Existence Theorems Of Positive Solutions Research Articles

Multiplicity of positive solutions for a nonlinear m-point BVP with sign changing nonlinearity on time scales

In this paper, by using fixed point theorems in cones, the existence of positive solutions is considered for a nonlinear m-point boundary value problem for the following second-order dynamic equation on time scales $$\begin{array}{l}(\phi(u^{\Delta}))^{\nabla}+a(t)f(t,u(t))=0,\quad t\in(0,T),\\\noalign{\vspace{2mm}}\displaystyle u(0)=\sum_{i=1}^{m-2}a_{i}u(\xi_{i}),\qquad \phi(u^{\Delta}(T))=\sum_{i=1}^{m-2}b_{i}\phi(u^{\Delta}(\xi_{i})),\end{array}$$ where φ:R⟶R is an increasing homeomorphism and positive homomorphism with φ(0)=0. The nonlinear term f may change sign. We obtain several existence theorems of positive solutions for the above boundary value problems. We should point out that the above equation we studied is same as that in Han and Jin (Communications in Nonlinear Science and Numerical Simulation, 2009), but the methods is different from Han and Jin (Communications in Nonlinear Science and Numerical Simulation, 2009), we generalize and improve the results (Han and Jin, Communications in Nonlinear Science and Numerical Simulation, 2009). As an application, a typical example to demonstrate our results is given.

Existence of positive solutions of nonlinear [formula omitted]-point BVP for an increasing homeomorphism and positive homomorphism on time scales

In this paper, we will consider the following multipoint boundary value problem for the following second-order dynamic equations on time scales ( ϕ ( u Δ ) ) ∇ + a ( t ) f ( t , u ( t ) ) = 0 , t ∈ ( 0 , T ) , u ( 0 ) = ∑ i = 1 m − 2 a i u ( ξ i ) , ϕ ( u Δ ( T ) ) = ∑ i = 1 m − 2 b i ϕ ( u Δ ( ξ i ) ) , where ϕ : R ⟶ R is an increasing homeomorphism and positive homomorphism and ϕ ( 0 ) = 0 . By using fixed point theorems, we obtain an existence theorem of positive solutions for the above boundary value problem, which includes and improve some related results in the relevant literature. As an application, an example to demonstrate our results is given.

Positive solutions of nonlinear m-point BVP for an increasing homeomorphism and homomorphism with sign changing nonlinearity on time scales

In this paper, several existence theorems of positive solutions are established for nonlinear m -point boundary value problem for the following dynamic equations on time scales ( ϕ ( u Δ ) ) ∇ + a ( t ) f ( t , u ( t ) ) = 0 , t ∈ [ 0 , T ] , ϕ ( u Δ ( 0 ) ) = ∑ i = 1 m − 2 a i ϕ ( u Δ ( ξ i ) ) , u ( T ) = ∑ i = 1 m − 2 b i u ( ξ i ) , where ϕ : R ⟶ R is an increasing homeomorphism and homomorphism and ϕ ( 0 ) = 0 . The nonlinear term f may change sign. As an application, an example to demonstrate our results is given.

Several Existence Theorems of Nonlinearm-Point BVP for an Increasing Homeomorphism and Homomorphism on Time Scales

Several existence theorems of positive solutions are established for nonlinearm-point boundary value problem for the following dynamic equations on time scales(ϕ(uΔ))∇+a(t)f(t,u(t))=0,t∈(0,T),ϕ(uΔ(0))=∑i=1m−2aiϕ(uΔ(ξi)),u(T)=∑i=1m−2biu(ξi), whereϕ:R→Ris an increasing homeomorphism and homomorphism andϕ(0)=0. As an application, an example to demonstrate our results is given.

Several existence theorems of nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales

In this paper, several existence theorems of positive solutions are established for nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, as an application, an example to demonstrate our results is given. The conditions we used in the paper are different from those in [H.R. Sun, W.T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524; H.R. Sun, W.T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica 49 (2006) 369–380 (in Chinese); Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math. Anal. Appl. 315 (2006) 144–153; Y. Wang, W. Ge, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian, Nonlinear Appl. 66 (6) (2007) 1246–1256].

On existence theorems of positive solutions to nonlinear singular differential equations

Suppose (i) f ( t , u ) : ( 0 , 1 ) × ( 0 , + ∞ ) → [ 0 , + ∞ ) is continuous and is increasing on u; (ii) there exists a function g : [ 1 , ∞ ) → ( 0 , + ∞ ) , g ( b ) < b and g ( b ) / b 2 is integrable on ( 1 , + ∞ ) such that f ( t , bu ) ⩽ g ( b ) f ( t , u ) , ∀ ( t , u ) ∈ ( 0 , 1 ) × ( 0 , + ∞ ) . Consider the singular boundary value problem (★) u ″ ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = au ( η ) , u ( 1 ) = 0 . Then a necessary and sufficient condition for Eq. (★) to have C [ 0 , 1 ] positive solutions is that 0 < ∫ 0 1 s ( 1 - s ) f ( s , 1 ) d s < ∞ , a necessary and sufficient condition for Eq. (★) to have C 1 [ 0 , 1 ] positive solutions is that 0 < ∫ 0 1 f ( s , 1 - s ) d s < ∞ . Also, the uniqueness, iterative methods and computational methods of C 1 [ 0 , 1 ] positive solutions have been studied. Our nonlinearity may be singular at t = 0 and/or t = 1 .

