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We study, according to topological approaches, the solvability of systems for biharmonic equations with Navier boundary conditions. New sublinear conditions for the existence of positive solutions are established. In addition, the most interesting part of this article is that the Leggett–Williams fixed point theorem is first used to analyze the existence of triple positive radial solutions for biharmonic elliptic systems. The uniqueness and multiplicity of positive radial solutions are also discussed. Two examples are presented to illustrate the applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a Hadamard-type fractional differential equation on the interval (1 , ∞) equipped with a new class of logarithmic type integro-initial conditions. We apply the Leggett–Williams fixed point theorem and the concept of iterative positive solutions to establish the existence of solutions for the problem at hand. Our results are new and enrich the literature on Hadamard-type fractional differential equations on the infinite domain. Examples illustrating the main results are presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity ( x ( t ) − c x ( t − δ ) ) ″ + a ( t ) g ( x ( t ) ) x ( t ) = λ b ( t ) f ( t , x ( t ) , x ( t − τ 1 ( t ) ) , x ′ ( t − τ 2 ( t ) ) ) $(x(t)-cx(t-\delta)){''}+a(t)g(x(t))x(t)=\lambda b(t)f(t,x(t),x(t-\tau_{1}(t)),x'(t-\tau_{2}(t)))$ . By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established.

In this article, the existence criteria of at least one or at least three positive solutions to the Caputo-type nonlinear fractional differential equation with integral boundary value conditions has been established. The method applied in this study is formulated by the well-known Guo-Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem. First, the Green's function for corresponding linear fractional differential equation of the main nonlinear fractional differential equation under same boundary value conditions has been constructed. Next, several essential properties of that Green's function have been proved. Finally, in cone spaces some new existence and multiplicity results for the Caputo-type nonlinear fractional differential equation with integral boundary value conditions are obtained. To support the analytic proof appropriate illustrative examples has also been discussed. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the existence of multiple positive solutions for boundary value problems of high-order Riemann–Liouville fractional differential equations involving the p-Laplacian operator. Not only new existence conclusions of two positive solutions are obtained by employing functional-type cone expansion-compression fixed point theorem, but also some sufficient conditions for existence of at least three positive solutions are established by applying the Leggett–Williams fixed point theorem. In addition, we demonstrate the effectiveness of the main result by using an example.

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This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity (x (t) − c x (t − δ)) ″ + a (t) g (x (t)) x (t) = λ b (t) f (t , x (t) , x (t − τ 1 (t)) , x ′ (t − τ 2 (t))) . By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple positive solutions for boundary value problems of high-order Riemann–Liouville fractional differential equations involving the p-Laplacian operator. Not only new existence conclusions of two positive solutions are obtained by employing functional-type cone expansion-compression fixed point theorem, but also some sufficient conditions for existence of at least three positive solutions are established by applying the Leggett–Williams fixed point theorem. In addition, we demonstrate the effectiveness of the main result by using an example. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by \[\begin{aligned} x^{\prime}(t)& = r(t)x(t) + \sum_{i=1}^{m} f_i(t,x(t)), \quad t \in [0,1],\\ \lambda x(0)& = x(1) + \sum_{j=1}^{n} \Lambda_j(\tau_j, x(\tau_j)),\quad \tau_j \in [0,1],\end{aligned}\] where \(r:[0,1] \rightarrow [0,\infty)\) is continuous; the nonlocal points satisfy \(0 \leq \tau_1 \lt \tau_2 \lt \ldots \lt \tau_n \leq 1\), the nonlinear function \(f_i\) and \(\tau_j\) are continuous mappings from \([0,1] \times [0,\infty) \rightarrow [0,\infty)\) for \(i = 1,2,\ldots ,m\) and \(j = 1,2,\ldots ,n\) respectively, and \(\lambda \gt 0\) is a positive parameter.

In this paper, we investigate the solvability of infinite-point boundary value problems for a class of higher-order factional differential equations on the half-line involving Riemann-Liouville derivatives. Under a new compactness criterion, some new results are obtained by means of the properties of the Green’s function and some appropriate fixed point theorems on cone. [ABSTRACT FROM AUTHOR]

This article concerns the existence of three non-negative solutions for two kinds of three-point boundary-value problems of nonlinear fractional differential equations, where the fractional derivative is taken in the Riemann-Liouville sense. Using Leggett-Williams fixed point theorem, we present some existence criteria and then illustrate our results with examples.

By applying Leggett-Williams fixed point theorem in a suitably constructed cone, we obtain the existence of at least three bounded positive solutions for a boundary value problem on the half line. Our result improves and complements some of the work in the literature.

