Coupled fractional differential systems with random effects in Banach spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
O. Zentar ◽  
M. Ziane ◽  
S. Khelifa
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Existence Of Solutions ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Random Differential Equations ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Vector Valued ◽  
Abstract Formulation

Abstract The purpose of this work is to investigate the existence of solutions for a system of random differential equations involving the Riemann–Liouville fractional derivative. The existence result is established by means of a random abstract formulation to Sadovskii’s fixed point theorem principle [A. Baliki, J. J. Nieto, A. Ouahab and M. L. Sinacer, Random semilinear system of differential equations with impulses, Fixed Point Theory Appl. 2017 2017, Paper No. 27] combined with a technique based on vector-valued metrics and convergent to zero matrices. An example is also provided to illustrate our result.

scholarly journals Existence Results for a System of Coupled Hybrid Differential Equations with Fractional Order

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Mohamed Hannabou ◽  
Khalid Hilal
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniqueness Result ◽  
Existence Results ◽  
Fractional Differential ◽  
Hybrid Differential Equations

This paper studies the existence of solutions for a system of coupled hybrid fractional differential equations. We make use of the standard tools of the fixed point theory to establish the main results. The existence and uniqueness result is elaborated with the aid of an example.

scholarly journals Terminal Value Problem for Differential Equations with Hilfer–Katugampola Fractional Derivative

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 672 ◽  
Author(s):  
Mouffak Benchohra ◽  
Soufyane Bouriah ◽  
Juan J. Nieto
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniqueness Of Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Terminal Value Problem

We present in this work the existence results and uniqueness of solutions for a class of boundary value problems of terminal type for fractional differential equations with the Hilfer–Katugampola fractional derivative. The reasoning is mainly based upon different types of classical fixed point theory such as the Banach contraction principle and Krasnoselskii’s fixed point theorem. We illustrate our main findings, with a particular case example included to show the applicability of our outcomes.

scholarly journals Existence Results for a System of Coupled Hybrid Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bashir Ahmad ◽  
Sotiris K. Ntouyas ◽  
Ahmed Alsaedi
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniqueness Result ◽  
Existence Results ◽  
Dirichlet Boundary Conditions ◽  
Fractional Differential

This paper studies the existence of solutions for a system of coupled hybrid fractional differential equations with Dirichlet boundary conditions. We make use of the standard tools of the fixed point theory to establish the main results. The existence and uniqueness result is elaborated with the aid of an example.

scholarly journals A note regarding extensions of fixed point theorems involving two metrics via an analysis of iterated functions

2020 ◽  
Vol 61 ◽  
pp. C15-C30
Author(s):  
Charles P Stinson ◽  
Saleh S Almuthaybiri ◽  
Christopher C Tisdell
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Nonlinear Dynamic Equations ◽  
Fractional Differential

