Compactness of Riemann–Liouville fractional integral operators

We obtain results on compactness of two linear Hammerstein integral operators with singularities, and apply the results to give new proof that Riemann–Liouville fractional integral operators of order $\alpha\in (0,1)$ map $L^{p}(0,1)$ to $C[0,1]$ and are compact for each $p\in \bigl(\frac{1}{1-\alpha},\infty\bigr]$. We show that the spectral radii of the Riemann–Liouville fractional operators are zero

Theories of Fixed Point Index and Applications

This thesis is devoted to the study of theories of fixed point index for generalized and weakly inward maps of condensing type and weakly inward A-proper maps. In Chapter 1 we recall some basic concepts such as cones, wedges, measures of noncompactness and theories of fixed point index for compact and gamma-condensing self-maps. We also give some new results and provide new proofs for some known results. In Chapter 2 we study approximatively compact sets giving examples and proving new results. The concept of an approximatively compact set is of importance in defining our index for a generalized inward map since there exists upper semicontinuous multivalued metric projections onto the approximatively compact convex set. We also introduce the concept of an M1-set which will play an important role in defining our fixed point index for generalized inward maps of condensing type since there exists continuous single-valued metric projections onto an Ml-closed convex set. Many examples of M1-closed convex sets are given. Weakly inward sets and weakly inward maps are studied in detail. New properties and examples on such sets and maps are given. We also introduce the new concept of generalized inward sets and generalized inward maps. The class of generalized inward maps strictly contain the class of weakly inward maps. Several necessary and sufficient conditions for a map to be generalized inward and examples of generalized inward maps are given. In Chapter 3 we define a fixed point index for a generalized inward compact map defined on an approximatively compact convex set and obtain many new fixed point theorems and nonzero fixed point theorems. In particular, norm-type expansion and compression theorems for weakly inward continuous maps in finite dimensional Banach spaces are obtained, which have not been considered previously. In Chapter 4 we define a fixed point index for a generalized inward maps of condensing type defined on an M1-closed convex set and obtain many new fixed point theorems and nonzero fixed point theorems. We also apply the abstract theory to some perturbed Volterra equations. In Chapter 5 we define a fixed point index for weakly inward A-proper maps. We obtain new fixed point theorems, nonzero fixed point theorem and results on existence of eigenvalues. We also give an application of the abstract theory to the existence of nonzero positive solutions of boundary value problems for second order differential equations

A new Bihari inequality and initial value problems of first order fractional differential equations

We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order α∈(0,1) . One novelty in this paper is that it is not assumed that f is continuous but that it satisfies an Lp -Carathéodory condition for some p>1α (detailed definitions are given in the paper). We prove existence on an interval [0, T] in cases where T can be arbitrarily large, called global solutions. The necessary a priori bounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist when f(t, u) grows at most linearly in u, and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature

Positive solutions of one-dimensional pp-Laplacian equations and applications to population models of one species

We prove new results on the existence of positive solutions of one-dimensional $p$-Laplacian equations under sublinear conditions involving the first eigenvalues of the corresponding homogeneous Dirichlet boundary value problems. To the best of our knowledge, this is the first paper to use fixed point index theory of compact maps to give criteria involving the first eigenvalue for one-dimensional $p$-Laplacian equations with $p\ne 2$. Our results generalize some previous results where either $p$ is required to be greater than $2$ or the nonlinearities satisfy stronger conditions. We shall apply our results to tackle a logistic population model arising in mathematical biology

Stability and phase portraits of susceptible-infective-removed epidemic models with vertical transmissions and linear treatment rates

We study stability and phase portraits of susceptible-infective-removed (SIR) epidemic models with horizontal and vertical transmission rates and linear treatment rates by studying the reduced dynamical planar systems under the assumption that the total population keeps unchanged. We find out all the ranges of the parameters involved in the models for the infection-free equilibrium and the epidemic equilibrium to be positive. The novelty of this paper lies in the demonstration and justification of the parameter conditions under which the positive equilibria are stable focuses or nodes. These phase portraits provide more detailed descriptions of behaviors and extra biological understandings of the epidemic diseases than local or global stability of the models. Previous results only discussed the stability of the SIR models with horizontal or vertical transmission rates and without treatment rates. Our results involving vertical transmission and treatment rates will exhibit the effect of the vertical transmissions and the linear treatment rates on the epidemic models

Fractional differential equations of Bagley-Torvik and Langevin type

Nonlinear fractional equations for Caputo differential operators with two fractional orders are studied. One case is a generalization of the Bagley-Torvik equation, another is of Langevin type. These can be confused as being the same but because fractional derivatives do not commute these are different problems. However it is possible to use some common methodology. Some new regularity results for fractional integrals of a certain type are proved. These are used to rigorously prove equivalences between solutions of initial value problems for the fractional derivative equations and solutions of the corresponding integral equations in the space of continuous functions. A novelty is that it is not assumed that the nonlinear term is continuous but that it satisfies the weaker L p-Carathéodory condition. Existence of solutions on an interval [0, T ] in cases where T can be arbitrarily large, so-called global solutions, are proved, obtaining the necessary a priori bounds by using recent fractional Gronwall and fractional Bihari inequalities

Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type

New criteria are established for the existence of multiple positive solutions of a Hammerstein integral equation of the form $$u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))ds \equiv Au(t)$$ where $k$ can have discontinuities in its second variable and $g \in L^{1}$. These criteria are determined by the relationship between the behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the principal (positive) eigenvalue of the linear Hammerstein integral operator $$Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)ds.$$ We obtain new results on the existence of multiple positive solutions of a second order differential equation of the form $$u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1],$$ subject to general separated boundary conditions and also to nonlocal $m$-point boundary conditions. Our results are optimal in some cases. This work contains several new ideas, and gives a {\it unified} approach applicable to many BVPs