If are two nonnegative functions in, respectively, then for any (2.9).
Theorem 3.1 Assume that (H1), (H2) hold and the following conditions are satisfied: (A1) There exist two nonnegative functions b ( t ), c ( t ) ∈ C [ 0, 1 ] with c ( t ) ≢ 0 and one continuous even function B : R → R + such that f ( t, x ) ≥ − b ( t ) − c ( t ) B ( x ) for all x ∈ R.
(f: [0, 1]times mathbb{R}rightarrow mathbb{R}) is continuous and there exist two nonnegative functions (c, din C[0,1]), (dnotequiv0) and (Bin C mathbb{R}, [0,+infty))) with begin{aligned}& lim_{|u|to+infty}frac{B u)}{|u|}=0 end{aligned} (1.4) such that begin{aligned}& f t, u geq-c(t)-d(t) B(u geq-cd t -d0,1], uinmat -d{R}.
(A3) (f: [0, 1]times mathbb{R}rightarrow mathbb{R}) is continuous and there exist two nonnegative functions (c, din C[0,1]), (dnotequiv0) and (Bin C mathbb{R}, [0,+infty))) with begin{aligned}& lim_{|u|to+infty}frac{B u)}{|u|}=0 end{aligned} (1.4) such that begin{aligned}& f t, u geq-c(t)-d(t) B(u geq-cd t -d0,1], uinmat -d{R}.
For a start, we impose the following assumptions: (A1) The function (f: Jtimes R^{n} times R^{n} rightarrow R^{n}) is continuous and there exist two nonnegative functions (L_{f}(cdot)) and (I_{f}(cdot)) such that begin{aligned} biglVert f t, x, u -f t, hat{x}, hat{u -f trVert leq L_{f}(t)Vert x-hat{x}Vert +I_{f}(t)Vert u-hat{u} bigrVertd{aleqned} for any (x,hat{x},u,hat{u}in R^{n}) and aL_{ftin J).
(A1) There exist two nonnegative functions:, such that., may be singular at.,, are continuous.
Let be two nonnegative integers such that.
(1.7) where λ, μ are two parameters and (f,g: [0,T]timesmathbb {R}rightarrowmathbb{R}) are two nonnegative continuous functions.
Let (g, h) and y be three nonnegative locally integrable functions on ((t_{0},infty)) such that (y') is locally integrable on ((t_{0},infty)), and which satisfy frac{dy}{dt}le gy+h and int_{t}^{t+r}g(s),dsle a_{1},qquad int_{t}^{t+r}h(s),dsle a_{2},qquad int _{t}^{t+r}y(s),dsle a_{3}, quad tge t_{0}, where (r,a_{1},a_{2},are3} ) are positive constants.
(1) We suppose also that f and g are two nonnegative real-valued functions such that ((f,g in L^{1}(a,b times L^{1}(a,b)).
