We study a nonlocal boundary value problem for the equation x''(t) + f(t,x(t),x'(t)) = 0, t ∈ [0,1]. By applying fixed point theorems on appropriate cones, we prove that this boundary value problem admits positive solutions with slope in a given annulus. It is remarkable that we do not assume f≥0. Here the sign of the function f may change.
We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.
Abstract. Applying the topological transversality method of Granas and the a priori bounds technique, we prove some existence theorems for diflerential inclusions of the form x" ∈ F(t, x, x'), x ∈ ℬ, where F is a Carathéodory multifunction with convex, compact values. No growth condition will be imposed on F.
The existence and multiplicity results are shown for certain types of problems with nonlinear boundary value conditions.
A differential equation of the form (q(t)k(u)u')' = F(t,u)u' is considered and solutions u with u(0) = 0 are studied on the halfline [0,∞). Theorems about the existence, uniqueness, boundedness and dependence of solutions on a parameter are given.
A nonlinear differential equation of the form (q(x)k(x)u')' = F(x,u,u') arising in models of infiltration of water is considered, together with the corresponding differential equation with a positive parameter λ, (q(x)k(x)u')' = λF(x,u,u'). The theorems about existence, uniqueness, boundedness of solution and its dependence on the parameter are established.
The problem of existence and asymptotic behavior of solutions of the quasilinear and quadratic singularly perturbed periodic boundary value problem as a small parameter at highest derivative tends to zero is studied.
The purpose of this note is to study existence of weak solutions for the quasilinear elliptic problem with Dirichlet boundary conditions.
We consider a quasilinear vector differential equation which involves the p-Laplacian and a maximal monotone map. The boundary conditions are nonlinear and are determined by a generally multivalued, maximal monotone map. We prove two existence theorems. The first assumes that the maximal monotone map involved is everywhere defined and in the second we drop this requirement at the expense of strengthening the growth hypothesis on the vector field. The proofs are based on the theory of operators of...
The method of quasilinearization for a periodic boundary value problem for nonlinear hybrid differential equations is studied. It is shown that the convergence is quadratic.