International Journal of Analysis and Applications ISSN 2291-8639 Volume 9, Number 2 (2015), 129-141 http://www.etamaths.com ITERATIVE SOLUTIONS OF NONLINEAR INTEGRAL EQUATIONS OF HAMMERSTEIN TYPE ABEBE R. TUFA, H. ZEGEYE∗ AND M. THUTO Abstract. Let H be a real Hilbert space. Let F, K : H → H be Lipschitz monotone mappings with Lipschtiz constants L1 and L2, respectively. Suppose that the Hammerstein type equation u + KFu = 0 has a solution in H. It is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein type equation. The results obtained in this paper improve and extend known results in the literature. 1. Introduction Let H be a real Hilbert space. A mapping A : D(A) ⊂ H → H is said to be L−Lipschitz if there exists L ≥ 0 such that ||Ax−Ay|| ≤ L||x−y||, for all x,y ∈ D(A).(1.1) A is called nonexpansive mapping if L = 1 and it is called contraction mapping if L < 1. It is easy to observe that the class of Lipschitz mappings includes the class of nonexpansive and hence the class of contraction mappings. A mapping A : D(A) ⊂ H → H is said to be γ− inverse strongly monotone if there exists a positive real number γ such that 〈x−y,Ax−Ay〉≥ γ||Ax−Ay||2, for all x,y ∈ D(A).(1.2) If A is γ−inverse strongly monotone, then it is Lipschitz continuous with Lipschitz constant 1 γ . A is said to be α-strongly monotone if for each x,y ∈ D(A) there exists α > 0 such that 〈x−y,Ax−Ay〉≥ α||x−y||2.(1.3) A mapping A : D(A) ⊂ H → H is called monotone if for each x,y ∈ D(A), the following inequality holds: 〈x−y,Ax−Ay〉≥ 0.(1.4) Evidently the set of γ-inverse strongly monotone and the set of α-strongly mono- tone mappings are included in the set of monotone mappings. 2010 Mathematics Subject Classification. 47H30, 47H05, 26A16. Key words and phrases. Hammerstein type equation; Lipschitz mapping; monotone mapping. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 129 130 TUFA, ZEGEYE AND THUTO A monotone mapping A : H → H is said to be maximal monotone if R(I + λA), the range of (I + λA), is H for every λ > 0, where I is the identity mapping on H. This is equivalent to saying that, a monotone mapping A is said to be maximal monotone if it is not properly contained in any other monotone mapping. For a maximal monotone mapping A and r > 0, a mapping Jr : R(I +rA) → D(A) given by Jr = (I + rA) −1 is called the resolvent of A. It is well known that the resolvent operator, Jr, is single valued and nonexpansive mapping. The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. Interests in monotone mappings stems mainly from the fact that many physically significant problems (see e.g [20]) can be modelled by initial value problems of the form: x′(t) + Ax(t) = 0,x(0) = x0,(1.5) where A is a monotone mapping in an Hilbert space H. Such evolution equation can be found in the heat, wave and Schrödinger equations. If x(t) is independent of t, the equation (1.5) reduces to Au = 0,(1.6) whose solutions correspond to the equilibrium points of the system (1.5). A variety of problems, for example, convex optimization, linear programming, and elliptic differential equations can be formulated as finding a zero of maximal monotone mappings. Consequently, many research efforts (see, e.g., Zarantonello [16], Minty [11], Kacurovskii [9] and Vainberg and Kacurovskii [14]) have been devoted to methods of finding appropriate solutions, if it exists, of equation (1.6) and then u + Au = 0.