The Exponential Functions of Discrete Fractional Calculus
Ferhan M. Atıcı
Western Kentucky University
Department of Mathematics
Bowling Green, KY 42101, USA
In this talk, we begin with proving some properties of the exponential functions of discrete fractional calculus along with some relations to the discrete Mittag-Leffler functions. We then introduce
sequential linear difference equations of fractional order with constant coefficients. Corresponding
characteristic equation is defined and considered in two cases where characteristic real roots are
same or distinct. We define a generalized Casoration for a set of discrete functions. As a consequence, for solutions, their nonzero Casoration ensures their linear independence. We introduce
completely monotonic functions on discrete domains. A sequence of results are obtained to prove
that the discrete Mittag-Leffler function, Fα,β (λ, x1 ), is discrete completely monotonic. We close
the talk with an application of exponential functions in modeling sigmoidal curves of tumor growth
of cancer.
Limit-point/Limit-circle Problem for Differential Equations
Miroslav Bartušek
Masaryk University
Department of Mathematics and Statistics
Brno, Czech Republic
One hundred years ago, H. Weyl initiated a problem where all solutions of a linear second order
differential equation are in L2 (limit-circle problem) or not in L2 (limit-point problem). We discuss
generalizations to second order and higher order differential equations. Main stress is devoted
to second order equations with p-Laplacian and damping. Asymptotic formulas for solutions are
derived and applied for obtaining sufficient and/or necessary conditions for solving the above given
problem.
1
Asymptotic Behavior of the Eigenvalues or
Toeplitz Integral Operators Associated with
the Hankel Transform
John V. Baxley
Wake Forest University
Department of Mathematics
Winston-Salem, NC 27109, USA
Let ν be a positive constant and let J(x) = x1/2−ν Jν−1/2 (x), 0 ≤ x < ∞, where Jα is the usual
Bessel function of order α. For a given real function F belonging to L1 (0, ∞), we define for u, v > 0,
R∞
RA
ρ(u, v) = 0 F (t)J(ut)J(vt)t2ν dt. For A ≥ 1, then (TA h)(x) = 0 h(y)ρ(x, y)y 2ν dy, 0 < x ≤ A,
is the corresponding (finite section) Toeplitz integral operator and TA : L2 (0, A; x2ν ) → L2 (0, A; x2ν )
is compact. We provide conditions on F sufficient to determine the asymptotic behavior of the
eigenvalues of TA as A → ∞. Of interest is the fact that the eigenvalues of a specific singular
ordinary differential operator in L2 (0, ∞; x2ν ) with compact inverse plays a critical role in the
asymptotic formula.
Lattice Boltzmann method and Simulation of Solutions
for Two-dimensional Sine-Gordon Equation
Huilin Lai1 and Xingwang Chen2
1
Fujian Normal University
School of Mathematics and Computer Science
Fuzhou, Fujian 350007, China
2
Savannah State University
Department of Engineering Technology and Mathematics
Savannah, GA 31404, USA
In this paper, a lattice Boltzmann model is developed to solve Two-dimensional Sine-Gordon
Equation. Through selecting equilibrium distribution function and an amending function properly,
the governing evolution equation can be recovered correctly according to our proposed scheme, in
which the Chapman-Enskog expansion is employed. The presented algorithm will be validated by
numerical experiments.The numerical results agrees well with our analytical solution which implies
the proposed method can be used to solve more general equations.
2
Chaos in a two-stage ecological model
Ross Chiquet
University of Louisiana at Lafayette
Department of Mathematics
Lafayette, LA 70503, USA
We develop a general discrete juvenile-adult population model composed of continuous, nonautonomous difference equations with strong nonlinear Ricker-type survivorship functions. We
begin the examination of our model’s dynamics by exploring the stability of the extinction equilibrium. Using bifurcation analysis, we explore the effects different birth rates have on our system
and which birthrates lead to chaos. We then make use of Lyapunov Exponents to show that our
system possesses sensitivity to initial conditions for certain birthrates. Finally, we examine our
model when the survivorship functions are mixed, strong nonlinearity for one age class and weak
for the other.
