Fixed Point Theorems For Contractions Research Articles

Integrable Solutions and Continuous Dependence of a Nonlinear Singular Integral Inclusion of Fractional Orders and Applications

Let E be a reflexive Banach space. In this article we study the existence of integrable solutions in the space of all lesbesgue integrable functions on E, L1([0,T],E), of the nonlinear singular integral inclusion of fractional orders beneath the assumption that the multi-valued function G has Lipschitz selection in E. The main tool applied in this work is the Banach contraction fixed point theorem. Moreover, the paper explores a qualitative property associated with these solutions for the given problem such as the continuous dependence of the solutions on the set of selections S1G(τ,x(τ)). As an application, the existence of integrable solutions of the two nonlocal and weighted problems of the fractional differential inclusion is investigated. We additionally provide an example given as a numerical application to demonstrate the effectiveness and value of our results.

On the Solution of Caputo Fractional High-Order Three-Point Boundary Value Problem with Applications to Optimal Control

This research paper establishes the existence and uniqueness of solutions for a non-integer high-order boundary value problem, incorporating the Caputo fractional derivative with a non-local type boundary condition. The analytical approach involves the introduction of the fractional Green’s function. To analyze our findings effectively, we apply the Banach contraction fixed point theorem as the primary principle. Furthermore, we illustrate our results through the presentation of various examples.

Stability of nonlinear impulsive higher order differential – fractional integral delay equations with nonlocal initial conditions

The aim of this paper is to investigate some types of stability such as generalized Hyers-Ulam- Rassias stability(G-H-U-R-stabile) and the relation with Hyers-Ulam(H-U-stabile) stable and Hyers-Ulam-Rassias stable (H-U-R-stabile) and generalized Hyers-Ulam stable (G-H-U- stable) to obtain which one guarantee to satisfy stability of equations included a nonlinear function some of them contains a delay time of solution and the other contain a vector of different order of derivatives for the solution to n-time and vector of fractional order of integrals with different fractional orders and that was the for using a claculse of fractional calculus to satisfies the issue of this techniques. Moreover, the nonlocal initial values for the proposal equation of nonlinear impulsive higher order differential – fractional integral delay time equations which are adding more interesting for nonlinear analytic object of nonlinear higher order integro – fractional order impulsive classes, and the impulsive difference of the equation has some necessary conditions to prove the results of solution to be stable with certain type has related with other types. The necessary and sufficient conditions which assumed on this nonlinear higher order integro-differential impulsive equation have been achieved the stability with interesting certain estimates obtain through the proving technique. Also the uniqueness of solution has been studied with same conditions was presented for stability and used for that issue a contraction fixed point theorem.

Almost Anti-Periodic Oscillation Excited by External Inputs and Synchronization of Clifford-Valued Recurrent Neural Networks

The main purpose of this paper was to study the almost anti-periodic oscillation caused by external inputs and the global exponential synchronization of Clifford-valued recurrent neural networks with mixed delays. Since the space consists of almost anti-periodic functions has no vector space structure, firstly, we prove that the network under consideration possesses a unique bounded continuous solution by using the contraction fixed point theorem. Then, by using the inequality technique, it was proved that the unique bounded continuous solution is also an almost anti-periodic solution. Secondly, taking the neural network that was considered as the driving system, introducing the corresponding response system and designing the appropriate controller, some sufficient conditions for the global exponential synchronization of the driving-response system were obtained by employing the inequality technique. When the system we consider degenerated into a real-valued system, our results were considered new. Finally, the validity of the results was verified using a numerical example.

Existence and stability results for nonlocal boundary value problems of fractional order

In this paper, we prove the existence and uniqueness of solutions for the nonlocal boundary value problem (BVP) using Caputo fractional derivative (CFD). We derive Green’s function and give some estimation for it to derive our main results. The main principles applied to investigate our results are based on the Banach contraction fixed point theorem and Schauder fixed point approach. We dwell in detail on some results concerning the Hyers-Ulam (H-U) type and generalized H-U (g-H-U) type stability also for problem we are considering. We justify our results with an illustrative example.

