A new approach to numerical solution of fixed-point problems and its application to delay differential equations
The article deals with the fixed point problem for an operator \(A\) acting in a metric space \((L,d)\) with an additional approximation structure. The family of spaces \(S_h\) and the families of operators \(r_h:L\to S_h\) and \(p_h:S_h\to L\) define a convergent approximation of \(L\) if for any \(x \in L\) \[ d(p_nr_nx,x)\big|_{h \to 0} \to 0 \] holds. The base of the author's arguments is the following statement (Theorem 2.1): If an operator \(A\) is continuous in \(L\) and the family of spaces \(S_h\) and the families of operators \(r_h\) and \(p_h\) define a convergent approximation of \(L\), then \(A\) possesses at least one fixed point if and only if there exists a non-negative function \(\varepsilon(h)\), \(\varepsilon(h) \to 0\) for \(h \to 0\), such that the operator \(A\) possesses \(\varepsilon(h)\)-fixed-points \(p_hx_h\) and the family \(\{p_hx_h\}\) is compact with \(h \to 0\). Recall that \(x \in L\) is an \(\varepsilon\)-fixed point of \(A\) if \(f(x,Ax)\leq \varepsilon\) and the family \(x^h\in L\) is compact with \(h\to 0\) if for any sequence \(x^{h_n}\) with \(h_n\to 0\) there exists a convergent subsequence of \(x^{h_n}\). As application the author considers the following boundary value problem \[ y''(x)= f(x,y(x),y'(x),y(\alpha(x))), \quad a\leq x\leq b, \] \[ y(x)= \varphi(x), \qquad \alpha_{\min} \leq x \leq a, \qquad y(b) = 0. \] Numerical realization of the last problem for three concrete triples of functions \((f,\alpha,\varphi)\) is also presented.
scientific article; zbMATH DE number 5688910
scientific article