Solvability for second order nonlinear impulsive boundary value problems

This work studies some two point impulsive boundary value problems composed by a fully differential equation, which higher order contains an increasing homeomorphism, by two point boundary conditions and impulsive e ects. We point out that the impulsive conditions are given via multivariate generalized functions, including impulses on the referred homeomorphism. The method used apply lower and upper solutions technique together with xed point theory. Therefore we have not only the existence of solutions but also the localization and qualitative data on their behavior. Moreover a Nagumo condition will play a key role in the arguments.

The paper deals with the existence of solutions for a general class of second-order nonlinear impulsive boundary value problems defined on an infinite interval. The main innovative aspects of the study are that the results are obtained under relatively mild conditions and the use of principal and nonprincipal solutions that were obtained in a very recent study. Additional results about the existence of bounded solutions are also provided, and theoretical results are supported by an illustrative example.

A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two‐point boundary value problem which depends on two parameters λ and α representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α = 0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the trivial solution is lost. The computations of λc required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α ≠ 0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each α ≠ 0 provided λ < λc. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.

In this paper we consider a boundary value problem for a fourth order nonlinear elliptic equation, which models a bending plate on nonlinear elastic foundation. Differently from other authors, here we propose a novel approach to investigation of solvability and numerical solution of the problem. Namely, we reduce it to an operator equation for the right hand side function and under some easily verified conditions we have established the existence and uniqueness of a solution. We have also constructed an iterative method for the solution of the problem. The theoretical results are illustrated on several examples, where the advantages in convergence rate of the method over other methods are shown.

In this work, a Second order non-linear elliptic boundary value problem with discontinuous coefficients arising in elastohydrodynamic lubrication is studied. Physically, it corresponds to a two dimensional model of the contact between an elastie rolling ball and a plane lubricated by a thin fluid film, when the viscosity is assumed to be constant. The model of Elrod-Adams is introduced in order to take into account the phenomenon of cavitation. A result of existence is obtained by means of regularization and fixed point techniques.

Higher Order Nonlinear Multi-Point Fractional Boundary Value Problems

E. R. El-Zahar ∗ and Y. S. Hamed † ∗ Department of Basic Engineering Science, Faculty of Engineering, Shebin El-Kom, Menofia University, Egypt. † Department of Engineering Mathematics, Faculty of Electronic Engineering, Menouf 32952, Menofia University, Egypt. ∗ Present address: Department of Mathematics, Faculty of science, Al-Kharj University, Al-Kharj, Saudi Arabia. † Present address: Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif, Saudi Arabia.

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Necessary and sufficient conditions for the existence of solutions of weakly nonlinear boundary-value problem of systems of integrodifferential equations with impulsive action at fixed points of time are obtained.Convergent iterative procedure of the solution and establish the connection between the necessary condition and the sufficient condition are presented.

Let T be a time scale. Let T be a time scale. We study the existence of positive solutions for the nonlinear three-point singular boundary value problem with higher-order p-Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with p-Laplacian operator are obtained.

We present sufficient conditions for the existence of at least twin or triple positive solutions of a nonlinear four-point singular boundary value problem with a -Laplacian dynamic equation on a time scale. Our results are obtained via some new multiple fixed point theorems.

An existence theorem for a class of nonlinear singular boundary value problems

In this paper, six functionals fixed point theorem is used to investigate the existence of at least three positive solutions for a nonlinear nth-order m-point p-Laplacian impulsive boundary value problem. As an application, we give an example to demonstrate our result.

The paper is concerned with the state and proof of the solvability theorem of unique state vector solution (SVS) of triple nonlinear hyperbolic boundary value problem (TNLHBVP), via utilizing the Galerkin method (GAM) with the Aubin theorem (AUTH), when the boundary control vector (BCV) is known. Solvability theorem of a boundary optimal control vector (BOCV) with equality and inequality state vector constraints (EINESVC) is proved. We studied the solvability theorem of a unique solution for the adjoint triple boundary value problem (ATHBVP) associated with TNLHBVP. The directional derivation (DRD) of the "Hamiltonian"(DRDH) is deduced. Finally, the necessary theorem (necessary conditions "NCOs") and the sufficient theorem (sufficient conditions" SCOs"), together denoted as NSCOs, for the optimality (OP) of the state constrained problem (SCP) are stated and proved.

In this paper, we investigate the existence of positive solutions of a class of fractional three-point boundary value problem with an advanced argument by using fixed-point index theory. Our results improve and extend some known results in the literature. Two examples are given to demonstrate the effectiveness of our results.