Article
Solvability of a secondorder multipoint boundary value problem at resonance
School of Mathematical Sciences, Xuzhou Normal University, Xuzhou, Jiangsu 221116, People’s Republic of China
Applied Mathematics and Computation (Impact Factor: 1.55). 02/2009; 208(1):2330. DOI: 10.1016/j.amc.2008.11.026 Source: DBLP
ABSTRACT
Based on the coincidence degree theory of Mawhin, we get a general existence result for the following secondorder multipoint boundary value problem at resonancex″(t)=f(t,x(t),x′(t))+e(t),t∈(0,1),x(0)=∑i=1mαix(ξi),x′(1)=∑j=1nβjx′(ηj),where f:[0,1]×R2→Rf:[0,1]×R2→R is a Carathéodory function, e∈L1[0,1],0<ξ1<ξ2<⋯<ξm<1,αi∈R,i=1,2,…,m,m⩾2e∈L1[0,1],0<ξ1<ξ2<⋯<ξm<1,αi∈R,i=1,2,…,m,m⩾2 and 0<η1<⋯<ηn<1,βj∈R,j=1,…,n,n⩾10<η1<⋯<ηn<1,βj∈R,j=1,…,n,n⩾1. In this paper, both of the boundary value conditions are responsible for resonance.

 "When L is linear, as is known, the coincidence degree theory of Mawhin[19]has played an important role in dealing with the existence of solutions for these problems. For more recent results, we refer the reader to[3,5,6,6,8,9,14,20,22,24,25]and the references therein. Moreover boundary value problems on the half line arise in many applications in physics such that in modeling the unsteady flow of a gas through semiinfinite porous media, in plasma physics, in determining the electrical potential in an isolated neutral atom, or in combustion theory. "
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ABSTRACT: The aim of this paper is the solvability of a class of higher order differential equations with initial conditions and an integral boundary condition on the half line. Using coincidence degree theory by Mawhin and constructing suitable operators, we prove the existence of solutions for the posed resonance boundary value problems.  [Show abstract] [Hide abstract]
ABSTRACT: This paper deals with the existence of solutions for the following nth order multipoint boundary value problem at resonance x((n))(t) = f (t, x(t), x'(t), x((n1)) (t)) + e (t), t is an element of (0, 1), x(0) = Sigma(i=1)(m2) alpha(i)x(xi(i)), x'(0) = (...) = x((n2)) (0) = 0, x(1) = x(eta), where f : [0, 1] x Rn > R is a continuous function, e is an element of L1[0, 1], alpha(i) is an element of R (1 less than or equal to i less than or equal to m2), 0 < xi(1) < xi(2) < (...) < xi(m) < 1 and 0 < eta < 1. An existence theorem is obtained by using the coincidence degree theory of Mawhin.  [Show abstract] [Hide abstract]
ABSTRACT: The structure of eigenvalues of −y″+q(x)y=λy, y(0)=0, and y(1)=∑k=1mαky(ηk), will be studied, where q∈L1([0,1],ℝ), α=(αk)∈ℝm, and 0<η1<⋯<ηm<1. Due to the nonsymmetry of the problem, this equation may admit complex eigenvalues. In this paper, a complete structure of all complex eigenvalues of this equation will be obtained. In particular, it is proved that this equation has always a sequence of real eigenvalues tending to +∞. Moreover, there exists some constant Aq>0 depending on q, such that when α satisfies ‖α‖≤Aq, all eigenvalues of this equation are necessarily real.
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