Besicovitch Almost Periodic Solutions of Abstract Semi-Linear Differential Equations with Delay

In this paper, first, we give a definition of Besicovitch almost periodic functions by using the Bohr property and the Bochner property, respectively; study some basic properties of Besicovitch almost periodic functions, including composition theorem; and prove the equivalence of the Bohr definition and the Bochner definition. Then, using the contraction fixed point theorem, we study the existence and uniqueness of Besicovitch almost periodic solutions for a class of abstract semi-linear delay differential equations. Even if the equation we consider degenerates into ordinary differential equations, our result is new.

Existence and controllability of Hilfer fractional neutral differential equations with time delay via sequence method

<abstract><p>This paper deals with the existence and approximate controllability outcomes for Hilfer fractional neutral evolution equations. To begin, we explore existence outcomes using fractional computations and Banach contraction fixed point theorem. In addition, we illustrate that a neutral system with a time delay exists. Further, we prove the considered fractional time-delay system is approximately controllable using the sequence approach. Finally, an illustration of our main findings is offered.</p></abstract>

The exponential tracking and disturbance rejection for the nonlinear wave equation via boundary control without any geometrical conditions on the domain

We consider the problem of exponential tracking and disturbance rejection for the nonlinear unstable gradient-dependent wave equation via boundary control without any geometrical conditions on the domain. By splitting the tracking problem into a linear exponentially stable wave equation and a nonlinear dynamical regulator wave equation, we construct an explicit feedback and feedforward boundary controller that involves the reference, the disturbance and the solutions of the two problems. If the controller is applied on the whole boundary and the nonlinearity is globally Lipschitz, then we prove that the wave average tracks a desired reference signal exponentially by using the wave Green's function and the Banach contraction fixed point theorem. If the controller is applied on only a part of the boundary, then the same result is proved for a linear unstable gradient-dependent wave equation, but it is open for the nonlinear wave equation. Furthermore, as a robust analysis, we prove that the change of tracking error continuously depends on the change of disturbance. As usual, the rate of convergence of tracking error is independent of the feedforward part of the controller. A numerical example is given to confirm our theoretical results.

Dynamics analysis of delayed Nicholson-type systems involving patch structure and asymptotically almost periodic environments

ABSTRACT This paper deals with a class of delayed Nicholson-type systems involving patch structure. First, we prove that the solution of the initial value problem with respect to the addressed system exists globally and is bounded. Second, we employ the contraction fixed point theorem and analytical techniques to establish the existence of a positive asymptotically almost periodic solution and its global attractivity. Finally, an example is arranged to illustrate the effectiveness and feasibility of the obtained results.

Uniqueness and Stability Results for Caputo Fractional Volterra-Fredholm Integro-Differential Equations

In this paper, we established some new results concerning the uniqueness and Ulam’s stability results of the solutions of iterative nonlinear Volterra-Fredholm integro-differential equations subject to the boundary conditions. The fractional derivatives are considered in the Caputo sense. These new results are obtained by applying the Gronwall–Bellman’s inequality and the Banach contraction fixed point theorem. An illustrative example is included to demonstrate the efficiency and reliability of our results

Boundary value problems for a class of stochastic nonlinear fractional order differential equations

Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \(D^\alpha u(t)=\lambda\sqrt{I^\beta[\sigma^2(t,u(t))]}\dot{w}(t)\ ,0< t< 1\) with boundary conditions \(u(0)=0,\,\,u'(0)=u'(1)=0,\) where \(\lambda>0\) is a level of the noise term, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(\dot{w}(t)\) is a generalized derivative of Wiener process (Gaussian white noise), \(D^\alpha\) is the Riemann-Liouville fractional differential operator of order \(\alpha\in (3,4)\) and \(I^\beta,\,\,\beta>0\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \(\alpha=2\) and \(\beta=0\) with \(u(0)=u(1)=0\) is also studied.

