Leggett-Williams Fixed Point Theorem Research Articles

Existence of Three Positive Solutions for Boundary Value Problem of Fourth Order with Sign-Changing Green’s Function

In this paper, we examine a fourth-order equation that has parameter dependency and boundary conditions in three different places. We prove some of the features of the relevant asymmetric Green’s function and infer its exact form. The resulting solutions are still positive and decreasing functions on the entire interval of the Green’s function definition, and they are concave in a specific subinterval, despite the fact that the function’s sign changes on the square of its definition. The fixed point theorem of Krasnoselskii is the foundation of the existence arguments. Next, using the Leggett–Williams fixed point theorem, it is concluded that there are at least three positive solutions. Lastly, an example is provided, to highlight the primary findings of the manuscript.

Positive Solutions to a System of Coupled Hadamard Fractional Boundary Value Problems

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem.

Existence and multiplicity of solutions of fractional differential equations on infinite intervals

In this research, we investigate the existence and multiplicity of solutions for fractional differential equations on infinite intervals. By using monotone iteration, we identify two solutions, and the multiplicity of solutions is demonstrated by the Leggett–Williams fixed point theorem.

Positive Solutions for a System of Fractional q-Difference Equations with Multi-Point Boundary Conditions

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q-difference equations that include fractional q-integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q-derivatives and fractional q-derivatives of various orders. The proofs of our principal findings employ a range of fixed-point theorems, including the Guo–Krasnosel’skii fixed-point theorem, the Leggett–Williams fixed-point theorem, the Schauder fixed-point theorem, and the Banach contraction mapping principle.

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.

Positive solutions for a class of fractional differential equations with infinite-point boundary conditions on infinite intervals

In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example.

On a System of Hadamard Fractional Differential Equations with Nonlocal Boundary Conditions on an Infinite Interval

Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main theorems, we employ the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem.

Positive Periodic Solutions for a First-Order Nonlinear Neutral Differential Equation with Impulses on Time Scales

In this article, we discuss the existence of a positive periodic solution for a first-order nonlinear neutral differential equation with impulses on time scales. Based on the Leggett–Williams fixed-point theorem and Krasnoselskii’s fixed-point theorem, some sufficient conditions are established for the existence of positive periodic solution. An example is given to show the feasibility and application of the obtained results. Since periodic solutions are solutions with symmetry characteristics, the existence conditions for periodic solutions also imply symmetry.

On the existence and multiplicity of positive solutions for a p-Laplacian fractional boundary value problem with an integral boundary condition

Aim of this work is to investigate existence and multiplicity of positive solutions of a fractional boundary value problem with an integral boundary condition with p-Laplacian operator. Necessary and sufficient conditions are presented to obtain existence and multiplicity results. Main tools are Krasnoselskii, Schaefer and Leggett-Williams fixed point theorems. Two examples are given to illustrate our results.

Multiple Positive Solutions for Fractional Boundary Value Problems with Integral Boundary Conditions and Parameter Dependence

In this paper, we consider the existence of multiple positive solutions to boundary value problems of nonlinear fractional differential equation with integral boundary conditions and parameter dependence. To obtain our results, we used a functional-type cone expansion-compression fixed point theorem and the Leggett–Williams fixed point theorem. Examples are included to illustrate the main results.

Positive solutions for a system of Hadamard fractional $ (\varrho_{1}, \varrho_{2}, \varrho_{3}) $-Laplacian operator with a parameter in the boundary

<abstract><p>In this paper, we are gratified to explore existence of positive solutions for a tripled nonlinear Hadamard fractional differential system with $ (\varrho_{1}, \varrho_{2}, \varrho_{3}) $-Laplacian operator in terms of the parameter $ (\sigma_{1}, \sigma_{2}, \sigma_{3}) $ are obtained, by applying Avery-Henderson and Leggett-Williams fixed point theorems. As an application, an example is given to illustrate the effectiveness of the main result.</p></abstract>

On the boundary value problems of piecewise differential equations with left-right fractional derivatives and delay

In this paper, we study the multi-point boundary value problems for a new kind of piecewise differential equations with left and right fractional derivatives and delay. In this system, the state variables satisfy the different equations in different time intervals, and they interact with each other through positive and negative delay. Some new results on the existence, no-existence and multiplicity for the positive solutions of the boundary value problems are obtained by using Guo–Krasnoselskii’s fixed point theorem and Leggett–Williams fixed point theorem. The results for existence highlight the influence of perturbation parameters. Finally, an example is given out to illustrate our main results.

Existence of positive solutions for the nonlinear fractional boundary value problems with p-Laplacian

The monotone iterative technique, theory of fixed point index in a cone and the Leggett-Williams fixed point theorem are applied to investigate the existence and multiplicity of positive solutions for four boundary value problems of nonlinear fractional differential equations with a p-Laplacian point operator and infinite delay. Moreover, examples are presented to illustrate a vast applicability of our main results.

Leggett-Williams fixed point theorem type for sums of operators and application in PDEs

Leggett-Williams fixed point theorem type for sums of operators and application in PDEs

EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS FOR A SYSTEM OF NONLINEAR FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEMS WITH <i>P</i> -LAPLACIAN OPERATOR

<abstract> In this paper, we deal with a coupled system of nonlinear fractional multi-point boundary value problems with $ p $-Laplacian operator. The existence and multiplicity of positive solutions are obtained by employing Leray-Schauder alternative theory, Leggett-Williams fixed point theorem and Avery-Henderson fixed point theorem. As an application, two examples are given to illustrate the effectiveness of our main results. </abstract>

Three Positive Solutions for Boundary Value Problem of Fractional q-Differential Equation

In this paper, we study the boundary value problem of a Riemann-Liouville fractional q-difference equation. By applying the Leggett-Williams fixed point theorem and the properties of the Green&amp;rsquo;s function, three positive solutions are obtained.

Multiplicity results for fourth order problems related to the theory of deformations beams

The main purpose of this paper is to establish the existence and multiplicity of positive solutions for a fourth-order boundary value problem with integral condition. By using a new technique of construct a positive cone, we apply the Krasnoselskii compression/expansion and Leggett-Williams fixed point theorems in cones to show our multiplicity results. Finally, a particular case is studied, and the existence of multiple solutions is proved for two different particular functions.

Positive Solutions to a Coupled Fractional Differential System with p-Laplacian Operator

In this paper, we consider a fractional differential system with multistrip and multipoint mixed boundary conditions involving p-Laplacian operator and fractional derivatives. The existence result of positive solutions is established by the Leggett-Williams fixed point theorem. Also, an example is presented to illustrate our main result.

Multiplicity of positive radial solutions of singular Minkowski-curvature equations

This paper is about a singular Dirichlet problem involving the mean curvature operator in Minkowski space. We get the existence of triple and arbitrarily many positive radial solutions by using the Leggett-Williams fixed point theorem. Recent results in the literature are significantly improved.

Solutions to nonlinear second-order three-point boundary value problems of dynamic equations on time scales

In this paper, we consider existence criteria of three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales. To show our main results, we apply the well-known Leggett-Williams fixed point theorem. Moreover, we present some results for the existence of single and multiple positive solutions for boundary value problems on time scales, by applying fixed point theorems in cones. The conditions we used in the paper are different from those in [Dogan A. On the existence of positive solutions for the one-dimensional $ p $-Laplacian boundary value problems on time scales. Dynam Syst Appl 2015; 24: 295-304].