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Basic results on hybrid differential equations. (English) Zbl 1206.34020
Summary: An existence theorem for hybrid nonlinear differential equations is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental differential inequalities are also established which are utilized to prove the existence of extremal solutions and a comparison result.

34A38Hybrid systems
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34A40Differential inequalities (ODE)
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text: DOI
[1] B.C. Dhage, V. Lakshmikantham, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations (submitted for publication)
[2] Dhage, B. C.: On ${\alpha}$-condensing mappings in Banach algebras, Math. student 63, 146-152 (1994) · Zbl 0882.47033
[3] Dhage, B. C.: A nonlinear alternative in Banach algebras with applications to functional differential equations, Nonlinear funct. Anal. appl. 8, 563-575 (2004) · Zbl 1067.47070
[4] Dhage, B. C.: Fixed point theorems in ordered Banach algebras and applications, Panamer. math. J. 9, No. 4, 93-102 (1999) · Zbl 0964.47026
[5] Dhage, B. C.: A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook math. J. 44, 145-155 (2004) · Zbl 1057.47062
[6] Heikkilä, S.; Lakshmikantham, V.: Monotone iterative technique for nonlinear discontinues differential equations, (1994) · Zbl 0804.34001
[7] Lakshmikantham, V.; Leela, S.: Differential and integral inequalities, (1969) · Zbl 0177.12403