Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem

The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems for the class of β−G, ψ−G contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process, some recent results have been generalized. As an application, we establish a set of conditions for the existence of a class of fractional integrals taking the parametric Riemann–Liouville formula. Moreover, we introduce numerical solutions of the class by using the set of fixed points.

On a Volterra Integral Equation with Delay, via w-Distances

The paper deals with a Volterra integral equation with delay. In order to apply the w-weak generalized contraction theorem for the study of existence and uniqueness of solutions, we rewrite the equation as a fixed point problem. The assumptions take into account the support of w-distance and the complexity of the delay equation. Gronwall-type theorem and comparison theorem are also discussed using a weak Picard operator technique. In the end, an example is provided to support our results.

On Some Fixed Point Results in E − Fuzzy Metric Spaces

In the existing literature, Banach contraction theorem as well as Meir-Keeler fixed point theorem were extended to fuzzy metric spaces. However, the existing extensions require strong additional assumptions. The purpose of this paper is to determine a class of fuzzy metric spaces in which both theorems remain true without the need of any additional condition. We demonstrate the wide validity of the new class.

Fuzzy Meir-Keeler’s Contraction and Characterization

In this study, a fuzzy Meir-Keeler’s contraction theorem for complete FMS based on George and Veeramani idea is established. Then, we characterize fuzzy Meir-Keeler’s contractions as contractive types induced by functions called fuzzy L -function. Moreover, we show that the converse of it is true. Finally, we bring some examples and corollaries certify our results and new improvement.

Application of a fixed point theorem on infinite cartesian product to an infinite system of differential equations

"In the paper Operators on infinite dimensional cartesian product, (Analele Univ. Vest Timişoara, Mat. Inform., 48 (2010), 253–263), by I. A. Rus and M. A. Şerban, the authors give a generalization of the Fibre contraction theorem on infinite dimensional cartesian product. In this paper we give an application of this abstract result to an infinite system of differential equations. "

Set-valued problems under bounded variation assumptions involving the Hausdorff excess

<p style='text-indent:20px;'>In the very general framework of a (possibly infinite dimensional) Banach space <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula>, we are concerned with the existence of bounded variation solutions for measure differential inclusions</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE100"> \begin{document}$ \begin{equation} \begin{split} &amp;dx(t) \in G(t, x(t)) dg(t),\\ &amp;x(0) = x_0, \end{split} \end{equation}\;\;\;\;\;\;(1) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ dg $\end{document}</tex-math></inline-formula> is the Stieltjes measure generated by a nondecreasing left-continuous function.</p><p style='text-indent:20px;'>This class of differential problems covers a wide variety of problems occuring when studying the behaviour of dynamical systems, such as: differential and difference inclusions, dynamic inclusions on time scales and impulsive differential problems. The connection between the solution set associated to a given measure <inline-formula><tex-math id="M3">\begin{document}$ dg $\end{document}</tex-math></inline-formula> and the solution sets associated to some sequence of measures <inline-formula><tex-math id="M4">\begin{document}$ dg_n $\end{document}</tex-math></inline-formula> strongly convergent to <inline-formula><tex-math id="M5">\begin{document}$ dg $\end{document}</tex-math></inline-formula> is also investigated.</p><p style='text-indent:20px;'>The multifunction <inline-formula><tex-math id="M6">\begin{document}$ G : [0,1] \times X \to \mathcal{P}(X) $\end{document}</tex-math></inline-formula> with compact values is assumed to satisfy excess bounded variation conditions, which are less restrictive comparing to bounded variation with respect to the Hausdorff-Pompeiu metric, thus the presented theory generalizes already known existence and continuous dependence results. The generalization is two-fold, since this is the first study in the setting of infinite dimensional spaces.</p><p style='text-indent:20px;'>Next, by using a set-valued selection principle under excess bounded variation hypotheses, we obtain solutions for a functional inclusion</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE102"> \begin{document}$ \begin{equation} \begin{split} &amp;Y(t)\subset F(t,Y(t)),\\ &amp;Y(0) = Y_0. \end{split} \end{equation}\;\;\;\;(2) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown that a recent parametrized version of Banach's Contraction Theorem given by V.V. Chistyakov follows from our result.</p>

Extension of fixed point results in intuitionistic fuzzy b metric space

The aim of this paper is to define the notion of intuitionistic fuzzy b metric space (in short, IFbMS) along with some useful results. We establish some important Lemmas in order to study the Cauchy sequence in IFbMS. To further develop the work, we establish some fixed point theorems and study the existence of unique fixed point of some self mappings in IFbMS. We also develop the concept of Ćirić quasi-Contraction theorem in IFbMS. Examples are provided to validate the non-triviality of the results.

Banach contraction theorem on fuzzy cone b-metric space

In the present paper the notion of fuzzy cone b -metric space has been introduced. Here we have defined fuzzy cone b -contractive mapping, and Banach contraction theorem for single mapping and pair of mappings has been proved in the setting of fuzzy cone b -metric space.