The purpose of this work is to advance the current state of mathematical knowledge regarding fixed point theorems of functions. Such ideas have historically enjoyed many applications, for example, to the qualitative and quantitative understanding of differential, difference and integral equations. Herein, we extend an established result due to Rus [Studia Univ. Babes-Bolyai Math., 22, 1977, 40–42] that involves two metrics to ensure wider classes of functions admit a unique fixed point. In contrast to the literature, a key strategy herein involves placing assumptions on the iterations of the function under consideration, rather than on the function itself. In taking this approach we form new advances in fixed point theory under two metrics and establish interesting connections between previously distinct theorems, including those of Rus [Studia Univ. Babes-Bolyai Math., 22, 1977, 40–42], Caccioppoli [Rend. Acad. Naz. Linzei. 11, 1930, 31–49] and Bryant [Am. Math. Month. 75, 1968, 399–400]. Our results make progress towards a fuller theory of fixed points of functions under two metrics. Our work lays the foundations for others to potentially explore applications of our new results to form existence and uniqueness of solutions to boundary value problems, integral equations and initial value problems. References Almuthaybiri, S. S. and C. C. Tisdell. ``Global existence theory for fractional differential equations: New advances via continuation methods for contractive maps''. Analysis, 39(4):117–128, 2019. doi:10.1515/anly-2019-0027 Almuthaybiri, S. S. and C. C. Tisdell. ``Sharper existence and uniqueness results for solutions to third-order boundary value problems, mathematical modelling and analysis''. Math. Model. Anal. 25(3):409–420, 2020. doi:10.3846/mma.2020.11043 Banach, S. ``Sur les operations dans les ensembles abstraits et leur application aux equations integrales''. Fund. Math., 3:133–181 1922. doi:10.4064/fm-3-1-133-181 Brouwer, L. E. J. ``Ueber Abbildungen von Mannigfaltigkeiten''. Math. Ann. 71:598, 1912. doi:10.1007/BF01456812 Bryant, V. W. ``A remark on a fixed point theorem for iterated mappings'' Am. Math. Month. 75: 399–400, 1968. doi:10.2307/2313440 Caccioppoli, R. ``Un teorema generale sullesistenza de elemente uniti in una transformazione funzionale''. Rend. Acad. Naz. Linzei. 11:31–49, 1930. Goebel, K., and W. A. Kirk. Topics in metric fixed point theory. Cambridge University Press, 1990, doi:10.1017/CBO9780511526152 Leray, J., and J. Schauder. ``Topologie et equations fonctionnelles''. Ann. Sci. Ecole Norm. Sup. 51:45–78, 1934. doi:10.24033/asens.836 O'Regan, D. and R. Precup. Theorems of Leray–Schauder type and applications, Series in Mathematical Analysis and Applications, Vol. 3. CRC Press, London, 2002. doi:10.1201/9781420022209 Rus, I. A. ``On a fixed point theorem of Maia''. Studia Univ. Babes-Bolyai Math. 22:40–42, 1977. Schaefer, H. H. ``Ueber die Methode der a priori-Schranken''. Math. Ann. 129:415–416, 1955. doi:10.1007/bf01362380 Tisdell, C. C. ``When do fractional differential equations have solutions that are bounded by the Mittag-Leffler function?'' Fract. Calc. Appl. Anal. 18(3):642–650, 2015. doi:10.1515/fca-2015-0039 Tisdell, C. C. ``A note on improved contraction methods for discrete boundary value problems.'' J. Diff. Eq. Appl. 18(10):1773–1777, 2012. doi:10.1080/10236198.2012.681781 Tisdell, C. C. ``On the application of sequential and fixed-point methods to fractional differential equations of arbitrary order.'' J. Int. Eq. Appl. 24(2):283–319, 2012. doi:10.1216/JIE-2012-24-2-283 Ehrnstrom, M., Tisdell, C. C. and E. Wahlen. ``Asymptotic integration of second-order nonlinear difference equations.'' Glasg. Math. J. 53(2):223–243, 2011. doi:10.1017/S0017089510000650 Erbe, L., A. Peterson and C. C. Tisdell. ``Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales.'' Nonlin. Anal. 69(7):2303–2317, 2008. doi:10.1016/j.na.2007.08.010 Tisdell, C. C. and A. Zaidi. ``Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling.'' Nonlin. Anal. 68(11):3504–3524, 2008. doi:10.1016/j.na.2007.03.043 Tisdell, C. C. ``Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions.'' Int.. J. Math. Ed. Sci. Tech. 48(7):1087–1095, 2017. doi:10.1080/0020739X.2017.1298856 Tisdell, C. C. ``On Picard's iteration method to solve differential equations and a pedagogical space for otherness.'' Int. J. Math. Ed. Sci. Tech. 50(5):788–799, 2019. doi:10.1080/0020739X.2018.1507051 Zeidler, E. Nonlinear functional analysis and its applications. Springer-Verlag, New York, 1986. doi:10.1007/978-1-4612-4838-5

Nonlocal Differential Equations with Convolution Coefficients and Applications to Fractional Calculus

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christopher S. Goodrich
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Differential Equations ◽  
Fractional Integral ◽  
Fixed Point Theory ◽  
Conjugate Problem ◽  
Point Theory ◽  
Liouville Type ◽  
Model Case ◽  
Fractional Differential

