Asymptotic solutions of forced nonlinear second order differential equations and their extensions

Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear differential equations on a half-axis. In addition, we extend the methods and present new similar results for integral equations and Volterra-Stieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of Volterra-Stieltjes integral equations is also presented

Near smoothness of Banach spaces

The aim of this paper is to discuss the concept of near smoothness in some Banach sequence spaces

Set quantities and Tauberian operators

The concept of convexity plays an important role in the classical geometry of normed spaces and it is frequently used in several branches of nonlinear analysis. In recent years some papers that contain generalizations of the concept of convexity with the help of the measures of noncompactness have appeared. The Tauberian operators were introduced by Kalton and Wilansky (1976) and they appear in the literature with the aim of responding to some questions related with the summability and the factorization of operators; in the preservation by isomorphisms in Banach spaces, and so forth. In this paper we make the study of the Tauberian operators, not starting from the Euclidean distance, but by means of general set quantities

About a Problem for Loaded Parabolic-Hyperbolic Type Equation with Fractional Derivatives

An existence and uniqueness of solution of local boundary value problem with discontinuous matching condition for the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals have been investigated. The uniqueness of solution is proved by the method of integral energy and the existence is proved by the method of integral equations. Let us note that, from this problem, the same problem follows with continuous gluing conditions (at λ=1); thus an existence theorem and uniqueness theorem will be correct and on this case

Riesz transforms and multipliers for the Bessel-Grushin operator

We establish that the spectral multiplier $\frak{M}(G_{\alpha})$ associated to the differential operator $$G_{\alpha}=- \Delta_x +\sum_{j=1}^m{{\alpha_j^2-1/4}\over{x_j^2}}-|x|^2 \Delta_y \; \text{on} (0,\infty)^m \times \R^n,$$ which we denominate Bessel-Grushin operator, is of weak type $(1,1)$ provided that $\frak{M}$ is in a suitable local Sobolev space. In order to do this we prove a suitable weighted Plancherel estimate. Also, we study $L^p$-boundedness properties of Riesz transforms associated to $G_{\alpha}$, in the case $n=1$.Comment: 33 page

Compactness Conditions in the Study of Functional, Differential, and Integral Equations

We discuss some existence results for various types of functional, differential, and integral equations which can be obtained with the help of argumentations based on compactness conditions. We restrict ourselves to some classical compactness conditions appearing in fixed point theorems due to Schauder, Krasnosel’skii-Burton, and Schaefer. We present also the technique associated with measures of noncompactness and we illustrate its applicability in proving the solvability of some functional integral equations. Apart from this, we discuss the application of the mentioned technique to the theory of ordinary differential equations in Banach spaces

Applicable Analysis and Discrete Mathematics EXISTENCE OF SOLUTIONS FOR HYBRID FRACTIONAL PANTOGRAPH EQUATIONS

In this paper, we study the existence of the hybrid fractional pantograph equation where α, µ, σ ∈ (0, 1) and D α 0 + denotes the Riemann-Liouville fractional derivative. The results are obtained using the technique of measures of noncompactness in the Banach algebras and a fixed point theorem for the product of two operators verifying a Darbo type condition. Some examples are provided to illustrate our results

EXISTENCE OF SOLUTIONS FOR HYBRID FRACTIONAL PANTOGRAPH EQUATIONS

In this paper, we study the existence of the hybrid fractional pantograph equation where α, µ, σ ∈ (0, 1) and D α 0 + denotes the Riemann-Liouville fractional derivative. The results are obtained using the technique of measures of noncompactness in the Banach algebras and a fixed point theorem for the product of two operators verifying a Darbo type condition. Some examples are provided to illustrate our results