(1.7) One important generalization of equation (1.7) is the so-called equation of Ham- merstein type (see e.g., [8]), where a nonlinear integral equation of Hammerstein type is one of the form: (1.8) u(x) + ∫ Ω k(x,y)f(y,u(y))dy = h(x), where dy is a σ-finite measure on the measure space Ω, the real kernel k is defined on Ω×Ω, f is a real-valued function defined on Ω×R and is, in general, nonlinear and h is a given function on Ω. If we now define a mapping K by Kv(x) := ∫ Ω k(x,y)v(y)dy; x ∈ Ω, and the so-called superposition or Nemytskii mapping by Fu(y) := f(y,u(y)) then, the integral equation (1.8) can be put in operator theoretic form as follows: (1.9) u + KFu = 0, where, without loss of generality, we have taken h ≡ 0. Given h in the function s- pace H, the integral equation then asks for some u in H such that (I +KF)(u) = h. We note that if K and F are monotone, then A := I + KF need not be necessarily monotone. SOLUTION OF HAMMERSTEIN TYP EQUATION 131 Equations of Hammerstein type play a crucial role in the theory that arise in differ- ential equations, for instance, elliptic boundary value problems whose linear parts possess Greens functions can, as a rule, be transformed into the form (1.9) (see e.g., [12], Chapter IV). Several existence and uniqueness theorems have been proved for equations of Ham- merstein type (see e.g., [1, 2, 3, 4, 7]). In general, equations of Hammerstein type (1.9) are nonlinear and there is no known standard method to find solutions for them. Consequently, methods of approximating solutions of such equations are of interest. In 2004, Chidume and Zegeye [6] used an auxiliary operator (in their proof), defined in a real Hilbert space in terms of K and F that is monotone whenever K and F are, and constructed an iterative procedure that converges strongly to the solution of Equation (1.9). In fact, they proved the following theorem. Theorem 1.1. ([6]) Let H be a real Hilbert space. Let F : D(F) ⊂ H → H,K : D(K) ⊂ H → H be bounded monotone mappings with R(F) ⊆ D(K) where D(F) and D(K) are closed convex subsets of H satisfying certain condition. Suppose the equation 0 = u+KFu has a solution in D(F). Let {λn} and {θn} be real sequences in (0, 1] satisfying the following conditions: (i) lim n→∞ θn = 0, (ii) ∑∞ n=1 λnθn = ∞, limn→∞ λn θn = 0, (iii) lim n→∞ ( θn−1 θn − 1 ) λnθn = 0. Let sequences {un} ⊆ D(F) and {vn} ⊆ D(K) be generated from u0 ∈ D(F) and v0 ∈ D(K), respectively by{ un+1 = PD(F) ( un −λn(Fun −vn + θn(un −w1)) ) , vn+1 = PD(K) ( vn −λn(Kvn + un + θn(vn −w2)) )(1.10) where w1 ∈ D(F) and w2 ∈ D(K) are arbitrary but fixed. Then, there exists d > 0 such that if λn ≤ d and λnθn ≤ d 2 for all n ≥ 0, the sequences {un} and {vn} con- verge strongly to u∗ and v∗, respectively in H, where u∗ is a solution of the equation 0 = u + KFu and v∗ = Fu∗. In 2012, Chidume and Djitte [5] introduced an iterative scheme and proved the following Theorem. Theorem 1.2. ([5]) Let H be a real Hilbert space. Let F,K : H → H be a bounded, monotone mapping and satisfy the range condition. Let {un} and {vn} be sequences in H defined iteratively from arbitrary u1,v1 ∈ H by{ un+1 = un −λn(Fun −vn) −λnθn(un −u1),n ≥ 1 vn+1 = vn −λn(Kvn + un) −λnθn(vn −v1),n ≥ 1 (1.11) where {λn} and {θn} are sequences in (0, 1) satisfying the following conditions. (i) lim n→∞ θn = 0, (ii) ∑∞ n=1 λnθn = ∞, λn = o(θn), (iii) limn→∞ ( θn−1 θn − 1 ) λnθn = 0. Suppose that u+KFu = 0 has a solution in H. Then, there exists a constant d0 > 0 such that if λn ≤ d0θn, for all n ≥ n0 for some n0 ≥ 1, then the sequence {un} converges to u∗, a solution of u + KFu = 0. 