Existence of positive solutions for singular boundary value problem on time scales

This paper deals with a class of boundary value problem of singular differential equations on time scales. The conditions we used here differ from those in the majority of papers as we know. An existence theorem of positive solutions is established by using the Krasnosel'skii fixed point theorem and an example is given to illustrate it.

New existence theorems of positive solutions for singular boundary value problems

In this paper, some nonexistence, existence and multiplicity of positive solutions are established for a class of singular boundary value problem. The authors also obtain the relation between the existence of the solutions and the parameter λ. The arguments are based upon the fixed point index theory and the upper and lower solutions method.

Singular boundary value problem on infinite time scale

This paper deals with a class of singular boundary value problems of differential equations on infinite time scale. An existence theorem of positive solutions is established by using the Schauder fixed point theorem and perturbation and operator approximation method, which resolves the singularity successfully and differs from those of some papers. In the end of the paper, an example is given to illustrate our main result.

Positive Solutions of Quasilinear Elliptic Equations

This paper is concerned with existence theorems for positive solutions of the Dirichlet problem for quasilinear elliptic differential equation containing a gradient term. Using the shooting method and the a priori estimates for the first zero, we obtain sufficient conditions for the existence of classical positive solutions of the problem in the ball.

Positive solution for three-point boundary value problems with sign changing nonlinearities

By constructing available operators, some new existence theorems of positive solutions are obtained for a class of three-point boundary value problems u″ + λh(t)f(t, u) = 0, 0 ≤ t ≤ 1, u(0) − βu′(0) = 0, u(1) = αu(η), where f is allowed to change sign, η ϵ (0, 1). The associated Green's function for the above problem is also given.

Abstract existence theorems of positive solutions for nonlinear boundary value problems

Let Ω⊂ R n be a bounded domain, H=L 2(Ω) , L: D( L)⊂ H→ H be a unbounded linear operator, and f: Ω ̄ × R→ R + be a continuous function. This paper deals with the existence of positive solutions of the operator equation Lu=f(x,u), u∈D(L), which is the abstract form of differential equations boundary value problems. The essential conditions on L and f guaranteeing the existence of positive solution are characterized. The discussion is based on the Krasnoselskii's fixed point theorem of cone mapping and the fixed point index theory in cones. The abstract results are applied to some differential equations boundary value problems, and some known results are extended and improved.

New existence theorems of positive solutions for Sturm–Liouville boundary value problems

By constructing available operators some new existence theorems of positive solutions are obtained for the Sturm–Liouville boundary value problem (p(t)u ′(t)) ′+λa(t)f(t,u(t))=0, α 1u(0)−β 1p(0)u ′(0)=0, α 2u(1)+β 2p(1)u ′(1)=0, where a( t)⩾0 and λ>0 is a parameter.

An existence theorem of positive solutions for elastic beam equation with both fixed end-points

By using the degree theory on cone an existence theorem of positive solution for a class of fourth-order two-point BVP’s is obtained. This class of BVP’s usually describes the deformation of the elastic beam with both fixed end-points.

Positive solutions of elliptic equations with discontinuous nonlinearities

In this paper an existence theorem of positive solutions to the Dirichlet problem for elliptic equations having nonlinear terms with an uncountable set of discontinuities is established. Some applications to special cases, such as problems with critical Sobolev growth, are also presented. The approach taken is strictly based on set-valued analysis and fixed point arguments.

Positive solutions to nonlinear singular second order boundary value problems

Existence theorems of positive solutions to a class of singular second order boundary value problems of the form y'' + f(x,y,y') = 0, 0 < x < 1, are established. It is not required that the function (x,y,z) → f(x,y,z) be nonincreasing in y and/or z, as is

Positive Solutions of Nonlinear Second Order Ordinary Differential Equations

We prove an existence theorem for positive solutions of nonlinear second order ordinary differential equations by using the perturbation method and the Schauder fixed point theorem.

Positive solutions of nonlinear second order ordinary differential equations

We prove an existence theorem for positive solutions of nonlinear second order ordinary differential equations by using the perturbation method and the Schauder fixed point theorem.

Impulsive integral equations in Banach spaces and applications

In this paper, we first use the fixed point theory to prove two existence theorems of positive solutions for the impulsive Fredholm integral Equations in Banach spaces. And then, we offer some applications to the two‐point boundary value problems for the second order impulsive differential equations in Banach spaces.

Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions

SynopsisIn this paper we characterise the levels of the functional (0.3) at which the Palais-Smale condition fails in the Sobolev space V(Ω) defined below. From this result we deduce an existence theorem for positive solutions to the mixed boundary problem (0.1)–(0.2) under geometrical assumptions on the domain Ω and the part of the boundary of Ω where a Neumann condition is prescribed.