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.

Ulam–Hyers–Mittag-Leffler stability for tripled system of weighted fractional operator with TIME delay

This study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. (Fractals 28:2040011 2020) in the frame of Chebyshev and Bielecki norms with time delay. The acquired results are obtained by using Banach fixed point theorems and the Picard operator (PO) method. Finally, a pertinent example of the results obtained is demonstrated.

Implicit functional differential equations with linear modification of the argument, via weakly Picard operator theory

"Let \mathbf{K}:=\mathbf{R}\text{ or }\mathbf{C},\text{ \ }0<\lambda <1 and f \in C([0,b] \times \textbf{K}^3,\textbf{K}). In this paper we use the weakly Picard operator theory technique to study the following functional-differential equation $$ y'(x)=f(x,y(x),y'(x),y(\lambda x)), x \in [0,b].$$ "

Existence and stability of solutions of $ \psi $-Hilfer fractional functional differential inclusions with non-instantaneous impulses

<abstract><p>In this paper, we prove two existence results of solutions for an $ \psi $-Hilfer fractional non-instantaneous impulsive differential inclusion in the presence of delay in an infinite dimensional Banah spaces. Then, by using the multivalued weakly Picard operator theory, we study the stability of solutions for the considered problem in the sense of $ \psi $-generalized Ulam-Hyers. To achieve our aim, we present a relation between any solution of the considered problem and the corresponding fractional integral equation. The given problem here is new because it contains a delay and non-instantaneous impulses effect. Examples are given to clarify the possibility of applicability our assumptions.</p></abstract>

A-contraction mappings of integral type in n-normed spaces

In this article, A-contraction type mappings in integral case are defined on a complete n-normed spaces and the existence of some fixed point theorems are proved in the complete n-normed spaces and given some results on Picard operator.