Alberto Cabada

Fixed points and approximate solutions for nonlinear operator equations

We study a nonlinear operator equation and obtain a sequence of approximate solutions by extending the monotone iterative technique to new situations. Some examples and applications are presented.

A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems

Abstract The purpose of this paper is to study a periodic boundary value problem for a nonlinear ordinary differential equation of second order. We generalize the monotone iterative method to cover some new situations and present a new approach considering the monotone iterates as orbits of a (discrete) dynamical system.

Existence and approximation of solutions for first-order discontinuous difference equations with nonlinear global conditions in the presence of lower and upper solutions☆

Abstract This paper is devoted to the study of the existence of solutions of first-order difference equations verifying nonlinear conditions that involve the global behavior of the solution. We prove that the existence of lower and upper solutions warrants the existence of such solutions lying in the sector formed by the mentioned functions. We also can prove that some classical results for differential equations are not true in general for this case.

The monotone method for first-order problems with linear and nonlinear boundary conditions

In this paper, we develop the monotone method for the first-order problem u(t) = f(t, u(t)) for a.e.t @? [a, b] when f is a Caratheodory function and u @? W^1^,^ ^1([a, b]). We consider the nonlinear boundary conditions L(u(a), u(b)) = 0, with L @? C(R^2, R) nondecreasing in x or nonincreasing in y, and the linear boundary conditions a0u(0) - b0u(T) = @l0, with a0, b0 and @l0 @? R. We prove the existence of solutions and the validity of the monotone method if there exists a lower solution @a and an upper solution @b, with either @a = = @b. For the linear conditions, we obtain eight new concepts of lower and upper solutions which generalize previous known cases.

Extremal solutions of second order nonlinear periodic boundary value problems

We study a periodic boundary value problem for a nonlinear ordinary differential equation of second order when the nonlinearity is given by a Caratheodory function. We generalize the monotone iterative method to cover the fully nonlinear case.

Existence of solution for discontinuous third order boundary value problems

Abstract In this paper, we obtain existence results for the problem u″′=q(u″) f(t,u) with boundary conditions u(a)=A, u(b)=B, u″(a)=C and u(a)=u(b), u′(a)=u′(b), u″(a)=C. We assume f a Caratheodory function, q∈L ∞ ( R ,(0,∞)) such that 1/q∈L loc ∞ ( R ,(0,∞)) and suppose the existence of lower and upper solutions. The existence of solution for the first considered conditions is obtained as a consequence of the fixed-points theorems. We obtain the solution of the second problem as a limit of solutions of the first case. For the first problem, the monotone method is developed.

Rapid convergence of the iterative technique for first order initial value problems

In this paper we study the convergence of the approximate solutions of the first order problem u(t)=f(t,w(t)); u(o)=uo. We prove that if there exists @?^kf/@?u^k, k >- 1, and it is a continuous function, then it is possible to construct two sequences of approximate solutions converging to a solution with rate of convergence of order k.

A generalized upper and lower solutions method for nonlinear second order ordinary differential equations

The purpose of this paper is to study a nonlinear boundary value nproblem of second order when the nonlinearity is a Caratheodory nfunction. It is shown that a generalized upper and lower solutions nmethod is valid, and the monotone iterative technique for finding the nminimal and maximal solutions is developed.

Approximate solutions to a new class of nonlinear diffusion problems

In this paper it is considered a new class of nonlinear differential equations that arises in the study of some diffusion processes. The authors present an existence and uniqueness result by constructing a sequence of approximate solutions. Theoretical and numerical aspects are considered.

A generalization of the method of upper and lower solutions for discontinuous first order problems with nonlinear boundary conditions

In this paper we prove the existence of extremal solutions for a discontinuous BVP between a lower and an upper solution which may be discontinuous in a finite set of points. Furthermore we present a monotone method with which we can explicitly construct a pair of sequences that converge to the extremal solutions, despite the discontinuities of the right hand side of the equation.