A New Study on the Fixed Point Sets of Proinov-Type Contractions via Rational Forms

In this paper, we presented some new weaker conditions on the Proinov-type contractions which guarantees that a self-mapping T has a unique fixed point in terms of rational forms. Our main results improved the conclusions provided by Andreea Fulga (On (ψ,φ)−Rational Contractions) in which the continuity assumption can either be reduced to orbital continuity, k−continuity, continuity of Tk, T-orbital lower semi-continuity or even it can be removed. Meanwhile, the assumption of monotonicity on auxiliary functions is also removed from our main results. Moreover, based on the obtained fixed point results and the property of symmetry, we propose several Proinov-type contractions for a pair of self-mappings (P,Q) which will ensure the existence of the unique common fixed point of a pair of self-mappings (P,Q). Finally, we obtained some results related to fixed figures such as fixed circles or fixed discs which are symmetrical under the effect of self mappings on metric spaces, we proposed some new types of (ψ,φ)c−rational contractions and obtained the corresponding fixed figure theorems on metric spaces. Several examples are provided to indicate the validity of the results presented.

Spectral Stability of the $${\overline{\partial }}-$$Neumann Laplacian: Domain Perturbations

We study spectral stability of the $${\bar{\partial }}$$ ∂ ¯ -Neumann Laplacian on a bounded domain in $${\mathbb {C}}^n$$ C n when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the $${\bar{\partial }}$$ ∂ ¯ -Neumann Laplacian on bounded pseudoconvex domains in $${\mathbb {C}}^n$$ C n , lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in $${\mathbb {C}}^n$$ C n .

Lower semi-continuity for $\mathcal A$-quasiconvex functionals under convex restrictions

We show weak lower semi-continuity of functionals assuming the new notion of a ``convexly constrained'' $\mathcal A$-quasiconvex integrand. We assume $\mathcal A$-quasiconvexity only for functions defined on a set $K$ which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent ``convexly constrained'' $\mathcal A$-free sequences. In a motivating example, the integrand is $-\det^{\frac{1}{d-1}}$ and the convex constraint is positive semi-definiteness of a matrix field.

MODIFIED PROXIMAL POINT ALGORITHM FOR MINIMIZATION AND FIXED POINT PROBLEM IN CAT(0) SPACES

In this paper, we study modified-type proximal point algorithm for approximating a common solution of a lower semi-continuous mapping and fixed point of total asymptotically nonexpansive mapping in complete CAT(0) spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved.

Iterative methods for optimization problems and image restoration

In this paper, we introduce a new accelerated iterative method for finding a common fixed point of a countable family of nonexpansive mappings in the Hilbert spaces framework. Using our main result, we obtain a new accelerated image restoration iterative method for solving a minimization problem in the form of the sum of two proper lower semi-continuous and convex functions. As applications, we apply our algorithm to solving image restoration problems.

Well-Posedness and Porosity for Symmetric Optimization Problems

In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric optimization problem possesses two solutions. In this paper, we strengthen this result using a porosity notion. We investigate the collection of all functions such that the corresponding optimization problem is well-posed and prove that its complement is a σ-porous set.

Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain $${\mathscr {E}}(M)$$ E ( M ) on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$ M ⊆ R d can be defined and study essential properties of p. We prove that if M is a $$C^1$$ C 1 submanifold of $${{\mathbb {R}}}^d$$ R d satisfying a Lipschitz condition on the tangent spaces, then $${\mathscr {E}}(M)$$ E ( M ) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is $$C^2$$ C 2 or if the topological skeleton of $$M^c$$ M c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a $$C^k$$ C k -submanifold M with $$k\ge 2$$ k ≥ 2 , the projection map is $$C^{k-1}$$ C k - 1 on $${\mathscr {E}}(M)$$ E ( M ) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion $$M\subseteq {\mathscr {E}}(M)$$ M ⊆ E ( M ) is that M is a $$C^1$$ C 1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with $$M\subseteq {\mathscr {E}}(M)$$ M ⊆ E ( M ) , then M must be $$C^1$$ C 1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between $${\mathscr {E}}(M)$$ E ( M ) and the topological skeleton of $$M^c$$ M c .

Aplikasi Pemetaan terkait Semi-Inner Produk pada Ruang Bernorma Real

In this article, we give the properties of mappings associated with the upper semi-inner product , lower semi-inner product and Lumer semi-inner product which generate the norm on a real normed space. Furthermore, we establish applications to the Birkhoff orthogonality and characterization of best approximants.

C0-stability of topological entropy for contactomorphisms

Topological entropy is not lower semi-continuous: small perturbation of the dynamical system can lead to a collapse of entropy. In this note we show that for some special classes of dynamical systems (geodesic flows, Reeb flows, positive contactomorphisms) topological entropy at least is stable in the sense that there exists a nontrivial continuous lower bound, given that a certain homological invariant grows exponentially.