Abstract
In this paper, we develop the theory of differential equations with linear perturbations of second type on time scales. An existence theorem for differential equations with linear perturbations of second type on time scales is given under mathscr{D}-Lipschitz conditions. Some fundamental differential inequalities on time scales, which are utilized to investigate the existence of extremal solutions, are also presented. The comparison principle on differential equations with linear perturbations of second type on time scales is established. Our results in this paper extend and improve some well-known results.
Highlights
In this paper, we discuss the following differential equations with linear perturbations of second type on time scales:[u(t) – f (t, u(t))] = g(t, u(t)), t ∈ J, (1)u(t0) = u0, where f, g ∈ Crd(J × R, R)
Some fundamental differential inequalities on time scales, which are utilized to investigate the existence of extremal solutions, are presented
5 Comparison theorems on time scales The main problem of the DITS is to estimate a bound for the solution set for the DITS related to DETS (1)
Summary
An existence theorem for DETS (1) is given under D-Lipschitz conditions. We place DETS (1) in the space Crd(J, R) of rd-continuous functions defined on J. D -functions have been widely used in the theory of nonlinear differential and integral equations for proving the existence results via fixed point methods. The following fixed point theorem in Banach algebra due to Dhage [16] is useful in the proofs of our main results.
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