132 TUFA, ZEGEYE AND THUTO More recently, Zegeye and Malonza [17] introduced a method which contains an auxiliary operator, defined in an Hilbert space in terms of K and F which, under certain conditions, is monotone whenever K and F are, and whose zeros are solu- tions of equation (1.9). They proved the following Theorem. Theorem 1.3. ([17]) Let H be a real Hilbert space. Let F : H → H and K : H → H be continuous and bounded monotone operators. Let E := H × H with norm ||z||2E = ||u|| 2 H + ||v|| 2 H, for z = (u,v) ∈ E and let a map T : E → E defined by Tz = T(u,v) := (Fu−v,u + Kv) be γ−inverse strongly monotone. Let a sequence {xn} be generated by:   x0 = w ∈ E chosen arbitrarily, wn = xn −γnTxn, xn+1 = αnw + βnxn + λnwn, (1.12) where αn,βn,γn,λn ∈ (0, 1) satisfy αn + βn + λn = 1 and limn→∞αn = 0,∑∞ n=1 αn = ∞; 0 < β ≤ βn,λn, for all n ≥ 0 and 0 < a0 ≤ γn ≤ γ, for some a0,β ∈ R. Then the sequence {xn} converges strongly to x∗ = [u∗,v∗] ∈ E, where u∗ is a solution of the equation 0 = u + KFu and v∗ = Fu∗. We observe that in Theorem 1.1 and Theorem 1.2, the convergence of the Schemes to the solution of the equation u+KFu = 0 is granted by the existence of a constant which is not clear how it is calculated. In Theorem 1.3, the auxiliary operator T is used in the iteration scheme and the condition imposed on T , which is γ−inverse strongly monotone, is strong. These lead us to the following question. Question: Is it possible to construct an iterative scheme which converges strongly to a solution of Hammerstein type equation (1.9) which does not require the exis- tence of a constant and does not involve an auxiliary operator? It is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammer- stein type equation (1.9). Our theorems provide an affirmative answer to the above question in Hilbert spaces. The results obtained in this paper improve and extend the results in this direction. 2. Preliminaries Let H be a real Hilbert space and C be a nonempty, closed and convex subset of H. It is well known that for every point x ∈ H, there exists a unique nearest point in C, denoted by PCx, i.e, ||x−PCx|| ≤ ||x−y|| for all y ∈ C.(2.1) The mapping PC is called the metric projection of H onto C and characterized by the following property (see, e.g., [13]): PCx ∈ C and 〈x−PCx,PCx−y〉≥ 0, for all x ∈ H,y ∈ C.(2.2) In the sequel we shall make use of the following lemmas. SOLUTION OF HAMMERSTEIN TYP EQUATION 133 Lemma 2.1. [13] Let H be a real Hilbert space and A : H → H be a monotone mapping. Then, A is maximal monotone if and only if R(I + rA) = H for some r > 0. Lemma 2.2. [20] Let H be a real Hilbert space. If A : H → H is monotone and continuous, then A is maximal monotone. Lemma 2.3. [18] Let H be a real Hilbert space. Then for all xi ∈ H and αi ∈ [0, 1] for i = 0, 1, 2, 3, ...,n such that α0 + α1 + α2 + ... + αn = 1 the following equality holds: ||α0x0 + α1x1 + ... + αnxn||2 = n∑ i=1 αi||xi||2 − ∑ 1≤i,j≤n αiαj||xi −xj||2. Lemma 2.4. Let H be a real Hilbert space. Then, for any given x,y ∈ H, the following inequality holds: ||x + y||2 ≤ ||x||2 + 2〈y,x + y〉. Lemma 2.5. [15] Let {an} be a sequence of nonnegative real numbers satisfying the following relation: an+1 ≤ (1 −αn)an + αnδn,n ≥ n0, where {αn}⊂ (0, 1) and {δn}⊂ R satisfying the following conditions: lim n→∞ αn = 0, ∞∑ n=1 αn = ∞, and lim sup n→∞ δn ≤ 0. Then, lim n→∞ an = 0. Lemma 2.6. [19] Let H be a real Hilbert space and let A : H → H be a continuous monotone mapping. Then, N(A) = {x ∈ H : Ax = 0} is closed and convex. Lemma 2.7. [10] Let {an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani < ani+1, for all i ∈ N. Then, there exists a nondecreasing sequence {mk}⊂ N such that mk →∞ and the following properties are satisfied by all (sufficiently large) numbers k ∈ N: amk ≤ amk+1 and ak ≤ amk+1. In fact, mk = max{j ≤ k : aj < aj+1}. Lemma 2.8. [6] Let H be a real Hilbert space. Let E = H ×H with norm ||z||2E = ||u|| 2 H + ||v|| 2 H for z = (u,v) ∈ E. Then, E is a real Hilbert space and for w1 = (u1,v1),w2 = (u2,v2) ∈ E, we have that 〈w1,w2〉 = 〈u1,u2〉 + 〈v1,v2〉. Lemma 2.9. [6] Let C and D be nonempty subsets of a real Hilbert space H. Let F : C → H, K : D → H be monotone mappings. Let E = H × H with norm ||z||2E = ||u|| 2 H + ||v|| 2 H for z = (u,v) ∈ E. Define a mapping T : C × D → E by Tz = T(u,v) := (Fu−v,Kv + u). Then, T is monotone mapping. 3. Main Result We first prove the following lemma which will be used in the sequel. Lemma 3.1. Let C and D be nonempty subsets of a real Hilbert space H. Let F : C → H, K : D → H be monotone mappings. Let E = H × H with norm ||z||2E = ||u|| 2 H + ||v|| 2 H for z = (u,v) ∈ E. Define a mapping T : C × D → E by Tz = T(u,v) := (Fu−v,Kv + u). Then we have the following. 134 TUFA, ZEGEYE AND THUTO (a) If F and K are Lipschitz, then T is Lipschitz. (b) If F and K are maximal monotone, then T is maximal monotone. Proof. Since F and K are monotone, by Lemma 2.9, T is monotone mapping. (a) Let z1 = (u1,v1),z2 = (u2,v2) ∈ C × D and let L1 and L2 be Lipschitz constants of F and K, respectively. Then, we have ||Tz1 −Tz2||2 = ||(Fu1 −v1,Kv1 + u1) − (Fu2 −v2,Kv2 + u2)||2 = ||Fu1 −Fu2 − (v1 −v2)||2 + ||Kv1 −Kv2 + (u1 −u2)||2 ≤ ||Fu1 −Fu2||2 + 2||Fu1 −Fu2||||v1 −v2|| + ||v1 −v2||2 +||Kv1 −Kv2||2 + 2||Kv1 −Kv2||||u1 −u2|| + ||u1 −u2||2 ≤ ||Fu1 −Fu2||2 + ||Fu1 −Fu2||2 + ||v1 −v2||2 + ||v1 −v2||2 +||Kv1 −Kv2||2 + ||Kv1 −Kv2||2 + ||u1 −u2||2 + ||u1 −u2||2 ≤ 2||Fu1 −Fu2||2 + 2||v1 −v2||2 + 2||Kv1 −Kv2||2 + 2||u1 −u2||2 ≤ 2L21||u1 −u2|| 2 + 2||v1 −v2||2 + 2L22||v1 −v2|| 2 + 2||u1 −u2||2 ≤ 2(L21 + 1)||u1 −u2|| 2 + 2(L22 + 1)||v1 −v2|| 2 ≤ L2(||u1 −u2||2 + ||v1 −v2||2), where L = √ 2 max{ √ L21 + 1, √ L22 + 1}. Thus ||Tz1 −Tz2|| ≤ L||z1 −z2|| and hence T is Lipschitz mapping. (b) Let 0 < r < 1. Then, since F and K are maximal monotone we have that R(I + rF) = H and R(I + rK) = H. Moreover, the resolvent JFr = (I + rF)−1 of F and JKr = (I + rK) −1 of K are nonexpansive. Now, let h = (h1,h2) ∈ E. Define G := E → E by Gw = (JFr (h1 +rv),JKr (h2−ru)) for all w = (u,v) ∈ E. By the nonexpansiveness of JFr and JKr , we have ||Gw1 −Gw2|| ≤ r||w1 −w2|| , for all w1,w2 ∈ E. Thus, G is a contraction mapping. Then, by the Banach contraction principle, G has a unique fixed point say w∗ = (u∗,v∗) ∈ E. That is, Gw∗ = w∗, where u∗ = JFr (h1 + rv∗) and v∗ = JKr (h2 − ru∗). Thus, for every h = (h1,h2) ∈ E, there exists w∗ = (u∗,v∗) ∈ E such that (I + rT)(w∗) = h. Hence, R(I + rT) = E. Therefore, by Lemma 2.1, T is maximal monotone. � Now, consider the sequences {un}, {vn} ⊂ H and let u′n = F(un − γn(Fun − vn)),v ′ n = K(vn −γn(Kvn + un)). Then through out the rest of the paper, we use the following notations. i) tn = un −γn [ u′n −vn + γn(Kvn + un) ] , ii) sn = vn −γn [ v′n + un −γn(Fun −vn) ] . We now prove the following theorem. Theorem 3.2. Let H be a real Hilbert space. Let F,K : H → H be Lipschitz monotone mappings with Lipschtiz constants L1 and L2, respectively. Suppose that the equation 0 = u + KFu has a solution in H. Let ū, v̄ ∈ H and the sequences {un}, {vn}⊂ H be generated from arbitrary u0, v0 ∈ H by{ un+1 = αnū + (1 −αn)(anun + (1 −an)tn), vn+1 = αnv̄ + (1 −αn)(anvn + (1 −an)sn), (3.1) SOLUTION OF HAMMERSTEIN TYP EQUATION 135 where γn ⊂ [a,b] ⊂ (0, 1L), for L := √ 2 max{ √ L21 + 1, √ L22 + 1}, {an} ⊂ (0,r] ⊂ (0, 1) and {αn}⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. Then, the sequences {un} and {vn} converge strongly to u∗ and v∗ respectively, in H, where u∗ is the solution of 0 = u + KFu and v∗ = Fu∗. Proof. F and K are maximal monotone by Lemma 2.2. Now, let E := H × H be endowed with the norm ||z||2E = ||u|| 2 H + ||v|| 2 H, for z = (u,v) ∈ E. Define T : E → E by T(z) = T(u,v) := (Fu − v,Kv + u). Then, by Lemma 3.1, T is Lipschtiz and maximal monotone mapping. we also observe that u∗ is the solution of 0 = u + KFu if and only if z∗ = (u∗,v∗) is a solution of 0 = Tz for v∗ = Fu∗. Thus, N(T) = {z ∈ E : Tz = 0} 6= ∅. Now, for initial point z0 = (u0,v0) ∈ E, define the sequence {zn} by { xn = zn −γnTzn, zn+1 = αnw + (1 −αn)[anzn + (1 −an)(zn −γnTxn)], (3.2) where w = (ū, v̄). Observe that we have zn = [un,vn], where {un} and {vn} are sequences in (3.1). Let yn = zn−γnTxn and p ∈ N(T). Then, by the monotonicity of T, we have ||yn −p||2 = ||zn −γnTxn −p||2 −||zn −γnTxn −yn||2 = ||zn −p||2 −||zn −yn||2 + 2γn〈Txn,p−yn〉 = ||zn −p||2 −||zn −yn||2 + 2γn ( 〈Txn −Tp,p−xn〉 +〈Tp,p−xn〉 + 〈Txn,xn −yn〉 ) ≤ ||zn −p||2 −||zn −yn||2 + 2γn〈Txn,xn −yn〉 = ||zn −p||2 −||zn −xn||2 − 2〈zn −xn,xn −yn〉 −||xn −yn||2 + 2γn〈Txn,xn −yn〉 = ||zn −p||2 −||zn −xn||2 −||xn −yn||2 +2〈zn −γnTxn −xn,yn −xn〉.(3.3) But since xn = zn −γnTzn and T is Lipschitzian we obtain 〈zn −γnTxn −xn,yn −xn〉 = 〈zn −γnTzn −xn,yn −xn〉 + 〈γnTzn −γnTxn,yn −xn〉 ≤ 〈γnTzn −γnTxn,yn −xn〉≤ γnL||zn −xn||||yn −xn||.(3.4) Thus, from (3.3) and (3.4) we have that ||yn −p||2 ≤ ||zn −p||2 −||zn −xn||2 −||xn −yn||2 + 2Lγn||zn −xn||||yn −xn|| ≤ ||zn −p||2 −||zn −xn||2 −||xn −yn||2 +γnL(||zn −xn||2 + ||xn −yn||2) ≤ ||zn −p||2 + (γnL− 1)||zn −xn||2 + (γnL− 1)||xn −yn||2.(3.5) 136 TUFA, ZEGEYE AND THUTO Thus, from (3.2), Lemma 2.3, and (3.5) we have the following: ||zn+1 −p||2 = ||αnw + (1 −αn)[anzn + (1 −an)yn] −p||2 ≤ αn||w −p||2 + (1 −αn)||an(zn −p) + (1 −an)(yn −p)||2 ≤ αn||w −p||2 + (1 −αn) [ an||zn −p||2 + (1 −an)||yn −p||2 ] ≤ αn||w −p||2 + (1 −αn)an||zn −p||2 + (1 −αn)(1 −an) [ ||zn −p||2 +(γnL− 1)||zn −xn||2 + (γnL− 1)||xn −yn||2 ] . = αn||w −p||2 + (1 −αn)||zn −p||2 + (1 −αn)(1 −an)(3.6) ×(γnL− 1) [ ||zn −xn||2 + ||xn −yn||2 ] Now, since from the hypotheses, we have γn < 1 L for all n ≥ 1, the inequality (3.6) implies that ||zn+1 −p||2 ≤ αn||w −p||2 + (1 −αn)||zn −p||2.(3.7) Therefore, by induction we get that ||zn+1 −p||2 ≤ max{||z0 −p||2, ||w −p||2},∀n ≥ 0, which implies that {zn}, {xn}, and {yn} are bounded. Let z∗ = PN(T)w. Then, using (3.2), Lemma 2.4, Lemma 2.3, (3.5), (3.6) and the fact that γn < 1 L , we obtain the following: ||zn+1 −z∗||2 = ||αn(w −z∗) + (1 −αn) [ anzn + (1 −an)yn −z∗ ] ||2 ≤ (1 −αn)||anzn + (1 −an)yn −z∗||2 +2αn〈w −z∗,zn+1 −z∗〉 ≤ (1 −αn)an||zn −z∗||2 + (1 −αn)(1 −an)||yn −z∗||2 +2αn〈w −z∗,zn+1 −z∗〉 ≤ (1 −αn) ( an||zn −z∗||2 + (1 −an)[||zn −z∗||2 + (γnL− 1)(||zn −xn||2 +||xn −yn||2)] ) + 2αn〈w −z∗,zn+1 −z∗〉 = (1 −αn)||zn −z∗||2 + (1 −αn)(1 −an)(γnL− 1)(||zn −xn||2(3.8) +||xn −yn||2) + 2αn〈w −z∗,zn+1 −z∗〉 ≤ (1 −αn)||zn −z∗||2 + 2αn〈w −z∗,zn+1 −z∗〉 ≤ (1 −αn)||zn −z∗||2 + 2αn〈w −z∗,zn −z∗〉 + 2αn||zn+1 −zn||||w −z∗||.