Generalized Iterative Technique for Nonlinear
Riemann-Liouville Fractional Differential Equations
Zachary Denton
North Carolina A & T State University
Department of Mathematics
Greensboro, NC 27412, USA
Existence and comparison results of the linear and nonlinear Riemann-Liouville fractional differential equations and nonlinear systems of order q, 0 < q < 1, are recalled and modified where
necessary. A generalized iterative technique is developed for decomposed nonlinear fractional differential equations of order q, Using coupled upper and lower solutions. The decomposition consists
of two functions that are convex/concave and a third that is only Lipschitz. Uniform convergence
is proved via weighted sequences, and quadratic convergence is proved where possible.
3
Nonlinear oscillations: continuous versus discrete
Zuzana Došlá
Masaryk University
Department of Mathematics and Statistics
Brno, Czech Republic
This talk presents the recent results in the oscillatory and asymptotic theory for the nonlinear
differential and difference equations.
We give new oscillation criteria for equations with generalized one-dimensional p-Laplacian and
we present new integral and summation inequalities which enable to give the complete analysis of
the nonoscillatory solutions of second-order half-linear, super-linear and sub-linear equations.
We show the similarities and discrepancies between continuous and discrete case and we will
formulate some open problems in this theory.
This is a joint work with M. Cecchi and M. Marini.
A 2nth Order Problem With Symmetry
Abdulmalik Al Twaty1 and Paul Eloe2
1
University of Benghazi
Benghazi, Libya
2
University of Dayton
Dayton, Ohio 45469, USA
In this article we apply an extension of a Leggett-Williams type fixed point theorem to a twopoint boundary value problem for a 2n−th order ordinary differential equation. The fixed point
theorem employs concave and convex functionals defined on a cone in a Banach space. Inequalities
that extend the notion of concavity to 2n−th order differential inequalities are derived and employed
to provide the necessary estimates. Symmetry is employed in the construction of the appropriate
Banach space.
4
On Some Topics in Nonlinear Oscillation
Lynn Erbe1 and Allan Peterson1 and BaoGuo Jia2
1
University of Nebraska-Lincoln
Department of Mathematics
Lincoln, NE 68588 USA
2
Zhongshan University
Department of Mathematics
Guangzhou, China
In this paper we investigate the oscillation of certain second order nonlinear dynamic equations
including the generalized Emden-Fowler equation. Our results extend and improve a number of
known results for oscillation of second order dynamic equations. Some examples are given to
illustrate the main results.
Stability and Turing Instability
of a Two-innovation Diffusion System
Wenying Feng and Lu Zhang
Department of Computing & Information Systems
Department of Mathematics
Trent University
Peterborough, Ontario, K9J 7B8, Canada
I will present a dynamical system model that represents the process of two-innovation diffusion.
Both the continuous and discrete forms of the model will be discussed. Stability and bifurcation regions are compared. We will also show that Turing instability conditions are satisfied by
migration-diffusion of the discrete model and therefore Turing patterns can be produced from the
numerical simulation.
5
Positive solutions of a nonlocal fractional BVP
Gennaro Infante
Università della Calabria
Dipartimento di Matematica ed Informatica
Cosenza, Italy
We discuss the existence of multiple positive solutions for a nonlocal fractional problem recently
considered by Nieto and Pimental. Our approach relies on classical fixed point index.
On the inverse doping profile problem
Victor Isakov
Wichita State University
Department of Mathematics and Statistics
Wichita, KS 67260, USA
We consider the identification of a so-called doping profile (source term) in the system of non
linear elliptic equations modeling a semiconductor device. We give a simplification of the model
and find useful adjoint problems motivated by applications to inverse problems and by a presence
of certain small physical parameters As a result we obtain first uniqueness results for the so-called
p − n junction from a realistic industrial data. Proofs use potential theory, energy estimates, and
the Novikov’s method for the inverse problem of gravimetry.
