scholarly journals Three-Point Boundary Value Problems of Nonlinear Second-Orderq-Difference Equations Involving Different Numbers ofq

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Thanin Sitthiwirattham ◽  
Jessada Tariboon ◽  
Sotiris K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Difference Equations ◽  
Boundary Value ◽  
Point Boundary ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Uniqueness Results ◽  
Nonlinear Alternative

We study a new class of three-point boundary value problems of nonlinear second-orderq-difference equations. Our problems contain different numbers ofqin derivatives and integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative) and Leray-Schauder degree theory, some new existence and uniqueness results are obtained. Illustrative examples are also presented.

Existence of solutions for Caputo fractional boundary value problems with integral conditions

2017 ◽  
Vol 33 (2) ◽  
pp. 207-217
Author(s):  
WENJUN LIU ◽  
◽  
HEFENG ZHUANG ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Boundary Value ◽  
Integral Conditions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Nonlinear Alternative ◽  
Banach’S Contraction Principle ◽  
Fractional Boundary Value Problems

In this paper, we investigate the existence results for Caputo fractional boundary value problems with integral conditions. Our analysis relies on Banach’s contraction principle, Leray-Schauder nonlinear alternative, Boyed and Wong fixed point theorem, and Krasnoselskii’s fixed point theorem. As applications, some examples are provided to illustrate our main results.

scholarly journals Nonlocal Sequential Boundary Value Problems for Hilfer Type Fractional Integro-Differential Equations and Inclusions

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 615
Author(s):  
Nawapol Phuangthong ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon ◽  
Kamsing Nonlaopon
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Boundary Value ◽  
Case Examples ◽  
Uniqueness Results ◽  
Nonlinear Alternative ◽  
Differential Equations And Inclusions

In the present research, we study boundary value problems for fractional integro-differential equations and inclusions involving the Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray–Schauder in the single-valued case, while Martelli’s fixed point theorem, a nonlinear alternative for multivalued maps, and the Covitz–Nadler fixed point theorem are used in the inclusion case. Examples are presented to illustrate our results.

scholarly journals Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1879-1894
Author(s):  
Cholticha Nuchpong ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Boundary Value ◽  
Case Examples ◽  
Uniqueness Results ◽  
Nonlinear Alternative ◽  
Differential Equations And Inclusions

Abstract In this paper, we study boundary value problems of fractional integro-differential equations and inclusions involving Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray-Schauder in the single-valued case, while Martelli’s fixed point theorem, nonlinear alternative for multi-valued maps, and Covitz-Nadler fixed point theorem are used in the inclusion case. Examples illustrating the obtained results are also presented.

scholarly journals Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Nichaphat Patanarapeelert ◽  
Thanin Sitthiwirattham
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Integral Boundary ◽  
Uniqueness Results ◽  
Fractional Hahn Integral

The existence and uniqueness results of two fractional Hahn difference boundary value problems are studied. The first problem is a Riemann-Liouville fractional Hahn difference boundary value problem for fractional Hahn integrodifference equations. The second is a fractional Hahn integral boundary value problem for Caputo fractional Hahn difference equations. The Banach fixed-point theorem and the Schauder fixed-point theorem are used as tools to prove the existence and uniqueness of solution of the problems.

scholarly journals On the boundary value problems of piecewise differential equations with left-right fractional derivatives and delay

2021 ◽  
Vol 26 (6) ◽  
pp. 1087-1105
Author(s):  
Yuxin Zhang ◽  
Xiping Liu ◽  
Mei Jia
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
State Variables ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence And Multiplicity

In this paper, we study the multi-point boundary value problems for a new kind of piecewise differential equations with left and right fractional derivatives and delay. In this system, the state variables satisfy the different equations in different time intervals, and they interact with each other through positive and negative delay. Some new results on the existence, no-existence and multiplicity for the positive solutions of the boundary value problems are obtained by using Guo–Krasnoselskii’s fixed point theorem and Leggett–Williams fixed point theorem. The results for existence highlight the influence of perturbation parameters. Finally, an example is given out to illustrate our main results.

scholarly journals Application of Fixed-Point Theory for a Nonlinear Fractional Three-Point Boundary-Value Problem

