Leggett-Williams Fixed Point Theorem Research Articles

Multiplicity results for a boundary value problem with integral boundary conditions

The main purpose of this paper is to establish existence and multiplicity of positive solutions for a system of fourth-order boundary value problem with multi-point and integral conditions. To prove our results, we used Leggett–Williams fixed point theorem. An example is presented to illustrate our main results.

Existence Results for Fractional Differential Equations with Multistrip Riemann–Stieltjes Integral Boundary Conditions

This paper is concerned with the existence and multiplicity of the positive solutions for a fractional boundary value problem with multistrip Riemann–Stieltjes integral boundary conditions. Our results are based on the Leggett–Williams fixed point theorem. In the end, two examples are worked out to illustrate our main work.

Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional q-Difference Equation

A class of boundary value problems of Caputo fractional q-difference equation is introduced. Green’s function and its properties for this problem are deduced. By applying these properties and the Leggett-Williams fixed-point theorem, existence criteria of three positive solutions are obtained. At last, some examples are given to illustrate the validity of our main results.

Hyers–Ulam Stability of a Class of Nonlocal Boundary Value Problem Having Triple Solutions

This manuscript is concerned with the investigation of three-point boundary value problem (BVP) of nonlinear fractional differential equations (FDEs) as provided by $$\begin{aligned} \begin{aligned} -&\mathbf {D}^\omega w(t)=\varphi (t,w(t));\ t\in (0, 1),\ \omega \in (1, 2],\\ {}&\mathbf {D}^\theta w(0)=0, \ w(1)= \gamma \mathbf {D}^{\theta }w(\eta ),\ \theta \in (0, 1], \end{aligned} \end{aligned}$$ where usual Riemann–Liouville derivative is denoted by $$\mathbf {D}^q$$ with order $$\omega $$ and $$\gamma , \eta \in (0, 1)$$ . Also $$\varphi :[0,1]\times [0,\infty )\rightarrow [0,\infty )$$ is continuous nonlinear function. Thank to classical Leggett–Williams fixed point theorem, appropriate conditions are established for the existence and multiplicity results. Further some useful results about Hyers–Ulam stability for the obtained solutions are also provided. The establish theory is properly justified by providing two examples.

Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain

AbstractIn this paper, we investigate the existence of positive solutions for Hadamard type fractional differential system with coupled nonlocal fractional integral boundary conditions on an infinite domain. Our analysis relies on Guo-Krasnoselskii’s and Leggett-Williams fixed point theorems. The obtained results are well illustrated with the aid of examples.

Positive Solutions for an Impulsive Second-Order Nonlinear Boundary Value Problem

By using the Leggett–Williams fixed point theorem, we prove the existence of three positive solutions for a second-order nonlinear ordinary differential equation with modified argument, subject to integral boundary conditions and impulsive effects.

The Positive Solutions for Integral Boundary Value Problem of Fractional p-Laplacian Equation with Mixed Derivatives

In this paper, we consider a class of integral boundary value problems of fractional p-Laplacian equation, which involve both Riemann–Liouville fractional derivative and Caputo fractional derivative. By using the generalization of Leggett–Williams fixed point theorem, some new results on the existence of at least three positive solutions to the boundary value problems are obtained. Finally, some examples are presented to illustrate the extensive potential applications of our main results.

Existence results for nonlinear boundary value problems with m-point integral boundary condition

In this paper, we investigate the existence of positive solutions for the nonlinear $m$-point boundary value problems with integral boundary condition. By using fixed-point index theorem and Leggett-Williams fixed point theorem, the existence and multiplicity of positive solutions are obtained. As an application, two examples are given to demonstrate our results.

一类无穷区间上的分数阶微分方程边值问题解的存在性

利用Leray-Schauder非线性抉择定理以及Leggett-Williams不动点定理研究了一类无穷区间上的分数阶微分方程积分边值问题，获得了该边值问题至少存在一个无界解和三个正解的充分条件。最后给出了两个例子作为所获结果的应用。 By using the Leray-Schauder nonlinear alternative theorem and the Leggett-Williams fixed point theorem, a class of boundary value problem for fractional differential equations with integral conditions on an infinite interval is investigated. Some sufficient conditions on the existence of at least one unbounded solution and three positive solutions are established. At last, two examples are given to illustrate the results.

Positive Solutions of Fractional Differential Equations with p-Laplacian

The multiplicity of positive solution for a new class of four-point boundary value problem of fractional differential equations with p-Laplacian operator is investigated. By the use of the Leggett-Williams fixed-point theorem, the multiplicity results of positive solution are obtained. An example is given to illustrate the main results.

Existence of positive solutions for second-order impulsive differential equations with delay and three-point boundary value problem

We study the existence of three positive solutions to a type of three-point boundary value problems for second-order impulsive and delay differential equations, and we obtain the result that there exist at least three nonnegative symmetric positive solutions by means of a generalization of the Leggett-Williams fixed point theorem.