Pseudo almost periodic synchronization of Clifford-valued fuzzy cellular neural networks with time-varying delays on time scales

At present, the research on discrete-time Clifford-valued neural networks is rarely reported. However, the discrete-time neural networks are an important part of the neural network theory. Because the time scale theory can unify the study of discrete- and continuous-time problems, it is not necessary to separately study continuous- and discrete-time systems. Therefore, to simultaneously study the pseudo almost periodic oscillation and synchronization of continuous- and discrete-time Clifford-valued neural networks, in this paper, we consider a class of Clifford-valued fuzzy cellular neural networks on time scales. Based on the theory of calculus on time scales and the contraction fixed point theorem, we first establish the existence of pseudo almost periodic solutions of neural networks. Then, under the condition that the considered network has pseudo almost periodic solutions, by designing a novel state-feedback controller and using reduction to absurdity, we obtain that the drive-response structure of Clifford-valued fuzzy cellular neural networks on time scales with pseudo almost periodic coefficients can realize the global exponential synchronization. Finally, we give a numerical example to illustrate the feasibility of our results.

Existence of Nonoscillation Solutions of Second-order Neutral Dierential Equations

This paper is concerned with existence of nonoscillation solution for a family of second-order neutral differential equations with positive and negative coefficients. A sufficient conditions for existence of nonoscillation solution is obtained by contraction fixed point theorem, special case of the equation has also been studied.

Asymptotically almost periodic solutions for certain differential equations with piecewise constant arguments

It is well known that differential equations with piecewise constant arguments is a class of functional differential equations, which has fascinated many scholars in recent years. These delay differential equations have been successfully applied to diverse models in real life, especially in biology, physics, economics, etc. In this work, we are interested in the existence and uniqueness of asymptotically almost periodic solution for certain differential equation with piecewise constant arguments. Due to the particularity of the equations, we cannot use the traditional method to convert it into the difference equation with exponential dichotomy. Through constructing Cauchy matrix of the investigated system to find the corresponding Green matrix of the difference equation, we need the concept of exponential dichotomy and the Banach contraction fixed point theorem of the corresponding system. Then we give some sufficient conditions to obtain the existence and uniqueness of asymptotically almost periodic solutions for these systems.

On algebraic properties of soft real points

In this paper, we introduce and discuss soft single points, which proceed towards soft real points by using real numbers and subsets of set of real numbers. We also define the basic operations on soft real points and explore the algebraic properties. We observe that the set of all soft real points forms a ring. Moreover, we study the soft real point metric using soft real point and explore some of its properties. We then establish a soft real point contraction fixed point theorem using soft real point metric space. It is interesting to mention that these concepts may be helpful for researchers to navigate the ideas put forth in a soft metric extension of several important fixed point theorems for metric spaces deduced from comparable existing results.

The Existence and Global Exponential Stability of Almost Periodic Solutions for Neutral-Type CNNs on Time Scales

In this paper, neutral-type competitive neural networks with mixed time-varying delays and leakage delays on time scales are proposed. Based on the contraction fixed-point theorem, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions of neural networks under consideration. The results obtained are brand new, indicating that the continuous time and discrete-time conditions of the network share the same dynamic behavior. Finally, two examples are given to illustrate the validity of the results obtained.

Exact controllability of fractional neutral integro-differential systems with state-dependent delay in Banach spaces

Abstract In this manuscript, we have a tendency to execute Banach contraction fixed point theorem combined with resolvent operator to analyze the exact controllability results for fractional neutral integro-differential systems (FNIDS) with state-dependent delay (SDD) in Banach spaces. An illustration is additionally offered to exhibit the achieved hypotheses.

Biological consistency of an epidemic model with both vertical and horizontal transmissions

A system of nonlinear integro-differential equations is investigated. The model describes an age structured S.I.S system of a disease with horizontal and vertical transmission. Global well-posedness is proved in L1space. The method is based on the contraction fixed point theorem. We exhibit a closed subset of a Banach space, on which a certain mapping is a strict contraction.

Rational Contractions in b-Metric Spaces

In this paper, we prove fixed point theorems for contractions and generalized weak contractions satisfying rational expressions in complete b-metric spaces. Our results generalize several well-known comparable results in the literature.

Complete Controllability for Fractional Evolution Equations

The paper is concerned with the complete controllability of fractional evolution equation with nonlocal condition by using a more general concept for mild solution. By contraction fixed point theorem and Krasnoselskii's fixed point theorem, we obtain some sufficient conditions to ensure the complete controllability. Our obtained results are more general to known results.