scholarly journals Multiple solutions for fractional differential equations with nonlinear boundary conditions

2010 ◽  
Vol 59 (8) ◽  
pp. 2880-2886 ◽  
Author(s):  
Xiping Liu ◽  
Mei Jia
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Fractional Differential

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 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.

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 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 ) .

scholarly journals Existence results for a general class of sequential hybrid fractional differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rahmat Ali Khan ◽  
Shaista Gul ◽  
Fahd Jarad ◽  
Hasib Khan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
General Class ◽  
Nonlinear Boundary ◽  
Degree Theory ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Functions ◽  
Nonlinear Boundary Value Problems ◽  
Uniqueness Results ◽  
Fractional Differential

AbstractIn this paper, we study a class of nonlinear boundary value problems (BVPs) consisting of a more general class of sequential hybrid fractional differential equations (SHFDEs) together with a class of nonlinear boundary conditions at both end points of the domain. The nonlinear functions involved depend explicitly on the fractional derivatives. We study the necessary conditions required for the unique solution to the suggested BVP under the Caratheodory conditions using the technique of measure of noncompactness and degree theory. We also develop conditions for uniqueness results and also on stability analysis.

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