We obtain radially symmetric solutions of some nonlinear (geometric) partial differential equations via a rigorous computer-assisted method. We introduce all main ideas through examples, accessible to non-experts. The proofs are obtained by solving for the coefficients of the Taylor series of the solutions in a Banach space of geometrically decaying sequences. The tool that allows us to advance from numerical simulations to mathematical proofs is the Banach contraction theorem.

We consider a differential equation with a state-dependent delay motivated by a queueing process. The time delay is determined by an algebraic equation involving the length of the queue for which a discontinuous differential equation holds. The new type of state-dependent delay raises some problems that are studied in this paper. We formulate an appropriate framework to handle the system, and show that the solutions define a Lipschitz continuous semiflow in the phase space. The second main result guarantees the existence of slowly oscillating periodic solutions.

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Consider the delay differential equationẋ(t) = a r 0 x(t − s) dη(s) − g(x(t)) and the neutral type differential equationẏ(t) = a r 0ẏ (t − s) dµ(s) − g(y(t)) where a > 0, g : R → R is smooth, ug(u) > 0 for u = 0, s 0 g(u) du → ∞ as |s| → ∞, r > 0, η and µ are nonnegative functions of bounded variation on [0, r], η(0) = η(r) = 0, r 0 η(s) ds = 1, µ is nondecreasing, µ does not have a singular part, r 0 dµ = 1. Both equations can be interpreted as price models. Global asymptotic stability of y = 0 is obtained, in case a ∈ (0, 1), for the neutral equation by using a Lyapunov functional. Then this result is applied to get global asymptotic stability of x = 0 for the (non-neutral) delay differential equation provided a ∈ (0, 1). As particular cases, two related global stability conjectures are solved, with an affirmative answer.

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