On Fractional Semilinear Nonlocal Initial Value Problem with State Dependent Delay

• Published in: Differential equations and dynamical systems Vol. 32; no. 3; pp. 685 - 708
• Main Authors: Alam, Md Mansur, Dubey, Shruti
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.07.2024; Springer Nature B.V
• Subjects: Boundary value problems; Computer Science; Engineering; Existence theorems; Fixed points (mathematics); Initial conditions; Mathematics; Mathematics and Statistics; Original Research; Nonlocal conditions; 47D06; Analytic semigroup; Primary 34K37; State dependent delay; Secondary 49J27; Mild solutions; Caputo fractional derivative
• ISSN: 0971-3514; 0974-6870
• DOI: 10.1007/s12591-022-00600-3

In this paper, we consider a class of fractional order semilinear abstract Cauchy problem with state dependent delay subject to nonlocal initial conditions, and enlarge the existence theory with two different sets of assumptions. Under the first set of assumptions, we establish the existence of Hölder classical solution. Since the Hölder exponent appears as an exponent on the metric function in contraction inequality, it is not suitable to use Banach contraction mapping principle. Krasnoselskii’s fixed point theorem becomes effective to overcome this situation. Under the second set of assumptions, we obtain only the existence of mild solution using Schauder’s fixed point theorem. Few examples have been provided to illustrate our results.