scholarly journals Nonlocal Fractional Hybrid Boundary Value Problems Involving Mixed Fractional Derivatives and Integrals via a Generalization of Darbo’s Theorem

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ayub Samadi ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Boundary Value Problem ◽  
Theoretical Result ◽  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Existence Result ◽  
Boundary Value ◽  
Measures Of Noncompactness ◽  
Fractional Derivatives And Integrals ◽  
Mixed Fractional Derivatives

In this work, a new existence result is established for a nonlocal hybrid boundary value problem which contains one left Caputo and one right Riemann–Liouville fractional derivatives and integrals. The main result is proved by applying a new generalization of Darbo’s theorem associated with measures of noncompactness. Finally, an example to justify the theoretical result is also presented.

scholarly journals A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmed Nouara ◽  
Abdelkader Amara ◽  
Eva Kaslik ◽  
Sina Etemad ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Research Work ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Numerical Examples ◽  
Fractional Derivatives And Integrals ◽  
Fractional Differential

AbstractIn this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

scholarly journals An Existence Result of Positive Solutions for Fully Second-Order Boundary Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Yongxiang Li ◽  
Yaya Shang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Existence Result ◽  
Boundary Value ◽  
Second Order ◽  
Upper Solution ◽  
Superlinear Growth ◽  
Lower And Upper Solution

An existence result of positive solutions is obtained for the fully second-order boundary value problem  -u′′(t)=f(t,u(t),u′(t)),  t∈[0,1],  u(0)=u(1)=0,wheref:[0,1]×R+×R→Ris continuous. The nonlinearityf(t,x,y)may be sign-changing and superlinear growth onxandy. Our discussion is based on the method of lower and upper solution.

Weighted Estimates for Boundary Value Problems with Fractional Derivatives

2020 ◽  
Vol 20 (4) ◽  
pp. 609-630 ◽  
Author(s):  
Ivan P. Gavrilyuk ◽  
Volodymyr L. Makarov ◽  
Nataliya V. Mayko
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Dirichlet Boundary ◽  
Boundary Effect ◽  
Boundary Value ◽  
Grid Method ◽  
Weighted Estimates ◽  
Regularity Properties ◽  
Better Than

AbstractWe consider the Dirichlet boundary value problem for linear fractional differential equations with the Riemann–Liouville fractional derivatives. By transforming the boundary value problem to the integral equation, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of the boundary value problems by a grid method is discussed and weighted estimates considering the boundary effect are obtained. It is shown that the accuracy (the convergence rate) near the boundary is better than inside the domain due to the influence of the Dirichlet boundary condition.

scholarly journals Existence and uniqueness results for fractional Navier boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imed Bachar ◽  
Hassan Eltayeb
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Operator Equation ◽  
Real Function ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Alpha Beta

Abstract We establish the existence, uniqueness, and positivity for the fractional Navier boundary value problem: $$\begin{aligned} \textstyle\begin{cases} D^{\alpha }(D^{\beta }\omega )(t)=h(t,\omega (t),D^{\beta }\omega (t)), & 0< t< 1, \\ \omega (0)=\omega (1)=D^{\beta }\omega (0)=D^{\beta }\omega (1)=0, \end{cases}\displaystyle \end{aligned}$$ { D α ( D β ω ) ( t ) = h ( t , ω ( t ) , D β ω ( t ) ) , 0 < t < 1 , ω ( 0 ) = ω ( 1 ) = D β ω ( 0 ) = D β ω ( 1 ) = 0 , where $\alpha,\beta \in (1,2]$ α , β ∈ ( 1 , 2 ] , $D^{\alpha }$ D α and $D^{\beta }$ D β are the Riemann–Liouville fractional derivatives. The nonlinear real function h is supposed to be continuous on $[0,1]\times \mathbb{R\times R}$ [ 0 , 1 ] × R × R and satisfy appropriate conditions. Our approach consists in reducing the problem to an operator equation and then applying known results. We provide an approximation of the solution. Our results extend those obtained in (Dang et al. in Numer. Algorithms 76(2):427–439, 2017) to the fractional setting.

scholarly journals Boundary Value Problems for Hybrid Caputo Fractional Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 282 ◽  
Author(s):  
Zidane Baitiche ◽  
Kaddour Guerbati ◽  
Mouffak Benchohra ◽  
Yong Zhou
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Main Tool ◽  
Measures Of Noncompactness ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

In this paper, we discuss the existence of solutions for a hybrid boundary value problem of Caputo fractional differential equations. The main tool used in our study is associated with the technique of measures of noncompactness. As an application, we give an example to illustrate our results.

