Some new existence results for boundary value problems involving ψ-Caputo fractional derivative

This paper concerns the boundary value problem for a fractional differential equation involving a generalized Caputo fractional derivative in b−metric spaces. The used fractional operator is given by the kernel k(t, s) = ψ(t) − ψ(s) and the derivative operator 1/ψʹ(t) d/dt . Some existence results are obtained based on fixed point theorem of α-φ−Graghty contraction type mapping. In the end, we provide some illustrative examples to justify the acquired results.Publisher's Versio

Absolute retractivity of the common fixed points set of two multifunctions

In 1970, Schirmer discussed about topological properties of the fixed point set of multifunctions ([ Properties of the fixed point set of contractive multifunctions , Canad. Math. Bull. 13 (1970), 169-173]). Later, some authors continued this study by providing different conditions ([M.C. Alicu and O. Mark, Some properties of the fixed points set for multifunctions , Studia Univ. Babes-Bolyai Math. 25 (1980), 77-79] and [B. Ricceri, Une propriete topologique de l'ensemble des points fixes d'une contraction multivoque a valeurs convexes , Atti. Acc. Lincei Rend. 81 (1987), 283-286]). Recently, Sintamarian proved results on absolute retractivity of the common fixed points set of two multivalued operators ([ A topological property of the common fixed points set of two multivalued operators , Nonlinear Anal. 70 (2009), 452-456] and [ A topological property of the common fixed points set of two multivalued operators satisfying a Latif-Beg type condition , Fixed Point Theory 9 (2008), 561-573]). We shall present some results on absolute retractivity of the common fixed points set of two multifunctions by using different conditions