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).     

TWO POSITIVE SOLUTIONS TO THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: -u′′(t) = λf(t, u(t)) for all t ∈ (0, 1) subjecting to u(0) = 0 and αu(η) = u(1), where η∈ (0, 1), α∈ [0, 1), and λ is a positive parameter. The nonlinear term f(t, u) is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ* ∈ (0, +∞], such that the above problem has at least two positive solutions for any λ∈ (0, λ*) under certain conditions on the nonlinear term f.     

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.     

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‖=+∞.     

Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian

In this paper we consider the multiplicity of positive solutions for the one-dimensional p-Laplacian differential equation (?p^′(u^′))+q(t)f(t,u,u^′)=0, t∈(0,1), subject to some boundary conditions. By means of a fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to some multipoint boundary value problems.     

Existence of three positive solutions for boundary value problems with p-Laplacian

By using fixed point theorem, we study the following equation g(u^′′(t))+a(t)f(u)=0 subject to boundary conditions, where g(v)=|v|^p−2v with p>1; the existence of at least three positive solutions is proved.     

Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations

This paper studies the existence of a positive solution to the second-order periodic boundary value problem

Exact number of positive solutions for semipositone problems

This paper is concerned with the exact number of positive solutions for the boundary value problem ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf^′(u) change sign exactly once from negative to positive on (0,∞).     

Infinitely many mountain pass solutions on a kind of fourth-order Neumann boundary value problem

In this paper, the existence of infinitely many mountain pass solutions are obtained for the fourth-order boundary value problem (BVP) u⁽⁴⁾(t)-2u^″(t)+u(t)=f(u(t)),0<t<1, u^′(0)=u^′(1)=u^?(0)=u^?(1)=0, where f:R→R is continuous. The study of the problem is based on the variational methods and critical point theory. We prove the conclusion by using sub-sup solution method, Mountain Pass Theorem in Order Intervals, Leray-Schauder degree theory and Morse theory.     

Triple Solutions for Second-Order Three-Point Boundary Value Problems

We establish the existence of at least three positive solutions to the second-order three-point boundary value problem, u″ + f(t, u) = 0, u(0) = 0, αu(η) = u(1), where η: 0 lt; η < 1, 0 < α < 1/η, and f: [0, 1] × [0, ∞) → [0, ∞) is continuous. We accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works.     

Minimal Positive Solutions to Some Singular Second-Order Differential Equations

The existence of a minimal C¹[0, 1] positive solution is established for some second-order singular boundary value and initial value problems by new schemes, which are related to x′. Our nonlinearity may be singular at t = 0, 1, x = 0, or x′ = 0.     

Positive solutions for the one-dimensional p-Laplacian

In this paper, the criterion for the existence of at least one positive solution of the one-dimensional p-Laplacian (b(t)Φ(u)′ + c(t)f(u) = 0, are obtained, where Φ(u) = |u|^p−1u, p > 0 is a constant, and b(t) > 0 for t > 0. The method used in this paper is shooting method.     

A finite difference solution to a two-dimensional parabolic inverse problem

In this paper, we consider the problem of finding u = u(x, y, t) and p = p(t) which satisfy u_t = u_xx + u_yy + p(t)u + ? in R × [0, T], u(x, y, 0) = f(x, y), (x, y) ∈ R = [0, 1] × [0, 1], u is known on the boundary of R and u(x^∗, y^∗, t) = E(t), 0 < t ? T, where E(t) is known and (x^∗, y^∗) is a given point of R. Through a function transformation, the nonlinear two-dimensional diffusion problem is transformed into a linear problem, and a backward Euler scheme is constructed. It is proved by the maximum principle that the scheme is uniquely solvable, unconditionally stable and convergent in L_∞ norm. The convergence orders of u and p are of O(τ + h²). The impact of initial data errors on the numerical solution is also considered. Numerical experiments are presented to illustrate the validity of the theoretical results.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

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.     

Positive solutions for one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative

This paper presents sufficient conditions for the existence and multiplicity of positive solutions to the one-dimensional p-Laplacian differential equation (?p^′(u^′))+a(t)f(u,u^′)=0, subject to some boundary conditions. We show that it has at least one or two or three positive solutions under some assumptions by applying the fixed point theorem.     

Analysis of a semigroup approach in the inverse problem of identifying an unknown parameters

This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation u_t(x, t)  = u_xx + qu_x(x, t) + p(t)u(x, t), with Dirichlet boundary conditions u(0, t) = ψ₀, u(1, t) = ψ₁. The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[·]:P→H^1,2[0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only zero function, then the input-output mapping Φ[·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data u_x(0, t) = f(t) the unknown parameter p(t) at (x, t) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[·]:P→H^1,2[0,T] is given explicitly interms of the semigroup.     

Automatic Controls for Nonlinear Dynamical Systems with Lipschitzian Trajectories

We would like to investigate on the solution to the automatic control problem given by the differential equation y′(t) = f(t, y(t), w(t)) for a given initial function x in the initial domain D(x, ω, Y) for almost all t in the interval I, with controls given by w(t) = g(t, y(t), T(y)(t)), where T is a nonanticipating and Lipschitzian operator. The result will be generalized for a dynamical system y′(t) = f(t, y(t), T(y), u(t)).     

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.     

Exact multiplicity results for a class of two-point boundary value problems

This paper is concerned with the exact number of positive solutions for boundary value problems ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which the nonlinearity f is positive on (0,∞) and (p−1)f(u)−uf^′(u) changes sign from negative to positive.