(3.9) Now, we consider two cases. Case 1. Suppose that there exists n0 ∈ N such that {||zn −z∗||} is decreasing for all n ≥ n0. Then, we get that, {||zn − z∗||)} is convergent. Thus, from (3.8), the fact that γn < b < 1 L for all n ≥ 0 and αn → 0 as n →∞, we have that yn −xn → 0,zn −xn → 0 as n →∞.(3.10) Moreover, from (3.2), (3.10) and letting n →∞, we get zn+1 −zn = αn(w −zn) + (1 −αn)(1 −an)(yn −zn) → 0.(3.11) Furthermore, since {zn} is bounded subset of H, which is reflexive, we can choose a subsequence {znj} of {zn} such that znj ⇀ ẑ and lim sup n→∞ 〈w − z∗,zn − z∗〉 = SOLUTION OF HAMMERSTEIN TYP EQUATION 137 lim j→∞ 〈w−z∗,znj −z ∗〉. This together with (3.10) implies that ynj ⇀ ẑ and xnj ⇀ ẑ. Now, we show that ẑ ∈ N(T). But, since T is Lipschitz continuous, we have ||Tynj −Txnj||→ 0 as j →∞. Let (s,t) ∈ G(T). Then, we have t−Ts = 0 and hence we get 〈s−z,t−Ts〉 = 0, for all z ∈ E. On the other hand, since ynj = znj −γnjTxnj , we have 〈znj −γnjTxnj − ynj,ynj −s〉 = 0, and hence, 〈s−ynj, (ynj −znj )/γnj + Txnj〉 = 0. Thus, we get 〈s−ynj, t〉 = 〈s−ynj,Ts〉 = 〈s−ynj,Ts〉−〈s−ynj, (ynj −znj )/γnj + Txnj〉 = 〈s−ynj,Ts−Tynj〉 + 〈s−ynj,Tynj −Txnj〉 −〈s−ynj, (ynj −znj )/γnj〉 ≥ 〈s−ynj,Tynj −Txnj〉−〈s−ynj, (ynj −znj )/γnj〉. This implies that 〈s − ẑ, t〉 ≥ 0, as j → ∞. Then, maximality of T gives that ẑ ∈ N(T). Thus, from (2.2), we immediately obtain that lim sup n→∞ 〈w −z∗,zn −z∗〉 = lim j→∞ 〈w −z∗,znj −z ∗〉 = 〈w −z∗, ẑ −z∗〉≤ 0.(3.12) Hence, it follows from (3.9), (3.11), (3.12) and Lemma 2.5 that ||zn − z∗|| → 0 as n →∞. Consequently, zn → z∗ = (u∗,v∗) = PN(T)w. Case 2. Suppose that there exists a subsequence {ni} of {n} such that ||zni −z ∗|| < ||zni+1 −z ∗||, for all i ∈ N. Then, by Lemma 2.7, there exist a nondecreasing sequence {mk}⊂ N such that mk →∞, and ||zmk −z ∗|| ≤ ||zmk+1 −z ∗|| and ||zk −z∗|| ≤ ||zmk+1 −z ∗||,(3.13) for all k ∈ N. Now, from (3.8), the fact that γn < 1L for all n ≥ 0 and αn → 0 as n → ∞, we get that ymk −xmk → 0,zmk −xmk → 0 as k → ∞. Thus, following the method in Case 1, we obtain lim sup k→∞ 〈w −z∗,zmk −z ∗〉≤ 0.(3.14) Now, replacing zn by zmk in (3.9), we have that ||zmk+1 −z ∗||2 ≤ (1 −αmk||zmk −z ∗||2 + 2αmk〈w −z ∗,zmk −z ∗〉, +2αmk||zmk+1 −zmk||.||w −z ∗||,(3.15) and hence (3.13) and (3.15) imply that αmk||zmk −x ∗||2 ≤ 2αmk〈w −z ∗,zmk −z ∗〉 + 2αmk||zmk+1 −zmk||.||w −z ∗||. But the fact that αmk > 0 implies that ||zmk −z ∗||2 ≤ 2〈w −z∗,zmk −z ∗〉 + 2||zmk+1 −zmk||.||w −z ∗||. Thus, using (3.14) and (3.11) we get that ||zmk −z ∗||→ 0 as k →∞. This together with (3.15) implies that ||zmk+1−z ∗||→ 0 as k →∞. But ||zk−z∗|| ≤ ||zmk+1−z ∗|| for all k ∈ N gives that xk → z∗. Therefore, from the above two cases, we can conclude that {zn} converges strongly to a point z∗ = (u∗,v∗) = PN(T)w, where u∗ is the solution of 0 = u + KFu and v∗ = Fu∗. The proof is complete. � 138 TUFA, ZEGEYE AND THUTO If, in Theorem 3.2, we assume that F is γ1-inverse strongly monotone and K is γ2-inverse strongly monotone, then both F and K are Lipschitz with Lipschitz con- stant L′ = max{ 1 γ1 , 1 γ2 } and hence we get the following corollary. Corollary 3.3. Let H be a real Hilbert space. Let F : H → H be γ1-inverse strongly monotone and K : H → H be