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

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

Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer

We give an existence theorem for some functional-integral equations which includes many key integral and functional equations that arise in nonlinear analysis and its applications. In particular, we extend the class of characteristic functions appearing in Chandrasekhar's classical integral equation from astrophysics and retain existence of its solutions. Extensive use is made of measures of noncompactness and abstract fixed point theorems such as Darbo's theorem

Solvability of a recursive functional equation in the sequence Banach space

The main aim of this paper is to study the existence of solutions of the following recursive functional equation x(n) = f(n, x(n), x(n - 1)) in the space l2, under general assumptions. The main tools of our existence theorem are the characterization of the relatively compact sets in the space l2 and Schauder Fixed point theorem. Moreover, our functional equation has as particular cases some integral equations of Urysohn type. Finally, we present some examples where our theorem can be applied