On generalized fractional integration by parts formulas and their applications to boundary value problems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Rafał Kamocki
Keyword(s):  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Fractional Integration ◽  
Integration By Parts ◽  
Caputo Derivatives ◽  
Mixed Fractional Derivatives ◽  
The Right

Abstract In the paper, we derive a formula of integration by parts for the left-sided Hilfer derivative. Moreover, we present a formula of such a type for the right-sided Hilfer derivative. These results generalize the well-known rules of integration by parts for the Riemann–Liouville and Caputo derivatives. Finally, we use these rules to investigate some boundary value problems involving mixed fractional derivatives.

scholarly journals Existence theory for nonlocal boundary value problems involving mixed fractional derivatives

2019 ◽  
Vol 24 (6) ◽  
Author(s):  
Bashir Ahmad ◽  
K. Ntouyas ◽  
Ahmed Alsaedi
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Existence Theory ◽  
Nonlinear Alternative ◽  
Mixed Fractional Derivatives ◽  
Differential Equations And Inclusions

In this paper, we develop the existence theory for a new kind of nonlocal three-point boundary value problems for differential equations and inclusions involving both left Caputo and right Riemann–Liouville fractional derivatives. The Banach and Krasnoselskii fixed point theorems and the Leray–Schauder nonlinear alternative are used to obtain the desired results for the singlevalued problem. The existence of solutions for the multivalued problem concerning the upper semicontinuous and Lipschitz cases is proved by applying nonlinear alternative for Kakutani maps and Covitz and Nadler fixed point theorem. Examples illustrating the main results are also presented.

On the solvability of the degenerate Noetherian difference-algebraic boundary value problem

2019 ◽  
Vol 33 ◽  
pp. 204-217
Author(s):  
Sergei Chuiko ◽  
Yaroslav Kalinichenko ◽  
Nikita Popov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Oscillations ◽  
Bounded Operator ◽  
Boundary Value ◽  
Homogeneous Part ◽  
Solvability Conditions ◽  
Engineering Control ◽  
Generalized Green Operator

The original conditions of solvability and the scheme of finding solutions of a linear Noetherian difference-algebraic boundary-value problem are proposed in the article, while the technique of pseudoinversion of matrices by Moore-Penrose is substantially used. The problem posed in the article continues to study the conditions for solvability of linear Noetherian boundary value problems given in the monographs of A.M. Samoilenko, A.V. Azbelev, V.P. Maximov, L.F. Rakhmatullina and A.A. Boichuk. The study of differential-algebraic boundary-value problems is closely related to the investigation of boundary-value problems for difference equations, initiated in the works of A.A. Markov, S.N. Bernstein, Y.S. Bezikovych, O.O. Gelfond, S.L. Sobolev, V.S. Ryabenkyi, V.B. Demidovych, A. Halanai, G.I. Marchuk, A.A. Samarskyi, Yu.A. Mytropolskyi, D.I. Martyniuk, G.M. Vainiko, A.M. Samoilenko and A.A. Boichuk. On the other hand, the study of boundary-value problems for difference equations is related to the study of differential-algebraic boundary-value problems initiated in the papers of K. Weierstrass, N.N. Lusin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, N.A. Perestiyk, V.P. Yakovets, A.A. Boichuk, A. Ilchmann and T. Reis. The study of differential-algebraic boundary value problems is also associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, control theory, motion stability theory. The general case of a linear bounded operator corresponding to the homogeneous part of a linear Noetherian difference-algebraic boundary value problem has no inverse is investigated. The generalized Green operator of a linear difference-algebraic boundary value problem is constructed in the article. The relevance of the study of solvability conditions, as well as finding solutions of linear Noetherian difference-algebraic boundary-value problems, is associated with the widespread use of difference-algebraic boundary-value problems obtained by linearizing nonlinear Noetherian boundary-value problems for systems of ordinary differential and difference equations. Solvability conditions are proposed, as well as the scheme of finding solutions of linear Noetherian difference-algebraic boundary value problems are illustrated in detail in the examples.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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