On the solvability of distributional and impulsive systems of Cauchy problems (Q435881)

This article discusses the solvability of the initial value problem for a system of nonlinear distributional differential equations of the form NEWLINE\[NEWLINEy_i'=f_i(y_1,\dots,y_m),\;\;\;y_i(a)=c_i,\;\;i\in\{1,\dots,m\}\eqno(1)NEWLINE\]NEWLINE on a real interval \([a,b]\). The components \(y_1,\dots,y_m\) are assumed to be in the space \({\mathcal B}[a,b]\) of functions which are bounded on \([a,b]\), left-continuous on \((a,b]\) and right-continuous at \(a\). Their derivatives (and also the values of \(f_i\)) are distributions on \([a,b]\).NEWLINENEWLINELet \(g\) be a distribution on \([a,b]\). If there exists a function \(G\in {\mathcal B}[a,b]\) whose distributional derivative is \(g\), the author defines the left-continuous primitive integral of \(g\) as \(\int_s^t g=G(t)-G(s)\) whenever \([s,t]\subseteq [a,b]\). Then, the system (1) can be rewritten in the integral form NEWLINE\[NEWLINEy_i(t)=c_i+\int_a^t f_i(y_1,\dots,y_m),\;\;\;t\in[a,b],\;\;i\in\{1,\dots,m\}.NEWLINE\]NEWLINE Using suitable fixed point theorems, the author is able to obtain sufficient conditions for the uniqueness and existence of solutions of (1). Other topics discussed in the paper include continuous dependence of solutions on the initial values, existence of the smallest and greatest solution of (1) which lie between a given subsolution and a given supersolution, and the existence of minimal and maximal solutions of the system NEWLINE\[NEWLINEy_i'=f_i(y_1,\dots,y_m),\;\;\;y_i(a)=0,\;\;i\in\{1,\dots,m\}NEWLINE\]NEWLINE on \([a,b]\).NEWLINENEWLINEAs an important special case, the author considers the system of impulsive distributional equations NEWLINE\[NEWLINEy_i'=\sum_{\lambda\in\Lambda_i}I_i(\lambda,y_1,\dots,y_m)\delta_\lambda+g_i(y_1,\dots,y_m),\;\;\;y_i(a)=c_i,\;\;i\in\{1,\dots,m\},NEWLINE\]NEWLINE where \(\delta_\lambda\) is the Dirac distribution at \(\lambda\in(a,b)\), and \(\Lambda_1,\dots,\Lambda_m\) are well-ordered subsets of \((a,b)\).NEWLINENEWLINEThe final section discusses higher-order distributional equations of the form NEWLINE\[NEWLINEy^{(m)}=g(y,y',\dots,y^{(m-1)}).NEWLINE\]NEWLINE Under the assumption that \(y^{(m-1)}\in{\mathcal B}[a,b]\), this equation can be transformed to the form (1). Again, an interesting special case is the impulsive distributional equation NEWLINE\[NEWLINEy^{(m)}(t)=\sum_{\lambda\in\Lambda} I(\lambda,y,y',\dots,y^{(m-1)})\delta_\lambda(t)+p(t,y,y',\dots,y^{(m-1)}).NEWLINE\]NEWLINE Throughout the whole paper, the theory is well illustrated by a number of examples.