γ2-inverse strongly monotone mappings. Suppose that the equation 0 = u + KFu has a solution in H. Let ū, v̄ ∈ H and the sequences {un},{vn}⊂ H be generated from arbitrary u0 and v0 in H by{ un+1 = αnū + (1 −αn)(anun + (1 −an)tn), vn+1 = αnv̄ + (1 −αn)[anvn + (1 −an)sn), (3.16) where γn ⊂ [a,b] ⊂ (0, 1L), for L := √ 2((L′)2 + 1), {an} ⊂ (0,r] ⊂ (0, 1) and {αn} ⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. Then, the sequences {un} and {vn} converge strongly to u∗ and v∗ respectively, where u∗ is the solution of the equation 0 = u + KFu and v∗ = Fu∗. If, in Theorem 3.2, we assume that F is Lipschitz α1-strongly monotone with Lips- chitz constant L1 and K is Lipschitz α2-strongly monotone with Lipschitz constant L2, then one can show that F is α1 L21 -inverse strongly monotone and K is α2 L22 -inverse strongly monotone and hence we get the following corollary. Corollary 3.4. Let H be a real Hilbert space. Let F : H → H be Lipschitz α1- strongly monotone and K : H → H be Lipschitz α2-strongly monotone mappings. Suppose that the equation 0 = u + KFu has a solution in H. Let ū, v̄ ∈ H and the sequences {un},{vn}⊂ H be generated from arbitrary u0 and v0 in H by{ un+1 = αnū + (1 −αn)(anun + (1 −an)tn), vn+1 = αnv̄ + (1 −αn)[anvn + (1 −an)sn), (3.17) where γn ⊂ [a,b] ⊂ (0, 1L), for L := √ 2((L′′)2 + 1) and L′′ = max{L 2 1 α1 , L22 α2 }, {an}⊂ (0,r] ⊂ (0, 1) and {αn} ⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and∑ αn = ∞. Then, the sequences {un} and {vn} converge strongly to u∗ and v∗ respectively, where u∗ is the solution of the equation 0 = u + KFu and v∗ = Fu∗. If, in Theorem 3.2, we assume that F = I, an identity mapping on H, then F is Lipschitz monotone with Lipschitz constant L1 = 1 and the sequences {tn} and {sn} reduce to: i) t′n = (1 −γn)un + (1 −γn)γnvn −γ2nKvn, ii) s′n = (1 −γ2n)vn + (γn − 1)γnun −γnv′n, where v′n = K(vn −γn(Kvn + un)) and hence we get the following corollary. Corollary 3.5. Let H be a real Hilbert space. Let K : H → H be Lipschitz monotone mapping with Lipschtiz constant L2. Suppose that the equation 0 = u + Ku has a solution in H. Let ū, v̄ ∈ H and the sequences {un},{vn} ⊂ H be generated from arbitrary u0 and v0 in H by{ un+1 = αnū + (1 −αn)(anun + (1 −an)t′n), vn+1 = αnv̄ + (1 −αn)[anvn + (1 −an)s′n), (3.18) SOLUTION OF HAMMERSTEIN TYP EQUATION 139 where γn ⊂ [a,b] ⊂ (0, 1L), for L := √ 2 max{ √ 2, √ L22 + 1}, {an} ⊂ (0,r] ⊂ (0, 1) and {αn} ⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. Then, the sequences {un} and {vn} both converge strongly to u∗, where u∗ is the solution of the equation 0 = u + Ku. If, in Theorem 3.2, we assume that K = I, an identity mapping on H, then K is Lipschitz monotone with Lipschitz constant L2 = 1 and the sequences {tn} and {sn} reduce to: i) t′′n = (1 −γ2n)un + (1 −γn)γnvn −γnu′n, ii) s′′n = (1 −γn)vn + (γn − 1)γnun + γ2nFun, where u′n = F(un −γn(Fun −vn)) and hence we get the following corollary. Corollary 3.6. Let H be a real Hilbert space. Let F : H → H be Lipschitz monotone mapping with Lipschtiz constant L1. Suppose that the equation 0 = u+Fu has a solution in H. Let ū, v̄ ∈ H and the sequences {un}, {vn} ⊂ H be generated from arbitrary u0 and v0 in H by{ un+1 = αnū + (1 −αn)(anun + (1 −an)t′′n), vn+1 = αnv̄ + (1 −αn)[anvn + (1 −an)s′′n), (3.19) where γn ⊂ [a,b] ⊂ (0, 1L), for L := √ 2 max{ √ 2, √ L21 + 1}, {an} ⊂ (0,r] ⊂ (0, 1) and {αn} ⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. Then, the