Existence of solutions to a class of nonlinear second order two-point boundary value problems

In this paper, the existence and multiplicity results of solutions are obtained for the second order two-point boundary value problem −u^″(t)=f(t,u(t)) for all t∈[0,1] subject to u(0)=u^′(1)=0, where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Positive and sign-changing solutions for fourth-order BVPs with parameters

This paper is concerned with the existence and multiplicity of positive and sign-changing solutions of the fourth-order boundary value problem u ⁽⁴⁾(t)=λ f(t,u(t),u ^′′(t)), 0<t<1,?u(0)?=?u(1)=u ^′′(0)=u ^′′(1)?=0, where f:[0,1]×?→? is continuous, λ∈? is a parameter. By using the fixed-point index theory of differential operators, it is proved that the above boundary value problem has positive, negative and sign-changing solutions for λ being different intervals. As an example, the boundary value problem u ⁽⁴⁾(t)+?η u ^′′(t)??ζu(t)=?λ f(t,u(t)), ?0<t<1,?u(0)=?u(1)=?u ^′′(0)=?u ^′′(1)=0 is also considered and some obtained results are the complement of the known results.     

Sign-changing solutions on a kind of fourth-order Neumann boundary value problem

In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u^″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u^′(0)=u^′(1)=u^?(0)=u^?(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions.     

Triple positive solutions for a class of third-order p-Laplacian singular boundary value problems

In this work, we study the existence of triple positive solutions for one-dimensional p-Laplacian singular boundary value problems $$\begin{array}{l}(\phi_p(y''(t)))'+f(t)g(t,\,y(t),\,y'(t),\,y''(t))=0,\quad 0<t<1,\\[3pt]ay(0)-by'(0)=0,\qquad cy(1)+dy'(1)=0,\qquad y''(0)=0,\end{array}$$ where φ _p (s)=|s| ^p?2 s,?p>1, g:[0,?1]×[0,?+∞)×R ²?[0,?+∞) and f:(0,?1)?[0,?+∞) are continuous. The nonlinear term f may be singular at t=0 and/or t=1. Firstly, Green’s function for the associated linear boundary value problem is constructed. Then, by making use of a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of triple positive solutions to the above boundary value problem. The interesting point is that the nonlinear term g involved with the first-order and second-order derivatives explicitly.     

Existence of positive solutions of Neumann boundary value problem via a cone compression-expansion fixed point theorem of functional type

This paper is devoted to study the existence of positive solutions of second-order boundary value problem $$-u''+m^2u=h(t)f(t,u),\quad t\in (0,1)$$ with Neumann boundary conditions $$u'(0)=u'(1)=0,$$ where m>0, f∈C([0,1]×?⁺,?⁺), and h(t) is allowed to be singular at t=0 and t=1. The arguments are based only upon the positivity of the Green function, a fixed point theorem of cone expansion and compression of functional type, and growth conditions on the nonlinearity f.     

Positive solutions of the nonlinear fourth-order beam equation with three parameters

This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation u(4)(t)+ηu^″(t)−ζu(t)=λf(t,u(t)), 0<t<1, u(0)=u(1)=u^″(0)=u^″(1)=0, where is continuous and ζ, η and λ are parameters. We show that there exists a such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0<λ<λ^*, λ=λ^* and λ>λ^*, respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution uλ(t) of the BVP depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ(t)‖=0 and limλ→+∞‖uλ(t)‖=+∞ for any t∈[0,1].     

Existence and uniqueness of positive solutions for some singular boundary value problems with linear functional boundary conditions

By using the upper and lower solution method and fixed point theory, we investigate some nonlinear singular second-order differential equations with linear functional boundary conditions. The nonlinear term f(t, u) is nonincreasing with respect to u, and only possesses some integrability. We obtain the existence and uniqueness of the C[0,1] positive solutions as well as the C ¹[0, 1] positive solutions.     

