KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION

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1 Fixed Point Theory, 13(2012), No. 1, nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd Physics Chngzhou University Chngzhou , Chin E-mil: Abstrct. A new fixed point theorem on nonliner expnsive opertors s defined in this rticle is firstly pointed out. Subsequently, we estblish severl fixed point theorems on the sum of A + B, where A is compct opertor, B is nonliner expnsive opertor. The results obtined generlize nd improve the corresponding results of Avrmescu nd Xing in ppers [C. Avrmescu, C. Vldimirescu, Some remrks on Krsnoselskii s fixed point theorem, Fixed Point Theory, 4 (2003) 3 13, T. Xing, R. Yun, A clss of expnsive-type Krsnosel skii fixed point theorems, Nonliner Anl. 71 (2009) ]. As pplictions, the existence theorem of nonnegtive solutions for clss of nonliner integrl eqution is discussed. Key Words nd Phrses: Fixed point theorem, nonliner expnsive mpping, nonliner integrl eqution Mthemtics Subject Clssifiction: 47H10, 45G Introduction In mny problems of nlysis, one encounters opertors which my be split in the form A + B, where A is compct opertor nd B is contrction in some sense, nd itself hs neither of these properties. Thus neither the Schuder fixed point theorem nor the Bnch contrction principle pplies directly in this cse, nd it becomes desirble to develop fixed point theorems for such situtions. An erly theorem of this type ws given by Krsnosel skii [1] (or see [2]). Theorem A. Let M be closed convex nonempty subset of Bnch spce X. Suppose tht A nd B mp M into X such tht (i) A is continuous nd A(M) is contined in compct set; (ii) B is contrction with constnt α < 1; (iii) Ax + By M( x, y M). Then the opertor eqution Au + Bu = u hs solution in M. Since then there hve ppered lrge number of ppers contributing generliztions or modifictions of the Theorem A nd their pplictions. One of the min fetures of such generliztions is the dopting of generlized forms of the Bnch contrction principle or the Schuder fixed point theorem. See [3] nd the references therein. Here the opertor eqution Au + Bu = u my lso be considered s perturbtion of Au = u. In [1], Krsnosel skii sserted the existence of solution of 285

2 286 FULI WANG AND FENG WANG the perturbed eqution under the condition of the perturbtive item B being contrction. However, in 2003, Avrmescu nd Vldimirescu firstly considered tht the perturbtive item B s n expnsion rther thn contrction, in [4] they obtined the following result. Theorem B. Let (X, ) be Bnch spce, nd let M = {x X : x < r}. Suppose tht A nd B mp M into X such tht (i) A is continuous, A(M) is contined in compct set; (ii) B is n expnsive mpping (i.e. there exists constnt h > 1 such tht Bx By h x y, for ll x, y X), Bθ = θ nd I B is surjective; (iii) sup Ax r(h 1). x M Then the opertor eqution Au + Bu = u hs solution in M. Recently, Xing nd Yun [5] lso considered B s n expnsion nd obtined the following result. Theorem C. Let (X, ) be Bnch spce, nd let M be nonempty closed convex subset of X. Suppose tht A nd B mp M into X such tht (i) A is continuous, A(M) is contined in compct subset of X; (ii) B is n expnsive mpping; (iii) x M implies Ax + B(M) M. Then the opertor eqution Au + Bu = u hs solution in M. The purpose of this pper is to improve nd generlize the Theorems B nd C. Being directly motivted by [4 5], in this pper, we will define new clss of mppings which is sid to be nonliner expnsion, nd prove new fixed point theorem on nonliner expnsions. As min results in present pper we will give severl Krsnosel skii type fixed point theorems for nonliner expnsion, nd n ppliction will be lso discussed. The reminder of this pper is orgnized s follows. In Section 2, we define new clss of nonliner expnsive mppings nd prove fixed point theorem on which. In Section 3, we give the results on Krsnosel skii type fixed point theorems for nonliner expnsion. In Section 4, the existence of nonnegtive solutions of clss of nonliner integrl eqution is discussed. 2. On nonliner expnsion In [6], F.E. Browder proved tht the constnt α in Bnch contrction principle cn be replced by the use of nondecresing nd right continuous function. Inspirtion by this ide we now investigte the nonliner expnsion in present section. Definition 2.1. Let (X, d) be metric spce, nd let M be certin subset of X. The mpping T : M X is sid to be nonliner expnsion, if there exists function φ : [0, + ) [0, + ) stisfying (i) φ is nondecresing, (ii) φ(t) > t for ech t > 0, (iii) φ is right continuous, such tht d(t x, T y) φ(d(x, y)), for ll x, y X. (2.1) Remrk 2.2. Note tht if one tkes φ(t) = ht with h > 1, the nonliner expnsion in (2.1) reduces to n expnsion with constnt h.

