Order Nonlinear Boundary Value Problems Research Articles

A smart nonstandard finite difference scheme for second order nonlinear boundary value problems

A new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems. An artificial parameter is introduced in the scheme. Symbolic computation is proposed for the construction of the scheme. Local truncation error of the method is discussed. Numerical examples are illustrated. Numerical results show that the method is very effective.

Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces

Solutions for a class of degenerate, nonlinear, nonlocal boundary value problems, arising in nano boundary layer fluid flows over a stretching surface, are obtained. Viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface are considered. Using the Schauder fixed point theorem, existence and uniqueness results are established. The effects of the slip parameter k and the suction parameter a on the fluid velocity and on the tangential stress are investigated and discussed (through numerical results). We find that for fluid flows at nanoscales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter k .

APPROXIMATE SOLUTIONS TO MHD SQUEEZING FLUID FLOW

In this paper, a steady axisymmetric MHD flow of two dimensional incompressible fluids is studied under the influence of a uniform transverse magnetic field. The governing equations are reduced to nonlinear boundary value problem by applying the integribility conditions. Optimal Homotopy Asymptotic Method (OHAM) is applied to obtain solution of reduced fourth order nonlinear boundary value problem. For comparison, the same problem is also solved by Variational Iteration Method (VIM).

SINGULAR INTEGRAL EQUATIONS AND APPLICATIONS TO NONLINEAR CONJUGATE PROBLEMS

In this paper, we establish the existence of multiple positive solutions for singular integral equations. The proof is based on a general existence principle established using a nonlinear alternative principle of Leray--Schauder type and a well-known fixed point theorem in cones. As an application, we consider higher order nonlinear singular conjugate boundary value problems. Some recent results in the literature are generalized and improved.

ON SOME PROBLEMS WITH NONLOCAL INTEGRAL CONDITION

We study the second order nonlinear boundary value problems with non‐local integral conditions and construct the Fučík type spectrum for these problems.

SINGULAR INTEGRAL EQUATIONS AND APPLICATIONS TO NONLINEAR CONJUGATE PROBLEMS

In this paper, we establish the existence of multiple positive solutions for singular integral equations. The proof is based on a general existence principle established using a nonlinear alternative principle of Leray--Schauder type and a well-known fixed point theorem in cones. As an application, we consider higher order nonlinear singular conjugate boundary value problems. Some recent results in the literature are generalized and improved.

Solving Second Order Non-Linear Boundary Value Problems by Four Numerical Methods

Solving Second Order Non-Linear Boundary Value Problems by Four Numerical Methods

The Vallée-Poussin problem for higher order nonlinear hyperbolic equations

For higher order nonlinear hyperbolic equations nonclassical boundary value problems of Vallée-Poussin type are studied, and unimprovable in a sense sufficient conditions of solvability, unique solvability and well-posedness are established.

Positive solutions of second order nonlinear difference boundary value problems

We study a class of second order nonlinear difference boundary value problems with separated boundary conditions. A series of criteria are obtained for the existence of one, two, arbitrary number, and even an infinite number of positive solutions. A theorem for the nonexistence of positive solutions is also derived. Several examples are given to demonstrate the applications. Our results improve and supplement several results in the literature.

Solvability for second order nonlinear impulsive boundary value problems

In this paper, we are concerned with the solvability for a class of second order nonlinear impulsive boundary value problem. New criteria are established based on Schaefer's fixed-point theorem. An example is presented to illustrate our main result. Our results essentially extend and complement some previous known results.

Nodal solutions of second order nonlinear boundary value problems

We study the nonlinear boundary value problem (BVP) consisting of the equation −(p(t)y′)′ + q(t)y = w(t)f(y) on [a, b] and a general separated boundary condition (BC). By comparing it with a linear Sturm--Liouville problem (SLP) we obtain conditions for the existence and nonexistence of nodal solutions of this problem. More specifically, let λn, n = 0, 1, 2, . . ., be the nth eigenvalue of the corresponding linear SLP. Then the BVP has a pair of solutions with exactly n zeros in (a, b) if λn is in the interior of the range of f(y)/y; and does not have any solution with exactly n zeros in (a, b) if λn is outside this range. These conditions become necessary and sufficient when f(y)/y is monotone on (−∞, 0) and on (0, ∞). We also discuss the changes of the number of different types of nodal solutions as the equation or the BC changes. Our results are obtained without assuming the global existence and uniqueness of solutions of the corresponding initial value problems.

