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.

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.

Controllability results of fractional integro-differential equation with non-instantaneous impulses on time scales

In this work, we investigate the controllability results of a fractional integro-differential equation with non-instantaneous impulses on time scales. Banach contraction theorem and the non-linear functional analysis have been used to establish these results. In support, a numerical example with simulation for different time scales is given to validate the obtained analytical outcomes.

Existence results of noninstantaneous impulsive fractional integro-differential equation

Existence of mild solution for noninstantaneous impulsive fractional order integro-differential equations with local and nonlocal conditions in Banach space is established in this paper. Existence results with local and nonlocal conditions are obtained through operator semigroup theory using generalized Banach contraction theorem and Krasnoselskii’s fixed point theorem, respectively. Finally, illustrations are added to validate derived results.

Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order0<α≤1with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order2<α≤3is proved and the ordinary difference Lyapunov inequality then follows asαtends to2from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.

On Some Qualitative Properties of Mild Solutions of Nonlocal Semilinear Functional Differential Equations

In the present paper, we investigate the qualitative properties such as existence, uniqueness and continuous dependence on initial data of mild solutions of first and second order nonlocal semilinear functional differential equations with delay in Banach spaces. Our analysis is based on semigroup theory and modified version of Banach contraction theorem.

ANALYSIS ON FRACTALS IN FUZZY METRIC SPACES

In this paper, we investigate the fractals generated by the iterated function system of fuzzy contractions in the fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard fuzzy metric spaces by using the fuzzy Banach contraction theorem. In addition to that, we discuss some results on fuzzy fractals such as Collage Theorem and Falling Leaves Theorem in the standard fuzzy metric spaces with respect to the standard Hausdorff fuzzy metrics.

Impulsive functional-differential equations with nonlocal conditions

The existence, uniqueness, and continuous dependence of a mild solution of an impulsive functional-differential evolution nonlocal Cauchy problem in general Banach spaces are studied. Methods of fixed point theorems, of aC0semigroup of operators and the Banach contraction theorem are applied.

Existence and uniqueness of solutions of a semilinear functional-differential evolution nonlocal Cauchy problem

Two theorems about the existence and uniqueness of mild and classical solutions of a semilinear functional-differential evolution nonlocal Cauchy problem in a general Banach space are proved. Methods of semigroups and the Banach contraction theorem are applied.