International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 1 (2017), 32-40 http://www.etamaths.com SOME RESULTS ON FIXED POINT THEOREMS IN BANACH ALGEBRAS DIPANKAR DAS1, NILAKSHI GOSWAMI1 AND VISHNU NARAYAN MISHRA2,3,∗ Abstract. Let X be a Banach algebra and D be a nonempty subset of X. Let (T1, T2) be a pair of self mappings on D satisfying some specific conditions. Here we discuss different situations for existence of solution of the operator equation u = T1uT2u in D. Similar results are established in case of reflexive Banach algebra X with the subset D. Again, considering a bounded, open and convex subset B in a uniformly convex Banach algebra X with three self mappings T1, T2, T3 on B, we derive the conditions for existence of solution of the operator equation u = T1uT2u + T3u in B. Application of some of these results to the tensor product is also shown here with some examples. 1. Introduction In 1988, Dhage initiated application of fixed point theorems in Banach algebras. Many papers ( [4], [5], [6]) of Dhage deals with the study of non-linear integral equations via fixed point theorems in Banach algebras. In 2010, Amar et al. [1], introduced a class of Banach algebras satisfying certain sequential conditions and gave applications of non-linear integral equations using fixed point theorems under certain conditions. In 2012, Pathak and Deepmala [24], defined P-Lipschitzian maps and derived some fixed points theorem of Dhage on a Banach algebra with examples. In [12], Kilbas et al. gave many applications in the field of integral equations. In [20] different application of convergent sequence can be seen. Many researchers viz., Mishra et al. ( [13], [14], [15], [16], [17], [18]), Deepmala ( [8], [9]), Mishra [19] etc., proved some results concerning the existence of solutions for some nonlinear functional- integral equations in Banach algebra and some interesting results. In 1982, Hadzic [11] proved a generalization of Rzepecki fixed point theorem for the sum of operators in Hausdorff topological vector space. In ( [25], [26], [27]), Vijayaraju proved the existance of fixed points for asymptotic 1-set contraction mappings in real Banach spaces and also for the sum of two mappings in reflexive Banach spaces. In this paper, for a Banach algebra X with a subset D, we take a pair of self-mappings (T1,T2) on D and study the conditions under which the operator equation u = T1uT2u has a solution in D. An application of the results to the tensor product of Banach algebras is also discussed here with some suitablle examples. Also, give an apllication for nonlinear functional-integral equation. Preliminaries Def. 1 [3] Let X be a Banach space and f be a continuous (not necessarily linear) mapping of X into itself. The mapping f is said to be completely continuous if the image under f of each bounded set of X is contained in a compact set. Def. 2 Let X be a Banach