Order Three-point Boundary Value Problem Research Articles

Existence of Positivity of the Solutions for Higher Order Three-Point Boundary Value Problems involving p-Laplacian

The present study focusses on the existence of positivity of the solutions to the higher order three-point boundary value problems involving $p$-Laplacian
 $$[\phi_{p}(x^{(m)}(t))]^{(n)}=g(t,x(t)),~~t \in [0, 1],$$
 $$
 \begin{aligned}
 x^{(i)}(0)=0, &\text{~for~} 0\leq i\leq m-2,\\
 x^{(m-2)}(1)&-\alpha x^{(m-2)}(\xi)=0,\\
 [\phi_{p}(x^{(m)}(t))]^{(j)}_{\text {at} ~ t=0}&=0, \text{~for~} 0\leq j\leq n-2,\\
 [\phi_{p}(x^{(m)}(t))]^{(n-2)}_{\text {at} ~ t=1}&-\alpha[\phi_{p}(x^{(m)}(t))]^{(n-2)}_{\text {at} ~ t=\xi}=0,
 \end{aligned}
 $$
 where $m,n\geq 3$, $\xi\in(0,1)$, $\alpha\in (0,\frac{1}{\xi})$ is a parameter.
 The approach used by the application of Guo--Krasnosel'skii fixed point theorem to determine the existence of positivity of the solutions to the problem.

Consistency of an Infinite system of third order three-point boundary value problem in the $bv_0$ space by the theory of measure of noncompactness

Several authors have examined the solvability conditions for an infinite system of differential equations in different Banach spaces using the concept of measure of noncompactness. In all these studies, they have considered differential equations where the boundary conditions are defined on two points. In this paper, we have studied the solvability conditions for an infinite system of third-order three point boundary value problem in the sequence space of bounded variation $bv_0$ with the help of the theory of measure of noncompactness and have given a suitable example to illustrate the result.

Multiple positive solutions for a coupled system of $p$-Laplacian fractional order three-point boundary value problems

In this paper, we attempt to establish the existence of at least three positive solutions for a coupled system of $p$-Laplacian fractional order boundary value problems \begin{aligned}\begin{cases} D_{a^+}^{\beta _1}(\phi _{p}(D_{a^+}^{\alpha _1}u(t)))=f_1(t, u(t), v(t)) & \quad a\lt t\lt b,\\ D_{a^+}^{\beta _2}(\phi _{p}(D_{a^+}^{\alpha _2}v(t)))=f_2(t, u(t), v(t)) & \quad a\lt t\lt b, \end{cases}\end{aligned} with the boundary conditions \begin{aligned}\begin{cases} u^{(j)}(a)=0,\ j=0,1,2,\ldots , \ u{''}(b)=\delta u{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _1}u(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _1} u(b)) =\vartheta \phi _{p}( D_{a^+}^{\alpha _1}u(\eta )),\\ v^{(j)}(a)=0,\ j=0,1,2,\ldots , \ v{''}(b)=\delta v{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _2}v(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _2} v(b))=\vartheta \phi _{p}( D_{a^+}^{\alpha _2}v(\eta )), \end{cases}\end{aligned} by applying the five functional fixed point theorem.

Existence of Positive Solutions for Higher Order Three-Point Boundary Value Problems on Time Scales

In this paper, by using the four functionals fixed point theorem, Avery-Henderson fixed point theorem and the five functionals fixed point theorem, respectively, we investigate the conditions for the existence of at least one, two and three positive solutions to nonlinear higher order three-point boundary value problems on time scales.

Existence results for a class of generalized fractional boundary value problems

In this paper, we study a class of generalized fractional order three-point boundary value problems that involve fractional derivative defined in terms of weight and scale functions. Using several fixed point theorems, the existence and uniqueness results are obtained.

Existence of nontrivial solutions for superlinear three-point boundary value problems

In this paper, we investigate the existence of nontrivial solutions for some superlinear second order three-point boundary value problems by applying new fixed point theorems in ordered Banach spaces with the lattice structure derived by Sun and Liu.

Higher order nonlinear dynamic systems on time scales

Abstract In this paper, we consider a nonlinear system of higher order three-point boundary value problems on time scales. The Schauder fixed point theorem is used to investigate the existence of solutions of nonlinear dynamic systems on time scales. Furthermore, we establish the criteria for the existence of at least one, two and three positive solutions for higher order nonlinear dynamic systems on time scales by using the four functionals fixed point theorem, the Avery–Henderson fixed point theorem and the five functionals fixed point theorem, respectively.

Existence of positive solutions for Riemann–Liouville fractional order three-point boundary value problem

In this paper, we study the following fractional order three-point boundary value problem [Formula: see text] where [Formula: see text], are the standard Riemann–Liouville fractional order derivatives with [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]: [Formula: see text] is continuous. By using several well-known fixed-point theorems in a cone, the existence of at least one and two positive solutions is obtained. Some examples are presented to illustrate the main results.

Unbounded solutions of third order three-point boundary value problems on a half-line

Abstract We consider the following third order three-point boundary value problem on a half-line: x'''(t)+q(t)f(t,x(t),x'(t),x''(t)) = 0, t ∈ (0,+∞), x'(0) = A, x(η) = B, x''(+∞) = C, where η ∈ (0,+∞), but fixed, and f : [0,+∞) × ℝ3 → ℝ satisfies Nagumo's condition. We apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory, to establish existence theory for at least one unbounded solution, and at least three unbounded solutions. To demonstrate the usefulness of our results, we illustrate two examples.