In this paper, we investigate nonlinear fractional differential equations of arbitrary order with advanced arguments \begin{equation*}\left\{\begin {array}{ll} D^\alpha_{0^+} u(t) +a(t)f(u(\theta(t)))=0,&03\,\, (n\in\mathbb{N}),~D^\alpha_{0^+}$ is the standard Riemann-Liouville fractional derivative of order $\alpha,$ $f: [0,\infty)\to [0,\infty),$ $a: [0,1]\to (0,\infty)$ and $\theta: (0,1)\to (0,1]$ are continuous functions. By applying fixed point index theory and Leggett-Williams fixed point theorem, sufficient conditions for the existence of multiple positive solutions to the above boundary value problem are established.

This article presents two modifications of the Leggett-Williams fixed point theorem, and two applications of these results to a terminal and to a boundary value problem for ordinary differential equations.

We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by... where r : [0, 1] → [0,∞) is continuous; the nonlocal points satisfy 0 ≤T1 2 < . . . < Tn ≤ 1, the nonlinear function fi and Tj are continuous mappings from [0, 1] × [0,∞) → [0,∞) for i = 1, 2, . . ., m and j = 1, 2, . . ., n respectively, and λ > 0 is a positive parameter. [ABSTRACT FROM AUTHOR]

A class of 3rd-order nonlocal boundary value problems (BVPs) with Riemann-Stieltjes integral conditions were studied. The existence of positive solutions to BVPs was explored via perturbed Hammerstein integral equations. Through the construction of the Green functions and discussion on their properties, the existence criterion for at least 3 or 2n - 1 positive solutions was obtained by means of the generalization of the Leggett-Williams fixed point theorem. The results generalize and improve some known results of the latest literatures, and fully reflect the influence of nonlinear terms involving derivatives on the existence of positive solutions. An example was also included to illustrate the main results. [ABSTRACT FROM AUTHOR]

By using Leggett-Williams fixed point theorem and Hölders inequality, the existence of three positive solutions for the fourth-order impulsive differential equations with integral boundary conditions x(4)(t) = (t)f(t,x(t)), 0 < t < 1, t ≠ tk, Δx|t=tk = Ik(tk, x(tk)), Δx'|t=tk = 0, k = 1,2,..., m, x(0) = ∫0¹ g(s)x(s)ds, x'(1) = 0, x''(0) = ∫0¹ = h(s)x''(s) ds, x'''(1) = 0 is considered, where (t) is Lp-integrable. Our results cover a fourth-order boundary value problem without impulsive effects and are compared with some recent results. [ABSTRACT FROM AUTHOR]

By applying the Leggett-Williams fixed point theorem in a suitably constructed cone, we obtain the existence of at least three unbounded positive solutions for a boundary value problem on the half line. Our result improves and complements some of the work in the literature. [ABSTRACT FROM AUTHOR]

Abstract: Concerned is a generalized Laplacian boundary value problem with a positive parameter. First we apply the Leggett–Williams fixed point theorem to establish sufficient conditions on the existence of at least three positive solutions for the parameter belonging to an explicit interval. Then, under a little bit stronger assumptions, we show that there are at least three positive symmetric solutions for the parameter in an open interval. The obtained results are new even for Laplacian boundary value problems and they are illustrated with an example. [Copyright &y& Elsevier]

Abstract: By using the Leggett–Williams fixed point theorem, this paper investigates the existence of at least three nontrivial radial convex solutions of systems of Monge–Ampère equations. [Copyright &y& Elsevier]

Abstract: In this paper, we study multiple nonnegative solutions for the fractional differential equation with integral boundary conditions By means of Leggett–Williams fixed point theorem, some new results on the existence of at least three nonnegative solutions are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, using the Leggett–Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equations with impulses and parameters on time scales where , are parameters, is an -periodic time scale, and both of them are -periodic functions, are-periodic functions, are nondecreasing with respect to their second arguments and -periodic with respect to their first arguments, respectively; and there exist two positive constants such that for all and is bounded, is a constant. [Copyright &y& Elsevier]

In this paper, we consider the existence of multiple positive solutions for the 2n-th order m-point boundary value problems: {x(2n)(t) = f(t, x(t), x" (t), ⋯ , x(2(n-1))(t)), 0 ≤ t ≤ 1, x(2i+1)(0) = Σj=1m-2 αijx(2i+1)(ξj), x(2i)(1) = Σj=1m-2 βijx(2i)(ξj), 0 ≤ i ≤ n - 1, where αij , βij (0 ≤ i ≤ n - 1, 1 ≤ j ≤ m - 2) ∈ [0,∞), Σj=1m-2 αij Σj=1m-2 βij ∈ (0, 1), 0 < ξ1 < ξ2 < . . . < ξm-2 < 1. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. [ABSTRACT FROM AUTHOR]

