By V. Lakshmikantham; S. Leela

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1) R"1-1. 1. Let v E C [ [ t ,, to a), R " ] ; ( t , v ( t ) )E E, and v;(t) exists for t E [tn , to a). 1) if + ~ %J,+(t)< g,(t, 4 t ) ) > + 74,d t ) > g,(t, 4 t ) ) hold for t E [t,, , to a). If v(t) satisfies the reversed inequalities, it is said to be a k over ( n - k) under-function. , K = 0 or k = n. We require that the function g(t, u ) should satisfy certain monotonic properties, which are listed below. 2. , k , j f p , and nonincreasing in uQ. , n, j # q . Evidently, the particular cases k = n and k = 0 in the mixed quasimonotone property correspond to quasi-monotone nondecreasing and quasi-monotone nonincreasing properties of the function g(t, u), respectively.

21) -km as t,. -+ t l - . 21) cannot be true. 4. 1, exists on [to, t, v ] , v > 0, and t , v < to for sufficiently small E > 0. I. I. ) ri(to, €1. 25), and this proves the existence of r l ( t ) on [to to 9 + a). 20) is true for t Furthermore, E [t, , t, + a). 1 now gives that ri(t) G ~ ( t ) , t~ [ t o , to + a). 18), as is desired. 5. 4 hold; m E C [ [ t o, to a), R ] such that ( t , m ( t ) , o) E E, t E [to , to a ) , and m(t,) < uo . 26) + a ) - 5'. Then, for all < r ( t ) , t E [to , to + a ) , m(t) ZI < Y(t), t E [to, t" -4- a).

I n either case, r(t) is said to be a minimax solution. A k max ( n - k) mini-solution reduces to a maximal solution when k = n and to a minimal solution when k = 0. Similarly, a k mini ( n - k) max-solution coincides with a minimal solution and a maximal solution when k = n and k = 0, respectively. ~ As minimax solutions include both maximal and minimal solutions as special cases, we consider below the existence problem for minimax solutions only. 1. Let g E CIRo , R"], where Ro : t o < < t < to + a, /I u - uo II < 6, and 11 g(t, u)II M on R, .