Asymptotic Stability of Solutions of a Nonlinear Integral Equation
Muhammad N. Islam
University of Dayton
Department of Mathematics
Dayton, OH 45469-2316 USA
In this paper we study the existence of asymptotically stable solutions of a nonlinear Volterra
integral equation. Schauder’s fixed point theorem is used as the primary mathematical tool.
6
Computing the Positive Solutions of the Discrete
Third-Order Three-Point Right Focal BVPs
Jun Ji and Bo Yang
Department of Mathematics and Statistics
Kennesaw State University
Kennesaw, GA 30144, USA
In this talk, we are concerned with the following discrete third-order three-point boundary value
problem:
∆3 yi = λai yi+1 for i = 0, 1, 2, · · · , n − 2,
(1)
2
y0 = ∆yp = ∆ yn−1 = 0,
(2)
where p and n are positive integers under the assumption that
(H1) n ≥ 2 and p are fixed integers such that (n − 1)/2 < p ≤ n − 1, and
Pn−2
(H2) ai ≥ 0 for 0 ≤ i ≤ n − 2 and i=0 ai > 0, 0 ≤ i ≤ n − 2 .
Note that the problem (1)-(2) is equivalent to the matrix equation of the form
(−D + λA)y = 0
where y = (y1 , y2 , y3 , · · · , yn )T , A = diag(a0 , a1 , a2 , · · · , an−2 , 0), and D is an n × n matrix given
by





D=




3
−1
0
···
0
0
0
−3
3
−1
···
0
0
0
1
0
0
−3
1
0
3 −3
1
··· ··· ···
0
0
0
0
0
0
0
0
0
···
···
···
···
···
···
···
0
0
0
···
0
0
−1
0 ···
0 ···
0 ···
··· ···
0 ···
0 ···
1 ···
0
0
0
0
0
0
0
0
0
0
0
0
··· ··· ··· ···
−1
3 −3
1
0 −1
2 −1
0
0
0
0





.




The existence of positive solutions to the discrete third-order three-point boundary value problems was
recently established in the literature. In this talk, we will propose an algorithm for the computation of
positive solutions. The proposed algorithm is iterative in nature. At each iteration of the algorithm, a
structured system of linear equations is solved through a Crout-like factorization technique by exploring the
pattern of zero elements in LU decomposition of the coefficient matrix and by avoiding the computation of
these zeros. We will show that the solver proposed for this problem requires only (8n−3p−9) multiplications
or divisions at each iteration. The proposed method will be utilized to find positive solutions of the thirdorder three-point boundary value problems of differential equation. We will demonstrate the efficiency of
the method through a numerical example.
7
On positive solutions of a discrete fourth order problem
John R. Graef1 , Lingju Kong1 , Min Wang1 , and Bo Yang2
1
University of Tennessee at Chattanooga
Department of Mathematics
Chattanooga, TN 37403, USA
2
Kennesaw State University
Department of Mathematics and Statistics
Kennesaw, GA 30144, USA
We study a class of nonlinear discrete fourth order Lidstone boundary value problems with dependence
on two parameters. The existence, uniqueness, and dependence of positive solutions on the parameters
are discussed. Two sequences are constructed so that they converge uniformly to the unique solution of
the problems. One example is included in the paper. Numerical computations of the example confirm our
theoretical results. Recent results in the literature are extended and improved.
The 21st Century Challenges: Differential Equations
G.S. Ladde
University of South Florida
Department of Mathematics and Statistics
Tampa, FL 33620-5700, USA
Recently, there are tremendous efforts underway to have multidisciplinary activities or interactions at
all level of the components of the higher educational institutions. In this work, we briefly provide the
mathematical education and research directions to prepare our 21st century work force to meet the demanding
challenges in the rapidly changing environment. By out-lining the role and scope of the multidisciplinary
activities or interactions, several illustrations will be provided for the adjustment to meet the expected needs
and the challenges in the areas of teaching and the research trends in the modeling, methods and analysis,
in particular, differential equations.
