Existence-uniqueness Theorem Research Articles

Delay differential equations with fractional differential operators: Existence, uniqueness and applications to chaos

<abstract><p>In this study, we consider a chaotic model in which fractional differential operators and the delay term are added. Using the Carathéodory existence-uniqueness theorem for this chaotic model modified with the Caputo fractional derivative, we show that the solution of the associated system exists and is unique. We consider the chaotic model with a delay term with Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives and present a numerical algorithm for these models. We then present the numerical solution of chaotic models with delay terms by using piecewise differential operators, where fractional, classical and stochastic processes can be used. We present the numerical solution of chaotic models with delay terms, as modified by using piecewise differential operators. The graphical representations of these models are simulated for different values of the fractional order.</p></abstract>

The existence–uniqueness and exponential estimate of solutions for stochastic functional differential equations driven by G‐Brownian motion

The objective of this article is to present the analysis of stochastic functional differential equations in the G‐framework (G‐SFDEs) with monotone type conditions. The existence–uniqueness theorem has been proved. The exponential and estimates have been obtained. We have derived that the Lyapunov exponent of the solution is less than .

Symmetry-Based Algorithms for Invertible Mappings of Polynomially Nonlinear PDE to Linear PDE

This paper is a sequel to our previous work where we introduced the MapDE algorithm to determine the existence of analytic invertible mappings of an input (source) differential polynomial system (DPS) to a specific target DPS, and sometimes by heuristic integration an explicit form of the mapping. A particular feature was to exploit the Lie symmetry invariance algebra of the source without integrating its equations, to facilitate MapDE, making algorithmic an approach initiated by Bluman and Kumei. In applications, however, the explicit form of a target DPS is not available, and a more important question is, can the source be mapped to a more tractable class? We extend MapDE to determine if a source nonlinear DPS can be mapped to a linear differential system. MapDE applies differential-elimination completion algorithms to the various over-determined DPS by applying a finite number of differentiations and eliminations to complete them to a form for which an existence-uniqueness theorem is available, enabling the existence of the linearization to be determined among other applications. The methods combine aspects of the Bluman–Kumei mapping approach with techniques introduced by Lyakhov, Gerdt and Michels for the determination of exact linearizations of ODE. The Bluman–Kumei approach for PDE focuses on the fact that such linearizable systems must admit a usually infinite Lie sub-pseudogroup corresponding to the linear superposition of solutions in the target. In contrast, Lyakhov et al. focus on ODE and properties of the so-called derived sub-algebra of the (finite) dimensional Lie algebra of symmetries of the ODE. Examples are given to illustrate the approach, and a heuristic integration method sometimes gives explicit forms of the maps. We also illustrate the powerful maximal symmetry groups facility as a natural tool to be used in conjunction with MapDE.

A Novel Numerical Method for Solving Volterra Integro-Differential Equations

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods.

R-order of convergence for the improved multi-point Chebyshev-like methods under generalized continuity condition

ABSTRACT In this paper, the convergence for improved multi-point Chebyshev-like methods is considered. Compared to the results of Chebyshev method with second derivative in reference [M.A. Hernández, M.A. Salanova, J. Comput. Appl. Math. 126 (2000), pp. 131–143], the R-order is maximised and the Hölder continuity condition is also relaxed. Moreover, these improved methods do not need second derivatives, when the computation of second derivative is large, these improved methods will be more efficient than Chebyshev method. Under the relaxed condition, an existence-uniqueness theorem is proved. The R-order of convergence for these improved methods is also analysed.

An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities

Let C be a nonempty closed convex subset of a real Hilbert space mathcal{H} with inner product langle cdot , cdot rangle , and let f: mathcal{H}rightarrow mathcal{H} be a nonlinear operator. Consider the inverse variational inequality (in short, operatorname{IVI}(C,f)) problem of finding a point xi ^{*}in mathcal{H} such that \t\t\tf(ξ∗)∈C,〈ξ∗,v−f(ξ∗)〉≥0,∀v∈C.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ f\\bigl(\\xi ^{*}\\bigr)\\in C, \\quad \\bigl\\langle \\xi ^{*}, v-f \\bigl(\\xi ^{*}\\bigr)\\bigr\\rangle \\geq 0, \\quad \\forall v\\in C. $$\\end{document} In this paper, we prove that operatorname{IVI}(C,f) has a unique solution if f is Lipschitz continuous and strongly monotone, which essentially improves the relevant result in (Luo and Yang in Optim. Lett. 8:1261–1272, 2014). Based on this result, an iterative algorithm, named the alternating contraction projection method (ACPM), is proposed for solving Lipschitz continuous and strongly monotone inverse variational inequalities. The strong convergence of the ACPM is proved and the convergence rate estimate is obtained. Furthermore, for the case that the structure of C is very complex and the projection operator P_{C} is difficult to calculate, we introduce the alternating contraction relaxation projection method (ACRPM) and prove its strong convergence. Some numerical experiments are provided to show the practicability and effectiveness of our algorithms. Our results in this paper extend and improve the related existing results.