3 KRASNOSEL SKII TYPE FIXED POINT THEOREM 287 The min result in this section is the following fixed point theorem on nonliner expnsive mppings. Theorem 2.3. Let M be closed subset of complete metric spce (X, d). Assume tht mpping T : M X is nonliner expnsion nd T (M) M, then T hs unique fixed point u M. Proof. Uniqueness is cler from (2.1), so we need only prove existence. Let µ = inf{d(x, T x) : x M}. Now, we prove µ = 0. Suppose tht this is not the cse. For ny ɛ > 0, we cn find x M such tht µ d(x, T x) µ + ɛ. Since T (M) M, there is y M such tht x = T y, nd so µ d(y, T y). By virtue of nondecresing property of φ, we hve φ(µ) φ(d(y, T y)) d(t y, T 2 y) = d(x, T x) < µ + ɛ, so φ(µ) µ, contrction. Thus µ = 0. In fct, it hs just been confirmed tht there is sequence {x n } in M stisfying lim d(x n, T x n ) = 0. We next show tht such {x n } is Cuchy sequence. Suppose n tht this is not the cse. We cn find two sequences of integers {m k } nd {n k } with m k > n k k, such tht α = lim inf d(x m k, x nk ) > 0. p, k p We define the sequence of rel numbers {α k } by α k = d(x mk, x nk ), k = 1, 2,..., then there is subsequence of α k which is nonincresing nd converges t α. Without loss of generlity, we might suppose tht it is {α k }. Thus φ(α k ) = φ(d(x mk, x nk )) d(t x mk, T x nk ) d(t x mk, x mk ) + d(x mk, x nk ) + d(x nk, T x nk ) = d(t x mk, x mk ) + d(t x nk, x nk ) + α k. Let k in (2.2), we obtin φ(α) α, contrdiction. Therefore {x n } is Cuchy sequence, nd (2.2) lim x n = x M, n for X is complete nd M is closed. Since M T (M), there exist {u n } nd u in M such tht x n = T u n nd x = T u. Finlly, we prove u is fixed point. For ny ɛ > 0, there is nture number N such tht d(x n, T x n ) < ɛ 3 nd d(x n, x) < ɛ 3, provided tht n > N. By the nonliner expnsion of T, we hve d(u, T u) d(u, u n ) + d(u n, x n ) + d(x n, T u) φ(d(u, u n )) + φ(d(u n, x n )) + d(x n, T u) d(t u, T u n ) + d(t u n, T x n ) + d(x n, T u) = 2d(x, x n ) + d(x n, T x n ) < ɛ,

4 288 FULI WANG AND FENG WANG which implies T u = u. Remrk 2.4. Unlike the formerly severl results on expnsions, we give n independent proof of Theorem 2.3 which do not rely upon the fixed point theorems on T 1. Corollry 2.5. Let M be closed subset in metric spce (X, d), nd let mpping T : M X be nonliner expnsive nd surjective. Then T hs unique fixed point in M. 3. Krsnosel skii type fixed point theorems for nonliner expnsion The min result of this section is the following Krsnosel skii type fixed point theorem under the condition of nonliner expnsion. Theorem 3.1. Let (X, ) be Bnch spce, nd let M be nonempty closed convex subset of X. Suppose tht A nd B mp M into X such tht (i) A is continuous, A(M) is contined in compct subset of X; (ii) B is nonliner expnsion with φ; (iii) Ax + B(M) M, for ny x M. Then there exists point u M such tht Au + Bu = u. Proof. For ny fixed x M, we define mpping T x : M X such tht Then T x is nonliner expnsion, since T x (z) = Ax + Bz. T x z 2 T x z 1 = Bz 2 Bz 1 φ( z 2 z 1 ), for ll z 1, z 2 M, nd T x (M) = Ax + B(M) M. By using Theorem 2.3, there is unique fixed point y M, i.e. y = Ax + By. From the bove discussing, new mpping S : M M is defined, such tht S(x) = y for ny x M. Next we will prove tht the mpping S is continuous nd S(M) is contined in compct subset of X. For sequence {x n } nd its limit x 0 in M, we set y n = S(x n ) nd y 0 = S(x 0 ) respectively. Now we show tht lim y n = y 0. Suppose tht this is n not the cse. We cn find subsequence y nk such tht η = lim inf y n k y 0 > 0. p, k p From the nonliner expnsion of mpping B, we hve φ( y nk y 0 ) By nk By 0 (Ax nk + By nk ) (Ax 0 + By 0 ) + Ax nk Ax 0 = y nk y 0 + Ax nk Ax 0. (3.1) Without loss of generlity, we might suppose tht { y nk y 0 } is nonincresing nd converges t η. Let k, by the continuity of A, from (3.1) we obtin φ(η) η, contrdiction, which implies y n y 0 s n. The continuity of S hs been proved. Furthermore, the inequlity φ( y k y ) y k y By k By y k y Ax k Ax,

