SOME FIXED POINT THEOREMS FOR ORDERED REICH TYPE ...

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d(x 0 , fx 0 ) = d(x n , fx n )d(x 0 , x 1 ) = d(x n , x n+1 ),d 0 = d nd 0 ≼ α n d 0 .As α = λ+µ1−δ < 1 (since λ + µ + δ < 1), the above inequality shows that d 0 = θ, i.e., d(x 0 , fx 0 ) = θ,so x 0 is a fixed point of f. Thus we assume that x n ≠ x m for all distinct n, m ∈ N.Again, as x n ⊑ x n+2 , we obtain from (I) and (4)d(x n , x n+2 ) = d(fx n−1 , fx n+1 )≼ λd(x n−1 , x n+1 ) + µd(x n−1 , fx n−1 ) + δd(x n+1 , fx n+1 )= λd(x n−1 , x n+1 ) + µd(x n−1 , x n ) + δd(x n+1 , x n+2 )≼ λ[d(x n−1 , x n ) + d(x n , x n+2 ) + d(x n+2 , x n+1 )] + µd(x n−1 , x n ) + δd(x n+1 , x n+2 )◭◭ ◭ ◮ ◮◮Go backFull ScreenCloseQuitso(5)= λ[d n−1 + d(x n , x n+2 ) + d n+1 ] + µd n−1 + δd n+1= (λ + µ)d n−1 + (λ + δ)d n+1 + λd(x n , x n+2 )≼ (λ + µ)α n−1 d 0 + (λ + δ)α n+1 d 0 + λd(x n , x n+2 )(λ + µ) + (λ + δ)α2d(x n , x n+2 ) ≼ α n−1 d 01 − λ≼ 2λ + µ + δ α n−1 d 0 ,1 − λwhere β = 2λ + µ + δ1 − λd(x n , x n+2 ) ≼ βα n−1 d 0 for all n ≥ 1,≥ 0.
For the sequence {x n } we consider d(x n , x n+p ) in two cases.If p is odd, say 2m + 1, then using rectangular inequality and (4), we obtaind(x n , x n+2m+1 ) ≼ d(x n+2m , x n+2m+1 ) + d(x n+2m−1 , x n+2m ) + d(x n , x n+2m−1 )= d n+2m + d n+2m−1 + d(x n+2m−1 , x n )≼ d n+2m + d n+2m−1 + d n+2m−2 + d n+2m−3 + · · · + d n≼ α n+2m d 0 + α n+2m−1 d 0 + α n+2m−2 d 0 + · · · + α n d 0◭◭ ◭ ◮ ◮◮so(6)= [α 2m + α 2m−1 + · · · + 1]α n d 0 ≼ αn1 − α d 0,d(x n , x n+2m+1 ) ≼αn1 − α d 0.If p is even, say 2m, then using rectangular inequality, (4) and (5), we obtaind(x n , x n+2m ) = d(x n+2m , x n )≼ d(x n+2m , x n+2m−1 ) + d(x n+2m−1 , x n+2m−2 ) + d(x n+2m−2 , x n )= d n+2m−1 + d n+2m−2 + d(x n+2m−2 , x n )≼ d n+2m−1 + d n+2m−2 + d n+2m−3 + d n+2m−4 + . . . + d n+2 + d(x n+2 , x n )Go backFull ScreenCloseQuitso(7)≼ α n+2m−1 d 0 +α n+2m−2 d 0 +α n+2m−3 d 0 +· · ·+α n+2 d 0 + βα n−1 d 0= [α 2m−1 + α 2m−2 + · · · + α 2 ]α n d 0 + βα n−1 d 0 ≼ αn1 − α d 0 + βα n−1 d 0d(x n , x n+2m ) ≼αn1 − α d 0 + βα n−1 d 0 .
Page 1 and 2: SOME FIXED POINT THEOREMS FOR ORDER
Page 3 and 4: ◭◭ ◭ ◮ ◮◮Go backFull Sc
Page 7: Then f has a fixed point. Furthermo
Page 11 and 12: Suppose that the set of fixed point
Page 15 and 16: The necessary and sufficient condit
Page 17: S. K. Malhotra, Dept. of Mathematic

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