Local Convergence and Dynamics of a Family of Iterative Methods for Multiple Roots of Nonlinear Equations

The local convergence of a family of iterative methods based on a parameter θ for multiple roots of nonlinear equations is established. It is established under the assumption that the derivative g(m+ 1) of the function g satisfies the Holder continuity condition. The existence–uniqueness theorem along with a posteriori error bounds are provided. The R-order convergence is shown to be equal to (2 + p), where p ∈ (0, 1]. The well-known methods of Chebyshev (θ = 0) and of Osada (θ = 1) belong to the family. A number of numerical examples are worked out. The radii of convergence balls for different values of the parameter θ are compared and tabulated. To the best of our knowledge, our work is the first attempt to establish the local convergence of a family of iterative methods based on a parameter θ for computing multiple roots of nonlinear equations. Moreover, its dynamical study is investigated and the parameter plane is constructed to find the best value of the parameter θ. The values of θ are investigated according to the character of the strange fixed point and the multiplicity of the root.

Analysis of Lossy Transmission Lines Terminated by Schottky Diode Circuit

In the present paper we consider a lossy transmission line terminated by a circuit corresponding to a Schottky diode. On the base of Kirchhoff's law boundary conditions are derived. Then a mixed problem for the lossy transmission line system is formulated. We reduce the mixed problem for the hyperbolic transmission line system to an initial value problem for a system of differential equations with delays on the boundary. We prove existence-uniqueness theorem for oscillatory solution. The paper ends with numerical example with real values of the Schottky diode parameters.

Semilocal convergence for a class of improved multi-step Chebyshev–Halley-like methods under extended conditions

Semilocal convergence for a class of improved multi-step Chebyshev–Halley-like methods is considered in this paper. Compared with the results for the Chebyshev method in Hernandez (J Comput Appl Math 126:131–143, 2000), the R-order of convergence is heightened and the Holder continuity of second derivative is also relaxed. Moreover, an existence-uniqueness theorem is proved under the extended conditions.

Well-Posedness of Mild Solutions to Stochastic Parabolic Partial Functional Differential Equations

In this paper, we study the well-posedness of mild solutions to stochastic parabolic partial functional differential equations with space–time white noise. Firstly, we establish an existence–uniqueness theorem under the global Lipschitz condition and the linear growth condition. Secondly, we show the existence–uniqueness property under the global/local Lipschitz condition but without assuming the linear growth condition. In particular, we consider the existence and uniqueness under the weaker condition than the Lipschitz condition. Finally, we obtain the nonnegativity and comparison theorems and utilize them to investigate the existence of nonnegative mild solutions under the linear growth condition without assuming the Lipschitz condition.

Existence–uniqueness and stability of reaction–diffusion stochastic Hopfield neural networks with S-type distributed time delays

Abstract The aim of this paper is to develop some theories of mild solutions to reaction–diffusion stochastic Hopfield neural networks with S-type distributed time delays driven by infinite dimensional Wiener processes. First, we transform the networks with infinite time delays defined on ( − ∞ , 0 ] into those with finite time delays defined on [ − τ , 0 ] by truncation method. Then, the existence–uniqueness theorem of the networks with finite time delays is established in terms of the generalized Halanay inequality. Next, the existence of equilibrium is demonstrated with homotopy invariance theorem and topological degree theory. And the stability is considered using M-function. Finally, with the help of approximation method, the well-posedness and stability of the original networks are obtained. An interesting example shows the effectiveness of our results.

Egg-eating predators in interaction with age-structured prey population

We investigate a predator-prey model for egg-eating predators in which the prey population is assumed to have an age structure. By the method of characteristics, this model reduces to a system of integral equations. Then a generalization of the Banach fixed-point theorem is used to show, under relatively mild conditions, the existence of a unique, global, weak solution to the population problem. Furthermore, this methodology allows us to generate a sequence of iterates, called the Picard iterates, that converges to the solution. Also, we strengthen the assumptions of the existence-uniqueness theorem to establish the validity of the corresponding conservation law in integral form. Thus we prove a result which shows the coexistence of both predator and prey species over a long time.