algebra and T1,T2 be two self mappings on X. Then T1,T2 are said to satisfy the nonvacuous condition if for every sequence {xn}⊂ X the operator equation limn→∞T1(u)T2(xn) = u, u ∈ X has one and only one solution (xn)0 in X. Def. 3 [21] T is demiclosed if {xn}⊂ D(T), xn → x and T(xn) → y (weakly) implies x ∈ D(T) and Tx = y. Def. 4 [21] T is closed if if {xn}⊂ D(T), xn → x and T(xn) → y implies x ∈ D(T) and Tx = y. Received 13th July, 2016; accepted 19th September, 2016; published 3rd January, 2017. 2010 Mathematics Subject Classification. 47B48; 46B28; 47A80; 47H10. Key words and phrases. Banach algebras; fixed points; projective tensor product. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 32 SOME RESULTS ON FIXED POINT THEOREMS IN BANACH ALGEBRAS 33 Def. 5 [22] T is said to be demicompact at a if for any bounded sequence {xn} in D such that xn − Txn → a as n → ∞, there exists a subsequence xni and a point b in D such that xni → b as i →∞ and b−T(b) = a Def. 6 ( [10], [23]) Let T : D → D be a mapping. (1) T is said to be uniformly L-Lipschitzian if there exists L > 0 such that, for any x,y ∈ D ‖Tnx−Tny‖6 L‖x−y‖ ∀ n ∈ N (2) T is said to be asymptotically nonexpansive if there exists a sequence bn ⊂ [1,∞) with bn → 1 such that, for any x,y ∈ D ‖Tnx−Tny‖6 bn‖x−y‖ ∀ n ∈ N Algebric tensor product: [2] Let X,Y be normed spaces over F with dual spaces X∗ and Y ∗ respectively. Given x ∈ X,y ∈ Y , Let x⊗y be the element of BL(X∗,Y ∗; F) (which is the set of all bounded bilinear forms from X∗ ×Y ∗ to F), defined by x⊗y(f,g) = f(x)g(y), (f ∈ X∗,g ∈ Y ∗) The algebraic tensor product of X and Y , X⊗Y is defined to be the linear span of {x⊗y : x ∈ X,y ∈ Y} in BL(X∗,Y ∗; F). Projective tensor norm: [2] Given normed spaces X and Y , the projective tensor norm γ on X⊗Y is defined by ‖u‖γ = inf{ ∑ i ‖xi‖‖yi‖ : u = ∑ i xi ⊗yi} where the infimum is taken over all (finite) representations of u. The completion of (X ⊗ Y,γ) is called projective tensor product of X and Y and it is denoted by X ⊗γ Y . Lemma 1: [28] Let X and Y be Banach spaces. Then γ is a cross norm on X⊗Y and ‖x⊗y‖γ = ‖x‖‖y‖ for every x ∈ X,y ∈ Y . Lemma 2: [2] X ⊗γ Y can be represented as a linear subspace of BL(X∗,Y ∗; F) consisting of all elements of the form u = ∑ i xi ⊗ yi where ∑ i‖xi‖‖yi‖ < ∞. Moreover, ‖u‖γ = inf{ ∑ i‖xi‖‖yi‖} over all such representations of u. Lemma 3: [2] Let X and Y be normed algebras over F. There exists a unique product on X ⊗ Y with respect to which X ⊗Y is an algebra and (a⊗ b)(c⊗d) = ac⊗ bd (a,c ∈ X,b,d ∈ Y ) Lemma 4: [2] Let X and Y be normed algebras over F. Then projective tensor norm on X ⊗Y is an algebra norm. Clearly, we can conclude that if X and Y are Banach algebras over F then X ⊗γ Y becomes a Banach algebra. 