5 KRASNOSEL SKII TYPE FIXED POINT THEOREM 289 implies tht for ny ɛ > 0, there is δ > 0 such tht y y k < ɛ, provided tht Ax Ax k < δ. Now let Ax 1,, Ax k be δ-net in A(M) nd y i = S(x i ), i = 1,, k. Then y 1,, y k re ɛ-net in S(M). This proves tht S(M) is contined in compct subset of X. By Schuder fixed point theorem, we know tht S hs fixed point u M such tht Au + Bu = u. Some different versions of Theorem 3.1 will be discussed subsequently. We introduce useful lemm in dvnce. Lemm 3.2. Let mpping T be nonliner expnsion on certin subset M in normed liner spce X. Then I T is injective nd (I T ) 1 is continuous, where I denotes the identity opertor. Proof. For ny x, y M nd x y, we hve (I T )x (I T )y T x T y x y φ( T x T y ) x y > 0. Hence (I T ) 1 is existent. For ny ξ, η (I T )(M), by (3.2), we hve (3.2) φ( (I T ) 1 ξ (I T ) 1 η ) (I T ) 1 ξ (I T ) 1 η ξ η, (3.3) which implies (I T ) 1 is continuous on (I T )(M). Theorem 3.3. Let (X, ) be Bnch spce, nd let M be nonempty closed convex subset of X. Suppose tht (i) A : M X is continuous, A(M) is contined in compct subset of X; (ii) B : X X is nonliner expnsion with φ; (iii) A(M) (I B)(X) nd (I B) 1 A(M) M. Then there exists point u M such tht Au + Bu = u. Proof. For ny fixed x M, by (iii), we hve Ax (I B)(X), so the mpping (I B) 1 A is well defined on M. By Lemm 3.2, (I B) 1 is continuous, thus (I B) 1 A(M) is contined in compct subset of X. By Schuder fixed point theorem, there is fixed point u M such tht Au + Bu = u. The following theorem my be considered s locl version of Theorem 3.1. Without loss of generlity, we cn dmit Bθ = θ, or else we cn use B 1 insted of B for B 1 x = Bx Bθ. Theorem 3.4. Let (X, ) be Bnch spce, nd let M r = {x X : x r}. Suppose tht (i) A : M r X is continuous, A(M r ) is contined in compct subset of X; (ii) B : X X is nonliner expnsion with φ, Bθ = θ, nd I B is surjective; (iii) σ = inf t (0,r] φ(t) t t > 0, sup x M r Ax σr. Then there exists point u M r such tht Au + Bu = u. Proof. By Theorem 3.3, it is only need to prove tht (I B) 1 A(M r ) M r. For ny x M r, we hve By Bθ = θ, we obtin σ φ( (I B) 1 Ax ) (I B) 1 Ax (I B) 1. (3.4) Ax (I B) 1 θ = θ.

