Dynamics and Ulam Stability for Fractional q-Difference Inclusions via Picard Operators Theory

In this manuscript, by using weakly Picard operators we investigate the Ulam type stability of fractional q-difference An illustrative example is given in the last section.

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>

Fractional q-Difference Inclusions in Banach Spaces

In this paper, we study a class of Caputo fractional q-difference inclusions in Banach spaces. We obtain some existence results by using the set-valued analysis, the measure of noncompactness, and the fixed point theory (Darbo and Mönch’s fixed point theorems). Finally we give an illustrative example in the last section. We initiate the study of fractional q-difference inclusions on infinite dimensional Banach spaces.