2. Main Results Theorem 1: Let D be a non-empty compact convex subset of a Banach Algebra X and let (T1,T2) be a pair of self-mappings on D such that (a) T1 and T2 are continuous, (b) T1uT2u ∈ D for all u ∈ D Then the operator equation u = T1uT2u has a solution in D. Proof. We define J : D → D by J(u) = T1uT2u. Let {qn} be a sequence in D converging to a point q. So, q ∈ D as D is closed. Now, ‖J(u) −J(v)‖ = ‖T1uT2u−T1vT2v‖ 6 ‖T1u−T1v‖‖T2u‖ + ‖T1v‖‖T2u−T2v‖ Since T1 and T2 are continuous so, J is continuous. By an application of Schauder’s fixed point theorem we have fixed point for J. Hence the operator equation u = T1uT2u has a solution. � 34 DAS, GOSWAMI AND MISHRA Corollary 1: Let DX, DY and DX ⊗DY be closed, convex and bounded subsets of Banach algebras X, Y and X ⊗γ Y respectively. Let (T1,T2) be a pair of self mappings on DX ⊗DY such that (a) T1 and T2 are completely continuous (b) T1uT2u ∈ DX ⊗DY for all u ∈ DX ⊗DY then the operator equation u = T1uT2u has a solution in DX ⊗DY . Example 1: Let Dl1 , DK and Dl1 ⊗DK be subsets of Banach algebras l1, K and l1 ⊗γ K respectively. Define Dl1 = {x ∈ Dl1 : ‖x‖6 M1} and DK = {y ∈ DK : ‖y‖6 M2} then clearly Dl1 , DK and Dl1 ⊗DK are closed, convex and bounded. We define T1,T2 : Dl1 ⊗γ DK → Dl1 ⊗γ DK by T1( ∑ i ai ⊗xi) = ∑ i{ ainxi n }n = T2( ∑ i ai ⊗xi), where ai = {ain}n. [ l1 ⊗γ X = l1(X) by [28]]. To show that T1 is compact: Let T1m : Dl1 ⊗γ DK → Dl1 ⊗γ DK be defined by T1m( ∑ i ai ⊗xi) = ∑ i {ai1xi, ai2xi 2 , ai3xi 3 , ......, aimxi m , 0, 0, 0, ....} Then each T1m is linear, bounded and compact [7]. Also, ‖(T1m −T1)( ∑ i ai ⊗xi)‖ = ‖ ∑ i {ai1xi, ai2xi 2 , ai3xi 3 , ......, aimxi m , 0, 0, 0, ....} − ∑ i {ai1xi, ai2xi 2 , ai3xi 3 , ......, aimxi m , aim+1xi m + 1 , ....}‖ = ‖ ∑ i {0, 0, ......, 0, aim+1xi m + 1 , aim+2xi m + 2 , ....}‖ 6 ∑ i ∞∑ j=m+1 1 j |aij|.|xi| < 1 m + 1 ∑ i ∞∑ j=m+1 |aij|.|xi| 6 1 m + 1 ∑ i ∞∑ j=1 |aij|.|xi| = 1 m + 1 ∑ i ‖ai‖.|xi| So, taking the projective tensor norm, ‖(T1m −T1)( ∑ i ai ⊗xi)‖ < 1 m + 1 ‖ ∑ i ai ⊗xi‖ Therefore, T1m → T1 and so, T1 is compact. Similarly, T2 is compact. Since every compact operator in Banach space is completely continuous, so T1 and T2 are completely continuous. Then, by Corollary 1, the operator equation has a solution. Theorem 2: Let X be a non-empty Banach Algebra and let T1,T2 be three self mappings on X such that (a) S is a homomorphism and it has a unique fixed point (b) T1S = ST1 and T2S = ST2 then the unique fixed point of S is a solution of the operator equation u = T1uT2u in X. Proof. Defne J : X → X by J(u) = T1uT2u. Let a be the unique fixed point of S. Now, J(S(u)) = T1(S(u))T2(S(u)) = S(T1(u))S(T2(u)) = S(T1uT2u) = S(Ju) Hence, S(Ja) = J(S(a)) = Ja so Ja = a as S has unique fixed point. Hence the operator equation u = T1uT2u has a solution. � SOME RESULTS ON FIXED POINT THEOREMS IN BANACH ALGEBRAS 35 Example 2: Given a closed and bounded interval I = [ 1 10 , 10 10 ] in R+ the set of real numbers, consider the nonlinear functional integral equation (in short FIE) x(s) = [x(α(s))]2[q(s) + ∫ s 0 