Available online at Advances in Fixed Point Theory, 2 (2012), No. 4, ISSN:

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1 Available online at Advances in Fixed Point Theory, 2 (2012), No. 4, ISSN: FIXED POINTS AND SOLUTIONS OF NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES A. EL-SAYED AHMED 1,2, AND ALAA KAMAL 3 1 Department of Mathematics, Sohag University Faculty of Science, City Sohag, Egypt 2 Current Address: Taif University, Faculty of Science, Mathematics Department Box 888 El-Hawiyah, El-Taif 5700, Saudi Arabia 3 Majmaah University Faculty of Science and Humanities, Mathematics Department Box 66 Ghat 11914, Saudi Arabia Abstract. In this paper, we obtain some common fixed point theorems in Banach spaces for two compatible mapping of type (T)/(I) and for weakly biased mappings. Also, we give applications for the solvability of certain non-linear functional equations. Keywords: Fixed points, compatible maps of type (T )/(I), weakly biased maps, operator equation AMS Subject Classification: 47H10; 54H25 1. Introduction The concept of compatible mappings of type (T) (type(i)) introduced by Pathak et al (see 10]). Definition ] Let I and T be a mappings from a normed space E into itself. The mappings I and T are said to be compatible mappings of type (T) if lim IT x n Ix n + lim IT x n T Ix n lim T Ix n T x n n n n Corresponding author Received April 5,

2 NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES 443 whenever {x n } is a sequence in E such that lim Ix n = lim T x n = t for some t E. n n Definition ] Let I and T be a mappings from a normed space E into itself. The mappings I and T are said to be compatible mappings of type (I) if lim T Ix n T x n + lim IT x n T Ix n lim IT x n Ix n n n n whenever {x n } is a sequence in E such that lim Ix n = lim T x n = t for some t E. n n Definition , 5] Let F and G be self-maps of a metric space (M, d). The pair {F, G} is G-biased if and only if whenever {x n } is a sequence in X and F x n, Gx n t X, then αd(gf x n, Gx n ) αd(f Gx n, F x n ) if α = lim inf and if α = lim sup. Definition , 5] Let F and G be self-maps of X. The pair {F, G} is weakly G-biased if and only if F p = Gp implies d(gf p, Gp) d(f Gp, F p). Clearly, every biased mappings are weakly biased mappings ( see proposition 1.1 in 4]). The paper is organized as follows: In Section 1, we explain some notations, concepts and the results as noted earlier which can be found in 1-13]. In Section 2, we prove a coincidence fixed point theorem for two compatible mappings of type (T)/ (I) in Banach spaces. Finally, we apply both definitions of compatible mappings of type (T)/ (I) and weakly G-baised to obtain solutions of nonlinear functional equations in Banach spaces as in Section 3.

3 444 A. EL-SAYED AHMED 1,2, AND ALAA KAMAL 3 2. A common fixed point theorem In this section, we obtain a common fixed point theorem in Banach spaces for two compatible mappings of type T. Now, we prove the following result. Our result is more general the correspondence one in 6] and 11]. Theorem 2.1. Let T and I be two compatible mappings of type (T )/(I) of a Banach space X into itself, satisfying the following conditions: (1) (1 k)i(x) + kt (X) I(X), where 0 < k < 1, (2) T x T y p Φ(max{ Ix Iy p, Ix T x p, Iy T y p, Ix T y p, Iy T x p }) for all x, y X, where p > 0, and the function Φ satisfies the following conditions: (a)φ : 0, ) 0, ) is nondecreasing and right continuous (b) For every t > 0, Φ(t) < t. If for some x 0 X, the sequence {x n } defined by (3) Ix n+1 = (1 c n )Ix n + c n T x n, for all n 0 with (i) 0 < c n 1 and (ii) lim n c n = h > 0 for n = 0, 1, 2,... converges to a point z in X and if I is continuous at z, then T and I have a unique common fixed point. Further, if I is continuous at T x, then T and I have a unique common fixed point at which T is continuous. Proof. Let z X such that lim x n = z. n Now since I is continuous at z. Then, we have that Ix n Iz as n, so from (3) we have T x n = Ix n+1 (1 c n )Ix n c n Iz (1 h)iz h = Iz as n.

