Lipschitz isomorphism and fixed point theorem for normed groups

For a complete normed abelian group G, we show that the mass of image of a rectifiable G-chain S under chain map f♯ induced by Lipschitz map f is controlled by the integral of Jacobian of f restricted on the support of S with respect to the associated radon measure μS.

Conditions for unique solvability of nonlinear simultaneous equations satisfying Lipschitz conditions and an application to nonlinear network equations are proposed. It is shown that the global invertibility of a Lipschitz continuous mapping f of $\mathbb{R}^n $ into itself and the Lipschitz continuity of $f^{ - 1} $ are verified by investigating the positivity of some principal minors of the Jacobian matrix of f in spite of the existence of nondifferentiable points of f. This is a generalization of previous works, especially Fujisawa and Kuh’s theorem, obtained for continuously differentiable or piecewise-linear mappings. This generalization is given by employing the Lebesgue integration of the Jacobian matrix over an open interval of $\mathbb{R}^n $. This result is applied to network equations, both resistive and dynamical; especially to the latter, the Lipschitz continuity of such inverse mappings is of great importance to guarantee the uniqueness of the solutions. In addition, network examples for which the above result is useful are given, and it is demonstrated that simple conditions for the unique solvability are obtained in terms of differential coefficients of network element characteristics.

In this paper we study the existence of continuous solutions and their constructions for a second order iterative functional equation, which involves iterate of the unknown function and a nonlinear term. Imposing Lipschitz conditions to those given functions, we prove the existence of continuous solutions on the whole $\mathbb{R}$ by applying the contraction principle. In the case without Lipschitz conditions we hardly use the contraction principle, but we construct continuous solutions on $\mathbb{R}$ recursively with a partition of $\mathbb{R}$.

In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α : H → R m \alpha : H \rightarrow \mathbb {R}^m is injective, with ( α ( x ) ) k = | ⟨ x , f k ⟩ | 2 (\alpha (x))_k = \lvert \langle x, f_k \rangle \rvert ^2 , where { f 1 , ⋯ , f m } \{f_1, \cdots , f_m\} is a frame for the Hilbert space H H , then there exists a left inverse map ω : R m → H \omega : \mathbb {R}^m \rightarrow H that is Lipschitz continuous. Additionally we obtain that the Lipschitz constant of this inverse map is at most 12 divided by the lower Lipschitz constant of α \alpha .

<abstract><p>We present Perov's type $ (\beta, F) $-contraction principle and examine the fixed points of the self-operators satisfying Perov's type $ (\beta, F) $-contraction principle in the context of vector-valued $ b $-metrics. A specific instance of the $ (\beta, F) $-contraction principle is the $ F $-contraction principle. We generalize a number of recent findings that are already in the literature and provide an example to illustrate the hypothesis of the main theorem. We apply the obtained fixed point theorem to show the existence of the solution to the delay integro-differential problem.</p></abstract>

We address the existence, uniqueness, and averaging principle for Caputo–Hadamard fractional dynamic systems with Dirichlet boundary conditions driven by Rosenblatt process and pure Lévy jumps. First, Lemma 4 establishes the equivalent integral equation representation of our system. Using this foundation, existence and uniqueness are proved by Banach's contraction principle under stochastic calculus, Lipschitz and finite energy conditions. Subsequently, under appropriate averaging assumptions, the system is averaged out with time scale ϵ. Mean-square convergence between original solution and its counterpart is verified by employing tools such as Wiener–Itô double integral, Cauchy–Schwarz, Doob's martingale, and Gronwall–Bellman inequalities. Eventually, computational example with numerical simulations is provided to support the theoretical results.

State observers for nonlinear systems are often designed for a canonical form of this system. However, this form may possess singular points, where the vector field is not defined or a Lipschitz condition is not fulfilled. This unpleasant behavior can possibly be avoided using an embedding into a higher dimensional space. A construction of such an embedding and the corresponding inverse map is discussed for polynomial systems using methods from algebraic geometry.

Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems

‎‎This contribution mainly focuses on some aspects of Lipschitz groups‎, ‎i.e.‎, ‎metrizable groups with Lipschitz multiplication and inversion map‎. ‎In the main result it is proved that metric groups‎, ‎with a translation-invariant metric‎, ‎may be characterized as particular group objects in the category of metric spaces and Lipschitz maps‎. ‎Moreover‎, ‎up to an adjustment of the metric‎, ‎any metrizable abelian group also is shown to be a Lipschitz group‎. ‎Finally we present a result similar to the fact that any topological nilpotent element $x$ in a Banach algebra gives rise to an invertible element $1-x$‎, ‎in the setting of complete Lipschitz groups‎.

The purpose of this paper is to present a regularization variant of the extragradient method for finding a common element of the solution sets for a variational inequality problem involving a \( \tilde k \)-Lipschitz continuous monotone mapping A and for a finite family of λi-inverse strongly-monotone operators {Ai}i = 1Nfrom a closed convex subset K into the Hilbert space H.

As is known to all, Lipschitz condition, which is very important to guarantee existence and uniqueness of solution for differential equations, is not frequently satisfied in real-world problems. In this paper, without the Lipschitz condition, we intend to explore a kind of novel coupled systems of fuzzy Caputo Generalized Hukuhara type (in short, gH-type) fractional partial differential equations. First and foremost, based on a series of notions of relative compactness in fuzzy number spaces, and using Schauder fixed point theorem in Banach semilinear spaces, it is naturally to prove existence of two classes of gH-weak solutions for the coupled systems of fuzzy fractional partial differential equations. We then give an example to illustrate our main conclusions vividly and intuitively. As applications, combining with the relevant definitions of fuzzy projection operators, and under some suitable conditions, existence results of two categories of gH-weak solutions for a class of fire-new fuzzy fractional partial differential coupled projection neural network systems are also proposed, which are different from those already published work. Finally, we present some work for future research.

In this paper we presents some theorems in 2 Banach spaces. Mathematics subject classification: 47H10, 54H25. I. Introduction: A large variety of the problems of analysis and applied mathematics reduce to finding solutions of non linear functional equations which can be formulated in terms of finding the fixed points of a non linear mapping. Fixed point theorems are very important tools for proving the existence and uniqness of the solutions to various differential, integral and partial differential equations and variational inequalities etc. representing phenomena arising in different fields. Therefore the fixed point methods specially Banach's contraction principle provides a powerful tool for obtaining the solutions of these equations which were very difficult to solve by any other methods. Recently described about the application of Banach's contraction principle (2). Ghalar (4) introduced the concept of 2- Banach. Recently Badshah and Gupta (3), Yadava, Rajput and Bhardwaj (6) and Yadav, Rajput, Choudhary and Bhardwaj (7) also worked for Banach and 2-Banach spaces for non contraction mapping. In present paper we prove some fixed point theorems for non-contraction mappings, in 2-Banach spaces motivated by above, before starting the main result first we write some definitions

Following the idea of T.A. Burton, of progressive contractions, presented in some examples (T.A. Burton, \emph{A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions}, Fixed Point Theory, 20 (2019), No. 1, 107-113) and the forward step method (I.A. Rus, \emph{Abstract models of step method which imply the convergence of successive approximations}, Fixed Point Theory, 9 (2008), No. 1, 293-307), in this paper we give some variants of contraction principle in the case of operators with Volterra property. The basic ingredient in the theory of step by step contraction is $G$-contraction (I.A. Rus, \emph{Cyclic representations and fixed points}, Ann. T. Popoviciu Seminar of Functional Eq. Approxim. Convexity, 3 (2005), 171-178). The relevance of step by step contraction principle is illustrated by applications in the theory of differential and integral equations.

Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.

In this paper, we study the existence of asymptotic almost automorphic solution of fractional neutral integro-differential equation. We prove the result using fixed-point theorems. We show the result with Lipschitz condition and without Lipschitz condition on the forcing term. Finally, examples are given to illustrate the analytical findings.