Existence of solutions for fourth order three-point boundary value problems on a half-line

In this paper, we apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory to establish the existence of unbounded solutions for the following fourth order three-point boundary value problem on a half- line x 0000 (t) + q(t) f(t, x(t), x 0 (t), x 00 (t), x 000 (t)) = 0, t2 (0,+¥), x 00 (0) = A, x(h) = B1, x 0 (h) = B2, x 000 (+¥) = C, where h2 (0,+¥), but fixed, and f : (0,+¥) R 4 ! R satisfies Nagumo's condition. We present easily verifiable sufficient conditions for the existence of at least one solu- tion, and at least three solutions of this problem. We also give two examples to illustrate the importance of our results.

Positive solutions to iterative systems of fractional order three-point boundary value problems with Riemann-Liouville derivative

Positive solutions to iterative systems of fractional order three-point boundary value problems with Riemann-Liouville derivative

Solution of fourth order three-point boundary value problem using ADM and RKM

In this paper, a computational method is proposed, for solving linear and nonlinear fourth order three-point boundary value problem (BVP) and the system of nonlinear BVP. This method is based on the Adomian decomposition method (ADM) and the reproducing kernel method (RKM). The solution of linear fourth order three-point boundary value problem (BVP) is determined by the reproducing kernel method, and the solution of nonlinear fourth order three-point BVP is determined using the combination of Adomian decomposition method and reproducing kernel method. The approximate solutions are given in the form of series. Numerical results are shown to illustrate the accuracy of the present method.

Second order three-point boundary value problems in abstract spaces

In this paper we investigate the existence of solutions of the nonhomogeneous three-point boundary value problem $$\left\{ {\begin{array}{*{20}c} {x''(t) + a(t)x'(t) + b(t)x(t) + u(t)f(t,x(t)) = 0, a.e. on [0,1],} \\ {x(0) = 0, x(1) - \zeta x(\eta ) = c.} \\ \end{array} } \right.$$ . The coefficient functions a and b are continuous real-valued functions on [0, 1], η and ζ are some positive constants. Denote by E a Banach space and assume, that u belongs to an Orlicz space i.e., u(·) ∈ LM([0, 1],ℝ), where M is an N-function and c ∈ E.

Existence of Nontrivial Solutions for Unilaterally Asymptotically Linear Three-Point Boundary Value Problems

Using fixed point theorems in ordered Banach spaces with the lattice structure, we consider the existence of nontrivial solutions under the condition that the nonlinear term can change sign and study the existence of sign-changing solutions for some second order three-point boundary value problems. Our results improve and generalize on those in the literatures.

Positive solutions of an $n$th order three-point boundary value problem

Positive solutions of an $n$th order three-point boundary value problem

Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM

Aims: In this paper an approximate method for the solution of third-order differential equation with two and three point boundary condition is developed using iterative reproducing kernel method. Methods: The third order boundary value problem is converted into integro-differential equation of second order two point boundary value problem. The reproducing kernel method which takes the form of a convergent series with easily computable components is used for the solution of second order two point boundary value problem. Results: Six numerical examples are given to demonstrate the efficiency of the present method. The results obtained are better than the existing methods developed in [19,20,21,22,23]. Conclusions: In this paper, the solution of linear and nonlinear third order (two and three point) boundary value problem is determined. For the solution of third order three point boundary value problem reproducing kernel method is proposed and obtained a good accuracy in absolute errors. As the reproducing kernel method cannot solve the third order three-point boundary value problems directly, so the third order boundary value problem is converted to second order two point boundary value problem after absorbing the nonlocal condition at . The method developed is compared with those developed by Li et al. [19], El-Salam et al. [20], Khan and Aziz [21], Li and Wu [22] and Wu and Li [23]. As observed in Example 3, 4, 5 and 6 that the method obtained in this paper is better than [19]-[23]. Results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for third-order linear as well as nonlinear boundary value problem.

Existence of the solutions for a class of nonlinear fractional order three-point boundary value problems with resonance

A class of nonlinear fractional order differential equation is investigated in this paper, where is the standard Riemann-Liouville fractional derivative of order , , . Using intermediate value theorem, we obtain a sufficient condition for the existence of the solutions for the above fractional order differential equations.

Existence of Multiple Positive Solutions for 3nth Order Three-Point Boundary Value Problems on Time Scales

Existence of Multiple Positive Solutions for 3nth Order Three-Point Boundary Value Problems on Time Scales

A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space

In this paper, a new numerical algorithm is provided to solve nonlinear three-point boundary value problems in a very favorable reproducing kernel space which satisfies all boundary conditions. Its reproducing kernel function is discussed in detail. We also prove that the approximate solution and its first and second order derivatives all converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear second order three-point boundary value problems.

Existence results for second order three-point boundary value problems

The paper is devoted to the study of second order differential equations and systems with nonlinear three point boundary conditions. The existence of solutions is proved using fixed point theorems: Banach’s and Boyd-Wong’s contraction principles, Perov’s and Schauder’s fixed point theorems. Mathematics subject classification (2010): 34B10, 34B15.