非线性四阶周期边值问题的最优正解

该文使用锥不动点定理研究了四阶周期边值问题u⁽⁴)-m⁴u+F(t, u(τ(t)))=0, 0 < t < 2π, u⁽ⁱ⁾(0)=u⁽ⁱ⁾(2π),~ i=0,1, 2, 3, 这里 F: [0,2π ]×R⁺→R⁺ 和τ: [0, 2π]→[0, 2π] 是连续的, 0-7.     

Existence results for second order boundary value problems using the barrier strip technique

Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem ${x''=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,}Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem x"=f(t,x),  t ? (0,1),  x(0)=A,  x￠(1)=0,{x'=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,} where the scalar function f(t, x) may be singular at x = A.     

Positive and dead core solutions of singular BVPs for third-order differential equations

The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u^″))^′=λf(t,u,u^′,u^″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y.     

Eigenvalue problems for second-order nonlinear dynamic equations on time scales

This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation uΔΔ(t)+λa(t)f(u(σ(t)))=0, t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive. We show that there exists a λ^∗>0 such that the above boundary value problem has at least two, one and no positive solutions for 0<λ<λ^∗, λ=λ^∗ and λ>λ^∗, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ‖=0 and limλ→+∞‖uλ‖=+∞.     

Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.     

Existence of three positive pseudo-symmetric solutions for a one dimensional p-Laplacian

We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the three point boundary value problem, (g(u′))′+a(t)f(u)=0, u(0)=0, and u(ν)=u(1), where g(v)=|v|^p−2v, with p>1 and ν∈(0,1).     

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

We consider the classical nonlinear fourth-order two-point boundary value problem . In this problem, the nonlinear term h(t)f(t, u(t), u′(t), u″(t)) contains the first and second derivatives of the unknown function, and the function h(t)f(t, x, y, z) may be singular at t = 0, t = 1 and at x = 0, y = 0, z = 0. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.     

一类奇异非线性三点边值问题的正解

应用锥上的不动点定理，建立了奇异非线性三点边值问题(u″(t)+a(t)f(u)=0,0＜t＜1,αu(0)-βu′(0)=0,u(1)-ku(η)=0)正解的一个存在性定理．这里η∈(0,1)是一个常数，a∈C( (0,1),［0,+∞)),f∈C(［0,+∞),［0,+∞))     

Nonexistence and existence of positive solutions for?second order singular three-point boundary value problems with derivative dependent and sign-changing nonlinearities

This paper discusses the nonexistence and the existence of positive solutions for second order singular three-point boundary value problems when the nonlinear term f(t,x,y) is sign-changing and may be singular at t=0, x=0, y=0.     

Positive solution of singular Dirichlet boundary value problems for second order differential equation system

This paper investigates the existence of positive solutions of singular Dirichlet boundary value problems for second order differential system. A necessary and sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C¹[0,1]×C¹[0,1] positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems. Our nonlinearity fi(t,x₁,x₂) may be singular at x₁=0, x₂=0, t=0 and/or t=1, i=1,2.     

Existence and uniqueness of entire solutions for a reaction-diffusion equation

We consider entire solutions of u_t=u_xx-f(u), i.e. solutions that exist for all (x,t)∈R², where f(0)=f(1)=0<f^′(0). In particular, we are interested in the entire solutions which behave as two opposite wave fronts of positive speed(s) approaching each other from both sides of the x-axis and then annihilating in a finite time. In the case f^′(1)>0, we show that such entire solution exists and is unique up to space-time translations. In the case f^′(1)<0, we derive two families of such entire solutions. In the first family, one cannot be any space-time translation of the other. Yet all entire solutions in the second family only differ by a space-time translation.     

Eigenvalues and the One-Dimensional p-Laplacian

We consider the boundary value problem (?_p(u′))′ + λF(t, u) = 0, with p > 1, t ∈ (0, 1), u(0) = u(1) = 0, and with λ > 0. The value of λ is chosen so that the boundary value problem has a positive solution. In addition, we derive an explicit interval for λ such that, for any λ in this interval, the existence of a positive solution to the boundary value problem is guaranteed. In addition, the existence of two positive solutions for λ in an appropriate interval is also discussed.