On the other hand, the operator [[PSI].sub.k] : [U.sub.k] [right arrow] [U.sub.k] is completely continuous by Lemma 10; then by Schauder's fixed-point Theorem
11, each problem of (1)-(3) has a solution.
(4) In the paper titled "Existence of Solutions for a Class of Coupled Fractional Differential Systems with Nonlocal Boundary Conditions," by applying Schauder fixed-point theorem
and Leray-Schauder nonlinear alternative theory, the authors were concerned with the existence of solutions to coupled fractional differential systems with fractional integral boundary value conditions.
Motivated by the methods of [21-23] and the above works, we study the criteria of three positive solutions for a Caputo fractional q-difference equation with integral boundary value conditions by employing properties of Green's function and the Leggett-Williams fixed-point theorem
in this paper.
Yorke, "A constructive proof of the Brouwer fixed-point theorem
and computational results," SIAM Journal on Numerical Analysis, vol.
Burton, "A fixed-point theorem
of Krasnoselskii," Applied Mathematics Letters.
Engl, A general stochastic fixed-point theorem
for continuous random operators on stochastic domains.
Hence, by the Krasnoselkii fixed-point theorem
(Theorem 2.7), we can conclude that P has a fixed point z on [B.sub.r].
By using the Guo-Krasnoselskii fixed-point theorem
, Jin and Yin  proved the existence of one positive solutions for the following boundary value problem of one-dimensional p-Laplacian with delay
In this article we use Schaefer's fixed-point theorem
to deduce the existence of periodic solutions of infinite delay nonlinear Volterra difference equations of the form
by using a multiple fixed-point theorem
to obtain three symmetric positive solutions under growth conditions on f.
Given these assumptions, G-M-S invoke the Brouwer Fixed-Point Theorem
, which states that under these givens, the curve must touch the 45-degree line at some point.
Ammon's system also discovered a proof of Banach's fixed-point theorem
, a powerful theorem in higher analysis functions.