scholarly journals Existence results to a ψ- Hilfer neutral fractional evolution equation with infinite delay

2021 ◽  
Vol 8 (1) ◽  
pp. 101-124
Author(s):  
Fatemeh Norouzi ◽  
Gaston M. N’guérékata
Keyword(s):  
Mild Solution ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Contraction Mapping Principle ◽  
Alternative Theorem ◽  
Banach Contraction ◽  
Equivalent Equation ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

Abstract In this paper, we prove the existence and uniqueness of a mild solution to the system of ψ- Hilfer neutral fractional evolution equations with infinite delay H 𝔻0 αβ;ψ [x(t) − h(t, xt )] = A x(t) + f (t, x(t), xt ), t ∈ [0, b], b > 0 and x(t) = ϕ(t), t ∈ (−∞, 0]. We first obtain the Volterra integral equivalent equation and propose the mild solution of the system. Then, we prove the existence and uniqueness of solution by using the Banach contraction mapping principle and the Leray-Schauder alternative theorem.

scholarly journals Revisiting of some outstanding metric fixed point theorems via E-contraction

2018 ◽  
Vol 26 (3) ◽  
pp. 73-98
Author(s):  
Andreea Fulga ◽  
Erdal Karapınar
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Fixed Point Theorems ◽  
Contractive Mapping ◽  
Contraction Mapping Principle ◽  
Contraction Mappings ◽  
Banach Contraction ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

AbstractIn this paper, we introduce the notion of α-ψ-contractive mapping of type E, to remedy of the weakness of the existing contraction mappings. We investigate the existence and uniqueness of a fixed point of such mappings. We also list some examples to illustrate our results that unify and generalize the several well-known results including the famous Banach contraction mapping principle.

scholarly journals Existence and uniqueness of solutions for system of Hilfer–Hadamard sequential fractional differential equations with two point boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Warissara Saengthong ◽  
Ekkarath Thailert ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study existence and uniqueness of solutions for a system of Hilfer–Hadamard sequential fractional differential equations via standard fixed point theorems. The existence is proved by using the Leray–Schauder alternative, while the existence and uniqueness by the Banach contraction mapping principle. Illustrative examples are also discussed.

scholarly journals Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuqi Wang ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Novel Approach ◽  
Banach Contraction ◽  
Left And Right ◽  
Banach Contraction Mapping Principle

AbstractIn this article, the existence and uniqueness of solutions for a multi-point fractional boundary value problem involving two different left and right fractional derivatives with p-Laplace operator is studied. A novel approach is used to acquire the desired results, and the core of the method is Banach contraction mapping principle. Finally, an example is given to verify the results.

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 the Existence and Uniqueness of Solution to Fractional-Order Langevin Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ahmed Salem ◽  
Noorah Mshary
Keyword(s):  
Fractional Order ◽  
Langevin Equation ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solution ◽  
Main Concern ◽  
Banach Contraction ◽  
Fractional Orders ◽  
Banach Contraction Mapping Principle

In this work, we give sufficient conditions to investigate the existence and uniqueness of solution to fractional-order Langevin equation involving two distinct fractional orders with unprecedented conditions (three-point boundary conditions including two nonlocal integrals). The problem is introduced to keep track of the progress made on exploring the existence and uniqueness of solution to the fractional-order Langevin equation. As a result of employing the so-called Krasnoselskii and Leray-Schauder alternative fixed point theorems and Banach contraction mapping principle, some novel results are presented in regarding to our main concern. These results are illustrated through providing three examples for completeness.

scholarly journals Existence Results for a Class of Semilinear Fractional Partial Differential Equations with Delay in Banach Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Bo Zhu ◽  
Baoyan Han ◽  
Xiangyun Lin
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Mild Solutions ◽  
Contraction Mapping Principle ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Banach Contraction ◽  
Banach Contraction Mapping Principle ◽  
Equations With Delay ◽  
Banach Contraction Mapping

In this paper, we consider a class of nonlinear time fractional partial differential equations with delay. We obtain the existence and uniqueness of the mild solutions for the problem by the theory of solution operator and the general Banach contraction mapping principle. We need not extra conditions to ensure the contraction constant 0<k<1. Therefore, under some general conditions, we obtain our main 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.

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