Existence Results Of Positive Solutions Research Articles

Positive solutions of a nonlinear impulsive equation with periodic boundary conditions

In this paper, we investigate nonlinear second order differential equations subject to linear impulse conditions and periodic boundary conditions. Sign properties of an associated Green’s function are exploited to get existence results for positive solutions of the nonlinear boundary value problem with impulse. Upper and lower bounds for positive solutions are also given. The results obtained yield periodic positive solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.

Positive Solutions for a Class of n Th-Order m-Point Singular Boundary Value Problem

In this paper,by using Krasnosel'skill fixed point theorem,the existence result of positive solutions is established for the following nth-order m-point singular boundary value problem: where h(t) can be singular at t=0,t=1,f(t,v_0,v_1,…,v_(n-2)) may also be singular at v_i=0(i=0,1,…,n-2).

Existence results of positive solutions to fractional differential equation with integral boundary conditions

In this paper we consider the existence, multiplicity, and nonexistence of positive solutions to fractional differential equation with integral boundary conditions. Our analysis relies on the fixed point index.

On positive solutions of a nonlocal fractional boundary value problem

In this paper, we investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation. Firstly, we give Green’s function and prove its positivity; secondly, the uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions; thirdly, by means of the fixed point index theory, we obtain some existence results of positive solution. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.

Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations

In this paper, we study a linear Sturm–Liouville impulsive problem and a nonlinear Sturm–Liouville impulsive problem. By applying a new approach via critical point theory and variational methods, the existence results of positive solutions are obtained.

Existence results of positive solutions of Kirchhoff type problems

In the present paper, we use variational methods to prove two existence results of positive solutions of the following Kirchhoff type problems { − ( a + b ∫ Ω | ∇ u | 2 ) Δ u = f ( x , u ) , in Ω ; u = 0 , on ∂ Ω . One deals with the asymptotic behaviors of f near zero and infinity and the other deals with 4-superlinear of f at infinity.

The Existence of Positive Solution to Three-Point Singular Boundary Value Problem of Fractional Differential Equation

We investigate the existence of positive solution to nonlinear fractional differential equation three-point singular boundary value problem:Dqu(t)+f(t,u(t))=0,0<t<1,u(0)=0,u(1)=αD(q−1)/2u(t)|t=ξ, where1<q≤2is a real number,ξ∈(0,1/2],α∈(0,+∞)andαΓ(q)ξ(q−1)/2<Γ((q+1)/2),Dqis the standard Riemann-Liouville fractional derivative, andf∈C((0,1]×[0,+∞),[0,+∞)),lim⁡t→+0f(t,⋅)=+∞(i.e.,fis singular att=0). By using the fixed-point index theory, the existence result of positive solutions is obtained.

The existence of positive solution for singular boundary value problems of first order differential equation on unbounded domains

By constructing a special cone and using cone compression and expansion fixed point theorem, this paper presents some existence results of positive solutions of singular boundary value problem on unbounded domains for a class of first order differential equation. As applications of the main results, two examples are given at the end of this paper.

Existence results for second-order system with impulse effects via variational methods

In this paper, we investigate the positive solutions of a second-order system with impulse effects. By using critical point theory the existence result of positive solutions is obtained.

Existence results of positive solutions to boundary value problem for fractional differential equation

In this paper, we consider the existence, multiplicity, and nonexistence of positive solutions to some class of boundary vale problem for fractional differential equation of high order. Our analysis relies on the fixed point index.

POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

We investigate the positive solution of nonlinear fractional differential equation with semi-positive nonlinearity \[ \begin{cases} D^\alpha_{0^+}u(t) + f(t, u(t)) = 0,\,\,\,\,\, 0 < t < 1,\\ u(0) = u'(1) = u''(0) = 0 \end{cases} \] where \(2 < \alpha\leq 3\) is a real number, \(D^\alpha_{0^+}\) is the Caputo's differentiation, and \(f : [0; 1] \times [0, \infty) \rightarrow (-\infty , \infty)\). By use of Krasnosel'skii fixed point theorem, the existence results of positive solution are obtained.

