For a locally convex space with the topology given by a family {p(┬; α)} α ∈ ω of seminorms, we study the existence and uniqueness of fixed points for a mapping defined on some set. We require that there exists a linear and positive operatorK, acting on functions defined on the index set Ω, such that for everyu,

Under some additional assumptions, one of which is the existence of a fixed point for the operator, we prove that there exists a fixed point of. For a class of elements satisfyingK ⁿ(p)u;┬))(α) → 0 asn → ∞, we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms.

中文翻译：

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局部凸空间中的不动点定理

对于局部凸空间，其拓扑由半范式{ p（┬;α）}α∈ω给出，我们研究了在某些集合上定义的映射的不动点的存在性和唯一性。我们要求存在一个线性和正算子K，作用于在索引集Ω上定义的函数，使得对于每个u，

在某些附加假设下（其中之一是运算符存在不动点），我们证明存在的不动点。对于满足n →∞的满足K ⁿ（p）u ;┬））（α）→0的一类元素，我们证明了不动点是唯一的。该类尤其包括证明不动点存在的类。我们通过证明Banach空间中一阶和二阶非线性微分方程解的存在性和唯一性来考虑几种应用。我们还考虑了带有非线性项的伪微分方程。