Differential Equations and Inclusions of Fractional Order with Impulse Effects in Banach Spaces

The present paper is concerned with the existence of solutions, in infinite- dimensional Banach spaces, for impulsive differential inclusions and equations of order q∈(1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in (1,2)$$\end{document}, with anti-periodic conditions and involving the Caputo derivative whether in the generalized sense (via the Riemann–Liouville fractional derivative,) or in the normal sense and whether the lower limit on each impulsive subinterval (ti,ti+1],i=0,1,…,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (t_{i} ,t_{i+1}],\, i=0,1,\ldots ,m$$\end{document} is keep at zero or is set at ti\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{i}$$\end{document}. Using the technique of fixed point and the properties of the measure of noncompactness, existence results are obtained. Moreover, we derive in reflexive Banach spaces, by using a new version weakly convergent criteria in the space of piecewise continuous functions, an existence result of solutions without assuming any condition on the multivalued function in terms of measure of noncompactness. Some examples will be given to illustrate the obtained results.