# CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS OF FIRST ORDER BOUNDARY VALUE PROBLEMS WITH DELAY AND NONLINEAR NONLOCAL BOUNDARY CONDITIONS AND APPLICATION TO HEMATOPOIESIS

TitleCONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS OF FIRST ORDER BOUNDARY VALUE PROBLEMS WITH DELAY AND NONLINEAR NONLOCAL BOUNDARY CONDITIONS AND APPLICATION TO HEMATOPOIESIS
Publication TypeJournal Article
Year of Publication2017
Secondary TitleCommunications in Applied Analysis
Volume21
Issue1
Start Page39
Pagination18
Date Published01/2017
Type of Workscientific: mathematics
ISSN1083-2564
AMS34B08, 34B10, 34B15, 34B18
Abstract

In this paper,  existence criteria for   positive solutions of the following nonlinear first order boundary value problem with delay and  nonlinear nonlocal boundary condition
\begin{eqnarray*}
x^{\prime}(t) & = & r(t)x(t) + p(t) \sum_{i=1}^{m} f_i(t,x(\alpha_{i}(t))), \,\,\, t \in [0,1],\\
\lambda x(0) & = & x(1) + \sum_{j=1}^{n} \Lambda_j(\tau_j, x(\tau_j)),\,\,\, \tau_j \in [0,1],
\end{eqnarray*}
are established using  Leray-Schauder theorem and Leggett-Williams fixed point theorem. These results are employed to  provide a complete existence criteria for positive solutions of the following  boundary value problem associated with the well known Hematopoiesis  model%\vfill\mbox{}\newpage
\begin{align}
x^{\prime}(t)  = & r(t)x(t) + p(t)\frac{x^{n}(t-\alpha)}{1+x^{m}(t-\alpha)}, \,\,\, t \in [0,1], \nonumber\\
\lambda x(0)   = &  x(1)  +  \Lambda(\tau,x(\tau)),\,\,\, \tau \in [0,1] \nonumber
\end{align}
where $m$ and $n$ are nonnegative parameters.

DOI10.12732/caa.v21i1.4
Short TitleExistence of Positive Solutions
Alternate JournalCAA
Refereed DesignationRefereed
References

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