCHITZ FUNCTIONS

In this study, some inequalities of Hermite Hadamard type obtained for p-convex functions are given for Lipschitz mappings. Also, some applications for special means have been given.

On the numerical approximation of vectorial absolute minimisers in $$L^\infty $$

Let $$\Omega $$ Ω be an open set. We consider the supremal functional $$\begin{aligned} \text {E}_\infty (u,{\mathcal {O}})\, {:}{=}\, \Vert \text {D}u \Vert _{L^\infty ( {\mathcal {O}} )}, \ \ \ {\mathcal {O}} \subseteq \Omega \text { open}, \end{aligned}$$ E ∞ ( u , O ) : = ‖ D u ‖ L ∞ ( O ) , O ⊆ Ω open , applied to locally Lipschitz mappings $$u : \mathbb {R}^n \supseteq \Omega \longrightarrow \mathbb {R}^N$$ u : R n ⊇ Ω ⟶ R N , where $$n,N\in \mathbb {N}$$ n , N ∈ N . This is the model functional of Calculus of Variations in $$L^\infty $$ L ∞ . The area is developing rapidly, but the vectorial case of $$N\ge 2$$ N ≥ 2 is still poorly understood. Due to the non-local nature of (1), usual minimisers are not truly optimal. The concept of so-called absolute minimisers is the primary contender in the direction of variational concepts. However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when $$n,N\ge 2$$ n , N ≥ 2 . We present numerical experiments aimed at understanding the behaviour of minimisers through a new technique involving p-concentration measures.

Some fixed point results on $\mathcal{N}_{b}$-cone metric spaces over Banach algebra

The main aim of this paper is to introduce the concept of $\mathcal{N}_{b}$ N b -cone metric spaces over a Banach algebra as a generalization of $\mathcal{N}$ N -cone metric spaces over a Banach algebra and b-metric spaces. Also, we study some coupled common fixed point theorems for generalized Lipschitz mappings in this framework. Finally, we give an example and an application to the existence of solutions of integral equations to illustrate the effectiveness of our generalizations. Some results in the literature are special cases of our results.

Cone Metric Spaces over Topological Modules and Fixed Point Theorems for Lipschitz Mappings

In this paper, we introduce the concept of cone metric space over a topological left module and we establish some coincidence and common fixed point theorems for self-mappings satisfying a condition of Lipschitz type. The main results of this paper provide extensions as well as substantial generalizations and improvements of several well known results in the recent literature. In addition, the paper contains an example which shows that our main results are applicable on a non-metrizable cone metric space over a topological left module. The article proves that fixed point theorems in the framework of cone metric spaces over a topological left module are more effective and more fertile than standard results presented in cone metric spaces over a Banach algebra.