Abstract The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is - A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) ⁢ u ′′ ⁢ ( t ) = λ ⁢ f ⁢ ( t , u ⁢ ( t ) ) , t ∈ ( 0 , 1 ) , q ≥ 1 , -A((b*u^{q})(1))u^{\prime\prime}(t)=\lambda f(t,u(t)),\quad t\in(0,1),\,q\geq 1, is considered. Due to the coefficient A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) {A((b*u^{q})(1))} appearing in the differential equation, the equation has a coefficient containing a convolution term. By choosing the kernel b in various ways, specific nonlocal coefficients can be recovered such as nonlocal coefficients equivalent to a fractional integral of Riemann–Liouville type. The results rely on the use of a nonstandard order cone together with topological fixed point theory. Applications to fractional differential equations are given, including a problem related to the ( n - 1 , 1 ) {(n-1,1)} -conjugate problem.

scholarly journals On Antiperiodic Boundary Value Problems for Higher-Order Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Afrah Assolami
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Existence Results ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

We study an antiperiodic boundary value problem of nonlinear fractional differential equations of orderq∈(4,5]. Some existence results are obtained by applying some standard tools of fixed-point theory. We show that solutions for lower-order anti-periodic fractional boundary value problems follow from the solution of the problem at hand. Our results are new and generalize the existing results on anti-periodic fractional boundary value problems. The paper concludes with some illustrating examples.

scholarly journals Applications of Fixed Point Theory to Investigate a System of Fractional Order Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Zareen A. Khan ◽  
Israr Ahmad ◽  
Kamal Shah
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Of View ◽  
Existence Theory ◽  
Solution Stability ◽  
Main Work ◽  
Fractional Differential ◽  
Hadamard Derivative

We investigate a nonlinear system of pantograph-type fractional differential equations (FDEs) via Caputo-Hadamard derivative (CHD). We establish the conditions for existence theory and Ulam-Hyers-type stability for the underlying boundary value system (BVS) of FDE. We use Krasnoselskii’s and Banach’s fixed point theorems to obtain the desired results for the existence of solution. Stability is an important aspect from a numerical point of view we investigate here. To justify the main work, relevant examples are provided.

scholarly journals Boundary Value Problem for Caputo–Fabrizio Random Fractional Differential Equations

2020 ◽  
Vol 6 (2) ◽  
pp. 218-230
Author(s):  
Fouzia Bekada ◽  
Saïd Abbas ◽  
Mouffak Benchohra
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Random Operators ◽  
Ulam Stability ◽  
Fractional Differential ◽  
Stability Results

AbstractThis article deals with some existence of random solutions and Ulam stability results for a class of Caputo-Fabrizio random fractional differential equations with boundary conditions in Banach spaces. Our results are based on the fixed point theory and random operators. Two illustrative examples are presented in the last section.

scholarly journals Terminal Value Problem for Implicit Katugampola Fractional Differential Equations in b -Metric Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Salim Krim ◽  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Erdal Karapinar
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fractional Differential ◽  
Terminal Value Problem ◽  
Type Contraction ◽  
Implicit Fractional Differential Equations

This manuscript deals with a class of Katugampola implicit fractional differential equations in b -metric spaces. The results are based on the α − φ -Geraghty type contraction and the fixed point theory. We express an illustrative example.

scholarly journals Existence of solutions or nonlinear nth-order differential equations and inclusions with nonlocal and integral boundary conditions via fixed point theory

Filomat ◽  
2014 ◽  
Vol 28 (10) ◽  
pp. 2149-2162 ◽  
Author(s):  
Bashir Ahmad ◽  
Sotiris Ntouyas ◽  
Hamed Alsulami
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Fixed Point Theory ◽  
Integral Boundary Conditions ◽  
Point Theory ◽  
Integral Boundary ◽  
Differential Equations And Inclusions

In this paper, a class of boundary value problems of nonlinear nth-order differential equations and inclusions with nonlocal and integral boundary conditions is studied. New existence results are obtained by means of some fixed point theorems. Examples are given for the illustration of the results.

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