Afterward, some fixed point results for generalized weak contraction mappings were proved by Choudhury et al. [2] by using some control function along with the notion of an altering distance function.
Li and Wang [4] obtained a minimax inequality by using some scalarization functions.
Simsek et al. gave some new results for Bernstein and q-Bernstein polynomials by using some special functions, polynomials, and numbers ([21, 22], and [23]).
However, for some discussion on this notion and Theorem 1.1, the reader can refer to the recent paper of Gopal et al. [23], where analogous results are proved by using some control functions.
In this paper, we introduce the notion of ((alpha,theta,k -contraction multi-valpha,theta,k -contractionh some fixed point results for such mappings by using somappingsol functions due to Jleli et and (J. Inestablishpl. 2014:439, 2014) in metric somees and fixedsh some interesting exampointto illustresultsr main results.
By using some exact penalty function such as l 1 penalty function (see [1, 9, 20 22]), the minimizer of the corresponding penalty problem must be a minimizer of the original problem when ε is sufficiently small.
Since numerical experimental results indicate that the number of nodes that scattered in the working domain can affect final solutions dramatically, a calibration scheme is introduced into the proposed modeling system beforehand by using some referred known functions.
We write (1.9) as an equivalent integral equation and then, by using some properties of its Green function, we are able to get a corresponding Lyapunov-type inequality.
We write (1.12) as an equivalent integral equation and then, by using some properties of its Green function and the Guo-Krasnoselskii fixed point theorem, we can obtain our first result asserting existence of nontrivial positive solutions to problem (1.12).
The purpose of this paper is to establish sufficient conditions on the existence of positive solutions for fractional q-difference system (1.1) by using some properties of the Green function and some fixed-point theorems such as the Banach contraction principle, Krasnoselskii's fixed-point theorem, and the Leray-Schauder nonlinear alternative.
