Three-point Boundary Conditions Research Articles

Exploring existence, uniqueness, and stability in nonlinear fractional boundary value problems with three-point boundary conditions

This study investigates the analysis of the existence, uniqueness, and stability of solutions for a Ψ-Caputo three-point nonlinear fractional boundary value problem using the Banach contraction principle and Sadovskii’s fixed point theorem. We demonstrate the practical implications of our analytical advancements for each situation, illustrating how the components of the fractional boundary value problem emerge in real-life occurrences. Our work significantly enhances the field of applied mathematics by offering analytical solutions and valuable insights.

Topology degree results on a G-ABC implicit fractional differential equation under three-point boundary conditions.

This research manuscript aims to study a novel implicit differential equation in the non-singular fractional derivatives sense, namely Atangana-Baleanu-Caputo ([Formula: see text]) of arbitrary orders belonging to the interval (2, 3] with respect to another positive and increasing function. The major results of the existence and uniqueness are investigated by utilizing the Banach and topology degree theorems. The stability of the Ulam-Hyers ([Formula: see text]) type is analyzed by employing the topics of nonlinear analysis. Finally, two examples are constructed and enhanced with some special cases as well as illustrative graphics for checking the influence of major outcomes.

Three Point Boundary Value Problems for Generalized Fractional Integro Differential Equations

This article deals with some existence results for a class of boundary value problem with three-point boundary conditions involving a nonlinear θ-Caputo fractional proportional integro differential equation. By means of some standard fixed point theorems, sufficient conditions for the existence of solutions are presented. Additionally, some applications of the main results are demonstrated.

Study of Hybrid Problems under Exponential Type Fractional-Order Derivatives

In this investigation, we develop a theory for the hybrid boundary value problem for fractional differential equations subject to three-point boundary conditions, including the antiperiodic hybrid boundary condition. On suggested problems, the third-order Caputo–Fabrizio derivative is the fractional operator applied. In this regard, the corresponding hybrid fractional integral equation is obtained by the Caputo–Fabrizio operator’s properties with the Green function’s aid. Then, we apply Dhage’s nonlinear alternative to the Schaefer type to prove the existence results. Finally, two examples are provided to confirm the validity of our main results.

On Coupled Systems of Hilfer-Hadamard Sequential Fractional Differential Equations with Three-Point Boundary Conditions

On Coupled Systems of Hilfer-Hadamard Sequential Fractional Differential Equations with Three-Point Boundary Conditions

Existence Results for a Fractional Differential Inclusion of Arbitrary Order with Three-Point Boundary Conditions

This paper studies existence of solutions for a new class of fractional differential inclusions of arbitrary order with three-point fractional integral boundary conditions. Our results are based on Bohnenblust-Karlin’s fixed point theorem.

Multiple Positive Solutions for a System of Fractional Order BVP with p-Laplacian Operators and Parameters

In this paper, we investigate the existence of positive solutions to a system of fractional differential equations that include the (r1,r2,r3)-Laplacian operator, three-point boundary conditions, and various fractional derivatives. We use a combination of techniques, including cone expansion and compression of the functional type, and the Leggett–Williams fixed point theorem, to prove the existence of positive solutions. Finally, we provide two examples to illustrate our main results.

Solvability of an infinite system of Langevin fractional differential equations in a new tempered sequence space

In this article, we establish a new tempered sequence space related to tempered sequence $$\ell _{p}^{\alpha },$$ $$p\ge 1$$ and obtain the Hausdorff measure of noncompactness of this space. By using this measure of noncompactness with Darbo’s fixed point theorem, we investigate the existence results for an infinite system of Langevin fractional differential equations involving a generalized derivative of two distinct fractional orders with three-point boundary conditions in this new sequence space. To illustrate the obtained results of these sequence spaces, a numerical example is provided.

Existence and stability analysis for Caputo generalized hybrid Langevin differential systems involving three-point boundary conditions

This research inscription gets to grips with two novel varieties of boundary value problems. One of them is a hybrid Langevin fractional differential equation, whilst the other is a coupled system of hybrid Langevin differential equation encapsuling a collective fractional derivative known as the ψ-Caputo fractional operator. Such operators are generated by iterating a local integral of a function with respect to another increasing positive function Ψ. The existence of the solutions of the aforehand equations is tackled by using the Dhage fixed point theorem, whereas their uniqueness is handled using the Banach fixed point theorem. On the top of this, the stability within the scope of Ulam–Hyers of solutions to these systems are also considered. Two pertinent examples are presented to corroborate the reported results.