g(t,x(β(t)))dt]2 (2.1) for all s ∈ I , where α,β : I → I,q : I → R+ and g : I ×R+ → R+ are continuous. By a solution of the FIE (1) we mean a continuous function x : I → R+ that satisfies FIE (1) on I. Let X = C(I,R+) be a Banach algebra of all continuous real-valued functions on I with the norm ‖x‖ = sups∈I |x(s)|. We shall obtain the solution of FIE (1) under some suitable conditions on the functions involved in (1). Suppose that the function g satisfy the condition |g(s,x)|6 1 −q,‖q‖ < 1 for all s ∈ I and x ∈ R+. Consider the two mappings T1,T2 : X → X defined by T1x(s) = [x(α(s))] 2, s ∈ I and T2x(s) = [q(s) + ∫ s 0 g(t,x(β(t)))dt]2,s ∈ I Then the FIE (1) is equivalent to the operator equation x(s) = T1x(s)T2x(s), s ∈ I. Let S : X → X defined by S(y) = √ y, y ∈ X, where √ y(t) = √ y(t), (positive squareroot) t ∈ I . Clearly, S is a homomorphism and it has a unique fixed point 1, where 1(s) = 1, s ∈ I. It is obvious that T1S = ST1 and T2S = ST2. So, 1 is a solution of FIE(1). Theorem 3: Let D be a non-empty compact convex subset of a Banach Algebra X and let T1,T2 : D → D be two continuous self maps such that T1 and T2 satisfies nonvacuous condition, then there exists a solution of the operator equation u = T1uT2u in D. Proof. We define J : D → D by J(xn) = (xn)0. First we show that J is continuous. Let {yn}n be a sequence in D such that yn → y as n → ∞. Since, T1 and T2 satisfies nonvacuous condition so we have J(yn) = (yn) 0 = lim n→∞ T1(yn) 0T2(yn) ⇒ lim n→∞ J(yn) = lim n→∞ T1( lim n→∞ J(yn))T2(yn) So, limn→∞J(yn) is a solution of the equation limn→∞T1(u)T2(xn) = u, u ∈ X. Now, lim n→∞ J(yn) = ( lim n→∞ yn) 0 = (y)0 = J(y) Therefore, J is continuous. For u ∈ D, Ju = u0 = T1(u0)T2(u). Clearly, we get J has a fixed point by Schauder’s theorem, say α in D. Therefore, α = J(α) = T1αT2α. Thus, α is a solution of the equation u = T1uT2u ∈ D � Theorem 4: Let D be a nonempty closed bounded and convex subset of a weakly compact Banach algebra X. Let T1 : D → D and T2 : D → X be two mappings such that (a) T1 satisfies asymptotically nonexpansive mapping and limn→∞[sup‖T1x−Tn1 x‖ : x ∈ D] = 0 (b) T2 is completely continuous and M = ‖T2(D)‖ < 1 (c) I −T1 �T2 is demiclosed and Tn1 uT2v ∈ D for u,v ∈ D and n ∈ N then there exists a solution of the operator equation u = T1uT2u(= (T1 �T2)u) in D. Proof. First we show that I − T1 � T2 is closed. Let c ∈ I −T1 �T2. Then there exists a sequence {cn}⊆ I −T1 �T2 such that cn → c as n →∞. Since cn ∈ I −T1 �T2 so cn = (I −T1 �T2)zn for some zn ∈ X. Since X is weakly compact so for every sequence {zn} in D there exists weakly convergence subsequence {zni} i.e., zni → z as n →∞. Now, zni −T1 �T2zni → c as n →∞ Since I −T1 �T2 is demiclosed so c = (I −T1 �T2)z. Therefore c ∈ I −T1 �T2. Hence I −T1 �T2 is closed. 