scholarly journals EXISTENCE AND STABILITY ANALYSIS OF SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATION WITH NONLOCAL INTEGRAL AND ANTI-PERIODIC-TYPE BOUNDARY CONDITIONS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040006 ◽  
Author(s):  
AMITA DEVI ◽  
ANOOP KUMAR ◽  
THABET ABDELJAWAD ◽  
AZIZ KHAN
Keyword(s):  
Boundary Conditions ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Banach Contraction ◽  
Existence And Stability ◽  
Fractional Differential ◽  
Type Boundary ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

In this paper, we deal with the existence and uniqueness (EU) of solutions for nonlinear Langevin fractional differential equations (FDE) having fractional derivative of different orders with nonlocal integral and anti-periodic-type boundary conditions. Also, we investigate the Hyres–Ulam (HU) stability of solutions. The existence result is derived by applying Krasnoselskii’s fixed point theorem and the uniqueness of result is established by applying Banach contraction mapping principle. An example is offered to ensure the validity of our obtained results.

scholarly journals Hilfer–Hadamard Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions

2021 ◽  
Vol 5 (4) ◽  
pp. 195
Author(s):  
Bashir Ahmad ◽  
Sotiris K. Ntouyas
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Existence Results ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

This paper is concerned with the existence and uniqueness of solutions for a Hilfer–Hadamard fractional differential equation, supplemented with mixed nonlocal (multi-point, fractional integral multi-order and fractional derivative multi-order) boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying the fixed point theorems due to Krasnoselskiĭ and Schaefer and Leray–Schauder nonlinear alternatives. We demonstrate the application of the main results by presenting numerical examples. We also derive the existence results for the cases of convex and non-convex multifunctions involved in the multi-valued analogue of the problem at hand.

ON HADAMARD FRACTIONAL CALCULUS

Fractals ◽  
2017 ◽  
Vol 25 (03) ◽  
pp. 1750033 ◽  
Author(s):  
LI MA ◽  
CHANGPIN LI
Keyword(s):  
Fractional Calculus ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Fractional Operators ◽  
Initial Value ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping ◽  
Derivative Order

This paper is devoted to the investigation of the Hadamard fractional calculus in three aspects. First, we study the semigroup and reciprocal properties of the Hadamard-type fractional operators. Then, the definite conditions of certain class of Hadamard-type fractional differential equations (HTFDEs) are proposed through the Banach contraction mapping principle. Finally, we prove a novel Gronwall inequality with weak singularity and analyze the dependence of solutions of HTFDEs on the derivative order and the perturbation terms along with the proposed initial value conditions. The illustrative examples are presented as well.

scholarly journals Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Asma ◽  
Sana Shabbir ◽  
Kamal Shah ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Stability Results ◽  
Rassias Stability ◽  
Banach Contraction Mapping ◽  
Implicit Fractional Differential Equations

AbstractSome fundamental conditions and hypotheses are established to ensure the existence, uniqueness, and stability to a class of implicit boundary value problems (BVPs) with Atangana–Baleanu–Caputo type derivative and integral. The required results are established by utilizing the Banach contraction mapping principle and fixed point theorem of Krasnoselskii. In addition, various types of stability results including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability are formulated for the problem under consideration. Pertinent examples are given to justify the results we obtain.

scholarly journals Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Systems with Nonlocal Coupled Fractional Integral Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 174
Author(s):  
Chanakarn Kiataramkul ◽  
Weera Yukunthorn ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Boundary Conditions ◽  
Fractional Integral ◽  
Integral Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Integral Boundary ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle

In this paper, we initiate the study of existence of solutions for a fractional differential system which contains mixed Riemann–Liouville and Hadamard–Caputo fractional derivatives, complemented with nonlocal coupled fractional integral boundary conditions. We derive necessary conditions for the existence and uniqueness of solutions of the considered system, by using standard fixed point theorems, such as Banach contraction mapping principle and Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.

scholarly journals A Self-Adjoint Coupled System of Nonlinear Ordinary Differential Equations with Nonlocal Multi-Point Boundary Conditions on an Arbitrary Domain

2021 ◽  
Vol 11 (11) ◽  
pp. 4798
Author(s):  
Hari Mohan Srivastava ◽  
Sotiris K. Ntouyas ◽  
Mona Alsulami ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Coupled System ◽  
Contraction Mapping Principle ◽  
Arbitrary Domain ◽  
Banach Contraction ◽  
Coupled Boundary Conditions ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

The main object of this paper is to investigate the existence of solutions for a self-adjoint coupled system of nonlinear second-order ordinary differential equations equipped with nonlocal multi-point coupled boundary conditions on an arbitrary domain. We apply the Leray–Schauder alternative, the Schauder fixed point theorem and the Banach contraction mapping principle in order to derive the main results, which are then well-illustrated with the aid of several examples. Some potential directions for related further researches are also indicated.