sequences {un} and {vn} converge strongly to u∗ and −u∗, respectively, where u∗ is the solution of the equation 0 = u + Ku. We note that the method of proof of Theorem 3.2 provides the following theorem for approximating the unique minimum norm point of solution of the Hammerstein type equation. Theorem 3.7. Let H be a real Hilbert space. Let F,K : H → H be Lipschitz monotone mappings with Lipschtiz constants L1 and L2, respectively. Suppose that the equation 0 = u + KFu has a solution in H. Let the sequences {un},{vn} ⊂ H be generated from arbitrary u0 and v0 in H by{ un+1 = (1 −αn)(anun + (1 −an)tn), vn+1 = (1 −αn)(anvn + (1 −an)sn), (3.20) where γn ⊂ [a,b] ⊂ (0, 1L), for L := √ 2 max{ √ L21 + 1, √ L22 + 1}, {an} ⊂ (0,r] ⊂ (0, 1) and {αn}⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. Then, the sequence {zn} = {(un,vn)} converges strongly to the unique minimum norm point z∗ = (u∗,v∗) in H × H, where u∗ is a solution of 0 = u + KFu and v∗ = Fu∗. Remark 3.8. Theorem 3.2 improves Theorem 3.4 of Chidume and Zegeye [6] and Theorem 3.1 of Chidume and Djitte [5] in the sense that the convergence of our scheme does not require the existence of a constant number. Remark 3.9. Theorem 3.2 extends Theorem 3.4 of Zegeye and Malonza [17] in the sense that our scheme, which does not involve the auxiliary mapping, provides strong convergence to a solution of Hammerstein type equation for a more general class of monotone mappings. Our theorems provide an affirmative answer to the above question in Hilbert spaces. 140 TUFA, ZEGEYE AND THUTO 4. Numerical example Now, we give an example of Lipschitz monotone mappings satisfying conditions of Theorem 3.2 and some numerical experiment result to explain the conclusion of the theorem. Let H = R with absolute value norm. Let F,K : R → R be defined by Fx = 3x and Kx = 2x− 14.(4.1) Clearly, F and K are Lipschitz maximal monotone mappings with constants 3 and 2, respectively. Furthermore, we observe that u∗ = 2 is the solution of u+KFu = 0. Now if we take, αn = 1 n+100 , γn = 1 n+200 + 0.01, an = 1 n+100 + 0.01, and w = (ū, v̄) = (1, 0), we observe that the conditions of Theorem 3.2 are satisfied and Scheme (3.1) reduces to{ un+1 = αnū + (1 −αn)(anun + (1 −an)tn), vn+1 = αnv̄ + (1 −αn)(anvn + (1 −an)sn), (4.2) where tn = (1 − 3γn + 8γ2n)un + (1 − 5γn)γnvn + 14γ2n, and sn = (3γ 2 n − 2γn + 1)vn + (5γn − 1)γnun − 28γ2n + 14γn. Thus, for (u0,v0) = (1, 3), (un,vn) converges strongly to (u ∗,v∗) = (2, 6) = PN(T)(w), where 2 is the solution of u + KFu = 0 and 6 = F(2). See the fol- lowing Table and Figure. n 1 101 1001 2001 3001 4001 5001 6001 7001 7901 un 1.0000 1.5552 1.9025 1.9470 1.9636 1.9723 1.9776 1.9812 1.9838 1.9856 vn 3.0000 5.0504 5.7888 5.8853 5.9213 5.9400 5.9516 5.9594 5.9650 5.9689 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1 2 3 4 5 6 iterations, n u n, v n u 0 =1, u n v 0 =3, v n Figure 1 SOLUTION OF HAMMERSTEIN TYP EQUATION 141 References [1] H. Brezis and F. Browder, Nonlinear integral equations and systems of Hammerstein type, Advances in Math., 18 (1975), 115-147. [2] H. Brezis and F. Browder, Existence theorems for nonlinear integral equations of Hammer- stein type, Bull. Amer. Math. Soc. 81 (1975), 73-78. [3] F. E. Browder, D. G. de Figueiredo and P. Gupta, Maximal monotone operators and a nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 76 (1970), 700- 705. 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