Generalised solutions for fully nonlinear PDE systems and existence–uniqueness theorems

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of Distributions to PDEs and is not based on either integration by parts or on the maximum principle. Instead, our starting point builds on the probabilistic representation of derivatives via limits of difference quotients in the Young measures over a toric compactification of the space of jets. After developing some basic theory, as a first application we consider the Dirichlet problem and we prove existence–uniqueness–partial regularity of solutions to fully nonlinear degenerate elliptic 2nd order systems and also existence of solutions to the ∞-Laplace system of vectorial Calculus of Variations in L∞.

Semilocal convergence of a continuation method in Banach spaces

This paper is concerned with the semilocal convergence of a continuation method between two third-order iterative methods, namely, the Halley’s and the convex acceleration of Newton’s method, also known as the Super-Halley’s method. This convergence analysis is discussed using the recurrence relations approach. This approach simplifies the analysis and leads to improved results. The convergence analysis is established under the assumption that the second Frechet derivative satisfies Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bound is derived in terms of a real parameter α ∈ [0, 1]. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences [15], we observed that our analysis gives better results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Halley’s method (α = 0) and the convex acceleration of Newton’s method (α = 1), respectively, with improved results.

Convergence of a parameter based iterative method for solving nonlinear equations in Banach spaces

In this paper, we introduce a new class of a family of third order iterative methods which depend on a real parameter for solving nonlinear equations in Banach spaces. The semilocal convergence of iterative method under the assumption that the second order Fŕechet derivative of the operator satisfies the Holder continuity condition and the R-order \(2+p\) of the iteration family is established. An existence-uniqueness theorem and error estimate in terms of a parameter \(\alpha \in [0,1]\) are provided for these iterations using a technique based on a new system of recurrence relations. Numerical application of nonlinear Hammerstein integral equations of the mixed type is provided.

Functional delay fractional equations

Abstract In this paper we discuss functional delay fractional equations. A Banach fixed point theorem is applied to obtain the existence (uniqueness) theorem. We also discuss such problems when a delay argument has a form α(t) = αt, 0 < α < 1, by using the method of successive approximations. Some existence results are also formulated in this case. An example illustrates the main result.

Semilocal convergence on a family of root-finding multi-point methods in Banach spaces under relaxed continuity condition

In this paper, we consider the semilocal convergence on a family of root-finding multi-point methods. Compared with the results in reference (Hernandez, M.A., Salanova, M.A., J. Comput. Appl. Math. 126, 131---143 3), these multi-point methods do not require the second derivative, Holder continuity condition is relaxed, and the R-order is also enhanced. We prove an existence-uniqueness theorem of the solution. The R-order for these multi-point methods is at least 6 + q with relaxed continuous second derivative, where q?[0,1].

Local convergence of a parameter based iteration with Hölder continuous derivative in Banach spaces

The local convergence analysis of a parameter based iteration with Holder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives. An existence-uniqueness theorem along with the derivation of error bounds for the solution is established. Different numerical examples including nonlinear Hammerstein equation are solved. The radii of balls of convergence for them are obtained. Substantial improvements of these radii are found in comparison to some other existing methods under similar conditions for all examples considered.

Global well-posedness and stability of the mild solutions for a class of stochastic partial functional differential equations

The aim of this paper is to develop some theories of stochastic partial functional differential equations (SPFDEs) driven by infinite dimensional Wiener processes under the quasi-local Lipschitz condition and the restriction growth condition. Firstly, we establish the existence-uniqueness theorem of the global mild solutions for SPFDEs by using the intercept technique. Then, we discuss asymptotic behavior of the solutions. Furthermore, some criteria of exponential stability in the mean square are obtained by using Lyapunov method. An example is provided to show the effectiveness of the theoretical results.

About a Problem for Loaded Parabolic-Hyperbolic Type Equation with Fractional Derivatives

An existence and uniqueness of solution of local boundary value problem with discontinuous matching condition for the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals have been investigated. The uniqueness of solution is proved by the method of integral energy and the existence is proved by the method of integral equations. Let us note that, from this problem, the same problem follows with continuous gluing conditions (atλ=1); thus an existence theorem and uniqueness theorem will be correct and on this case.