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 [9] 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.