8
The Eu0 space and u0 -positive operators
Kunquan Lan
Ryerson University
Department of Mathematics
Toronto, Ontario, Canada M5B 2K3
In this talk, we introduce the Eu0 space and wedge Pu0 by a wedge P satisfying P 6= −P and provide
some properties of Eu0 and Pu0 . As applications of these properties, we prove a necessary and sufficient
condition for an operator to be u0 -positive.
Backward Bifurcation in a Mathematical Model
for HIV Infection in vivo with Anti-Retroviral Treatment
Michael Y. Li1 and Liancheng Wang2
1
University of Alberta
Department of Mathematical and Statistical Sciences
Edmonton, Alberta, T6G 2G1 Canada
2
Kennesaw State University
Department of Mathematical and Statistical Sciences
Kennesaw, GA 30144, USA
Anti-retroviral treatments (ART) such as HAART have been used to control the replication of HIV virus
in HIV-positive patients. In this paper, we study an in-host model of HIV infection with ART and carry out
mathematical analysis of the global dynamics and bifurcations of the model in different parameter regimes.
Among our discoveries is a parameter region for which backward bifurcation can occur.Biologically, the
catastrophic behaviors associated with backward bifurcations may explain the sudden rebound of HIV viral
load when ART is stopped, and possibly provide an explanation for the viral blips during ART suppression
of HIV.
9
Existence and uniqueness of solutions of boundary
value problems by solution matching
Johnny Henderson and Xueyan Sherry Liu
Baylor University
Department of Mathematics
Waco, TX 76798, USA
We are concerned with the existence and uniqueness of solutions to boundary value problems on an
interval [a, c] for the nth order ordinary differential equation y (n) = f (x, y, y 0 , . . . , y (n−1) ) for n ≥ 3, by
matching solutions on [a, b] with solutions on [b, c]. Monotonicity conditions on f are imposed for distinct
cases that arise.
Generalizing the Discrete and Continuous
Domains of Boundary Data Smoothness for Solutions
of Second Order Boundary Value Problems
Jeffrey W. Lyons
Nova Southeastern University
Division of MST
Fort Lauderdale, FL 33314 USA
In this talk, we ascertain the smoothness of the boundary conditions for solutions of the dynamic boundary value problem y ∆∆ = f (t, y, y ∆ ), y(t1 ) = y1 , y(t2 ) = y2 on the time scale hZ with respect to the
boundary data. Afterward, we show that the results of two recent papers can be attained by (1) setting h=1
which yields a difference equation on Z and (2) letting h limit to zero which yields a differential equation on
R.
10
Nonoscillation for superlinear Emden-Fowler
differential equations
Mauro Marini
University of Florence
Dept. of Mathematics and Informatics “U. Dini”
Florence 50139, Italy
Consider the second order Emden-Fowler type differential equation
0
a(t)|x0 |α sgn x0 + b(t)|x|β sgn x = 0
R ∞ −1/α
where a, b are continuous functions for t ≥ 0, a(t) > 0, b(t) ≥ 0 and 0 < α < β. When Ya =
a
is
divergent, it is known that this equation can have three possible types of nononscillatory solutions (subdominant, intermediate and dominant solutions). The open problem on their possible coexistence is solved and
sufficient conditions for the existence of intermediate solutions are presented too.
These results have been achieved in a joint research with Zuzana Dosla (University of Brno).
Positive Solutions of a Nonlinear Third Order
Boundary Value Problem
Jeffrey T. Neugebauer
Eastern Kentucky University
Richmond, KY 40475
We classify extremal points of the third order boundary value problem
u000 + p(x)u = 0, 0 ≤ x ≤ b,
satisfying the boundary conditions
u(0) = u0 (η) = u00 (b) = 0,
where η and β are fixed, and 0 < 1/2 < η < b. These results are then used to show the existence of a positive
solution of the third order nonlinear boundary value problem
u000 + f (x, u) = 0
satisfying the same boundary conditions.
