Abstract: In this paper, we prove some fixed point theorems of $\alpha$-nonexpansive mapping in Banach spaces which introduced by K. Goebel and M. A. Pineda (2007) [1]. In the first theorem of main result, we prove the existence of common fixed point sets of $\alpha$-nonexpansive mapping and continuous mapping. In the second theorem in our…

# Author: Results in Fixed Point Theory and Applications

## Existence and stability of fractional implicit differential equations with complex order -by D Vivek, K. Kanagarajan and E. M. Elsayed

Abstract: In this paper, we study existence results for nonlocal initial value problems for fractional implicit differential equations with complex order. The Krasnoselkii's fixed point theorem and Banach contraction principle are used for proving the main results. Moreover, we discuss the Ulam-Hyers stability. Keywords: Fractional implicit differential equations, Nonlocal condition, Complex order, Existence. DOI: https://doi.org/10.30697/rfpta-2018-27 ______…

## Common fixed point results in metric spaces endowed with a graph -by Binayak S. Choudhury, Nikhilesh Metiya, Debashis Khatua

Abstract: In this paper we establish some common fixed point results for a pair of mappings in a metric space having the additional structure of a directed graph. The mappings are assumed to satisfy certain almost $G$-contractions without and with rational terms. Each of our results is illustrated with example. The approach here is a blending…

## Fixed point theorems for $(\alpha, \phi)$-contractive mappings of rational type in complex valued metric spaces with applications -by Priya Shahi, Jatinderdeep Kaur and S. S. Bhatia

Abstract: A very interesting approach in the theory of contractive mappings was recently given by Samet et al. (Nonlinear Anal. 75, 2012, 2154-2165) by using the concept of $\alpha$-$\psi$ contractive type mappings in metric spaces. The purpose of this paper is to give the generalized version of $\alpha$-$\psi$ contractive type mappings in complex valued metric spaces.…

## Common best proximity points for generalized $\alpha-\phi-$Geraghty proximal contractions -by Konrawut Khammahawong, Poom Kumam and Yeol Je Cho

Abstract: In this paper, we establish some common best proximity point theorems for generalized $\alpha-\phi-$Geraghty proximal contraction mappings in complete metric spaces. Moreover, we give some examples to illustrate our main results. Our results improve and extend various results given by some authors in literature. Keywords: Common best proximity point; fixed point; $\alpha$-proximal admissible mapping; $\alpha-\phi-$Geraghty proximal…

## Approximating coupled coincidence points by $\alpha$-dense curves -by Gonzalo García

Abstract: We present a new iterative procedure converging, under suitable conditions, to a coupled coincidence point of two given mappings. Our main tool will be the so called $\alpha$-dense curves, which allow us to construct such procedure in a stable way, in the specified sense, and with controlled approximation error. To justify our result, we will…

## Existence and stability results for Langevin equations with Hilfer fractional derivative -by S. Harikrishnan, K. Kanagarajan and E. M. Elsayed

Abstract: In this paper, we study two Hilfer fractional derivatives of the form \begin{equation}D^{\alpha_1, \beta} \left(D^{\alpha_2, \beta} + \lambda \right) x(t) = f(t, x(t)), \ t\in J:=(a,b], \label{ee}\end{equation}\begin{equation}I^{1-\gamma} x(a) = x_a, \gamma = (\alpha_1+\alpha_2)(1-\beta) + \beta,\end{equation} where $\ D^{\alpha_1, \beta}, D^{\alpha_2, \beta}$ is two Hilfer fractional derivatives, $\alpha_1$ and $\alpha_2$ are the order of derivatives,…