Contraction Theorem Research Articles

Fixed point theorems for (φ-ψ) contractions in partially ordered metric-like spaces using new auxiliary functions

In this paper, we introduce a class of new auxiliary functions and establish certain fixed point theorems under (?-?) contractive conditions in partially ordered metric-like spaces. Our work generalizes some results in the literature and assumes some as particular case. Examples are provided to support the useability of our results.

Fixed point theorems for contractions in semicomplete semimetric spaces

We introduce the concept of semicompleteness on semimetric space, which is weaker than completeness. We prove fixed point theorems for contractions in semicomplete semimetric spaces. Also, we generalize JachymskiMatkowski-Swia¸ tkowski’s fixed point theorem in semimetric spaces.

Well-Posedness of a Mathematical Model for Alzheimer's Disease

We consider the existence and uniqueness of solutions of an initial-boundary value problem for a coupled system of PDE's arising in a model for Alzheimer's disease. Apart from reaction diffusion equations, the system contains a transport equation in a bounded interval for a probability measure which is related to the malfunctioning of neurons. The main ingredients to prove existence are: the method of characteristics for the transport equation, a priori estimates for solutions of the reaction diffusion equations, a variant of the classical contraction theorem, and the Wasserstein metric for the part concerning the probability measure. We stress that all hypotheses on the data are not suggested by mathematical artefacts, but are naturally imposed by modelling considerations. In particular the use of a probability measure is natural from a modelling point of view. The nontrivial part of the analysis is the suitable combination of the various mathematical tools, which is not quite routine and requires various technical adjustments.

Computer-assisted proofs for radially symmetric solutions of PDEs

We obtain radially symmetric solutions of some nonlinear (geometric) partial differential equations via a rigorous computer-assisted method. We introduce all main ideas through examples, accessible to non-experts. The proofs are obtained by solving for the coefficients of the Taylor series of the solutions in a Banach space of geometrically decaying sequences. The tool that allows us to advance from numerical simulations to mathematical proofs is the Banach contraction theorem.

A Classes of Variational Inequality Problems Involving Multivalued Mappings

The main objective of the Variational inequality problem is to study some functional analytic tools, projection method and fixed point theorems and then exploiting these to study the existence of solutions and convergence analysis of iterative algorithms developed for some classes of Variational inequality problem. The main objective of this paper is to study the existence of solutions of some classes of Variational inequalities using fixed point theorems for multivalued and using Banach contraction theorem we prove the existence of a unique solution of multi value Variational inequality problem.

Fixed point theorems for contractions of rational type in complete metric spaces

Fixed point theorems for contractions of rational type in complete metric spaces

Solvability for a class of evolution equations of fractional order with nonlocal conditions on the half-line

In this paper, we get a new form equivalent integral equation for a class of evolution equations of fractional order with nonlocal conditions on the half-line. With the aid of it, the uniqueness of the mild solution is obtained by the Banach contraction theorem. Also, we present the existence and uniqueness theorem of positive mild solutions by the monotone iterative method without assumption of lower and upper solutions.

Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < α < 5/2

Fractional difference operators with discrete-Mittag-Leffler kernels of order α > 1 are defined and their corresponding fractional sum operators are confirmed. We prove existence and uniqueness theorems for the discrete fractional initial value problems in the frame of discrete Caputo (ABC) and Riemann (ABR) operators by using Banach contraction theorem. Then, we prove Lyapunov type inequality for a Riemann type fractional difference boundary value problem of order 2 < α < 5∕2 within discrete Mittag-Leffler kernels, where the limiting case α → 2+ results in the ordinary difference Lyapunov inequality. Examples are given to clarify the applicability of our results and an application about the discrete fractional Sturm-Liouville eigenvalue problem is analyzed.

Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation u⁗+βu″+eu−1=0 for all parameter values β∈[0.5,1.9]. For each β, a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.

Fractional operators with exponential kernels and a Lyapunov type inequality

In this article, we extend fractional calculus with nonsingular exponential kernels, initiated recently by Caputo and Fabrizio, to higher order. The extension is given to both left and right fractional derivatives and integrals. We prove existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial value problems by using Banach contraction theorem. Then we prove Lyapunov type inequality for the Riemann type fractional boundary value problems within the exponential kernels. Illustrative examples are analyzed and an application about Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

On Some Fixed Point Results in LP Space

The Banach fixed point theorem or contraction theorem of a complete metric space into itself .It state conditions sufficient for the existence and uniqueness of a fixed point

A characterization of proximal normal structure via proximal diametral sequences

We introduce a concept of pointwise cyclic relatively nonexpansive mapping involving orbits to investigate the existence of best proximity points using a geometric property defined on a nonempty and convex pair of subsets of a Banach space X, called weak proximal normal structure. Examples are given to support our main conclusions. We also introduce a notion of proximal diametral sequence and establish a characterization of proximal normal structure and show that every nonempty and convex pair in uniformly convex in every direction Banach spaces has weak proximal normal structure. As an application, we give a new existence theorem for cyclic contractions in reflexive Banach spaces without strictly convexity condition.

$G$-asymptotic contractions in metric spaces with a graph and fixed point results

In this paper, we discuss the existence and uniqueness of fixed points for $G$-asymptotic contractions in metric spaces endowed with a graph. The result given here is a new version of Kirk's fixed point theorem for asymptotic contractions in metric spaces endowed with a graph. The given result here is a generalization of fixed point theorem for asymptotic contraction from metric s paces to metric spaces endowed with a graph.

Optimal approximate solution theorems for Geraghty's proximal contractions in partially ordered sets via w-distances

Optimal approximate solution theorems for Geraghty's proximal contractions in partially ordered sets via w-distances

A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order alphain[0,1] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo (ABC) and Riemann (ABR) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order 2<alphaleq3 in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

Best proximity point theorems for proximal cyclic contractions

The purpose of this article is to compute a global minimizer of the function $$x\longrightarrow d(x, Tx)$$ , where T is a proximal cyclic contraction in the framework of a best proximally complete space, thereby ensuring the existence of an optimal approximate solution, called a best proximity point, to the equation $$Tx=x$$ when T is not necessarily a self-mapping.

A generalization of Istrățescu’s fixed point theorem for convex contractions

In this paper we prove a generalization of Istr\u{a}\c{t}escu's theorem for convex contractions. More precisely, we introduce the concept of iterated function system consisting of convex contractions and prove the existence and uniqueness of the attractor of such a system. In addition we study the properties of the canonical projection from the code space into the attractor of an iterated function system consisting of convex contractions.

Existence of Mori fibre spaces for 3-folds in char p

We prove the following results for projective klt pairs of dimension 3 over an algebraically closed field of characteristic p>5: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc.

On mild solutions of a semilinear mixed Volterra-Fredholm functional integrodifferential evolution nonlocal problem in Banach spaces

The existence, uniqueness and continuous dependence on initial data of mild solutions of nonlocal mixed Volterra-Fredholm functional integrodifferential equations with delay in Banach spaces has been discussed and proved in the present paper. The results are established by using the semigroup theory and modified version of Banach contraction theorem.

Remarks on curvature in the transportation metric

According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the “hyperbolic” toric Kahler–Einstein equation e Φ = detD 2Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D 2Φ, e Φ dx) has a non-positive Bakry–Emery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry–Emery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.