Existence and uniqueness of solutions for stochastic differential equations of fractional-order 
                
                    
                                    
                        q
                        &gt;
                        1
                    
                
                $q &gt; 1$
             with finite delays

Zhang, Xianmin; Agarwal, Praveen; Liu, Zuohua; Peng, Hui; You, Fang; Zhu, Yaping

doi:10.1186/s13662-017-1169-3

Cited by 28 publications

(13 citation statements)

References 19 publications

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“…Author details 1 Department of Mathematics, The Government Sadiq College Women University Bahawalpur, 63100 Bahawalpur, Pakistan. 2 Department of Mathematics, The Islamia University of Bahawalpur, 63100 Bahawalpur, Pakistan. 3 Department of Mathematics, Cankaya University, 06530 Ankara, Turkey.…”

Section: Lemma 20mentioning

confidence: 99%

“…They used these inequalities to formulate some fractional integral inequalities of Chebyshev type. Zhang et al [2] gave the sufficient conditions for the existence and uniqueness of solutions for fractional differential systems by applying the generalized Gronwall inequality. Saoudi et al [3] used inequalities to present the existence of solutions to the boundary value problem for the nonlinear fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

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A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

et al. 2021

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In this article, we present the continuity analysis of the 3D models produced by the tensor product scheme of $(m+1)$ ( m + 1 ) -point binary refinement scheme. We use differences and divided differences of the bivariate refinement scheme to analyze its smoothness. The $C^{0}$ C 0 , $C^{1}$ C 1 and $C^{2}$ C 2 continuity of the general bivariate scheme is analyzed in our approach. This gives us some simple conditions in the form of arithmetic expressions and inequalities. These conditions require the mask and the complexity of the given refinement scheme to analyze its smoothness. Moreover, we perform several experiments by using these conditions on established schemes to verify the correctness of our approach. These experiments show that our results are easy to implement and are applicable for both interpolatory and approximating types of the schemes.

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Section: Lemma 20mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

et al. 2021

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“…Recently, many researchers have investigated sufficient conditions for the existence, uniqueness, and stability of solutions for the nonlinear fractional Langevin equations involving various types of fractional derivatives and by using different types of methods such as standard fixed point theorems, Leray–Schauder theory, variational methods, or monotone iterative technique combined with the method of upper and lower solutions, and so forth. For more details, see previous works 12–33 …”

Section: Introductionmentioning

confidence: 99%

ψ$$ \psi $$‐Caputo type time‐delay Langevin equations with two general fractional orders

Aydın

Mahmudov

2023

Math Methods in App Sciences

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In the present paper, first, a ψ$$ \psi $$‐delayed Mittag–Leffler type function is introduced, which generalizes the existing delayed Mittag–Leffler type function. Second, by means of ψ$$ \psi $$‐delayed Mittag–Leffler type function, an exact analytical solution formula to non‐homogeneous linear delayed Langevin equations involving two distinct ψ$$ \psi $$‐Caputo type fractional derivatives of general orders is obtained. Moreover, existence and uniqueness, stability of solution to nonlinear delayed Langevin fractional differential equations is obtained in the weighted space. Numerical and simulated examples are shared to exemplify the theoretical findings.

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“…The existence and uniqueness result is given and we also give the moment growth bound estimate on the solution of the above equation. Similar models have been considered for Caputo derivatives [22][23][24] where existence and uniqueness results were studied. To the best of our knowledge, we are the first to consider this model for the conformable fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation

Omaba

Nwaeze

2019

Fractal Fract

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We study a class of conformable time-fractional stochastic equation T a α,t u(The initial condition u(x, 0) = u 0 (x), x ∈ R is a non-random function assumed to be non-negative and bounded, T a α,t is a conformable time-fractional derivative, σ : R → R is Lipschitz continuous andẆ t a generalized derivative of Wiener process. Some precise condition for the existence and uniqueness of a solution of the class of equation is given and we also give an upper bound estimate on the growth moment of the solution. Unlike the growth moment of stochastic fractional heat equation with Riemann-Liouville or Caputo-Dzhrbashyan fractional derivative which grows in time like t c 1 exp(c 2 t), c 1 , c 2 > 0; our result also shows that the energy of the solution (the second moment) grows exponentially in time for t ∈ [a, T], T < ∞ but with at most c 1 exp(c 2 (t − a) 2α−1 ) for some constants c 1 , and c 2 .

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Existence and uniqueness of solutions for stochastic differential equations of fractional-order q > 1 $q > 1$ with finite delays

Cited by 28 publications

References 19 publications

A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

ψ$$ \psi $$‐Caputo type time‐delay Langevin equations with two general fractional orders

Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation

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