36 DAS, GOSWAMI AND MISHRA For u,v ∈ D, we define Jn : D → D by Jn(u) = qnTn1 uT2v. where qn = (1 − 1 n ) bn and {bn}→ 1 as n → ∞. Now, ‖Jn(u) −Jn(p)‖ = ‖qnTn1 uT2v −qnT n 1 pT2v‖ = qn‖T2v‖‖T n 1 u−T n 1 v‖ 6 qnbnM‖u−p‖ = (1 − 1 n )M‖u−p‖6 M‖u−p‖ Since Jn is contraction and so it has unique fixed point Kn(v) ∈ D (say), where Kn(v) = Jn(Kn(v)) = qnT n 1 (Kn(v))T2v. Now, for any v,y ∈ D we have ‖Kn(v) −Kn(y)‖ = ‖qnTn1 (Knv)T2v −qnT n 1 (Kny)T2y‖ 6 qn‖Tn1 (Knv) −T n 1 (Kny)‖‖T2v‖ + qn‖T n 1 (Kny)‖‖T2v −T2y‖ (2.2) For fixed a ∈ D, we have ‖Tn1 (u)‖ = ‖T n 1 (u) −T n 1 (a) + T n 1 (a)‖ 6 bn‖u−a‖ + ‖Tn1 (a)‖ = d(say) < ∞ From equation (2), we have ‖Kn(v) −Kn(y)‖6 dqn 1 −M ‖T2v −T2y‖ So, Kn is completely continuous as T2 is completely continuous. By Schauder’s fixed point theorem Kn has a fixed point xn, say in D. Hence xn = Knxn = Jn(xn) = qnT n 1 (xn)T2xn. Now, xn −Tn1 xnT2xn = (qn − 1)T n 1 xnT2xn → 0 as n →∞ (2.3) ‖xn −T1xnT2xn‖6 ‖xn −Tn1 xnT2xn‖ + ‖T n 1 xnT2xn −T1xnT2xn‖ = ‖xn −Tn1 xnT2xn‖ + ‖T2xn‖‖T n 1 xn −T1xn‖ → 0 as n →∞ (by (3) and condition (a)) So, 0 ∈ I −T1 �T2 as I −T1 �T2 is closed. Hence there exists a point r such that 0 = (I −T1 �T2)r. Hence the theorem follows. � Theorem 5: Let D be a nonempty closed bounded and convex subset of a reflexive Banach algebra X. Let T1 : D → D and T2 : D → X be two mappings such that (a) T1 satisfies uniformly L−Lipschitzian mapping and limn→∞[sup‖T1x−Tn1 x‖ : x ∈ D] = 0 (b) T2 is completely continuous and M = ‖T2(D)‖ such that LM < 1 (c) I −T1 �T2 is demiclosed and Tn1 uT2v ∈ D for u,v ∈ D and n ∈ N then there exists a solution of the operator equation u = T1uT2u(= (T1 �T2)u) in D. Theorem 6: Let DX, DY and DX ⊗DY be closed bounded and convex subsets of a Banach algebras X, Y and X ⊗γ Y respectively. Let (T1,T2) be a pair of self mappings on DX ⊗DY such that (a) T1 satisfies uniformly L−Lipschitzian mapping and limn→∞[sup‖T1x − Tn1 x‖ : x ∈ DX ⊗ DY ] = 0 (b) T2 is completely continuous and M = ‖T2(DX ⊗DY )‖ such that LM < 1 (c) if {xn}⊂ DX ⊗DY with xn −T1xnT2xn → 0 as n →∞ then there exists b ∈ DX ⊗DY such that 0 = (I −T1 �T2)b and Tn1 uT2v ∈ DX ⊗DY for u,v ∈ DX ⊗DY and n ∈ N then there exists a solution of the operator equation u = T1uT2u in DX ⊗DY . Example 3: Let Dl1 , DR and Dl1 ⊗DR be subsets of Banach algebras l1, R and l1 ⊗γ R respectively. Define Dl1 = {x ∈ Dl1 : ‖x‖6 1} and DR = {y ∈ DR : ‖y‖6 1} then clearly Dl1 , DR and Dl1 ⊗DR are bounded closed and convex. SOME RESULTS ON FIXED POINT THEOREMS IN BANACH ALGEBRAS 37 We define T1 : Dl1 ⊗γ DR → Dl1 ⊗γ DR is defined by T1( ∑ i ai ⊗xi) = T1( ∑ i {(ain )xi}n) = T1(u), (say) = −u where if u = {y1,y2, ...} then −u = {−y1,−y2, ...}. It is easy to see that T1 satisfies uniformly L−Lipschitzian (where L = 1) whether n is odd or even. But lim n→∞ [sup‖T1x−Tn1 x‖ : x ∈ Dl1 ⊗DR] = 0 only when n is odd. Hence condition (a) of Theorem 6 is satisfied. Now, let T2 : Dl1⊗γDR → Dl1⊗γDR be defined by T2( ∑ i ai⊗xi) = 1 2 ∑ i{ ainxi n }n, where ai = {ain}n. Clearly, condition (b) of Theorem 6 is satisfied with M = ‖T2(Dl1 ⊗ DR)‖ 6 1 2 hence LM < 1. Proceeding as in Theorem 4, we have, for {xn}⊂ Dl1 ⊗DR xn −T1xnT2xn → 0 as n →∞. Now we can take b as the constant sequence {0, 0, 0, ...