Fixed point theorems for A-contraction mappings of integral type

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1 Available olie at J. Noliear Sci. Appl. 5 (212, Research Article Fixed poit theorems for A-cotractio mappigs of itegral type Matu Saha a, Debashis Dey b, a Departmet of Mathematics, The Uiversity of Burdwa, Burdwa-71314, West Begal, Idia. b Koshigram Uio Istitutio, Koshigram-71315, Burdwa, West Begal, Idia. This paper is dedicated to Professor Ljubomir Ćirić Commuicated by Professor V. Beride Abstract I the preset paper, we prove aalogues of some fixed poit results for A-cotractio type mappigs i itegral settig. c 212 NGA. All rights reserved. Keywords: fixed poit, geeral cotractive coditio, itegral type. 21 MSC: Primary 54H25; Secodary 47H1. 1. Itroductio ad Prelimiaries Fixed poit theory plays a crucial part i oliear fuctioal aalysis ad is useful for provig the existece theorems for oliear differetial ad itegral equatios. First importat result o fixed poits for cotractive type mappig was give by S. Baach [3] i I the geeral settig of complete metric space, this theorem rus as follows ( see Theorem 2.1,[8] or, Theorem 1.2.2,[17]. Theorem 1.1. (Baach cotractio priciple Let (X, d be a complete metric space, c (, 1 ad f : X X be a mappig such that for each x, y X, d(fx, fy cd(x, y (1.1 the f has a uique fixed poit a X, such that for each x X, lim f x = a. Correspodig author addresses: matusaha@yahoo.com (Matu Saha, debashisdey@yahoo.com (Debashis Dey Received

2 M. Saha, D. Dey, J. Noliear Sci. Appl. 5 (212, A elemetary accout of the Baach cotractio priciple ad some applicatios, icludig its role i solvig oliear ordiary differetial equatios, is i [6]. The cotractio mappig theorem is used to prove the iverse fuctio theorem i [([15], pp A beautiful applicatio of cotractio mappigs to the costructio of fractals (iterpreted as fixed poits i a metric space whose poits are compact subsets of the plae is i ([16], Chap. 5. After the classical result by Baach, Kaa [9] gave a substatially ew cotractive mappig to prove the fixed poit theorem. Sice the there have bee may theorems emerged as geeralizatios uder various cotractive coditios. Such coditios ivolve liear ad oliear expressios (ratioal, irratioal, ad geeral type. The itrested reader who wats to kow more about this matter is recommeded to go deep ito the survey articles by Rhoades ([12], [13], [14] ad Biachii [4], ad ito the refereces therei A-cotractio O the otherhad, Akram et al.[2] itroduced a ew class of cotractio maps, called A-cotractio, which is a proper superclass of Kaa s[9], Biachii s[4] ad Reich s[1] type cotractios. Akram et al.[2] defied A-cotractios as follows: Let a o-empty set A cosistig of all fuctios α : R 3 + R + satisfyig: (A 1 : α is cotiuous o the set R 3 + of all triplets of o-egative reals(with respect to the Euclidea metric o R 3. (A 2 : a kb for some k [, 1 wheever a α (a, b, b or a α (b, a, b or a α (b, b, a for all a, b. Defiitio 1.2. A self-map T o a metric space X is said to be A-cotractio, if it satisfies the coditio d (T x, T y α (d (x, y, d (x, T x, d (y, T y for all x, y X ad some α A. Example 1.3. Let a self-map T o a metric space (X, d satisfyig d (T x, T y β max {d (T x, x + d (T y, y, d (T y, y + d (x, y, d (T x, x + d (x, y} for all x, y X ad some β [, 1 2, is a A-cotractio. (see [2] for detail ad compariso with other cotractio maps. I 22, A.Braciari[5] aalyzed the existece of fixed poit for mappig T defied o a complete metric space (X, d satisfyig a geeral cotractive coditio of itegral type i the followig theorem. Theorem 1.4. (Braciari Let (X, d be a complete metric space, c (, 1 ad let T : X X be a mappig such that for each x, y X, d(t x,t y c d(x,y (1.2 where ϕ : [, + [, + is a Lesbesgue-itegrable mappig which is summable (i.e. with fiite itegral o each compact subset of [, +, o-egative, ad such that for each ɛ >, >, the T has a uique fixed poit a X such that for each x X, lim T x = a. After the paper of Braciari, a lot of research works have bee carried out o geeralizig cotractive coditios of itegral type for differet cotractive mappigs satisfyig various kow properties. A fie work has bee doe by Rhoades[11] extedig the result of Braciari by replacig the coditio (1.2 by the followig d(t x,t y [d(x,t y+d(y,t x] max{d(x,y,d(x,t x,d(y,t y, c 2 } (1.3