4 NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES 445 Now we shall show that T z = Iz. Form (2) we have T x n T z p Φ(max{ Ix n Iz p, Ix n T x n p, Iz T z p, Ix n T z p, Iz T x n p }). Taking the limit as n yields Iz T z p Φ(max{0, 0, Iz T z p, Iz T z p, 0}). If T z Iz p > 0, then one obtains the contradiction Iz T z p < Iz T z p. Therefore, T z = Iz i.e, z is a coincidence point of T and I. compatible mappings of type (T) if Now since T and I are lim IT x n Ix n + lim IT x n T Ix n lim T Ix n T x n n n n whenever {x n } is a sequence in X such that Hence using (2), lim Ix n = lim T x n = t, for some t X. n n (4) T 2 z T z p Φ(max{ IT z Iz p, IT z T 2 z p, Iz T z p, IT z T z p, Iz T 2 z p }), or or T 2 z T z p IT z T Iz p ( T Iz Iz + IT z Iz ) p T 2 z T z T 2 z T z IT z Iz, which implies that, IT z Iz 0, and so (5) IT z = Iz = T z.

5 446 A. EL-SAYED AHMED 1,2, AND ALAA KAMAL 3 Substitute from (5) to (4), we obtain that T 2 z T z p Φ(max{0, Iz T 2 z p, 0, 0, Iz T 2 z p }) Φ(max{0, T z T 2 z p, 0, 0, T z T 2 z p }) Φ( T 2 z T z p ) T 2 z T z p which is a contradiction. Therefore, we obtain that T 2 z = T z = IT z, i.e., T z is a common fixed point of T and I. Let v be a another common fixed point of T and I. By (2), we have u v p = T u T v p Φ(max{ Iu Iv p, Iu T u p, Iv T v p, Iu T v p, Iv T u p }), = Φ(max{ u v p, 0, 0, u v p, v u p }) which implies that u = v. This completing the proof of the theorem. Remark 2.1. Similar result of Theorem 2.1, can be obtained if we used the concept of weakly G-biased mappings instead of compatible mappings of type (T )/(I). 3. Application to operator equations By using a contraction condition more general than that used in 11] and replacing weakly compatible mappings by compatible mappings of type (T)/ (I), we investigate the solvability of certain nonlinear functional equations in Banach spaces. Theorem 3.1. Let {f n } be sequence of elements in a Banach space X. Let ν n be the unique solution of the equation u T Iu = f n, where T, I : X X satisfying the following conditions (h 1 ) T and I are compatible mapping of type (T) (h 2 ) T 2 = I 2 = I, where I denotes the identity mapping,

6 NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES 447 (h 3 ) T x T y 2 q max{ Ix Iy 2, Ix T x 2, Iy T y 2, Ix T y 2, Iy T x 2 } for all x, y X, where q (0, 1). If f n 0 as n, then the sequence {ν n } converges to the solution of the equation u = T u = Iu. Proof. We will show that {ν n } is a Cauchy sequence. ν n ν m 2 ] 2 ν n T Iν n + T Iν n T Iν m + T Iν m ν m 2 ] ν n T Iν n + T Iν m ν m ] + 2 ν n T Iν n + T Iν m ν m ] T Iν n ν n + ν n ν m + ν m T Iν m + T Iν n T Iν m 2 2 ] ν n T Iν n + T Iν m ν m ] + 2 ν n T Iν n + T Iν m ν m ] T Iν n ν n + ν n ν m + ν m T Iν m { +q max I 2 ν n I 2 ν m 2, I 2 ν n T Iν n 2, I 2 ν m T Iν m 2, I 2 ν n T Iν m 2, I 2 ν m T Iν n 2 } 2 ] ν n T Iν n + T Iν m ν m ] + 2 ν n T Iν n + T Iν m ν m ] 2 T Iν n ν n + ν n ν m + ν m T Iν m { 2 +q max ν n ν m 2, ν n T Iν n 2, ν m T Iν m 2, ν n ν m + ν m T Iν m ], ] 2 } ν m ν n + ν n T Iν n. On letting n and using the hypothesis, we obtain ν n ν m 2 q ν n ν m 2,