scholarly journals On the Solvability of Caputo -Fractional Boundary Value Problem Involving -Laplacian Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hüseyin Aktuğlu ◽  
Mehmet Ali Özarslan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Laplacian Operator ◽  
Fractional Boundary Value Problem ◽  
Solvability Conditions ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

We consider the model of a Caputo -fractional boundary value problem involving -Laplacian operator. By using the Banach contraction mapping principle, we prove that, under some conditions, the suggested model of the Caputo -fractional boundary value problem involving -Laplacian operator has a unique solution for both cases of and . It is interesting that in both cases solvability conditions obtained here depend on , , and the order of the Caputo -fractional differential equation. Finally, we illustrate our results with some examples.

scholarly journals Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces

2019 ◽  
Vol 3 (2) ◽  
pp. 27 ◽  
Author(s):  
Ayşegül Keten ◽  
Mehmet Yavuz ◽  
Dumitru Baleanu
Keyword(s):  
Banach Spaces ◽  
Fractional Derivatives ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Kernel Functions ◽  
Banach Contraction ◽  
Exponential Decay Law ◽  
Nonlocal Cauchy Problem ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.

scholarly journals On a functional equation arising in mathematical biology and theory of learning

2015 ◽  
Vol 24 (1) ◽  
pp. 9-16
Author(s):  
VASILE BERINDE ◽  
◽  
ABDUL RAHIM KHAN ◽  
◽  
Keyword(s):  
Functional Equation ◽  
Mathematical Biology ◽  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Approximation Result ◽  
Banach Contraction ◽  
The Right ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

V. Istrat¸escu [Istr ˘ at¸escu, V. I., ˘ On a functional equation, J. Math. Anal. Appl., 56 (1976), No. 1, 133–136] used the Banach contraction mapping principle to establish an existence and approximation result for the solution of the functional equation ϕ(x) = xϕ((1 − α)x + α) + (1 − x)ϕ((1 − β)x), x ∈ [0, 1], (0 < α ≤ β < 1), which is important for some mathematical models arising in biology and theory of learning. This equation has been studied by Lyubich and Shapiro [A. P. Lyubich, Yu. I. and Shapiro, A. P., On a functional equation (Russian), Teor. Funkts., Funkts. Anal. Prilozh. 17 (1973), 81–84] and subsequently, by Dmitriev and Shapiro [Dmitriev, A. A. and Shapiro, A. P., On a certain functional equation of the theory of learning (Russian), Usp. Mat. Nauk 37 (1982), No. 4 (226), 155–156]. The main aim of this note is to solve this functional equation with more general arguments for ϕ on the right hand side, by using appropriate fixed point tools.

scholarly journals Maia type fixed point theorems for Ćirić-Prešić operators

2018 ◽  
Vol 10 (1) ◽  
pp. 18-31
Author(s):  
Margareta-Eliza Balazs
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Contraction Mapping ◽  
Fixed Point Theorems ◽  
Acta Math ◽  
Contraction Mapping Principle ◽  
Contraction Condition ◽  
Banach Contraction ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

Abstract The main aim of this paper is to obtain Maia type fixed point results for Ćirić-Prešić contraction condition, following Ćirić L. B. and Prešić S. B. result proved in [Ćirić L. B.; Prešić S. B. On Prešić type generalization of the Banach contraction mapping principle, Acta Math. Univ. Comenian. (N.S.), 2007, v 76, no. 2, 143–147] and Luong N. V. and Thuan N. X. result in [Luong, N. V., Thuan, N. X., Some fixed point theorems of Prešić-Ćirić type, Acta Univ. Apulensis Math. Inform., No. 30, (2012), 237–249]. We unified these theorems with Maia’s fixed point theorem proved in [Maia, Maria Grazia. Un’osservazione sulle contrazioni metriche. (Italian) Rend. Sem. Mat. Univ. Padova 40 1968 139–143] and the obtained results are proved is the present paper. An example is also provided.

scholarly journals A Fixed Point Result with a Contractive Iterate at a Point

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 606 ◽  
Author(s):  
Badr Alqahtani ◽  
Andreea Fulga ◽  
Erdal Karapınar
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Continuity Condition ◽  
Ulam Stability ◽  
Banach Contraction ◽  
Iteration Number ◽  
The Given ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

In this manuscript, we define generalized Kincses-Totik type contractions within the context of metric space and consider the existence of a fixed point for such operators. Kincses-Totik type contractions extends the renowned Banach contraction mapping principle in different aspects. First, the continuity condition for the considered mapping is not required. Second, the contraction inequality contains all possible geometrical distances. Third, the contraction inequality is formulated for some iteration of the considered operator, instead of the dealing with the given operator. Fourth and last, the iteration number may vary for each point in the domain of the operator for which we look for a fixed point. Consequently, the proved results generalize the acknowledged results in the field, including the well-known theorems of Seghal, Kincses-Totik, and Banach-Caccioppoli. We present two illustrative examples to support our results. As an application, we consider an Ulam-stability of one of our results.

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