Existence of symmetric solutions for a class of BVP with integral boundary conditions

In this paper, we study the symmetric solutions of second-order BVP with integral boundary conditions. By using a generalized Leggett-Williams fixed point theorem and some other techniques, we obtain sufficient conditions for the existence of symmetric positive solutions for the system. Meanwhile, an example is devoted to demonstrate our results in the end.

Positive solutions to semi-positone fourth-order $$\phi $$ ϕ -Laplacian BVPs

This work is devoted to proving existence of positive solutions to a fourth-order semipositone \(\phi \)-Laplacian boundary problem. The nonlinearity may have time-singularity and change sign. Existence results are proved using the Krasnosel’skii and the Leggett-Williams fixed point theorems and examples of applications are provided.

Existence of positive periodic solutions in shifts δ± for a nonlinear first order functional dynamic equation on time scales

Let T ? R be a periodic time scale in shifts ?? with period P ? [t0,?)T. In this paper we consider the nonlinear functional dynamic equation of the form x?(t) = a(t)x(t)- ?b(t) f (x(h(t))), t ? T. By using the Krasnoselski?, Avery-Henderson and Leggett-Williams fixed point theorems, we present different sufficient conditions for the nonexistence and existence of at least one, two or three positive periodic solutions in shifts ?? of the above problem on time scales. We extend and unify periodic differential, difference, h-difference and q-difference equations and more by a new periodicity concept on time scales.

Existence and multiplicity of positive solutions for a class of fractional differential equations with three-point boundary value conditions

In this paper, we consider the nonlinear three-point boundary value problem of fractional differential equations $$D^{\alpha}_{0^{+}}u(t)+a(t)f\bigl(t,u(t)\bigr)=0, \quad 0< t< 1, 2< \alpha\leq3, $$ with boundary conditions $$u(0)=0,\qquad D^{\beta}_{0^{+}}u(0)=0,\qquad D^{\beta}_{0^{+}}u(1)=bD^{\beta}_{0^{+}}u( \xi),\quad 1\leq\beta\leq2, $$ involving Riemann-Liouville fractional derivatives $D^{\alpha}_{0^{+}}$ and $D^{\beta}_{0^{+}}$ , where $a(t)$ maybe singular at $t=0$ or $t=1$ . We use the Banach contraction mapping principle and the Leggett-Williams fixed point theorem to obtain the existence and uniqueness of positive solutions and the existence of multiple positive solutions. We investigate the above fractional differential equations without many preconditions by the fixed point index theory and obtain the existence of a single positive solution. Some examples are given to show the applicability of our main results.

Positive solutions for higher order differential equations with integral boundary conditions

In this paper, we consider the existence of at least three positive solutions for the 2nth order differential equations with integral boundary conditions $$ \left \{ \textstyle\begin{array}{l} x^{(2n)}(t)=f(t, x(t), x''(t),\ldots,x^{(2(n-1))}(t)), \quad 0\leq t\leq 1, x^{(2i)}(0)=\int_{0}^{1}k_{i}(s)x^{(2i)}(s) \,\mathrm{d}s,\qquad x^{(2i)}(1)=0, \quad 0\leq i\leq n-1, \end{array}\displaystyle \right . $$ where $(-1)^{n}f>0$ is continuous, and $k_{i}(t)\in L^{1}[0,1]$ ( $i=0,1,\ldots,n-1$ ) are nonnegative. The associated Green’s function for the higher order differential equations with integral boundary conditions is first given, and growth conditions are imposed on f which yield the existence of multiple positive solutions by using the Leggett-Williams fixed point theorem.

Triple positive solutions for two classes of delayed nonlinear fractional FDEs with nonlinear integral boundary value conditions

This paper is concerned with two classes of delayed nonlinear fractional functional differential equations (FDEs) with nonlinear Riemann-Stieltjes integral boundary value conditions. By employing the well-known Leggett-Williams fixed point theorem and a generalization of Leggett-Williams fixed point theorem, some new sufficient criteria are established to guarantee the existence of at least triple positive solutions. As applications, some interesting examples are presented to illustrate our main results.

Existence of three positive solutions for boundary value problem with fractional order and infinite delay

Abstract In this paper we investigate the existence three positives solutions by using Leggett-Williams fixed point theorem in cones for three boundary value problem with fractional order and infinite delay.

Existence of Positive Solutions for Second-Order Impulsive Boundary Value Problems on Time Scales

In this paper, by using Leray–Schauder fixed-point theorem, Avery–Henderson fixed-point theorem and Leggett–Williams fixed-point theorem, respectively, we investigate the conditions for the existence of at least one, two and three positive solutions to nonlinear second-order impulsive boundary value problems on time scales.

On the existence of positive solutions for the second-order boundary value problem

This paper is concerned with the existence of positive solutions to a second order boundary value problem. By imposing growth conditions on f and using a generalization of the Leggett–Williams fixed point theorem, we prove the existence of at least three symmetric positive solutions.