scholarly journals Positive Solutions of a Fractional Boundary Value Problem with Sequential Derivatives

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1489
Author(s):  
Alexandru Tudorache ◽  
Rodica Luca
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Derivatives ◽  
Fixed Point Index ◽  
Index Theory ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential ◽  
Derivatives Of

We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions cover some symmetry cases for the unknown function. In the proof of our main existence result, we use an application of the Krein-Rutman theorem and two theorems from the fixed point index theory.

Positive Solutions for a Semipositone Singular Riemann–Liouville Fractional Differential Problem

2019 ◽  
Vol 20 (7-8) ◽  
pp. 823-831 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Rodica Luca
Keyword(s):  
Positive Solutions ◽  
Fractional Derivatives ◽  
Fixed Point Index ◽  
Index Theory ◽  
Multiple Positive Solutions ◽  
Differential Problem ◽  
Height Functions ◽  
Fixed Point Index Theory ◽  
Fractional Differential ◽  
Bounded Sets

AbstractWe study the existence of multiple positive solutions for a nonlinear singular Riemann–Liouville fractional differential equation with sign-changing nonlinearity, subject to Riemann–Stieltjes boundary conditions which contain fractional derivatives. In the proof of our main theorem, we use various height functions of the nonlinearity of equation defined on special bounded sets, and two theorems from the fixed point index theory.

scholarly journals Existence and multiplicity of positive solutions for a singular Riemann-Liouville fractional differential problem

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 3931-3942
Author(s):  
Rodica Luca
Keyword(s):  
Positive Solutions ◽  
Fractional Derivatives ◽  
Fixed Point Index ◽  
Index Theory ◽  
Point Index ◽  
Differential Problem ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

We investigate the existence and multiplicity of positive solutions for a nonlinear Riemann-Liouville fractional differential equation with a nonnegative singular nonlinearity, subject to Riemann-Stieltjes boundary conditions which contain fractional derivatives. In the proofs of our main results, we use an application of the Krein-Rutman theorem and some theorems from the fixed point index theory.

scholarly journals On a singular Riemann–Liouville fractional boundary value problem with parameters

2021 ◽  
Vol 26 (1) ◽  
pp. 151-168
Author(s):  
Alexandru Tudorache ◽  
Rodica Luca
Keyword(s):  
Fixed Point ◽  
Fixed Point Index ◽  
Nonlocal Boundary ◽  
Index Theory ◽  
Nonlocal Boundary Conditions ◽  
Principal Characteristic ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Semipositone Problem ◽  
Fractional Differential

We investigate the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a positive parameter subject to nonlocal boundary conditions, which contain fractional derivatives and Riemann–Stieltjes integrals. The nonlinearity of the equation is nonnegative, and it may have singularities at its variables. In the proof of the main results, we use the fixed point index theory and the principal characteristic value of an associated linear operator. A related semipositone problem is also studied by using the Guo–Krasnosel’skii fixed point theorem.

Positive solutions for a system of nonlocal fractional boundary value problems

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Johnny Henderson ◽  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Existence Results ◽  
Hammerstein Integral Equations ◽  
Existence Of Positive Solutions ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

AbstractWe investigate the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to multipoint boundary conditions. Existence results for systems of nonlinear Hammerstein integral equations are also presented. Some nontrivial examples are included.

scholarly journals Multiplicity of Positive Solutions for Fractional Differential Equation withp-Laplacian Boundary Value Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Sabbavarapu Nageswara Rao
Keyword(s):  
Fixed Point ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Laplacian Operator ◽  
Multiple Positive Solutions ◽  
Fixed Integer ◽  
Fixed Point Index Theory ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

We investigate the existence of multiple positive solutions of fractional differential equations withp-Laplacian operatorDa+β(ϕp(Da+αu(t)))=f(t,u(t)),  a<t<b,uja=0,  j=0,1,2,…,n-2,u(α1)(b)=ξu(α1)(η),ϕp(Da+αu(a))=0=Da+β1(ϕp(Da+αu(b))), whereβ∈(1,2],α∈(n-1,n],  n≥3,ξ∈(0,∞),η∈(a,b),β1∈(0,1],α1∈{1,2,…,α-2}is a fixed integer, andϕp(s)=|s|p-2s,  p>1,  ϕp-1=ϕq,  (1/p)+(1/q)=1, by applying Leggett–Williams fixed point theorems and fixed point index theory.

scholarly journals Positive Solutions for a Class of Fourth-Orderp-Laplacian Boundary Value Problem Involving Integral Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Yan Sun
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlinear Term ◽  
Integral Boundary Conditions ◽  
Index Theory ◽  
Integral Conditions ◽  
Interesting Point ◽  
Integral Boundary ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

Under some conditions concerning the first eigenvalues corresponding to the relevant linear operator, we obtain sharp optimal criteria for the existence of positive solutions forp-Laplacian problems with integral boundary conditions. The main methods in the paper are constructing an available integral operator and combining fixed point index theory. The interesting point of the results is that the nonlinear term contains all lower-order derivatives explicitly. Finally, we give some examples to demonstrate the main results.

Existence, multiplicity and non-existence of positive solutions for two-point boundary-value problems with strong singularity

2010 ◽  
Vol 140 (6) ◽  
pp. 1187-1196
Author(s):  
Chan-Gyun Kim
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Boundary ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Image Position

We study the existence, multiplicity and non-existence of positive solutions for the singular two-point boundary-value problemswhere $\varphi_{p}(s)=|s|^{p-2}s$, $p>1$, λ is a non-negative real parameter and f ∈ C((0, 1) × [0,∞), (0,∞)). Here, f(t, u) may be singular at t = 0 and/or 1. To obtain the main results we use the global continuation theorem and fixed-point index theory.

scholarly journals Positive solutions of singular nonlinear Sturm-Liouville boundary value problems

2004 ◽  
Vol 45 (4) ◽  
pp. 557-571
Author(s):  
Yan Sun ◽  
Lishan Liu ◽  
Yeol Je Cho
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Sturm Liouville

AbstractBy using fixed point index theory, we present the existence of positive solutions for a Sturm-Liouville singular boundary value problem with at least one positive solution. Our results significantly extend and improve many known results even for non-singular cases.

scholarly journals Positive Solutions for (k,n−k) Conjugate Multipoint Boundary Value Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Yulin Zhao
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Real Banach Space ◽  
Boundary Value ◽  
Index Theory ◽  
Zero Element ◽  
Singular Boundary ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Multipoint Boundary Value Problems

By means of the fixed point index theory of strict-set contraction operator, we study the existence of positive solutions for the multipoint singular boundary value problem(-1)n-kun(t)=f(t,ut),0<t<1,n≥2,1≤k≤n-1,u(0)=∑i=1m-2‍aiu(ξi),u(i)(0)=u(j)(1)=θ,1≤i≤k−1,0≤j≤n−k−1in a real Banach spaceE, whereθis the zero element ofE,0<ξ1<ξ2<⋯<ξm-2<1,ai∈[0,+∞),i=1,2,…,m-2.As an application, we give two examples to demonstrate our results.

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