Eigenvalue of fourth-order [formula omitted]-point boundary value problem with derivatives

This paper deals with the existence of positive solutions for the fourth-order nonlinear ordinary differential equation x ⁗ ( t ) = λ g ( t ) f ( t , x ( t ) , x ″ ( t ) ) , 0 < t < 1 , subject to the boundary conditions: α x ( 0 ) − β x ′ ( 0 ) = δ x ( 1 ) + γ x ′ ( 1 ) = 0 , x ″ ( 0 ) = x ″ ( 1 ) − ∑ i = 1 m − 2 η i x ″ ( ξ i ) = 0 , where α , β , γ , δ ≥ 0 are constants such that ρ = α δ + α γ + β δ > 0 , and ξ i ∈ ( 0 , 1 ) , η i ∈ [ 0 , + ∞ ) ( i = 1 , 2 , … , m − 2 ) . By means of a fixed-point theorem due to Krasnaselskii, some new existence results of positive solutions for the above multi-point boundary value problem are obtained, which improve the main results of Graef et al. [J.R. Graef, C. Qian, B. Yang, A three-point boundary value problem for nonlinear fourth-order differential equations, J. Math. Anal. Appl. 287 (2003) 217–233]. An example is given to demonstrate the main results of this paper.

Solvability for Two Classes of Higher-Order Multi-Point Boundary Value Problems at Resonance

Using the theory of coincidence degree, we establish existence results of positive solutions for higher-order multi-point boundary value problems at resonance for ordinary differential equation , with one of the following boundary conditions: , and , , where is a continuous function, , all the have not the same sign. We also give some examples to demonstrate our results.

Existence of Positive Solutions for Fourth-Order Three-Point Boundary Value Problems

We are concerned with the nonlinear fourth-order three-point boundary value problem , , , , . By using Krasnoselskii's fixed point theorem in a cone, we get some existence results of positive solutions.

Positive solutions of second-order [formula omitted]-point boundary value problems with changing sign singular nonlinearity

In this work, some new existence results of positive solutions for a class of singular m -point boundary value problems with changing sign nonlinearity are obtained, which improve on many known results.

Global bifurcation phenomena for singular one-dimensional p-Laplacian

In this paper, we present global existence results for the following problem (P λ ) { φ p ( u ′ ( t ) ) ′ + λ h ( t ) f ( u ( t ) ) = 0 , a.e. in ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where φ p ( x ) = | x | p − 2 x , p > 1 , λ a positive parameter and h a nonnegative measurable function on ( 0 , 1 ) which may be singular at t = 0 and/or t = 1 , and f ∈ C ( R + , R + ) with R + = [ 0 , ∞ ) . By applying the global bifurcation theorem and figuring the shape of unbounded subcontinua of solutions, we obtain many different types of global existence results of positive solutions. We also obtain existence results of sign-changing solutions for ( P λ ) when f is an odd symmetric function.

Positive solutions of singular dirichlet boundary value problems with sign-changing nonlinearities

By a new approach, we present a new existence result of positive solutions to the following Dirichlet boundary value problem, x ″ ( t ) + f ( t , x ( t ) ) = 0 , 0 < t < 1 , x ( 0 ) = 0 = x ( 1 ) It is remarkable that the result of this paper is not obtained by employing the fixed-point theorems in cone and the method of the lower and upper functions. Our nonlinearity may change sign and be singular at t = 0, 1, χ = 0.

Nodal radial solutions for a superlinear problem

The superlinear Dirichlet problem - Δ u = g ( u ) - λ is considered. Existence of Nodal radial solutions is proved for λ large enough without the classical condition on the increasing of g ( x ) depending on the dimension of the space [W. Dambrosio, On the multiplicity of radial solutions to superlinear Dirichlet problems in bounded domains, J. Differential Equations 196 (2004) 91–118; D. de Figueiredo, P.L. Lions, R.D. Nussbaum, A priory estimates and existence results of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982)]. The proof is based on a qualitative description of solutions and a new classification given in ([M. Rouaki, Topological degree and a nonlinear Dirichlet problem, Nonlinear Anal. 54 (2003) 801–817]).

Symmetric Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem

In this paper, we consider the following second-order three-point boundary value problem $$ \begin{array}{*{20}c} {{{u}\ifmmode{''}\else$''$\fi{\left( t \right)} + a{\left( t \right)}f{\left( {u{\left( t \right)}} \right)} = 0,}} & {{0 < t < 1,}} \\ {{u{\left( 0 \right)} - u{\left( 1 \right)} = 0,}} & {{{u}\ifmmode{'}\else$'$\fi{\left( 0 \right)} - {u}\ifmmode{'}\else$'$\fi{\left( 1 \right)} = u{\left( {1/2} \right)},}} \\ \end{array} $$ where a : (0, 1) → [0, ∞) is symmetric on (0, 1) and may be singular at t = 0 and t = 1, f : [0, ∞) → [0, ∞) is continuous. By using Krasnoselskii’s fixed point theorem in a cone, we get some existence results of positive solutions for the problem. The associated Green’s function for the three-point boundary value problem is also given.

On the existence of positive solutions of second order differential equations

Using the fixed point method we prove an existence result for positive solutions of nonlinear second order ordinary differential equations. An application to semilinear Schrodinger equations in exterior domains is also presented.