Abstract
In this paper, we develop the theory of differential equations with mixed perturbations of the second type on time scales. We give an existence theorem for differential equations with mixed perturbations of the second type on time scales under Lipschitz condition. We also present some fundamental differential inequalities on time scales, which are utilized to investigate the existence of extremal solutions. We establish the comparison principle for differential equations with mixed perturbations of the second type on time scales. Our results in this paper extend and improve some well-known results.
Highlights
In this paper, we discuss the following differential equations with mixed perturbations of the second type on time scales (DETS): ⎧ ⎨[u(t)–k(t,u(t)) f (t,u(t)) ] = g(t, u(t)),⎩u(t0) = u0, t ∈ J, (1)where f ∈ Crd(J × R, R \ {0}) and k, g ∈ Crd(J × R, R)
We present some fundamental differential inequalities on time scales (DITS), which are utilized to investigate the existence of extremal solutions
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 the DETS (1)
Summary
We give an existence theorem for the DETS (1) under Lipschitz conditions. We present some fundamental differential inequalities on time scales (DITS), which are utilized to investigate the existence of extremal solutions. For any v ∈ L1(J, R), the -differentiable function u is a solution of the DETS We will give the following existence theorem for the DETS (1).
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