FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

70 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa Tangwhere C 0 [a, b] denotes the space of all continuous functions on [a, b]. Wedefine the norms ‖·‖ f , ‖·‖ g , ‖·‖ĝ and the operator T f , ̂T f , T g , ̂T g as follows:‖y‖ f := K fT f y(t) :=̂T f y(t) :=∫ ba∫ bamax |y(t)|,a≤t≤bG(t, s)f(s, y(s)) ds,Ĝ(t, s)f(s, y(s)) ds, ∀ y(t) ∈ P;[‖y‖ g := max Kg |y(t)| + L g | C a Dt θ y(t)| ] ,a≤t≤bT g y(t) :=∫ baG(t, s)g ( s, y(s), C a D θ sy(s) ) ds, ∀ y(t) ∈ P 1 ,[‖y‖ĝ := max Kg |y(t)| + L g | RLa Dt θ y(t)| ] ,a≤t≤b̂T g y(t) :=∫ baĜ(t, s)g ( s, y(s), RLa D θ sy(s) ) ds, ∀ y(t) ∈ P 2 .Obviously, P, P 1 , P 2 are complete normed spaces with respect to ‖ · ‖ f ,‖ · ‖ g and ‖ · ‖ĝ, respectively. The operators T f , ̂T f , T g , ̂T g are continuous.Now (3.19) and (3.21) can be rewritten as y = T y and y = ̂T f y, for y ∈ P;(3.23), (3.24) can be rewritten as y = T g y and y = ̂T g y, respectively. Accordingto the contractive mapping principle, “finding a sufficient conditionfor existence and uniqueness of the solution for FBVP (3.1)–(3.2) (or (3.7)–(3.8) or (3.3)–(3.4) or (3.9)–(3.10))” is equivalent to “finding a sufficientcondition under which T f (or ̂T f or T or ̂T g ) is a contractive mapping ofP(or P 1 or P 2 )”. Then, the following theorems are true.Theorem 3.1. (1) Let f be a continuous function on [a, b] × R andsatisfy the Lipschitz condition (3.5).(1.1) If2K f (b − a) γ< 1, (3.27)Γ(γ + 1)then there exists a unique solution for FBVP (3.1)–(3.2) in P.(1.2) If(γ − 1) γ−1 (b − a) γK fγ γ < 1, (3.28)Γ(γ + 1)then there exists a unique solution for FBVP (3.7)–(3.8) in the space P.(2) Let g be a continuous function on [a, b] × R × R and satisfy theLipschitz condition (3.6).