6 290 FULI WANG AND FENG WANG It follows from (3.3) nd (3.4) tht σ (I B) 1 Ax φ( (I B) 1 Ax ) (I B) 1 Ax Ax σr, which mens (I B) 1 Ax r, i.e. (I B) 1 A(M r ) M r. Remrk 3.5. Note tht Theorems B nd C in Section 1 respectively follow s specil cse of Theorem 3.4 nd Theorem 3.1 if we choose φ(t) = ht with h > Applictions In this section, we will pply Theorem 3.1 to the following nonliner integrl eqution u(t) = (1 + t 2 )e u(t) K(t, s, u(s))ds, t [, b], (4.1) where K : [, b] [, b] R R is continuous function. Let X = C[, b]. Then X is Bnch spce with the sup norm u = sup u(t). 0 t 1 Denote M = {x X : x r, x(t) 0, t [, b]}, where r > 0. Theorem 4.1. Suppose tht 1 + t 2 b K(t, s, x(s)) (1 + t2 )e r r, (t, s, x) [, b] [, b] M. (4.2) b Then the integrl eqution (4.1) hs t lest one nonnegtive solution u C[, b]. Proof. We cn esily convert (4.1) into the following opertor eqution x(t) = (Ax)(t) + (Bx)(t), t [, b], where the opertors A nd B re defined by (Ax)(t) = K(t, s, x(s))ds, t [, b], x M; (4.3) (Bx)(t) = (1 + t 2 )e x(t), t [, b], x M. (4.4) We next verify tht the mppings A nd B stisfy the conditions (i)-(iii) of Theorem 3.1. Firstly, using the sme method s in [7 8], the mpping A : M X is continuous, nd A(M) is contined in compct subset of X. Secondly, the mpping B : M X is nonliner expnsive. In fct, for ny x, y C[, b], we hve (Bx)(t) (Bx)(t) = (1 + t 2 ) e x(t) e y(t) e x(t) e y(t) = e min{x(t),y(t)} (e x(t) y(t) 1) e x(t) y(t) 1 x(t) y(t) [x(t) y(t)]2, t [, b]. Setting φ(t) = t t2, we obtin Bx By φ( x y ), i.e. B is nonliner expnsion. Finlly, for ny x, z M, we my define (z(t) + y(t) = ln K(t, s, x(s))ds) 1 + t 2, t [, b]. (4.5)

7 KRASNOSEL SKII TYPE FIXED POINT THEOREM 291 Consequently for t [, b], from (4.2), we get nd (z(t) + y(t) = ln K(t, s, x(s))ds) 1 + t 2 ( z(t)+ y(t) = ln ( r+ ln ln(e r ) = r. Thus y M. By (4.5), (4.3) nd (4.4), we hve z(t) = 1 0 ( ln K(t, s, x(s))ds 1+t 2 ) K(t, s, x(s))ds 1+t 2 ) K(t, s, x(s))ds 1 + t 2 ) 0, K(t, s, x(s))ds + (1 + t 2 )e y(t) = (Ax)(t) + (Bx)(t), t [, b], which implies Ax + B(M) M. Thus ll conditions of Theorem 3.1 re stisfied nd the ssertion follows. Remrk 4.2. Krsnosel skii fixed point theorem is interesting in view of the fct tht it hs wide rnge of pplictions to nonliner integrl equtions of mixed type for proving the existence of solutions. However, s fr s the uthors know, the existence of nonnegtive solutions for clss of nonliner integrl eqution s (4.1) hs nerly not been involved by former results. In certin sense, we cn interpret clss of equtions such s (4.1) s follows: if compct opertor hs the fixed point property, then this property cn be inherited for some perturbtion, provided tht this perturbtion is nonliner expnsive. Acknowledgements. The second uthor ws supported finncilly by the Nturl Science Foundtion of Chngzhou University (No. JS201008) nd Ntionl Nturl Science Foundtion of Chin (No ). References [1] M.A. Krsnosel skii, Some problems of nonliner nlysis, Amer. Mth. Trnsl., 10(2)(1958), no. 2, [2] D.R. Smrt, Fixed Point Theorems, Cmbridge University Press, Cmbridge, 31, [3] S. Prk, Generliztions of the Krsnoselskii fixed point theorem, Nonliner Anl., 47(2007), [4] C. Avrmescu, C. Vldimirescu, Some remrks on Krsnoselskii s fixed point theorem, Fixed Point Theory, 4(2003), [5] T. Xing, R. Yun, A clss of expnsive-type Krsnosel skii fixed point theorems, Nonliner Anl., 71(2009), [6] F.E. Browder, On the convergence of successive pproximtions for nonliner functionl equtions, Indgt. Mth., 30(1968), [7] P. Drbek, J. Milot, Methods of Nonliner Anlysis, Birkuser Verlg AG., 261, [8] E. Zeidler, Applied Functionl Anlysis (Applied Mthemticl Sciences 108), Springer-Verlg, Received: Octobedr 4, 2010; Accepted: Jnury 26, 2011.

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