About nondecreasing solutions for first order neutral functional differential equations

• Main Authors: Alexander Domoshnitsky, A. Maghakyan, S. Yanetz
• Format: Article
• Language: English
• Published: University of Szeged 2012-05-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: neutral equations; cauchy functions; positivity; nondecreasing solutions; maximum principles
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=1089

Conditions that solutions of the first order neutral functional differential equation \[ (Mx)(t)\equiv x^{\prime }(t)-(Sx^{\prime })(t)-(Ax)(t)+(Bx)(t)=f(t), t\in \lbrack 0,\omega ], \] are nondecreasing are obtained. Here $A:C_{[0,\omega ]}\rightarrow L_{[0,\omega ]}^{\infty }$ ,$\;B:C_{[0,\omega ]}\rightarrow L_{[0,\omega]}^{\infty }$ and $S:L^{\infty }{}_{[0,\omega ]}\rightarrow L_{[0,\omega]}^{\infty }$ are linear continuous operators, $A$ and $B$ are positive operators, $C_{[0,\omega ]}$ is the space of continuous functions and $L_{[0,\omega ]}^{\infty }$ is the space of essentially bounded functions defined on $[0,\omega ]$. New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.