Eigenvalue comparisons for a class of boundary value problems of second order difference equations

We consider the boundary value problem for second order difference equation

     

Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations

In this paper, two Chebyshev-like third order methods free from second derivatives are considered and analyzed for systems of nonlinear equations. The methods can be obtained by having different approximations to the second derivatives present in the Chebyshev method. We study the local and third order convergence of the methods using the point of attraction theory. The computational aspects of the methods are also studied using some numerical experiments including an application to the Chandrasekhar integral equations in Radiative Transfer.     

Eigenvalue comparisons for second order difference equations with Neumann boundary conditions

The boundary value problem for second order difference equation

     

Recurrence relations for a Newton-like method in Banach spaces

The convergence of iterative methods for solving nonlinear operator equations in Banach spaces is established from the convergence of majorizing sequences. An alternative approach is developed to establish this convergence by using recurrence relations. For example, the recurrence relations are used in establishing the convergence of Newton's method [L.B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, 1979] and the third order methods such as Halley's, Chebyshev's and super Halley's [V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method, Computing 44 (1990) 169–184; V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Halley method, Computing 45 (1990) 355–367; J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000) 227–236; J.M. Gutiérrez, M.A. Hernández, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997) 171–183; J.M. Gutiérrez, M.A. Hernández, Recurrence relations for the Super–Halley method, Comput. Math. Appl. 7(36) (1998) 1–8; M.A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–445 [10]].     

Periodic solutions of non-linear discrete Volterra equations with finite memory

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.     

Blow up oscillating solutions to some nonlinear fourth order differential equations

We give strong theoretical and numerical evidence that solutions to some nonlinear fourth order ordinary differential equations blow up in finite time with infinitely many wild oscillations. We exhibit an explicit example where this phenomenon occurs. We discuss possible applications to biharmonic partial differential equations and to the suspension bridges model. In particular, we give a possible new explanation of the collapse of bridges.     

On the global convergence of path-following methods to determine all solutions to a system of nonlinear equations

In this paper we prove a general theorem stating a sufficient condition for the inverse image of a point under a continuously differentiable map from ℝ ⁿ to ℝ ^k to be connected. This result is applied to the trajectories generated by the Newton flow. Several examples demonstrate the applicability of the results to nontrivial problems. This work was supported by the Deutsche Forschungsgemeinschaft.     

Applying carvalho's method to find periodic solutions of difference equations

In this paper we apply the method initially developed in [1] for differential-difference equations, to the case of difference equations, in order to find 2 and 3-periodic solutions of some equations that often appear in the literatures as are for instance the case of Applications 2,5 which are examples of population growth models, and Application 4, which is a standard example of nonlinear higher order scalar difference equation depending on two parameters (see, Kocik and Ladas [3]).     

Asymptotic behavior of bounded solutions to a nonhomogeneous second-order evolution equation of monotone type

By using previous results of Djafari Rouhani [B. Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale university, 1981, part I, pp. 1-76; B. Djafari Rouhani, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990) 465-476; B. Djafari Rouhani, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990) 226-235] for dissipative systems, we study the asymptotic behavior of solutions to the following system of second-order nonhomogeneous evolution equations:

     

Improving order and efficiency: Composition with a modified Newton’s method

In this paper a zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to compose a given iterative method with a modified Newton’s method that introduces just one evaluation of the function. To carry out this procedure some classical methods with different orders of convergence are used to obtain new methods that can be generalized in Banach spaces.     

On two perturbation estimates of the extreme solutions to the equations X ± A*XA = Q

Two perturbation estimates for maximal positive definite solutions of equations X + A*X⁻¹A = Q and X − A*X⁻¹A = Q are considered. These estimates are proved in [Hasanov et al., Improved perturbation Estimates for the Matrix Equations X ± A*X⁻¹A = Q, Linear Algebra Appl. 379 (2004) 113-135]. We derive new perturbation estimates under weaker restrictions on coefficient matrices of the equations. The theoretical results are illustrated by numerical examples.     

Periodic solutions for a class of higher-dimension functional differential equations with impulses

By employing a fixed-point theorem in cones, we establish some criteria for existence of positive periodic solutions of a class of n$n$-dimension periodic functional differential equations with impulses. We also give some applications to several biomathematical models and new results are obtained.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

Optimal order results for a class of regularization methods using unbounded operators

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.     

On the computation of parameter derivatives of solutions of linear difference equations

A method is given to compute the parameter derivatives of recessive solutions of second-order inhomogeneous linear difference equations. The case of difference equations in which all solutions have the same rate of growth is also discussed.