scholarly journals On the Parametrization of Caputo-Type Fractional Differential Equations with Two-Point Nonlinear Boundary Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 707 ◽  
Author(s):  
Nazım I. Mahmudov ◽  
Sedef Emin ◽  
Sameer Bawanah
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Nonlinear Boundary ◽  
Sufficient Conditions ◽  
Nonlinear Boundary Conditions ◽  
Successive Approximations ◽  
Limit Function ◽  
Fractional Differential ◽  
Linear Restrictions

In this paper, we offer a new approach of investigation and approximation of solutions of Caputo-type fractional differential equations under nonlinear boundary conditions. By using an appropriate parametrization technique, the original problem with nonlinear boundary conditions is reduced to the equivalent parametrized boundary-value problem with linear restrictions. To study the transformed problem, we construct a numerical-analytic scheme which is successful in relation to different types of two-point and multipoint linear boundary and nonlinear boundary conditions. Moreover, we give sufficient conditions of the uniform convergence of the successive approximations. Also, it is indicated that these successive approximations uniformly converge to a parametrized limit function and state the relationship of this limit function and exact solution. Finally, an example is presented to illustrate the theory.

Multiple Solutions of Nonlinear Fractional Differential Equations with p-Laplacian Operator and Nonlinear Boundary Conditions

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Yiliang Liu ◽  
Liang Lu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Upper And Lower Solutions ◽  
Nonlinear Boundary Conditions ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

AbstractIn this paper, we deal with multiple solutions of fractional differential equations with p-Laplacian operator and nonlinear boundary conditions. By applying the Amann theorem and the method of upper and lower solutions, we obtain some new results on the multiple solutions. An example is given to illustrate our results.

scholarly journals Existence and Uniqueness of the Solutions for Fractional Differential Equations with Nonlinear Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Xiping Liu ◽  
Fanfan Li ◽  
Mei Jia ◽  
Ertao Zhi
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Conditions ◽  
Iterative Sequence ◽  
Fractional Differential

We study the existence and uniqueness of the solutions for the boundary value problem of fractional differential equations with nonlinear boundary conditions. By using the upper and lower solutions method in reverse order and monotone iterative techniques, we obtain the sufficient conditions of both the existence of the maximal and minimal solutions between an upper solution and a lower solution and the uniqueness of the solutions for the boundary value problem and present the iterative sequence for calculating the approximate analytical solutions of the boundary value problem and the error estimate. An example is also given to illustrate the main results.

scholarly journals Systems of differential equations with fully nonlinear boundary conditions

1997 ◽  
Vol 56 (2) ◽  
pp. 197-208 ◽  
Author(s):  
H.B. Thompson
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Neumann Boundary ◽  
Nonlinear Boundary Conditions ◽  
Fully Nonlinear ◽  
Systems Of Differential Equations ◽  
Special Cases

We give sufficient conditions involving f, g and ω in order that systems of differential equations of the form y″ = f(x, y, y′), x in [0, 1] with fully nonlinear boundary conditions of the form g((y(0), y(1)), (y′(0), y′(1))) = 0 have solutions y with (x, y) in . We use Schauder degree theory in a novel space. Well known existence results for the Picard, the periodic and the Neumann boundary conditions follow as special cases of our results.

scholarly journals USING A PRIOR ESTIMATE METHOD TO INVESTIGATE SEQUENTIAL HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040004
Author(s):  
GHAZALA NAZIR ◽  
KAMAL SHAH ◽  
THABET ABDELJAWAD ◽  
HAMMAD KHALIL ◽  
RAHMAT ALI KHAN
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Nonlinear Boundary Conditions ◽  
Growth Conditions ◽  
Nonlinear Functions ◽  
Prior Estimate ◽  
Estimate Method ◽  
Fractional Differential

In this paper, our main objective is to develop the conditions that assure the existence of solution to a system of boundary value problems (BVPs) of sequential hybrid fractional differential equations (SHFDEs). The problem is considered under the nonlinear boundary conditions. Nonlinear functions involved in the considered system of SHFDEs are continuous and satisfy the growth conditions. We convert the system of SHFDEs to the system of fixed points problem by using the technique of the topological degree theory also called prior estimate method. We establish sufficient conditions that guarantee the existence and uniqueness of positive solution to the system under consideration. Moreover, suitable results are also developed for the Hyers–Ulam stability analysis for the solution of the considered problem. An example is also included to reveal our main result.

scholarly journals Existence and uniqueness of nonlocal boundary conditions for Hilfer–Hadamard-type fractional differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmad Y. A. Salamooni ◽  
D. D. Pawar
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Nonlinear Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential

AbstractIn this paper, we use some fixed point theorems in Banach space for studying the existence and uniqueness results for Hilfer–Hadamard-type fractional differential equations $$ {}_{\mathrm{H}}D^{\alpha ,\beta }x(t)+f\bigl(t,x(t)\bigr)=0 $$ D α , β H x ( t ) + f ( t , x ( t ) ) = 0 on the interval $(1,e]$ ( 1 , e ] with nonlinear boundary conditions $$ x(1+\epsilon )=\sum_{i=1}^{n-2}\nu _{i}x(\zeta _{i}),\qquad {}_{\mathrm{H}}D^{1,1}x(e)= \sum_{i=1}^{n-2} \sigma _{i}\, {}_{\mathrm{H}}D^{1,1}x( \zeta _{i}). $$ x ( 1 + ϵ ) = ∑ i = 1 n − 2 ν i x ( ζ i ) , H D 1 , 1 x ( e ) = ∑ i = 1 n − 2 σ i H D 1 , 1 x ( ζ i ) .

Sign in / Sign up

Export Citation Format

Share Document