Integral equations, implicit functions, and fixed points

Abstract

Abstract. The problem is to show that (1) V (t, x) = S(t,t 0 H (t, s, x(s)) ds) has a solution, where V defines a contraction,Ṽ , and S defines a compact map,S. A fixed point of P ϕ =Sϕ + (I −Ṽ )ϕ would solve the problem. Such equations arise naturally in the search for a solution of f (t, x) = 0 where f(0, 0) = 0, but ∂f(0, 0)/∂x = 0 so that the standard conditions of the implicit function theorem fail. Now P ϕ =Sϕ + (I −Ṽ )ϕ would be in the form for a classical fixed point theorem of Krasnoselskii if I −Ṽ were a contraction. But I −Ṽ fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that I −Ṽ has enough properties that an extension of Krasnoselskii's theorem still holds and, hence, (1) has a solution. This substantially improves the classical implicit function theorem and proves that a general class of integral equations has a solution.

Nonlinear contractive conditions: A comparison and related problems

“…which was introduced by Matkowski [Ma77] We close our shortened history of studies on nonlinear contractive conditions with recalling the following definition given by Burton [Bu96] in 1996. We say that T is a large…”

Fixed point theorems in ordered 𝐿-spaces

“…Theorem 4.7 is an extension of Perov's fixed point theorem, while Theorem 4.8 generalizes Sehgal and Bharucha-Reid's fixed point principle. It is worth noting that other fixed point results can be established by replacing the contraction condition on f with a generalized contraction assertion (see [6], [1], [10], etc. ).…”

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

“…We refer the readers to [1]- [6], [8]- [15] and references therein for a wealth of reference materials on the subject. More recently researchers have given special attentions to the study of equations in which the delay argument occurs in the derivative of the state variable as well as in the independent variable, so-called neutral differential equations.…”

“…Inspired and motivated by the works mentioned above and the papers [1]- [6], [8]- [15] and the references therein, we study the existence of periodic and nonnegative periodic solutions of the nonlinear neutral differential equation…”