Existence of Monotone Positive Solution for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.

POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRODINGER EQUATIONS

In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.

Existence of Positive Solutions for Systems of Second-order Nonlinear Singular Differential Equations with Integral Boundary Conditions on Infinite Interval

By using cone theory and the Monch fixed theorem combined with a monotone iterative technique,we investigate the existence of positive solutions for systems of secondorder nonlinear singular differential equations with integral boundary conditions on infinite interval and establish the existence theorem of positive solutions and iterative sequence for approximating the positive solutions.The results in this paper improve some known results.

Existence of Positive Solutions to Boundary Value Problem for a Nonlinear Fractional Differential Equation

In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES

In this article,we study the following critical problem involving the fractional Laplacian:{(−Δ)^α/2u−γu/|x|^α=λ|u|^q−2/|x|^s+|u|^2^∗α^(t)−2u/|x|^t in Ω,u=0 in R^N∖Ω,whereΩ⊂R^N(N>α)is a bounded smooth domain containing the origin,α∈(0,2),0≤s,t<α,1≤q<2,λ>0,2α^*(t)=2(N-t)/N-αis the fractional critical Sobolev-Hardy exponent,0≤γ<γH,and γH is the sharp constant of the Sobolev-Hardy inequality.We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques.

Positive Solutions for Fourth-order Delay Differential Equation of Boundary Value Problem with p-Laplacian

In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian(Φp(u))(t) + a(t)f(t, u(t- τ), u(t)) = 0, 0 < t < 1;u(0) = u(0) = 0, u(1) = αu(η);u(t) = 0,- τ≤ t ≤ 0.By using Schauder fixed-point theorem, some sufficient conditions are obtained which guarantee the fourth-order delay differential equation of boundary value problem with p-Laplacian has at least one positive solution. Some corresponding examples are presented to illustrate the application of our main results.

Existence of Multiple Positive Solutions for Second-order Three-point Boundary Value Problems on a Half-line

In this paper,we are concerned with the existence of multiple positive solutions to a second-order three-point boundary value problem on the half-line.The results are obtained by the Leggett-Williams fixed point theorem.

Positive Solutions for First-order PBVPs on Time Scales

In this paper, the author studies the following nonlinear dynamic equation {x△(t) = r(t)x(σ(t)) + f(t, x(σ(t))), t ∈ [0, T ], x(0) = x(σ(T )). By applying and improving the generalized form of Leggett-Williams fixed point theorem, sufficient conditions are established for the existence of positive solutions.

POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms:{−(a+b∫_(R^(3))|∇u|^(2)dx)Δu+V(x)u=Q(x)u^(p)+εf(x),u>0,x∈R^(3),u∈H^(1)(R^(3)).Here a,b are positive constants,V(x),Q(x)are positive radial potentials,1<p<5,ε>0 is a small parameter,f(x)is an external source term in L^(2)(R^(3))∩L^(∞)(R^(3)).

POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM

In this paper,we will study the nonlocal and nonvariational elliptic problem{−(1+a||u||_(q)^(αq))Δu=|u|^(p−1)u+h(x,u,∇_(u))inΩ,u=0 on∂Ω,(0.1)(1)where a>0,α>0,1<q<2^(∗),p∈(0,2^(∗)−1)∖{1}andΩis a bounded smooth domain in R^(N)(N≥2).Under suitable assumptions about h(x,u,∇u),we obtain\emph{a priori}estimates of positive solutions for the problem(0.1).Furthermore,we establish the existence of positive solutions by making use of these estimates and of the method of continuity.

Existence of Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function

In this article, by using a fixed point theorem, we study following fourth-order three-point BVP: where f ∈ C([0,1]×[0,+∞),[0,+∞)) α ∈ [0,6) and . The main point to emphasize is that although the corresponding Green’s function is changing signs, by applying the fixed point theorem, we can still obtain at least two positive solutions and degreased solutions under certain suitable conditions.