11
On Positivity Improving Semigroups and Resolvents
of Regular Sturm–Liouville Operators
Roger Nichols
University of Tennessee at Chattanooga
Department of Mathematics
Chattanooga, TN 37403, USA
In this talk, we consider self-adjoint extensions of the minimal operator associated with Sturm–Liouvilletype differential expressions,
0
1
− p[f 0 + sf ] + sp[f 0 + sf ] + qf on (a, b) ⊂ R,
τf =
r
where the coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p 6= 0, r > 0 a.e.
on (a, b), and p−1 , q, r, s ∈ L1loc ((a, b); dx), and f belongs to a suitable class of functions. In the case
where τ is regular on (a, b) (i.e., when (a, b) is a finite interval and L1loc ((a, b); dx) above may be replaced by
L1 ((a, b); dx)), all self-adjoint extensions of the minimal operator are characterized by boundary conditions
at a and b. Using integral operator techniques and the Beurling–Deny criterion, under the assumption
p > 0 a.e. on (a, b), we classify all boundary conditions leading to self-adjoint extensions which generate
a positivity preserving semigroup (equivalently, resolvent). Using a Krein-type resolvent identity, we show
that the semigroup and resolvent are actually positivity improving in this case. As a result, the eigenspace
corresponding to the smallest eigenvalue of such a self-adjoint extension is one-dimensional and spanned by
a positive function.
The content of this talk is based on various joint collaborations with Steve Clark (Missouri S&T),
Jonathan Eckhardt (Vienna), Fritz Gesztesy (Missouri) and Gerald Teschl (Vienna, ESI).
Global Attractivity of First Order Differential Equations with Delay
Seshadev Padhi
Birla Institute of Technology
Department of Applied Mathematics
Mesra Ranchi-835215, India
In this talk, we have obtained sufficient conditions for the global attractivity of solutions of of first order
delay differential equations of the form
x0 (t) + a(t)x(t − τ ) = 0
where a : [σ, ∞) → (0, ∞) and τ > 0 is a real number. The obtained result, are then applied to many
mathematical models to obtain new sufficient conditions on global attractivity.
12
Multiple Periodic Solutions for Some Difference
Equations Subjected to Allee Effects
Smita Pati
Birla Institute of Technology
Department of Applied Mathematics
Mesra, Ranchi, Jharkhand-835215, India
This paper deals with the existence of multiple positive periodic solutions to a nonautonomous scalar
difference equation subjected to Allee effects. Existence is established using Leggett-Williams multiple fixed
point theorem. This result is employed to find the minimum number of positive periodic solutions admitted
by a model representing dynamics of a renewable resource that is subjected to Allee effects in a seasonally
varying environment.
Nabla Fractional Calculus
Allan Peterson
University of Nebraska-Lincoln
Department of Mathematics
Lincoln, NE 68588 USA
This will be an introductory talk on the properties of nabla fractional sums and nabla fractional differences.
13
Global attractivity in a higher order nonlinear difference
equation and applications
Chuanxi Qian
Mississippi State University
Department of Mathematics and Statistics
Mississippi State, MS 39762, USA
Consider the following higher order nonlinear difference equation of the form
xn+1 = (1 − tn )xn + tn f (xn−k ), n = 0, 1, · · ·
(3)
where f is a continuous function mapping interval I = [a, b] into I, {tn } is an arbitrary sequence in [0, 1] and
k is a nonnegative integer. By a solution of Eq.(1), we mean a sequence {xn } which is defined for n ≥ −k
and which satisfies Eq.(1) for n ≥ 0. With Eq.(1) we associate an initial condition of the form
a−k , a−k+1 , · · · , a0 ∈ I.
(4)
Then IVP (1)-(2) has a unique solution {xn } valid for n ≥ 0 and
xn ∈ I, n = 0, 1, · · · .
In this presentation, we first introduce some background of the equation, and then talk about several sufficient
conditions for a fixed point of f to be a global attractor of all solutions of the equation. Applications to
some difference equations derived from mathematical biology are also shown.