3 M. Saha, D. Dey, J. Noliear Sci. Appl. 5 (212, for all x, y X with some c [, 1. I a very recet paper, Dey et al.[7] proved some fixed poit theorems for mixed type of cotractio mappigs of itegral type i complete metric space. Motivated ad ispired by these cosequet works, we itroduce the aalogues of some fixed poit results for A-cotractio mappigs i itegral settig which i tur geeralize several kow results. Also we have aalyzed the existece of fixed poit of mappig over two related metrics due to Theorem 4 of [1] i itegral settig. Our results substatially exted, improve, ad geeralize comparable results i the literature. 2. Mai results Theorem 2.1. Let T be a self-mappig of a complete metric space (X, d satisfyig the followig coditio: d(t x,t y d(x,y d(x,t x d(y,t y α,, (2.1 for each x, y X with some α A, where ϕ : [, + [, + is a Lesbesgue-itegrable mappig which is summable (i.e. with fiite itegral o each compact subset of [, +, o-egative, ad such that for each ɛ >, > (2.2 The T has a uique fixed poit z X ad for each x X, lim T x = z. Proof. Let x X be arbitrary ad, for brevity, defie x +1 = T x. For each iteger 1, from (2.1 we get, d(x,x +1 = d(t x 1,T x The by the axiom A 2 of fuctio α, d(x,x +1 k for some k [, 1 as α A. I this fashio, oe ca obtai d(x,x +1 ( d(x 1,x α, ( d(x 1,x α, d(x 1,x k d(x 1,x d(x 1,T x 1 d(x 1,x,, d(x,t x d(x,x+1 (2.3 d(x 2 k 2,x 1... Takig limit as, we get d(x k,x 1 d(x,x+1 lim = as k [, 1

4 M. Saha, D. Dey, J. Noliear Sci. Appl. 5 (212, which, from(2.2 implies that lim d(x, x +1 = (2.4 We ow show that {x } is a Cauchy sequece. Suppose that it is ot. The there exists a ɛ > ad subsequeces {m(p} ad {(p} such that m(p < (p < m(p + 1 with Now d(x m(p, x (p ɛ, d(x m(p, x (p 1 < ɛ (2.5 d(x m(p 1, x (p 1 d(x m(p 1, x m(p + d(x m(p, x (p 1 So by (2.4 ad (2.6 we get d(xm(p 1,x (p 1 lim p Usig (2.3, (2.5 ad (2.7 we get d(xm(p,x (p < d(x m(p 1, x m(p + ɛ (2.6 k (2.7 d(xm(p 1,x (p 1 k which is a cotradictio, sice k [, 1. Therefore, {x } is Cauchy, hece coverget. Call the limit z. From (2.1 we get d(t z,x+1 = Takig limit as, we get ( d(t z,z α o, d(t z,t x ( d(z,x α, d(z,t z So by the axiom A 2 of fuctio α, d(t z,z = k. =, d(z,t z, d(x,x +1 which, from (2.2, implies that d(t z, z = or, T z = z. Next suppose that w( z be aother fixed poit of T. The from (2.1 we have d(z,w = d(t z,t w ( d(z,w α, ( d(z,w = α, d(z,t z d(z,z ( d(z,w = α,,,, d(w,t w d(w,w

