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In this article, we establish some results on the existence of multiple positive solutions for a system of nonlinear fractional order infinite-point boundary value problems. The main tool is a fixed point theorem of the cone expansion and compression of functional type due to Avery, Anderson and O’Regan for at least one positive solution. We also prove that the boundary value problems has at least three positive solutions by applying the five functional fixed point theorem. And then, we establish the existence of at least 2k − 1 positive solutions to the fractional order boundary value problems for any arbitrary positive integer k. Publisher's Version

In this paper, a system of fractional elliptic equation is investigated, which involving fractional critical Sobolev-Hardy exponent and concave-convex terms. By means of variational methods and analytic techniques, the existence and multiplicity of positive solutions to the system is established.

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^n$$ is an open bounded set with continuous boundary, $$n>2s$$ with $$s\in (0,1),(-\Delta )^{s}$$ is the fractional Laplacian operator, $$\mu $$ is a positive real parameter, $$q\in [2, 2^*_s)$$ and $$2^*_s=2n/(n-2s)$$ is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of $$\Omega $$ . Precisely, we show that the problem has at least $$cat_{\Omega }(\Omega )$$ nontrivial solutions, provided that $$q=2$$ and $$n\geqslant 4s$$ or $$q\in (2, 2^*_s)$$ and $$n>2s(q+2)/q$$ , extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

The first and second order of accuracy stable difference schemes for the numerical solution of the mixed problem for the fractional parabolic equation are presented. Stability and almost coercive stability estimates for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations. WOS:000308762000001 2-s2.0-84866991995

In this paper, we study the existence of countably many positive solutions for nonlinear singular boundary value problem(ϕ(u′))′+a(t)f(u(t))=0,0

Let R 0,∞ and let Z denote the set of all integers. If a, b ∈ Z, a < b, then T a, b denotes the discrete interval {a, a 1, . . . , b}. LetΔu k u k 1 −u k be the forward difference operator. Let T,N ∈ Z, T < N, and let X stand for the space of functions u : T T − 1,N 1 → R equipped with the norm ‖u‖ max{|u k | : k ∈ T T − 1,N 1 }. Clearly, X is an N − T 3 dimensional Banach space. Denote byA the set of continuous maps γ : X → R. We say that α, β ∈ A are compatible if for each μ ∈ 0, 1 the problem Δ φ Δu k − 1 0, k ∈ T T,N , 1.1

In this paper, general existence theorems are presented for the singular equation-(ϕp(y′))′=q(t)f(t,y),0

In this paper, we obtain new existence results for multiple positive solutions of a delay singular differential boundary-value problem. Our main tool is the fixed point index method.

In this paper, we discuss the existence of multiple positive solutions for the fourth-order boundary value problem u^{(4)}(t) = f(t, u(t), u''(t)), 0 < t < 1, u(0) = u(1) = u''(0) = u''(1) = 0, where f : [0, 1] × [0,1) \rightarrow [0,1) is continuous. Existence theorems are established via the theory of fixed point index in cones.

In this paper, by a specially constructed cone and the fixed point index theory, we investigate the existence of multiple positive solutions for the following singular semipositone problem: y″+λf(t,y)=0, t∈(0,1), y(0)=y(1)=0. The nonlinear term f(t,y) may be singular at t=0, t=1, and y=0, also may be negative for some values of t and y; and λ is a positive parameter.

We study the existence of positive solutions of a nonlinear eigenvalue problem for a two-point boundary-value problem for a family of second-order disfocal ordinary differential equations. We apply a cone-theoretic fixed-point theorem and obtain both sufficient conditions for the existence of positive solutions and sufficient conditions for the existence of multiple positive solutions.

In this paper, we establish the existence of multiple positive solutions for singular integral equations. The proof is based on a general existence principle established using a nonlinear alternative principle of Leray--Schauder type and a well-known fixed point theorem in cones. As an application, we consider higher order nonlinear singular conjugate boundary value problems. Some recent results in the literature are generalized and improved.

This thesis concerns the study of some singular elliptic problems. In these problems the singularity is due to a nonlinear term that blows up on the boundary. More precisely, in Chapter 2, we investigate the question of multiplicity of solutions for a singular problem with critical growth in dimension $N \geq 3$. In Chapter 3, we investigate the validity of $C^1$ versus $W^{1,p}_0$ energy minimizers for a quasilinear elliptic singular problem. In Chapter 4, we present global bifurcation results for a semilinear elliptic singular problem with critical growth in dimension 2 with exponential growth.; L'objet de cette thèse concerne l' étude de quelques problèmes elliptiques singuliers. Dans les problémes que nous avons considérés, ce caractère singulier se caractérise par la présence d'une nonlinéarité qui explose au bord du domaine où le problème est posé. Plus précisément, dans le chapitre 2, nous abordons la question de la multiplicité de solutions pour un problème elliptique critique singulier en dimension $N\geq 3$. Dans le chapitre 3, Nous discutons la validité de la propriété $C^1$ versus $W^{1,p}_0 $ minimiseurs de l' énergie pour un problème quasilinéaire elliptique singulier. Enfin, dans le chapitre 4, nous présentons des résultats de bifurcation globale pour un problème semilinéaire elliptique singulier et critique en dimension 2 avec croissance sur-exponentielle.