Boundedness in Functional Difference Equations
Youssef N. Raffoul
University of Dayton
Department of Mathematics
Dayton, OH 45469-2316 USA
We consider a system of functional difference equations of the form
x(n + 1) = G(n, xn ), x(n) ∈ Rk
where G : Z+ × Rk → Rk is continuous in x and give necessary and sufficient conditions for uniform
boundedness of all solutions.
14
Generalized Monotone Iterative Method for
Caputo Fractional Differential Equations with
Anti-Periodic Boundary Conditions.
Diego Ramı́rez
Lamar University
Beaumont, TX 77710, USA
The purpose of this work is to develop a Monotone Method for the antiperiodic boundary value problem
with 0 < q < 1 on J = [0, T ],
c
Dq u(t)
u(0)
= f (t, u(t)) + g(t, u(t)),
= −u(T ),
where f (t, u) is increasing in u and g(t, u) is decreasing in u.
We will define coupled lower and upper solutions v0 (t) and w0 (t). Next we will construct two sequences
{vn (t)}, {wn (t)} which converge uniformly and monotonically to coupled minimal and maximal solutions ρ
and r, respectively; i.e. ρ and r satisfy the system
c
c
Dq ρ(t)
q
D r(t)
= f (t, ρ(t)) + g(t, r(t)),
ρ(0) = −r(T ),
= f (t, r(t)) + g(t, ρ(t)),
r(0) = −ρ(T ).
Our iterates are solutions of initial value problems.
15
Second-order impulsive differential inclusions
and nonsmooth critical point theory
Yu Tian
Beijing University of Posts and Telecommunications
School of Science
Beijing, 100876, P.R.China
A nonsmooth version of a three critical point theorem of Ricceri (due to Iannizzotto) is used to obtain
three anti-periodic solutions for a second-order impulsive differential inclusion with a perturbed nonlinearity
and two parameters, that is

 −(Φp (u0 (x)))0 + M Φp (u(x)) ∈ λF (u(x)) + µG(x, u(x)) in [0, T ] \ {x1 , x2 , · · · , xm },
−∆Φp (u0 (xk )) = Ik (u(xk )), k = 1, 2, · · · , m,
(5)

u(0) = −u(T ), u0 (0) = −u0 (T ),
where p > 1, T > 0, M ≥ 0, Φp (x) := |x|p−2 x, 0 = x0 < x1 < · · · < xm < xm+1 = T, ∆Φp (u0 (xk )) =
0 −
0 +
0 −
0
Φp (u0 (x+
k )) − Φp (u (xk )), where u (xk ) and u (xk ) denote the right and left limits, respectively, of u (x) at
x = xk , Ik ∈ C(R, R), k = 1, 2, · · · , m, λ, µ are positive parameters, and F is a multifunction defined on R
satisfying
(F1 ) F : R → 2R is upper semicontinuous (i.e., u.s.c.) with compact convex values;
(F2 ) min F, max F : R → R are Borel measurable;
(F3 ) |ξ| ≤ a(1 + |s|r−1 ) for all s ∈ R, ξ ∈ F (s), r > 1(a > 0),
G is a multifunction defined on [0, T ] × R, satisfying
(G1 ) G(x, ·) : R → 2R is u.s.c. with compact convex values for a.e. x ∈ [0, T ]\{x1 , x2 , · · · , xm },
(G2 ) min G, max G : [0, T ]\{x1 , x2 , · · · , xm } × R → R are Borel measurable;
(G3 ) |ξ| ≤ a(1 + |s|r−1 ) for a.e. x ∈ [0, T ], s ∈ R, ξ ∈ G(x, s), r > 1.
We shall prove that, whenever λ is large enough and µ is small enough, (5) admits at least three solutions.
Moreover, we shall achieve an estimate of the solutions’ norms independent of G, λ and µ. The results will
be obtained by defining the weak solutions, proving that the weak solution is just the solution of original
problem and verifying regular assumption since impulsive terms appear.
