Existence Results for Generalized Bagley-Torvik Type Fractional Differential Inclusions with Nonlocal Initial Conditions

Lizhen Chen,Gang Li

doi:10.1155/2018/2761321

Abstract

In this article, we prove the existence of solutions for the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions. It allows applying the noncompactness measure of Hausdorff, fractional calculus theory, and the nonlinear alternative for Kakutani maps fixed point theorem to obtain the existence results under the assumptions that the nonlocal item is compact continuous and Lipschitz continuous and multifunction is compact and Lipschitz, respectively. Our results extend the existence theorems for the classical Bagley-Torvik inclusion and some related models.

Highlights

IntroductionAs far as the author knows, there are few papers on the existence of the generalized BagleyTorvik type fractional differential inclusion (1) except for Ibrahim, Dong, and Fan [17]
In this article, we will consider the following generalized Bagley-Torvik type fractional differential inclusions: cDV1 z (t) − χcDV2 z (t) ∈ G (t, z (t)), t ∈ (0, 1] = B, (1)z (0) = h (z) where cDV1 and cDV2 are Caputo fractional derivatives with 0 < ]1 ≤ 1 and 0 < ]2 < ]1, χ ∈ R is a constant, and G is a multifunction.By introducing nonlocal conditions into the initial-value problems, Byszewski and Lakshmikantham [1] provided a more accurate model for the nonlocal initial valued problem since more information was incorporated in the experiment
We prove the existence of solutions for the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions

Summary

Introduction

As far as the author knows, there are few papers on the existence of the generalized BagleyTorvik type fractional differential inclusion (1) except for Ibrahim, Dong, and Fan [17]. They studied the following equation: Journal of Function Spaces cDθV (s) − acDδV (s) + g (s, V (s)) = 0; 0 < s < 1,. We shall be concerned with the existence of the generalized Bagley-Torvik type fractional differential inclusions (1) by employing the noncompactness measure of Hausdorff, fractional calculus, and the nonlinear alternative for Kakutani maps fixed point theorem, when h is compact continuous and Lipschitz continuous and G is compact and Lipschitz, respectively. The structure of this article is as follows: some preliminary knowledge is introduced in Section 2; some existence criteria are derived from (1) in Section 3; in the end, we use an example to illustrate an application of the main result

Preliminaries

Main Results

Example

Conclusion