5 M. Saha, D. Dey, J. Noliear Sci. Appl. 5 (212, So by the axiom A 2 of fuctio α, d(z,w = which, from (2.2, implies that d(z, w = or, z = w ad so the fixed poit is uique. Next theorem describes commo fixed poit of two self-maps o X havig two related metrics i itegral settig. This result geeralizes Theorem 4 of [1] i itegral setig. Theorem 2.2. Let X be a set with two metrics d ad δ satisfyig the followig coditios: (i d(x,y δ(x,y for all x, y X; (ii X is complete with respect to d; (iii S, T are self-maps o X such that T is cotiuous with respect to d ad δ(t x,sy δ(x,y δ(x,t x δ(y,sy α,, (2.8 for each x, y X with some α A, where ϕ : [, + [, + is a Lesbesgue-itegrable mappig which is summable (i.e. with fiite itegral o each compact subset of [, +, o-egative, ad such that for each ɛ >, > (2.9 The S ad T have a uique commo fixed poit z X. Proof. For each iteger, we defie x 2+1 = T x 2 x 2+2 = Sx 2+1 The from (2.8 we get, δ(x1,x 2 δ(t x,sx 1 = ( δ(x,x 1 α, ( δ(x,x 1 α, The by the axiom A 2 of fuctio α, δ(x1,x 2 k δ(x,x 1 for some k [, 1. Similarly oe ca show that δ(x2,x 3 k δ(x1,x 2 δ(x,t x δ(x,x 1 for some k [, 1. I geeral, for ay r N odd or eve, δ(xr,x r+1 k δ(xr 1,x r,, ad so for ay N odd or eve, oe ca easily obtai that δ(x,x +1 δ(x1,sx 1 δ(x1,x 2 (2.1 (2.11 (2.12 δ(x k,x 1 (2.13

6 M. Saha, D. Dey, J. Noliear Sci. Appl. 5 (212, The by the coditio (i of the theorem oe obtais d(x,x +1 Takig limit as, we get δ(x,x +1 d(x,x+1 lim = as k [, 1 which, from(2.9 implies that δ(x k,x 1 lim d(x, x +1 = (2.14 We ow show that {x } is a Cauchy sequece with respect to (X, d. So for ay iteger p >, d(x,x +p δ(x,x +p δ(x,x δ(x+p 1,x +p δ(x+1,x +2 δ(x k,x 1 δ(x + k +1,x 1 δ(x +...k +p 1,x 1 k 1 k δ(x,x 1 as sice k [, 1. Therefore, {x } is Cauchy. Hece by completeess of X, {x } coverges to some z X, i.e. d(x, z as for some z X. Sice T is give to be cotiuous with the respect to d we have d(x2+1,z = lim = lim d(t x2,z = lim d(t z,z So by (2.9 d(t z, z = i.e. T z = z. Now by (2.8 δ(z,sz = α( δ(t z,sz δ(z,z α(,, The by the axiom A 2 of fuctio α, δ(z,sz, δ(z,sz δ(z,t z, δ(z,sz k. = (2.15