16
Linear systems of linear differential equations
with R-symmetric coefficient matrices
William F. Trench
Trinity University
Department of Mathematics
San Antonio, Texas 78212-7200, USA
Let Cn×n (I) denote the set of continuous
n ×n matrices on an interval I. We say that R ∈ Cn×n (I)
L
k−1 `
−1
where ζ = e−2πi/k , d0 + d1 + · · · + dk−1 = n, and
is a nontrivial k-involution if R = P
`=0 ζ Id` P
L
k−1
P 0 = P `=0 U` with U` ∈ Cd` ×d` (I). We say that A ∈ Cn×n (I) is R-symmetric if R(t)A(t)R−1 (t) = A(t),
t ∈ I, and we show that if A is R-symmetric then solving x0 = A(t)x or x0 = A(t)x + f (t) reduces to solving
k independent d` × d` systems, 0 ≤ ` ≤ k − 1. We consider the asymptotic behavior of the solutions in the
case where I = [t0 , ∞).
Riemann Liouville and Caputo Fractional
Differential and Integral Inequalities with Applications
Aghalaya S. Vatsala1 and Donna S. Stutson
1
University of Louisiana at Lafayette
Department of Mathematics
Lafayette, Louisiana 70504, USA
Differential and integral inequalities has played a dominant role in the qualitative study of differential
and integral equations. In this work, we will study fractional differential and integral inequalities. The
fractional differential and integral equations will include the derivatives and integral of both the Riemann
Lioville type as well as the Caputo type. These inequalities are useful in proving theoretical existence and
uniqueness results for nonlinear fractional differential and integral equations. It is also useful in developing
iterative techniques which are both theoretical and computational. We can prove the existence and compute
the minimal and maximal solutions or coupled minimal and maximal solutions of the nonlinear fractional
equations by the iterative technique. Further, if uniqueness conditions are satisfied, we can prove the
existence of a unique solution. The unique solution can be computed numerically. Moreover, the interval of
existence is guaranteed by coupled lower and upper solutions. We will present some numerical examples as
an application of our theoretical results.
17
A Chebyshev spectral method for solving Riemann-Liouville
fractional boundary value problems
John R. Graef, Lingju Kong, and Min Wang
University of Tennessee at Chattanooga
Department of Mathematics
Chattanooga, TN 37403, USA
The authors derive a series of explicit formula to approximate the Riemann-Liouville derivative and integral of arbitrary order by shifted Chebyshev polynomials. Then the formulas are applied to solve boundary
value problems involving Riemann-Liouville derivatives.
Positive Solutions of Nonlinear Equations
via linear operator comparisons
Jeff Webb
University of Glasgow
School of Mathematics and Statistics
Glasgow, G12 8QQ, Scotland
We discuss a class of positive linear operators that is a modification of the class of u0 -positive linear
operators of Krasnosel’skiı̆. We use a new comparison theorem for this class to give some short proofs of
new fixed point index results for some nonlinear integral operators that arise from boundary value problems
with either local or nonlocal boundary conditions. In particular, for some types of boundary conditions,
especially nonlocal ones, when the Green’s function, the kernel of the integral operator, satisfies a stronger
positivity condition, we obtain a new existence result for multiple positive solutions under conditions which
depend solely on the principal eigenvalue of the corresponding linear operator.
18
Positive Solutions to a Third Order Three Point
Boundary Value Problem
Bo Yang
Kennesaw State University
Department of Mathematics and Statistics
Kennesaw, GA 30144, USA
We study a three point boundary value problem for a third order differential equation. Some new upper
estimates for positive solutions of the problem are obtained. Sufficient conditions for the existence and
nonexistence of positive solutions of the problem are established. These new existence and nonexistence
results improve those in the literature.
Stability by Schauders Fixed Point Theorem
for Nonlinear Delay and Fractional Differential Equations
Bo Zhang
Fayetteville State University
Department of Mathematics and Computer Science
Fayetteville, NC 28301, USA
In this paper we study a nonlinear scalar differential equation with variable delays and give conditions
to ensure that the zero solution is asymptotically stable by applying Schauders Fixed Point Theorem. These
conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient
functions. An asymptotic stability theorem with a necessary and sufficient condition is proved. The same
technique is also applied to some nonlinear fractional differential equations of Caputo type.
19