This article presents two modifications of the Leggett-Williams fixed point theorem, and two applications of these results to a terminal and to a boundary value problem for ordinary differential equations. [ABSTRACT FROM AUTHOR]

Abstract: Sufficient criteria are established for the existence of periodic solutions and three positive periodic solutions to a system with lossless transmission lines by using Mawhin’s continuation theorem and Leggett–Williams fixed point theorem. [Copyright &y& Elsevier]

In this paper, we study Sturm-Liouville boundary value problems for second-order functional differential equations on infinite intervals [-τ +∞). By using the Leggett-Williams fixed point theorem, we obtain sufficient conditions for the existence of multiple positive solutions of the functional boundary value problem. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to using the Leggett–Williams fixed point theorem to derive some sufficient conditions for the existence of triple positive solutions to a type of -Laplacian boundary value problems on time scales. The interesting point is that the nonlinear term depends on the first-order delta derivative explicitly. [Copyright &y& Elsevier]

Abstract: In this paper, a type of nonlinear neutral functional differential equations are considered. Some results on the existence of three positive periodic solutions are obtained by using the Leggett–Williams fixed point theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence of at least three positive solutions for the nonlinear th-order -point boundary value problem where . The associated Green’s function for the th-order -point boundary value problem is first given, and growth conditions are imposed on the nonlinearity which yield the existence of multiple positive solutions by using the Leggett–Williams fixed point theorem. [Copyright &y& Elsevier]

In this paper we investigate the existence of multiple positive solutions for nonlinear boundary value problems with integral boundary conditions. We shall rely on the Leggett-Williams fixed point theorem. [ABSTRACT FROM AUTHOR]

We establish the existence of at least three positive periodic solutions to the second order differential equation with periodic coefficients \[ \hspace*{6pc} -(p(t)x'(t))'+q(t)x(t)=f(t,x(t)),\quad t\in R,\hspace*{6.4pc}(\ast) \] where f is continuous with f(t + T, x) = f(t,x) for (t,x) ∊ R × R and T > 0, p, q are continuous and T-periodic with p > 0 and q ≥ 0. We accomplish this by making growth assumptions on f, which can apply to many more cases than those discussed in recent works. An example to illustrate the main result is given. [ABSTRACT FROM AUTHOR]

For the second-order m-point boundary value problemwhere ki>0 (i=1,2,…,m-2), 0<ξ1<ξ2<⋯<ξm-2<1, growth conditions are imposed on f which yield the existence of at least three positive solutions by using the Leggett–Williams fixed point theorem. The associated Green''s function for the m-point boundary value problem is also given. [Copyright &y& Elsevier]

We consider the following nonlinear impulsive delay differential equation with variable coefficients and forcing term: Using new methods different from those in Tang et al. (Appl. Math. Comput. 131 (2002) 373), Nieto (J. Comput. Appl. Math. Lett. 15 (2002) 489), Franco and Nieto (J. Comput. Appl. Math. 88 (1998) 144), Nieto (J. Math. Anal. Appl. 205 (1997) 423), He and Yu (J. Math. Anal. Appl. 272 (2002) 67), we establish stability theorems for (*) when e(t)≡0 under the assumption that there is {bk} such that bkx2⩽x(x+Ik(x))⩽x2 for k∈N and x∈R (Theorems 2.1–2.4), the existence results for three positive periodic solutions and nonexistence results for periodic solution of (*) when e(t)≡0 under the assumption exp∫lower limit 0, upper limit T a(u) du≠∏0⩽tk and bk>-1 for all k (Theorems 3.1–3.2) and the existence results for periodic solution of (*) at resonance, which is caused by impulses i.e. exp (∫lower limit 0, upper limit T a(u) du)=∏0⩽tk, under the assumption bk>-1 for all k (Theorems 4.2–4.3). Some results obtained improve and generalize the known theorems and some other results are new. Examples are presented to illustrate the main results. [Copyright &y& Elsevier]

The existence of triple positive solutions for a boundary-value problem governed by the Φ-Laplacian is investigated, when Φ is a so-called sup-multiplicative-like function (in a sense introduced in [22]) and the boundary conditions include nonlinear expressions at the end points (as in [21, 28]). The Leggett-Williams fixed point theorem in a cone is used. The results improve and generalize known results given in [21]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of at least three positive solutions for the 2nth order m-point boundary value problem y(2n)(t)=ft,y(t),y″(t),…,y(2(n−1))(t), 0&les;t&les;1,y(2i)(0)=0, y(2i)(1)=∑lower limit j=1, upper limit m−2 kijy(2i)(ξj), 0&les;i&les;n−1, where kij>0 (i=0,1,…,n−1; j=1,2,…,m−2), 0<ξ1<ξ2<⋯<ξm−2<1. The associated Green''s function for the higher order m-point boundary value problem 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. [Copyright &y& Elsevier]