Non-Lipschitz Homogeneous Volterra Integral Equations

Abstract

In this chapter we introduce a class of nonlinear Volterra integral equations (VIEs) which have certain properties that deviate from the standard results in the field of integral equations. Such equations arise from various problems in shock wave propagation with nonlinear flux conditions. The basic equation we will consider is the nonlinear homogeneous Hammerstein–Volterra integral equation of convolution type

$$\displaystyle u(t) = \int _0^t k(t-s) g(u(s))\,\mathrm {d}s. $$

When g(0) = 0, this equation has function u ≡ 0 as a solution (trivial solution). It is interesting to determine whether there exists a nontrivial solution or not. Classical results on integral equations are not to be applied here since most of them fail to assure the existence of other solution than the trivial one. Several characterizations of the existence of nontrivial solutions under different hypothesis on the kernel k and the nonlinearity g will be presented. We will also focus on the uniqueness of nontrivial solutions for such equations. In this regard, it is important to note that this equation can be written as a fixed point equation, so we shall also discuss the attracting character of the solutions with respect to the Picard iterations of the nonlinear integral operator defined by the RHS of the equation. Indeed we will give some examples for which those iterations do not converge to the nontrivial solutions for some initial conditions and we will study the attraction basins for such repelling solutions. Numerical estimation of the solutions is also discussed. Collocation methods have proven to be a suitable technique for such equations. However, classical results on numerical analysis of existence and convergence of collocation solutions cannot be considered here either since the non-Lipschitz character of the nonlinear operator prevents these results from being applied. New concepts on collocation solutions are introduced along with their corresponding results on existence and uniqueness of collocation solutions.