Existence and approximation of solution of three-point boundary value problems on time scales

The method of upper and lower solutions and the generalised quasilinearization technique for second order nonlinear three-point time scale boundary value problems of the type are developed. A monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained.

The exact number of solutions for the second order nonlinear boundary value problem

Abstract A second order nonlinear differential equation with homogeneous Dirichlet boundary conditions is considered. An explicit expression for the root functions for an autonomous nonlinear boundary value problem is obtained using the results of the paper [SOMORA, P.: The lower bound of the number of solutions for the second order nonlinear boundary value problem via the root functions method, Math. Slovaca 57 (2007), 141–156]. Other assumptions are supposed to prove the monotonicity of root functions and to get the exact number of solutions. The existence of infinitely many solutions of the boundary value problem with strong nonlinearity is obtained by the root function method as well.

Conformal metrics with constant $Q$-curvature for manifolds with boundary

In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M, g) with smooth boundary there exists a metric in the conformal class [g] of the background metric g with constant Q-curvature, zero T -curvature and zero mean curvature under generic conformally invariant assumptions. The problem is equivalent to solving a fourth order nonlinear elliptic boundary value problem (BVP) with boundary condition given by a third-order pseudodifferential operator, and homogeneous Neumann conditions. It has a variational structure, but since the corresponding Euler-Lagrange functional is in general unbounded from above and below, we need to use min-max methods combined with a new topological argument and a compactness result for the above BVP.

On some third order nonlinear boundary value problems: Existence, location and multiplicity results

We prove an Ambrosetti–Prodi type result for the third order fully nonlinear equation u ‴ ( t ) + f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) = s p ( t ) with f : [ 0 , 1 ] × R 3 → R and p : [ 0 , 1 ] → R + continuous functions, s ∈ R , under several two-point separated boundary conditions. From a Nagumo-type growth condition, an a priori estimate on u ″ is obtained. An existence and location result will be proved, by degree theory, for s ∈ R such that there exist lower and upper solutions. The location part can be used to prove the existence of positive solutions if a non-negative lower solution is considered. The existence, nonexistence and multiplicity of solutions will be discussed as s varies.

ON FUČIK SPECTRA FOR THIRD ORDER EQUATIONS

We construct the Fučik spectrum for some third order nonlinear boundary value problems. This spectrum differs essentially from the known Fučik spectra.

The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

Abstract We consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.

A Collocation Method for Global Approximation of General Second Order BVPs

For the numerical solution of the second order nonlinear two-point boundary value problems a family of polynomial global methods is derived.Numerical examples provide favorable comparisons with other existing methods.

Generalized approximations and rapid convergence of solutions of m-point boundary value problems

The upper and lower solutions method and the generalized quasilinearization technique for second order nonlinear m-point boundary value problem of the type - x ″ = f ( t , x , x ′ ) , t ∈ [ 0 , 1 ] δ x ( 0 ) - γ x ′ ( 0 ) = 0 , x ( 1 ) = ∑ i = 1 m - 2 α i x ( η i ) , η i ∈ ( 0 , 1 ) is developed. A monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained in the C 1 norm.

On positive periodic solutions of a nonautonomous neutral delay [formula omitted]-species competitive system

The purpose of this paper is to investigate a general neutral delay n -species competitive nonautonomous system. Different from Gaines and Mawhin’s continuation theorem [R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977] and abstract continuation theory for k -set contraction [W. Petryshyn, Z. Yu, Existence theorem for periodic solutions of higher order nonlinear periodic boundary value problems, Nonlinear Anal. 6 (1982) 943–969; R.E. Gaines, J.L. Mawhin, Lecture Notes in Mathematics, vol. 586, Springer-Verlag, Berlin, 1977], a new existence theorem is introduced. With the help of the theorem and some techniques, a set of sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution. As an application, we also examine some special cases which have been studied extensively in the literature. Some known results are improved and generalized.