} for which 0 = (I − T1 � T2)b and Tn1 uT2v ∈ Dl1 ⊗DR for u,v ∈ Dl1 ⊗DR. So, the condition (c) of Theorem 6 is satisfied. Hence the operator equation u = T1uT2u has a solution. Theorem 7: Let B be the bounded, open and convex subset with 0 ∈ B in a uniformly convex Banach algebra X. Let (T1,T2,T3) be three self mappings on B such that (a) T1 satisfies uniformly L−Lipschitzian mapping on B and limn→∞[sup‖T1x−Tn1 x‖ : x ∈ B] = 0 (b) T1 is demicompact on B and M = ‖T2(B)‖ such that LM < 1 (c) T2, T3 are completely continuous and T n 1 uT2v + T3v ∈ B for u,v ∈ B and n ∈ N then there exists a solution of the operator equation u = T1uT2u + T3u(= (T1 �T2)u + T3u) in B. Proof. Since T2 is a completely continuous, it is demicompact on B. Also T1 is demicompact by (b). So for a sequence {cn} ∈ B such that cn −T1cn → a, cn −T2cn → b as n → ∞ in B, there exists subsequence {cnk} such that cnk → c as k →∞, where c ∈ B. Since T1, T2 and T3 are continuous so T1cnk → T1c, T2cnk → T2c and T3cnk → T3c. Now we show that I −T1 �T2 −T3 is closed. Let z ∈ I −T1 �T2 −T3. Then for {zn}⊆ (I −T1 �T2 −T3)cn such that zn → z as n →∞. We have as in Theorem 4, cnk −T1 �T2cnk −T3cnk → z as n →∞ Since I −T1 �T2 −T3 is continuous so c ∈ I −T1 �T2 −T3. Hence I −T1 �T2 −T3 is closed. Define Jn : B → B by Jn(u) = qn(Tn1 uT2v + T3v), where {qn}→ 1 as n →∞. Now, ‖Jn(u) −Jn(p)‖6 qnLM‖u−p‖ Since Jn is contraction and so it has unique fixed point Knv ∈ B (say) Knv = Jn(Knv) = qn(T n 1 (Knv)T2v + T3v). Now, for any v,y ∈ B we have ‖Kn(v) −Kn(y)‖6 qn‖Tn1 (Knv) −T n 1 (Kny)‖‖T2v‖ + qn‖T n 1 (Kny)‖‖T2v −T2y‖ + ‖T3v −T3y‖ (2.4) For fixed a ∈ B, we have ‖Tn1 (u)‖6 L‖u−a‖ + ‖T n 1 (a)‖ = d(say) < ∞ From equation (4), we have ‖Kn(v) −Kn(y)‖6 dqn 1 −LM ‖T2v −T2y‖ + qn 1 −LM ‖T3v −T3y‖ 38 DAS, GOSWAMI AND MISHRA So, Kn is completely continuous as T2 and T3 are completely continuous. By Schauder’s fixed point theorem Kn has a fixed point xn, say in B. Hence xn = Knxn = Jn(xn) = qn(T n 1 (xn)T2xn + T3(xn)). Now, xn −Tn1 xnT2xn −T3xn = (qn − 1)(T n 1 xnT2xn + T3xn) → 0 as n →∞ (2.5) ‖xn −T1xnT2xn −T3xn‖6 ‖xn −Tn1 xnT2xn −T3xn‖ + ‖T2xn‖‖T n 1 xn −T1xn‖ → 0 as n →∞ (by (5) and condition (a)) Since, 0 ∈ I − T1 � T2 − T3 and I − T1 � T2 − T3 is closed. Hence there exists a point r such that 0 = (I −T1 �T2 −T3)r. Hence the theorem follows. � If 0 /∈ B in the above Theorem 7. Theorem 8: Let B be the bounded, open and convex subset in a uniformly convex Banach algebra X. Let (T1,T2,T3) be three self mappings on B such that (a) there exists r ∈ B such that r = T1c + T2c for some c ∈ B (b) all the conditions of above Theorem 7 then there exists a solution of the operator equation u = T1uT2u + T3u(= (T1 �T2)u + T3u) in B. Proof. Suppose that K = B − r = {x − r : x ∈ B}. Since B is open and bounded, so is K, and K = B −r and 0 ∈ K. Define (T1,T2,T3) are three self maps on K by T1(c−r) = T1c−r, T2(c−r) = T2c−r and T3(c−r) = T3c−r. Hence (T1,T2,T3) are three continuous self mappings in K and I −T1 �T2 −T3 is closed in K. Then (i) T1 satisfies uniformly L−Lipschitzian mapping and lim n→∞ [sup‖T1(x−r) −Tn1 (x−r)‖ : x−r ∈ K] = 0 (ii) Since T1 is demicompact in B, so T1 is demicompact in K. Also, LM < 1. Similarly, since T2 and T3 are completely continuous in B, so T2 and T3 are completely continuous in K. (iii) Clearly Tn1 (u−r)T2(v −r) + T3(v −r) ∈ K for u−r,v −r ∈ K and n ∈ N. Hence all the conditions of Theorem 7 satisfied so, there exists a solution m−r such that m−r = T1(m−r)T2(m−r) + T3(m−r) (2.6) T1(a−r)T2(a−r) + T3(a−r) = [(T1(a) −r][T2(a) −r] + T3(a) −r = T1aT2a + T3a−r − [r(T1a + T2a) −r2] without loss of generality if r = T1m + T2m, m ∈ B we have T1(m−r)T2(m−r) + T3(m−r) = T1(m)T2(m) + T3(m) −r Then from equation (6) we have a solution of the equation u = T1uT2u + T3u. � Theorem 9: Let B be the bounded, open and convex subset in a uniformly convex Banach algebra X. Let (T1,T2,T3) be three self mappings on B such that (a) T1 satisfies uniformly L−Lipschitzian mapping on B, there exists r ∈ D such that limn→∞[sup‖T1(x) −Tn1 (x)‖ : x ∈ B] = 0 and r = T1c + T2c for some c ∈ B (b) T2 and T3 are completely continuous M = ‖T2(B)‖ such that LM < 1 and Tn1 uT2v + T3v ∈ B for u,v ∈ B and n ∈ N (c) if {xn} ∈ B with xn − T1xnT2xn − T3xn → 0 as n → ∞ then there exists b ∈ B such that 0 = (I −T1 �T2 −T3)b. then there exists a solution of the operator equation u = T1uT2u + T3u(= (T1 �T2)u + T3u) in B. SOME RESULTS ON FIXED POINT THEOREMS IN BANACH ALGEBRAS 39 Acknowledgement The authors are extremely grateful to the anonymous learned referee(s) for their keen reading, valuable suggestion and constructive comments for the improvement of the manuscript. The corresponding au- thor VNM acknowledges that this project was supported by the Cumulative Professional Development Allowance (CPDA), SVNIT, Surat, Gujarat, India. 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Vijayraju, Fixed Point Theorems For a Sum of Two Mappings in Locally Convex Spaces, Int. J. Math. and Math. Sci., 17(4) (1994), 681-686. [28] A. Raymond Ryan, Introduction to Tensor Product of Banach Spaces, London, Springer -Verlag, 2002. 1Department of Mathematics, Gauhati University, Guwahati-781014, Assam, India 2Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technol- ogy, Ichchhanath Mahadev Dumas Road, Surat 395 007, Gujarat, India 3L. 1627 Awadh Puri Colony Beniganj, Phase -III, Opposite - Industrial Training Institute (I.T.I.), Ayo- dhya Main Road Faizabad 224 001, Uttar Pradesh, India ∗Corresponding author: vishnunarayanmishra@gmail.com, vishnu narayanmishra@yahoo.co.in 1. Introduction Preliminaries 2. Main Results Acknowledgement References