On nonlinear distributional and impulsive Cauchy problems
There are established qualitative results for the unique, smallest, greatest, minimal, and maximal solutions of the nonlinear distributional Cauchy problem \[ y'=f(y),\qquad y(a)=c.\tag{P} \] In the first part of the paper, a conversion of problem (P) into an integral equation is given; some related fixed point results are also presented. The next section is devoted to the study of conditions under which problem (P) is well-posed, that is, it has a unique solution that depends continuously on the initial value \(c\). There are also proved existence and comparison results for the smallest and greatest solutions of the Cauchy problem (P). The case when solutions also monotone is also treated in this paper. Existence results are established for minimal and maximal solutions of the Cauchy problem (P). The main tools in the proofs are fixed point results, which are established by means of ordinary and generalized iteration methods. Concrete examples are solved to illustrate the obtained results. Iteration methods, induction and/or Maple programming are used to determine solutions.
scientific article; zbMATH DE number 5987812
scientific article