7 448 A. EL-SAYED AHMED 1,2, AND ALAA KAMAL 3 which is a contradiction. It follows therefore that {ν n } is a Cauchy sequence in X. Hence it converges, say to ν in X. Also ν T Iν ν ν n + ν n T Iν n + T Iν n T Iν ν ν n + ν n T Iν n + {q max{ ν n ν 2, ν n T Iν n 2, ν T Iν 2, ν n ν + ν T Iν ] 2, ν n ν + ν n T Iν n ] 2 }} 1 2. Hence taking the limit as n, we get ν = T Iν, which from (h 2 ) implies that T ν = Iν. Since T and I are compatible mapping of type (T ), we have lim IT x n Ix n + lim IT x n T Ix n lim T Ix n T x n n n n whenever {x n } is a sequence in X such that Using (h 2 ) and (h 3 ), we obtain that lim Ix n = lim T x n = t, for some t X. n n v T v 2 = T 2 v T v 2 q max{ IT v Iv 2, IT v T 2 v 2, Iv T v 2, IT v T v 2, Iv T Iv 2 } = q max{ v T v 2, 0, 0, v T v 2, v T v 2 }, which implies that ν = T ν. Then Iν = IT ν = T Iν = ν, and ν is also a fixed point of I. Now, for the class of weakly G-biased mappings, we obtain the following theorem: Theorem 3.2. Let {f n } and {g n } be sequences of elements in a Banach space X. Let {ν n } be the unique solution of the system of equations u F Gu = f n and u HGu = g n, where F, G and H : X X satisfying the following conditions: (I) {F, G} and {H, G} are weakly G-biased pairs, (II) F 2 = G 2 = H 2 = I, where I denotes the identity mapping, (III) F x Hy 2 q max{ Gx Gy 2, Gx F x 2, Gy Hy 2, Gx F y 2, Gy Hx 2 } for all x, y X, where q (0, 1). If f n, g n 0 as n, then the sequence {ν n } converges to the solution of the equation u = F u = Gu = Hu.

8 NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES 449 Proof. We will show that {ν n } is a Cauchy sequence ν n ν m 2 ν n F Gν n + F Gν n HGν m + HGν m ν m ] 2 { ν n F Gν n + HGν m ν m } ν n F Gν n + HGν n ν m ] F Gν n ν n + ν n ν m + ν m HGν m ] + F Gν n HGν n 2 { ν n F Gν n + HGν m ν m } ν n F Gν n + HGν n ν m ] F Gν n ν n + ν n ν m + ν m HGν m ] + q max{ G 2 ν n G 2 ν m G 2 ν n F Gν n, G 2 ν n HGν m 2, G 2 ν n F Gν m 2, G 2 ν m HGν m 2 } ν n F Gν n + HGν m ν m ] ν n F Gν n + F Gν m ν m ] F Gν n ν n + ν n ν m + ν m HGν m ] +q max{ ν n ν m 2, ν n F Gν n 2, ν m HGν m 2, ν n ν m + ν m F Gν m ] 2, ν m ν n + ν n HGν n ] 2 }. Letting n with m > n, we have which implies that lim ν n ν m 2 q lim ν n ν m 2, m,n m,n lim ν n ν m 2 = 0. m,n Thus {ν n } is a Cauchy sequence in X. And converges to a point ν in X. Further; ν HGν ν ν n + ν n F Gν n + F Gν n HGν ν ν n + ν n F Gν n + {q max{ ν n ν 2, ν n F Gν n 2, ν HGν 2, ν n ν + ν F Gν ] 2, ν n ν + ν n HGν n ] 2 }} 1 2 Hence taking the limit as n, we get ν = HGν, which from (I) implies that Hν = Gν. Similarly, Gν = F ν. From (I), we have d(gf z, Gz) d(f Gz, F z) and d(ghz, Gz) d(hgz, Hz),

9 450 A. EL-SAYED AHMED 1,2, AND ALAA KAMAL 3 which implies d(g 2 z, Gz) d(f 2 z, F z) and d(g 2 z, Gz) d(h 2 z, Hz), so we obtain H 2 z = Hz = F 2 z = F z and F Gv = GF v = v = HGv = GHv. Using (II) and (III), v Hv 2 = F 2 v Hv 2 q max{ GF v Gv 2, GF v F 2 v 2, Gv Hv 2, GF v F v 2, Gv HF v 2 } = q max{ v Hv 2, 0, 0, v Hv 2, v Hv 2 }, which implies that v = Hv. Then Gv = GF v = F Gv = v, Gv = GHv = HGv = v, this completing the proof of the theorem. Corollary 3.1. Let {f n } be sequence of elements in a Banach space X. Let ν n be the unique solution of the equation u F Gu = f n, where F, G : X X satisfying the following conditions: (d 1 ) F and G are weakly G-biased mappings (d 2 ) F 2 = G 2 = I, where I denotes the identity mapping, (d 3 ) F x F y 2 q max{ Gx Gy 2, Gx F x 2, Gy F y 2, Gx F y 2, Gy F x 2 } for all x, y X, where q (0, 1). If f n 0 as n, then the sequence {ν n } converges to the solution of the equation u = F u = Gu.

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