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>

OCCURRENCE OF STABLE ROUGH FRACTIONAL INTEGRAL INCLUSION

In the present paper, we generalize the Fredholm type integral operator, by using the fractional rough kernel. We also deal with the Ulam-Hyers stability for rough fractional integral inclusion and utilize the weakly Picard operator method as well as the generalized Covitz-Nadler fixed point theorem.

Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory

Considerable attention has been recently given to the existence of solutions of initial or boundary value problems for fractional differential equations and inclusions with Hilfer fractional derivative. Motivated by these results, in this paper we will present existence, data dependence and Ulam stability results for some differential inclusions with Hilfer fractional derivative. The results follow as applications of the multi-valued weakly Picard operator theory. An example illustrates the main result of the paper.

Heuristic introduction to weakly Picard operator theory

In this paper we study the impact of weakly Picard operator theory, [see I. A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58 (2003), No. 1, 191–219] on the following problem: what can we do in order to find conditions under which a given operator is a weakly Picard operator?

Some Properties of Solutions of a Functional-Differential Equation of Second Order with Delay

Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results for the solutions of a system of functional-differential equations with delays are proved. The techniques used are Perov’s fixed point theorem and weakly Picard operator theory.

From a Dieudonne theorem concerning the Cauchy problem to an open problem in the theory of weakly Picard operators

Let (X, d) be a complete metric space and let f : X → X be a self operator. In this paper we study the following two problems: Problem 1. Let f be such that its fixed points set is a singleton, i.e., Ff = {x∗}. Under which conditions the next implication does hold: f is asymptotically regular ⇒ f is a Picard operator? Problem 2. Let f be such that, Ff 6= φ. Under which conditions the following implication does hold: f is asymptotically regular ⇒ f is a weakly Picard operator? The case of operators defined on a linear L∗-space is also studied.

Ulam-Hyers stability for partial differential equations

Using the weakly Picard operator technique, we will present some Ulam-Hyers stability results for some partial differential equations.

Properties of the solutions of those equations for which the Krasnoselskii iteration converges

Let (X, +, R, →) be a vectorial L-space, Y ⊂ X a nonempty convex subset of X and f : Y → Y be an operator with Ff := {x ∈ Y | f(x) = x} 6= ∅. Let 0 < λ < 1 and let fλ be the Krasnoselskii operator corresponding to f, i.e., fλ(x) := (1 − λ)x + λf(x), x ∈ Y. We suppose that fλ is a weakly Picard operator (see I. A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58 (2003), No. 1, 191-219). The aim of this paper is to study some properties of the fixed points of the operator f: Gronwall lemmas and comparison lemmas (when (X, +, R, →, ≤) is an ordered L-space) and data dependence (when X is a Banach space). Some applications are also given.