scholarly journals Anti-Periodic Boundary Value Problems for Nonlinear Langevin Fractional Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 443 ◽  
Author(s):  
Fang Li ◽  
Hongjuan Zeng ◽  
Huiwen Wang
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Periodic Boundary ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Periodic Boundary Value Problems ◽  
Fractional Differential ◽  
The Fixed Point Theorem

In this paper, we focus on the existence of solutions of the nonlinear Langevin fractional differential equations involving anti-periodic boundary value conditions. By using some techniques, formulas of solutions for the above problem and some properties of the Mittag-Leffler functions E α , β ( z ) , α , β ∈ ( 1 , 2 ) , z ∈ R are presented. Moreover, we utilize the fixed point theorem under the weak assumptions for nonlinear terms to obtain the existence result of solutions and give an example to illustrate the result.

A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications

2017 ◽  
Vol 8 (1) ◽  
pp. 386-454 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Periodic Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Impulsive Fractional Differential Equations ◽  
Periodic Boundary Value Problems ◽  
Fractional Differential

Abstract In this paper, we present a new method for converting boundary value problems of impulsive fractional differential equations to integral equations. Applications of this method are given to solve some types of anti-periodic boundary value problems for impulsive fractional differential equations. Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann–Liouville and Hadamard fractional derivatives with order {q\in(0,1)} , see Theorem 2, Theorem 4, Theorem 6 and Theorem 8. Then we obtain exact expression of piecewise continuous solutions of these fractional differential equations see Theorem 1, Theorem 2, Theorem 3 and Theorem 4. Finally, four classes of integral type anti-periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. See Theorems 4.1–4.4. We allow the nonlinearity {p(t)f(t,x)} in fractional differential equations to be singular at {t=0,1} and be involved a super-linear and sub-linear term. The analysis relies on Schaefer’s fixed point theorem. In order to avoid misleading readers, we correct the results in [28] and [65]. We establish sufficient conditions for the existence of solutions of an anti-periodic boundary value problem for impulsive fractional differential equation. The results in [68] are complemented. The results in [81] are corrected. See Lemma 5.1, Lemma 5.7, Lemma 5.10 and Lemma 5.13.

scholarly journals Multiple solutions for a class of boundary value problems of fractional differential equations with generalized Caputo derivatives

2021 ◽  
Vol 6 (12) ◽  
pp. 13119-13142
Author(s):  
Yating Li ◽  
◽  
Yansheng Liu
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Ulam Stability ◽  
Caputo Derivatives ◽  
Fractional Differential ◽  
The Fixed Point Theorem

<abstract><p>This paper is mainly concerned with the existence of multiple solutions for the following boundary value problems of fractional differential equations with generalized Caputo derivatives:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \hskip 3mm \left\{ \begin{array}{lll} ^{C}_{0}D^{\alpha}_{g}x(t)+f(t, x) = 0, \ 0&lt;t&lt;1;\\ x(0) = 0, \ ^{C}_{0}D^{1}_{g}x(0) = 0, \ ^{C}_{0}D^{\nu}_{g}x(1) = \int_{0}^{1}h(t)^{C}_{0}D^{\nu}_{g}x(t)g'(t)dt, \end{array}\right. $\end{document} </tex-math></disp-formula></p> <p>where $ 2 &lt; \alpha &lt; 3 $, $ 1 &lt; \nu &lt; 2 $, $ \alpha-\nu-1 &gt; 0 $, $ f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+}) $, $ g' &gt; 0 $, $ h\in C([0, 1], \mathbb{R}^{+}) $, $ \mathbb{R}^{+} = [0, +\infty) $. Applying the fixed point theorem on cone, the existence of multiple solutions for considered system is obtained. The results generalize and improve existing conclusions. Meanwhile, the Ulam stability for considered system is also considered. Finally, three examples are worked out to illustrate the main results.</p></abstract>

scholarly journals Existence of Solutions for Fractional Differential Equations with p-Laplacian Operator and Integral Boundary Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Jingli Xie ◽  
Lijing Duan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence Of Solutions ◽  
Integral Boundary Conditions ◽  
Laplacian Operator ◽  
Integral Boundary ◽  
Fractional Differential ◽  
Integral Boundary Value Problems ◽  
The Fixed Point Theorem

In this paper, we investigate a class of integral boundary value problems of fractional differential equations with a p-Laplacian operator. Existence of solutions is obtained by using the fixed point theorem, and an example is given to show the applicability of our main result.

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