Four-point Boundary Conditions Research Articles

A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order

This paper studies a nonlinear fractional differential equation of an arbitrary order with four-point nonlocal integral boundary conditions. Some existence results are obtained by applying standard fixed point theorems and Leray-Schauder degree theory. The involvement of nonlocal parameters in four-point integral boundary conditions of the problem makes the present work distinguished from the available literature on four-point integral boundary value problems which mainly deals with the four-point boundary conditions restrictions on the solution or gradient of the solution of the problem. These integral conditions may be regarded as strip conditions involving segments of arbitrary length of the given interval. Some illustrative examples are presented.

Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations

This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations. For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition. Singular perturbation problems arise in the heat transfer problems with large Peclet numbers. We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval. Our analysis relies on the method of lower and upper solutions.

Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations

This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations. For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition. Singular perturbation problems arise in the heat transfer problems with large Peclet numbers. We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval. Our analysis relies on the method of lower and upper solutions.

The iterative solutions of 2 nth-order nonlinear multi-point boundary value problems

The aim of this paper is to investigate the existence of iterative solutions for a class of 2 nth-order nonlinear multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as three- or four-point boundary condition, ( n + 2)-point boundary condition and 2( n − m)-point boundary condition. The existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique. A monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution. Two examples are presented to illustrate the results.

Positive solutions to a new kind Sturm–Liouville-like four-point boundary value problem

In this paper, we considered the following four-point boundary value problem x ″ ( t ) + h ( t ) f ( t , x ( t ) , x ′ ( t ) ) = 0 , 0 < t < 1 , x ′ ( 0 ) − α 1 x ( ξ ) = 0 , x ′ ( 1 ) + α 2 x ( η ) = 0 , where 0 < α 1 < 1 ξ , 0 < α 2 < 1 1 − η , 0 < ξ < η < 1 , α 1 α 2 η − α 1 α 2 ξ + α 1 + α 2 > 0 . By applying fixed-point theorems, we obtain a variety of existence results. In particular, our four-point boundary condition is a new kind Sturm–Liouville-like boundary condition, which has rarely been considered up to now.

Multiple Solutions of Boundary-Value Problems for Fourth-Order Differential Equations with Deviating Arguments

This paper considers fourth-order differential equations with four-point boundary conditions and deviating arguments. We establish sufficient conditions under which such boundary-value problems have positive solutions. We discuss such problems in the cases when the deviating arguments are delayed or advanced. In order to obtain the existence of at least three positive solutions, we use a fixed-point theorem due to Avery and Peterson. To the authors’ knowledge, this is a first paper where the existence of positive solutions of boundary-value problems for fourth-order differential equations with deviating arguments is discussed.

Solutions and Green's Functions for Boundary Value Problems of Second-Order Four-Point Functional Difference Equations

We consider the Green's functions and the existence of positive solutions for a second-order functional difference equation with four-point boundary conditions.

A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions

In this paper, we concert with the existence of positive solution for the following nonlinear singular differential system with four-point boundary conditions - x ″ = f ( t , y ) , - y ″ = g ( t , x ) , α x ( 0 ) - β x ′ ( 0 ) = δ x ( 1 ) + γ x ′ ( 1 ) = 0 , y ( 0 ) = ay ( ξ 1 ) , y ( 1 ) = by ( ξ 2 ) , where 0 < ξ 1 < ξ 2 < 1 , α , β , δ , γ , a , b are nonnegative constants such that ρ = β γ + α γ + α δ > 0 . By structuring upper and lower solution and using Schauder fixed point theorem, a necessary and sufficient condition for the existence of positive solutions is established. An example is worked out to illustrate our main result.

Four-Point Boundary-Value Problems for Differential–Algebraic Systems

We apply the monotone iterative method to nonlinear four-point boundary conditions for differential–algebraic systems with causal operators. Sufficient conditions under which such problems have solutions (extremal or unique) are given. An example illustrates the theoretical results.

Positive solutions of four-point singular boundary value problems

Existence of C 1 positive solutions for a class of second-order nonlinear singular equations of the type - x ″ ( t ) + λ x ′ ( t ) = f ( t , x ( t ) ) , t ∈ ( 0 , 1 ) , subjectto four-point boundary conditions of the type x ( 0 ) = ax ( η ) , x ( 1 ) = bx ( δ ) , 0 < η ⩽ δ < 1 , is established. Existence of C 1 positive solution is proved with the upper and lower solutions method. Examples are included to show the validity of our results. Finally, the method of quasilinearization is developed to approximate the solution. We show that under suitable conditions on f, there exists a sequence of solutions of linear problems that converges monotonically and quadratically to the solution of the original nonlinear problem.

Existence and iteration of positive solutions to a class of Sturm–Liouville-like [formula omitted]-Laplacian boundary value problems

This paper deals with the existence and iteration of positive solutions for the following p -Laplacian differential equation: ( ϕ p ( u ′ ( t ) ) ) ′ + q ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , subject to the following Sturm–Liouville-like four-point boundary condition: u ( 0 ) − α u ′ ( ξ ) = 0 , u ( 1 ) + β u ′ ( η ) = 0 where ξ , η ∈ ( 0 , 1 ) . By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions.

Existence results for a fourth-order ordinary differential equation with a four-point boundary condition

The fourth-order differential equation y ( 4 ) ( t ) − f ( t , y ( t ) , y ″ ( t ) ) = 0 , 0 ≤ t ≤ 1 , with the four-point boundary value problem y ( 0 ) = y ( 1 ) = 0 , a y ″ ( ξ 1 ) − b y ‴ ( ξ 1 ) = 0 , c y ″ ( ξ 2 ) + d y ‴ ( ξ 2 ) = 0 is studied in this work, where 0 ≤ ξ 1 < ξ 2 ≤ 1 . Some results on the existence of at least one positive solution to the above four-point boundary value problem are obtained by using the Krasnoselskii fixed point theorem.

Existence of solutions for fourth order differential equation with four-point boundary conditions

In this paper we investigate the existence of solutions of a class of four-point boundary value problems for a fourth order ordinary differential equation. Our analysis relies on a nonlinear alternative of Leray–Schauder type.

Positive solution of fourth order ordinary differential equation with four-point boundary conditions

In this work, the authors consider the fourth order nonlinear ordinary differential equation u ( 4 ) ( t ) = f ( t , u ( t ) ) , 0 < t < 1 , with the four-point boundary conditions u ( 0 ) = u ( 1 ) = 0 , a u ″ ( ξ 1 ) − b u ‴ ( ξ 1 ) = 0 , c u ″ ( ξ 2 ) + d u ‴ ( ξ 2 ) = 0 , where 0 ≤ ξ 1 < ξ 2 ≤ 1 . By means of the upper and lower solution method and fixed point theorems, some results on the existence of positive solutions to the above four-point boundary value problem are obtained.