Fibre Picard operators on gauge spaces and applications (Q957468)

This article deals with Picard and weakly Picard operators operators in a \(L\)-spaces (spaces with a convergence). A Picard operator in the space \(X\) is an arbitrary operator with the following two properties: (i) \(\text{Fix} \, T = \{x^*\}\) (i.e., \(T\) has a unique fixed point), (ii) \(T^n \to x^*\) as \(n \to \infty\) for all \(x \in X\). A weakly Picard operator in the space \(X\) is an operator so that \(T^nx\), for each \(x \in X\), converges to a fixed point of \(T\). The main result of this article is concerned with operators of type \(A(x,y) = (B(x),C(x,y))\), where \(x \in X\), \(y \in Y\), \(X\) is an \(L\)-space, \(Y\) is a sequentially complete gauge space with \(K\)-metric \(d_i(x,y)\), \(i \in I\). It is assumed that \(B\) is a Picard operator with a fixed point \(x^*\), \(C\) is an operator satisfying the condition \[ d_i(C(x,y_1),C(x,y_2)) \leq \alpha_i d_i(y_1,y_2), \qquad i \in I, \] with \(\alpha_i < 1\), for all \(x \in X\), \(y_1, y_2 \in Y\), and continuous with respect to \(x\). It is proved that \(A\) under these assumptions is also Picard. A similar statement for weakly Picard operators is also proved. Generalizations are possible with the condition \[ d_i(C(x,y_1),C(x,y_2)) \leq \varphi_i (d_i(y_1,y_2)), \qquad i \in I, \] with \(\varphi_i: \;{\mathbb R}_+ \to {\mathbb R}_+\), \(i \in I\), satisfying the following conditions: (i) \(\varphi\) is monotone increasing, (ii) there exist numbers \(k_0\), \(\alpha\), \(0 < \alpha < 1\), and a convergent series of nonnegative terms \(v_k\) such that \(\varphi^{k+1}(t) \leq \alpha \varphi^k(t) + v_k\) for each \(t > 0\) and each \(k \geq k_0\). In the end of the article, the functional-integral equations \[ x(t) = \int_{-t}^t K(t,s,x(s),x(h(s))) \, ds + g(t), \] \[ x(t,\lambda) = \int_{-t}^t K(t,s,x(s,\lambda),x(h(s),\lambda),\lambda) \, ds + g(t,\lambda) \] are considered.