Utilizing the theory of fixed point index for compact maps, we establish new results on the existence of positive solutions for a certain third order boundary value problem. The boundary conditions that we study are of nonlocal type, involve Stieltjes integrals and are allowed to be nonlinear.
The paper deals with the existence of multiple positive solutions for the boundary value problem (ϕ(p(t)u (n−1))(t))′ + a(t)f(t, u(t), u′ (t), . . . , u(n−2)(t)) = 0, 0 < t < 1, u (i) (0) = 0, i = 0, 1, . . . , n − 3, u (n−2)(0) = mP−2 i=1 αiu (n−2)(ξi), u(n−1)(1) = 0, where ϕ: R → R is an increasing homeomorphism and a positive homomorphism with ϕ(0) = 0. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.
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.
This paper shows that some characterizations of the harmonic majorization of the Martin function for domains having smooth boundaries also hold for cones.
This paper deals with the existence of positive ω-periodic solutions for the neutral functional differential equation with multiple delays (u(t) − cu(t − δ))′ + a(t)u(t) = f(t, u(t − τ1), . . . , u(t − τn)). The essential inequality conditions on the existence of positive periodic solutions are obtained. These inequality conditions concern with the relations of c and the coefficient function a(t), and the nonlinearity f(t, x1, . . . , xn). Our discussion is based on the perturbation method of positive operator and fixed point index theory in cones.
The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition u ′′ + g(t)f(t, u) = 0, t ∈ (0, 1), u(0) = αu(ξ) + λ, u(1) = βu(η) + µ. Criteria for the existence of nontrivial solutions of the problem are established. The nonlinear term f(t, x) may take negative values and may be unbounded from below. Conditions are determined by the relationship between the behavior of f(t, x)/x for x near 0 and ±∞, and the smallest positive characteristic value of an associated linear integral operator. The analysis mainly relies on topological degree theory. This work complements some recent results in the literature. The results are illustrated with examples.
We establish several inequalities for the spectral radius of a positive commutator of positive operators in a Banach space ordered by a normal and generating cone. The main purpose of this paper is to show that in order to prove the quasi-nilpotency of the commutator we do not have to impose any compactness condition on the operators under consideration. In this way we give a partial answer to the open problem posed in the paper by J. Bračič, R. Drnovšek, Y. B. Farforovskaya, E. L. Rabkin, J. Zemánek (2010). Inequalities involving an arbitrary commutator and a generalized commutator are also discussed.