This thesis concerns the study of some singular elliptic problems. In these problems the singularity is due to a nonlinear term that blows up on the boundary. More precisely, in Chapter 2, we investigate the question of multiplicity of solutions for a singular problem with critical growth in dimension $N \geq 3$. In Chapter 3, we investigate the validity of $C^1$ versus $W^{1,p}_0$ energy minimizers for a quasilinear elliptic singular problem. In Chapter 4, we present global bifurcation results for a semilinear elliptic singular problem with critical growth in dimension 2 with exponential growth.; L'objet de cette thèse concerne l' étude de quelques problèmes elliptiques singuliers. Dans les problémes que nous avons considérés, ce caractère singulier se caractérise par la présence d'une nonlinéarité qui explose au bord du domaine où le problème est posé. Plus précisément, dans le chapitre 2, nous abordons la question de la multiplicité de solutions pour un problème elliptique critique singulier en dimension $N\geq 3$. Dans le chapitre 3, Nous discutons la validité de la propriété $C^1$ versus $W^{1,p}_0 $ minimiseurs de l' énergie pour un problème quasilinéaire elliptique singulier. Enfin, dans le chapitre 4, nous présentons des résultats de bifurcation globale pour un problème semilinéaire elliptique singulier et critique en dimension 2 avec croissance sur-exponentielle.

In this paper we consider the following nonlinear third order two-point boundary value problem \begin{align*} & x'''(t)+f(t,x(t))=0,\quad a \lt t \lt b,\\ & x(a)=x''(a)=x(b)=0. \end{align*} By using the Leggett-Williams and Krasnosel'skii fixed-point theorems, we offer criteria for the existence of three positive solutions to the boundary value problem. Examples are also included to illustrate the results obtained.

By the use of a Guo-Krasnoselskii theorem in cones, existence of positive eigenvalues yielding nonnegative or positive solutions to an integral equation is studied. The results are applied to a variety of boundary value problems concerning ordinary differential equations. (C) 2009 Elsevier Ltd. All rights reserved. Nonlinear Analysis-Theory Methods & Applications

In this paper, the following fourth-order three-point boundary value problem with $p$-Laplacian operator is studied: $$ \left\{ \begin{array}{ll} (\phi_p(u''(t)))'' = a(t) f(u(t)), \quad t \in (0,1), \\ u(0) = \xi u(1), \; u'(1) = \eta u'(0), \\ u''(0) = \alpha_1 u''(\delta), \; u''(1) = \beta_1 u''(\delta), \end{array} \right. $$ where $\alpha_1, \; \beta_1 \ge 0$, $\xi \neq 1$, $\eta \neq 1$, $0 \lt \delta \lt 1$ and $\phi_p(z) = |z|^{p-2} z$ for $p \gt 1$. We impose growth conditions on $f$ which guarantee the existence of at least three positive solutions for the problem.

In this paper, we study a Palais-Smale condition in unbounded domains. Furthermore, we apply this result to prove that the semilinear elliptic equation in a Esteban-Lions domain with holes has multiple positive solutions.

Values of λ are determined for which there exist positive solutions of the system of dynamic equations, , , for , satisfying the boundary conditions, , where is a time scale. A Guo-Krasnosel'skii fixed point-theorem is applied.

This article concerns the nonlocal problem $$\displaylines{ -\Big(a-b\int_{\mathbb{R}^4}|\nabla u|^2\,dx\Big)\Delta u=|u|^2u+\mu f(x), \quad\text{in }\mathbb{R}^4,\cr u\in \mathcal{D}^{1,2}(\mathbb{R}^4), }$$ where a, b are positive constants, $\mu$ is a non-negative parameter, $f(x)\in L^{4/3}(\mathbb{R}^4)$ is a non-negative function. By using the variational method, the existence of multiple positive solutions are obtained.

The existence of positive solutions of a nonlocal boundary value problem for a second order differential equation is investigated. By assuming that the response function is quiet at zero, in a sense introduced here, and it satisfies some easy conditions, existence results for a countable set of positive solutions are given.