7 M. Saha, D. Dey, J. Noliear Sci. Appl. 5 (212, ad so by (2.9 Sz = z. Thus z is a commo fixed poit of S ad T. For the uiqueess, let w( z be aother commo fixed poit of S ad T i X. The by (2.8 δ(z,w = α( α( δ(t z,sw δ(z,w δ(z,w, δ(z,t z,, k. = as α A The by (2.9 we have δ(z, w = ad so z = w. If S = T, the the Theorem 2.2 gives as follows., δ(w,sw Corollary 2.3. Let X be a set with two metrics d ad δ satisfyig the followig coditios: (i d(x,y δ(x,y for all x, y X; (ii X is complete with respect to d; (iii T is a self-map o X such that T is cotiuous with respect to d ad δ(t x,t y δ(x,y δ(x,t x δ(y,t y α,, (2.16 for each x, y X with some α A, where ϕ : [, + [, + is a Lesbesgue-itegrable mappig which is summable (i.e. with fiite itegral o each compact subset of [, +, o-egative, ad such that for each ɛ >, The T has a uique fixed poit z X. > (2.17 We have aother similar result if we omit the coditio (ii of corollary 2.3 ad the cotiuity of T with respect to d is replaced by assumig the cotiuity at a poit. The we get the same coclusio uder much less restricted coditio. Theorem 2.4. Let X be a set with two metrics d ad δ satisfyig the followig coditios: (i d(x,y δ(x,y for all x, y X; (ii T is a self-map o X such that T is cotiuous at z X with respect to d ad δ(t x,t y δ(x,y δ(x,t x δ(y,t y α,, (2.18 for each x, y X with some α A, where ϕ : [, + [, + is a Lesbesgue-itegrable mappig which is summable (i.e. with fiite itegral o each compact subset of [, +, o-egative, ad such that for each ɛ >, > (2.19 (iii There exists a poit x X such that the sequece of iterates {T x } has a subsequece {T i x } covergig to z i (X, d. The T has a uique fixed poit z X.

8 M. Saha, D. Dey, J. Noliear Sci. Appl. 5 (212, Proof. Cosiderig the sequece {x } as defied by x +1 = T x for i.e. x 1 = T x, x 2 = T x 1 = T 2 x,..., x = T x ad proceedig as i the proof of theorem 2.2 we ca easily arrive at a coclusio that the sequece is Cauchy with respect to d. Sice the subsequece {x i } of the Cauchy sequece {x } coverges to z, therefore {x } coverges to z i X with respect to d i.e. lim x = z. Sice T is give to be cotiuous at z with the respect to d we have d(x+1,z = lim = lim d(t x,z = lim d(t z,z So by (2.9 d(t z, z = i.e. T z = z. Thus T has a fixed poit. Uiqueess of z is also very clear. Remark 2.5. O settig ϕ(t = 1 over [, + i each results, the cotractive coditio of itegral type trasforms ito a geeral cotractive coditio ot ivolvig itegrals. 3. Example ad applicatio Let X = {, 1, 2, 3, 4} ad d be the usual metric of reals. Let T : X X be give by T x = 2, if x = = 1, otherwise Agai let ϕ : R + R + be give by ϕ(t = 1 for all t R +. The ϕ : [, + [, + is a Lesbesgue-itegrable mappig which is summable o each compact subset of [, +, o-egative, ad such that for each ɛ >, >. Now as we kow from Example 1.3, a self-map T satisfyig d (T x, T y β max {d (T x, x + d (T y, y, d (T y, y + d (x, y, d (T x, x + d (x, y} for all x, y X ad some β [, 2 1, is a A-cotractio; we have d(t x,t y d(x,y d(x,t x α,, { d(t x,x+d(x,y = β max, d(t y,y+d(x,y } d(y,t y d(t x,x+d(t y,y, which is satisfied for all x, y X ad some β [, 1 2 (see Theorem 2, Akram et al.[2]. So all the axioms of Theorem 2.1 are satisfied ad 1, is of course a uique fixed poit of T. We also ca show the clear distictio betwee our result ad that of Braciari (cotractive coditio 1.2 ad that of Rhoades (cotractive coditio 1.3 Let us take x =, y = 1. The from coditio 1.2, we have d(t x,t y c d(x,y implies c 1 which is ot true. So T does ot satisfy the coditio 1.2 of Braciari. Agai for same x, y X, d(t x,t y [d(x,t y+d(y,t x] max{d(x,y,d(x,t x,d(y,t y, 1 = c = c max {1, 2,, 1} 2 } which implies c 1 2. Now if we take < c < 1 2, the coditio 1.3 of Rhoades does ot satisfy.

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