Existence of solution for a nonlinear fifth-order three-point boundary value problem

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.

In this paper, we study the existence of nontrivial solution for the fourth-order three- point boundary value problem having the following formu(4) (t) + f (t, u(t)) = 0, 0 < t < 1,u(0) = α(η), u'(0) = u''(0) = 0, u(1) = βu(η),where η ∈ (0, 1), α, β ∈ R, f ∈ C ([0, 1] × R, R). We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.

The existence results of at least one positive pseudo-symmetric solution for a nonlinear second-order three-point boundary value problem are established by means of the fixed point theorem due to Avery, Anderson and Krueger.

The main objective of this paper is to develop new spectral solutions for linear and nonlinear fifth-order boundary value problems. For this aim, a new operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The key idea for obtaining the suggested spectral solutions is to convert the fifth-order linear and nonlinear boundary value problems into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. A theoretical error bound is derived and the convergence analysis of the proposed expansion is proved. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the suggested algorithms.

Three positive solutions of a nonlinear three-point boundary value problem

Via the linking theorem, the existence of nontrivial solutions for a nonlinear elliptic Dirichlet boundary value problem with an inverse square potential is proved.

Positive solutions for nonlinear three-point boundary value problems on time scales

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.

Solution of nonlinear boundary value problems — I method of third order convergence for solution of nonlinear two-point boundary value problems single second order equation

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 apply a cone theoretic fixed point theorem and obtain conditions for the existence of positive solutions to the three-point nonlinear second order boundary value problemwith boundary conditions where and

In this paper, a nonlinear three-point boundary value problem of higher-order singular fractional differential equations is discussed. By applying the properties of Green function and some fixed point theorems for sum-type operator on cone, some new criteria on the existence and uniqueness of solutions are obtained. Moreover, two iterative sequences are given for uniformly approximating the positive solution, which are important for practical application. At last, we give two examples to illustrate the main results.

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

In this paper, we present efficient numerical algorithms for the approximate solution of nonlinear fourth-order boundary value problems. The first algorithm deals with the sinc–Galerkin method (SGM). The sinc basis functions prove to handle well singularities in the problem. The resulting SGM discrete system is carefully developed. The second method, the Adomian decomposition method (ADM), gives the solution in the form of a series solution. A modified form of the ADM based on the use of the Laplace transform is also presented. We refer to this method as the Laplace Adomian decomposition technique (LADT). The proposed LADT can make the Adomian series solution convergent in the Laplace domain, when the ADM series solution diverges in the space domain. A number of examples are considered to investigate the reliability and efficiency of each method. Numerical results show that the sinc–Galerkin method is more reliable and more accurate.

Positive solutions of nonlinear singular third-order two-point boundary value problem