Solvability of Sequential Fractional Differential Equation at Resonance

The sequential fractional differential equations at resonance are introduced subject to three-point boundary conditions. The emerged fractional derivative operators in these equations are based on the Caputo derivative of order that lies between 1 and 2. The vital target of the current contribution is to investigate the existence of a solution for the boundary value problem by using the coincidence degree theory due to Mawhin which is basically depending on the Fredholm operator with index zero and two continuous projectors. An example is given to illustrate the deduced theoretical results.

Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis

Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods.

On a Lyapunov-Type Inequality for Control of a ψ-Model Thermostat and the Existence of Its Solutions

In this paper, a new structure of an applied model of thermostat is defined using the generalized ψ-operators with three-point boundary conditions. Some useful properties of the relevant Green’s function are established, and based on these properties, the Lyapunov-type inequality is constructed for the given extended ψ-model thermostat with the help of Jensen’s inequality. By defining mild solutions for such an extended system, the existence and non-existence conditions are discussed.

A Coupled System of Caputo–Hadamard Fractional Hybrid Differential Equations with Three-Point Boundary Conditions

This article presents a study of the existence and uniqueness of solutions for a system of hybrid fractional differential equations involving fractional derivatives of the Caputo-Hadamard type with three-point hybrid boundary conditions. In addition to this, the “Hyres–Ulam” stability of the solutions for this type of equation is verified, and finally a numerical example was presented to support our theoretical results.

Blowup of Solutions to a Damped Euler Equation with Homogeneous Three-Point Boundary Condition

It is known that solutions to the inviscid Proudman-Johnson equation subject to a homogeneous three-point boundary condition can develop singularities in finite time. Given the regularizing effect that damping can have, in this paper we consider the possibility of finite-time singularity formation in solutions of the generalized inviscid Proudman-Johnson equation with damping subject to the same homogeneous three-point boundary condition. In particular, we will derive criteria the initial condition must satisfy in order for solutions to blowup in finite time under the effect of a smooth time-dependent damping term.

Applications of variational methods to some three-point boundary value problems with instantaneous and noninstantaneous impulses

In this paper, we study the multiple solutions for some second-order p-Laplace differential equations with three-point boundary conditions and instantaneous and noninstantaneous impulses. By applying the variational method and critical point theory the multiple solutions are obtained in a Sobolev space. Compared with other local boundary value problems, the three-point boundary value problem is less studied by variational method due to its variational structure. Finally, two examples are given to illustrate the results of multiplicity.

EXISTENCE AND UNIQUENESS SOLUTION FOR THREE-POINT HADAMARD-TYPE FRACTIONAL VOLTERRA BVP

In this paper we study the existence and uniqueness solution for a first kind fractional Volterra boundary value problem involving Hadamard type and three-point boundary conditions. Our analysis is based on Krasnoselskii-Zabreiko’s fixed point theorem and Banach contraction principle. As an application we discuss a Hadamard type boundary value problem with fractional integral boundary conditions. We emphasize that our results are new in the context of Hadamard fractional calculus and are well illustrated with the aid of examples.

Existence of solutions for a three-point Hadamard fractional resonant boundary value problem

Abstract This article focuses on the creation of an existence theorem for a fully nonlinear Hadamard fractional boundary value problem subject to special three-point boundary conditions. By making use of the coincidence degree theory, it is proved that our governing problem makes resonance, that is, the linear part of the differential operator is non-invertible (equally, the corresponding linear problem has at least one nontrivial solution). Constructing some hypotheses on the linear part of the differential operator, nonlinearities and boundary conditions, we give an existence criterion for at least one solution of the fractional-order resonant boundary value problem under study. At the end, a numerical example is presented to illustrate the obtained theoretical results.

Existence results for nonlinear sequential Caputo and Caputo-Hadamard fractional differential equations with three-point boundary conditions in Banach spaces

In this paper, we study the existence of solutions for nonlinear sequential Caputo and Caputo-Hadamard fractional differential equations with three-point boundary conditions by using measure of noncompactness combined with fixed point theorem of M?nch. An example illustrating the effectiveness of the theoretical results is presented.

On the Ulam-Hyers-Rassias stability of two structures of discrete fractional three-point boundary value problems: Existence theory

<abstract><p>We prove existence and uniqueness of solutions to discrete fractional equations that involve Riemann-Liouville and Caputo fractional derivatives with three-point boundary conditions. The results are obtained by conducting an analysis via the Banach principle and the Brouwer fixed point criterion. Moreover, we prove stability, including Hyers-Ulam and Hyers-Ulam-Rassias type results. Finally, some numerical models are provided to illustrate and validate the theoretical results.</p></abstract>

Nontrivial solutions of fourth order ordinary differential equations with nonlocal three-point boundary conditions

Nontrivial solutions of fourth order ordinary differential equations with nonlocal three-point boundary conditions