Existence of Positive Solutions to Semipositone Fractional Differential Equations

In this paper, by means of constructing a special cone, we obtain a sufficient condition for the existence of positive solution to semipositone fractional differential equation.

Positive Solution for a Singular Fourth-Order Differential System

In this paper, we investigate the existence of positive solutions for the singular fourth-order differential system u(4) = φu + f (t, u, u”, φ), 0 < t < 1, -φ” = μg (t, u, u”), 0 < t < 1, u (0) = u (1) = u”(0) = u”(1) = 0, φ (0) = φ (1) = 0;where μ > 0 is a constant, and the nonlinear terms f, g may be singular with respect to both the time and space variables. The results obtained herein generalize and improve some known results including singular and non-singular cases.

Existence and Multiplicity of Positive Solutions for a Singular Third-Order Three-Point Boundary Value Problem with a Parameter

In this paper, we investigate the existence of positive solutions for a singular third-order three-point boundary value problem with a parameter. By using fixed point index theory, some existence, multiplicity and nonexistence results for positive solutions are derived in terms of different values of λ.

Positive Solution for a Hadamard Fractional Singular Boundary Value Problem of Order μ ∈(2,3)

In this article, we establish the existence of positive solution for the following Hadamard fractional singular boundary value problem where is continuous and singular at t = a, t = b and x = 0. Further, is Hadamard fractional derivative of order μ. Moreover, the existence of positive solution has been established using fixed point index for a completely continuous map in a cone. Also, an example is included to show the validity of our result.

Existence of Positive Solutions to a Fractional-Kirchhoff System

LetΩbe a bounded smooth domain in RN(N≥3).Assuming that 0<s<1,1<p,q≤N+2s/N-2s with(p,q)≠(N+2s/N-2s,N+2s/N-2s),and a,b>0 are constants,we consider the existence results for positive solutions of a class of fractional elliptic system below,{(a+b[u]^(2)_(s))(-Δ)^(s)u=vp+h_(1)(x,u,v,▽u,▽v),x∈Ω,(-Δ)^(s)v=u^(q)+h_(2)(x,u,▽,▽u,▽v),x∈Q,u,v>0,x∈Ω,u=v=0,x∈RN\Ω.Under some assumptions of hi(x,u,v,▽u,▽v)(i=1,2),we get a priori bounds of the positive solutions to the problem(1.1)by the blow-up methods and rescaling argument.Based on these estimates and degree theory,we establish the existence of positive solutions to problem(1.1).

Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C

This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation Xm+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

Existence of Positive Solutions of Second-order Periodic Boundary Value Problems with Sign-Changing Green＇s Function

In this paper, we consider the existence of positive solutions of second-order periodic boundary value problem u′′+ (1/2+ ε)~2*u = λg(t)f(u), t∈[0, 2π], u(0) = u(2π), u′(0) = u′(2π),where 0 <ε <12, g : [0, 2π]→ R is continuous, f : [0, ∞)→R is continuous and λ > 0 is a parameter.

POSITIVE SOLUTIONS TO m-POINT BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH p-LAPLACIAN

In this paper,we are concerned with the existence of positive solutions to an m-point boundary value problem with p-Laplacian of nonlinear fractional differential equation.By means of Krasnosel'skii fixed-point theorem on a convex cone and Leggett-Williams fixed-point theorem,the existence results of solutions are obtained.

MULTIPLICITY OF POSITIVE SOLUTIONS TO MULTIPOINT BOUNDARY VALUE PROBLEM WITH ONE-DIMENSIONAL p-LAPLACIAN

In this paper, we consider a multipoint boundary value problem for one-dimensional p-Laplacian. Using a fixed point theorem due to Bai and Ge, we study the existence of at least three positive solutions to the boundary value problem. In this problem, the nonlinear term explicitly involves a first-order derivative, which is different from some previous ones.