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2 Journal of Applied Analysis

2 2021

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Existence of solutions of BVPs for fractional Langevin equations involving Caputo fractional derivatives

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Kiyamehr Z. , Baghani H.

Journal of Applied Analysis

2021

tom Vol. 27, nr 1

47--55

This article investigates a nonlinear fractional Caputo-Langevin equation Dβ(Dα + λ)x(t) = f(t, x(t)), 0 < t < 1, 0 < α ≤ 1, 1 < β ≤ 2, subject to the multi-point boundary conditions x(0) = 0, D2αx(1) + λDαx(1) = 0, x(1) =η∫0 x(τ) dτ for some 0 < η < 1, where Dα is the Caputo fractional derivative of order α, f : [0, 1] × ℝ → ℝ is a given continuous function, and λ is a real number. Some new existence and uniqueness results are obtained by applying an interesting fixed point theorem.

Fixed point theorems for a new generalization of contractive maps in incomplete metric spaces and its application in boundary value problems

80%

Ahmadi Z. , Lashkaripour R. , Baghani H.

Journal of Applied Analysis

2021

tom Vol. 27, nr 2

199--207

In this paper, we obtain some fixed point theorems for multivalued mappings in incomplete metric spaces. Moreover, as motivated by the recent work of Olgun, Minak and Altun [M. Olgun, G. Minak and I. Altun, A new approach to Mizoguchi-Takahashi type fixed point theorems, J. Nonlinear Convex Anal. 17 (2016), no. 3, 579-587],we improve these theorems with a new generalization contraction condition for multivalued mappings in incomplete metric spaces. This result is a significant generalization of somewell-known results in the literature. Also,we provide some examples to show that our main theorems are a generalization of previous results. Finally, we give an application to a boundary value differential equation.