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 526
Author(s):  
Ehsan Pourhadi ◽  
Reza Saadati ◽  
Sotiris K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Point Theorem ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Point Boundary ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Positive Solutions

Throughout this paper, via the Schauder fixed-point theorem, a generalization of Krasnoselskii’s fixed-point theorem in a cone, as well as some inequalities relevant to Green’s function, we study the existence of positive solutions of a nonlinear, fractional three-point boundary-value problem with a term of the first order derivative ( a C D α x ) ( t ) = f ( t , x ( t ) , x ′ ( t ) ) , a < t < b , 1 < α < 2 , x ( a ) = 0 , x ( b ) = μ x ( η ) , a < η < b , μ > λ , where λ = b − a η − a and a C D α denotes the Caputo’s fractional derivative, and f : [ a , b ] × R × R → R is a continuous function satisfying the certain conditions.

Difference equations in abstract spaces

1998 ◽  
Vol 64 (2) ◽  
pp. 277-284 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O'Regan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Difference Equations ◽  
Boundary Value ◽  
Existence Results ◽  
Second Order ◽  
Abstract Spaces

AbstractExistence results are presented for second order discrete boundary value problems in abstract spaces. Our analysis uses only Sadovskii's fixed point theorem.

scholarly journals EXISTENCE OF POSITIVE SOLUTIONS FOR SUPERLINEAR SEMIPOSITONE $m$-POINT BOUNDARY-VALUE PROBLEMS

2003 ◽  
Vol 46 (2) ◽  
pp. 279-292 ◽  
Author(s):  
Ruyun Ma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Positive Parameter ◽  
Point Boundary ◽  
Existence Of Positive Solutions ◽  
The Fixed Point Theorem

AbstractIn this paper we consider the existence of positive solutions to the boundary-value problems\begin{align*} (p(t)u')'-q(t)u+\lambda f(t,u)\amp=0,\quad r\ltt\ltR, \\[2pt] au(r)-bp(r)u'(r)\amp=\sum^{m-2}_{i=1}\alpha_iu(\xi_i), \\ cu(R)+dp(R)u'(R)\amp=\sum^{m-2}_{i=1}\beta_iu(\xi_i), \end{align*}where $\lambda$ is a positive parameter, $a,b,c,d\in[0,\infty)$, $\xi_i\in(r,R)$, $\alpha_i,\beta_i\in[0,\infty)$ (for $i\in\{1,\dots m-2\}$) are given constants satisfying some suitable conditions. Our results extend some of the existing literature on superlinear semipositone problems. The proofs are based on the fixed-point theorem in cones.AMS 2000 Mathematics subject classification: Primary 34B10, 34B18, 34B15

scholarly journals Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Athasit Wongcharoen ◽  
Bashir Ahmad ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Uniqueness Result ◽  
Existence Results ◽  
Inclusion Problem ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Nonlinear Alternative

We discuss the existence and uniqueness of solutions for the Langevin fractional differential equation and its inclusion counterpart involving the Hilfer fractional derivatives, supplemented with three-point boundary conditions by means of standard tools of the fixed-point theorems for single and multivalued functions. We make use of Banach’s fixed-point theorem to obtain the uniqueness result, while the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem are applied to obtain the existence results for the single-valued problem. Existence results for the convex and nonconvex valued cases of the inclusion problem are derived via the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem respectively. Examples illustrating the obtained results are also constructed. (2010) Mathematics Subject Classifications. This study is classified under the following classification codes: 26A33; 34A08; 34A60; and 34B15.

scholarly journals Triple Positive Solutions for Third-Orderm-Point Boundary Value Problems on Time Scales

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Jian Liu ◽  
Fuyi Xu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Time Scales ◽  
Point Theorem ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Triple Positive Solutions

We study the following third-orderm-point boundary value problems on time scales(φ(uΔ∇))∇+a(t)f(u(t))=0,t∈[0,T]T,u(0)=∑i=1m−2biu(ξi),uΔ(T)=0,φ(uΔ∇(0))=∑i=1m−2ciφ(uΔ∇(ξi)), whereφ:R→Ris an increasing homeomorphism and homomorphism andφ(0)=0,0<ξ1<⋯<ξm−2<ρ(T). We obtain the existence of three positive solutions by using fixed-point theorem in cones. The conclusions in this paper essentially extend and improve the known results.

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