Infinite first order differential systems with nonlocal initial conditions

Gennaro Infante,Fadila Madjidi,Petru Jebelean

doi:10.1186/s13661-015-0314-8

Gennaro Infante, Fadila Madjidi + Show 1 more

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https://doi.org/10.1186/s13661-015-0314-8

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Abstract

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear" growth then the system has at least one solution. The approach relies on the application, in a suitable Fr\'echet space, of the classical Schauder-Tychonoff fixed point theorem. We show that, as a special case, our approach covers the case of a system of a finite number of differential equations. An illustrative example of application is also provided.

Highlights

1 Introduction In the recent paper [ ] Bolojan-Nica and co-authors developed a technique that can be used to study the solvability of a system of N first order differential equations of the form
We mention that there exists a wide literature on differential equations subject to nonlocal conditions; we refer here to the pioneering work of Picone [ ], the reviews by Whyburn [ ], Conti [ ], Ma [ ], Ntouyas [ ] and Štikonas [ ], the papers by Karakostas and Tsamatos [, ] and by Webb and Infante [, ]
The methodology to treat initial value problems for infinite systems often relies on using the theory of differential equations in Banach spaces, but it is known that this approach can reduce the set of solutions and some of their specific properties

Summary

Introduction

For systems of a finite number of differential equations on noncompact intervals, subject to linear (or more general) conditions, we refer to the papers by Andres et al [ ], Cecchi et al [ , ], De Pascale et al [ ], Marino and Pietramala [ ] and Marino and Volpe [ ]. The methodology to treat initial value problems for infinite systems often relies on using the theory of differential equations in Banach spaces, but it is known that this approach can reduce the set of solutions and some of their specific properties. The case of initial value problems for infinite systems on non-compact intervals has been investigated in [ ], where the authors extended a comparison theorem due to Stokes [ ], valid for the scalar case.

Results

Conclusion