Nonlinear Equations Of Hammerstein Type Research Articles

Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose X is a real q -uniformly smooth Banach space and F , K : X → X are Lipschitz ϕ -strongly accretive maps with D ( K ) = F ( X ) = X . Let u ∗ denote the unique solution of the Hammerstein equation u + K F u = 0 . An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u ∗ . No invertibility assumption is imposed on K and the operators K and F need not be defined on compact subsets of X . Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.

Strong approximation of the solutions of a system of operator equations in Hilbert spaces

We give sharp conditions for the strong approximation of the solutions of a system of the form by the solutions of a system of difference equations of the form where F i are monotone operators in a Hilbert space and is the metric projection. Thus, the results complete (and correct) those by Chidume et al. [Chidume, C.E. and Zegeye, H., 2004, Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert spaces, Proceedings of the American Mathematical Society, 133(3), 851–858]. Also, we are interested in what happens when small perturbations of F i and q ij occur. Meanwhile, we generalize a numerical difference inequality due to Alber (see, e.g. Alber, Ya. I., 1983, Recurrence relations and variational inequalities, Soviet Mathematics Doklady, 27, 511–517) which is used to obtain approximation results.

On local regularization methods for linear Volterra equations and nonlinear equations of Hammerstein type

Local regularization methods allow for the application of sequential solution techniques for the solution of Volterra problems, retaining the causal structure of the original Volterra problem and leading to fast solution techniques. Stability and convergence of these methods was shown to hold on a large class of linear Volterra problems, i.e., the class of ν-smoothing problems for ν = 1, 2, … in Lamm (2005 Inverse Problems 21 785–803). In this paper, we enlarge the family of convergent local regularization methods to include sequential versions of classical regularization methods such as sequential Tikhonov regularization. In fact, sequential Tikhonov regularization was considered earlier by Lamm and Eldén (1997 SIAM J. Numer. Anal. 34 1432–50) but there the theory was limited to the class of discretized one-smoothing Volterra problems. An interesting feature of sequential classical regularization methods is that they involve two regularization parameters: the usual local regularization parameter r controls the size of the local problem while a second parameter α controls the amount of regularization to be applied in each subproblem. This approach suggests a wavelet type of regularization method with the parameter r controlling spatial resolution and α controlling frequency resolution. In this paper, we also show how the ‘future polynomial regularization’ method of Cinzori (2004 Inverse Problems 20 1791–806) can be viewed as a special case of the general framework of Lamm (2005) in the 1-smoothing case. In addition, we extend the results of Lamm (2005) to nonlinear Volterra problems of Hammerstein type and give numerical results to illustrate the effectiveness of the method in this case.

Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space

Let H be a real Hilbert space. Let F : D(F) C H → H, K: D(K) C H → H be bounded monotone mappings with R(F) C D(K), where D(F) and D(K) are closed convex subsets of H satisfying certain conditions. Suppose the equation 0 = u + KFu has a solution in D(F). Then explicit iterative methods are constructed that converge strongly to such a solution. No invertibility assumption is imposed on K, and the operators K and F need not be defined on compact subsets of H.

The solution by iteration of nonlinear equations of Hammerstein type

Let E be a real uniformly smooth Banach space. Suppose N is a set-valued (locally) accretive map with open domain D(N) C E , K is a single-valued map with domain D(K) C E such that Im(N) C D(K) and K~ exists and is (locally) strongly accretive on Im(K) . If the equation f e x + KNx has a solution x* E D(N) , the strong convergence of iteration methods of the Ishikawa and Mann types to this solution is established. Related theorems deal with the iterative solution of nonlinear equations of the form f G x + Tx for the case where T is accretive. Explicit convergence rates are established. MIRAMARE TRIESTE November 1996 Permanent address: Department of Mathematics and Computer Science, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria.

Iterative solutions of parameter-dependent nonlinear equations of hammerstein type

The unique solution of a general nonlinear Hammerstein-type equation is obtained based on the contraction mapping principle. Introducing a suitable weighted norm, the range of values of the involved parameter allowed by the contraction mapping principle is shown to be increased. Detailed construction of the suitable weight function is given. Application to a specific Hammerstein integral equation is considered supported by a numerical example.

A method of solving systems of nonlinear Hammerstein-type integral equations

In this a r t i c le we c o n s i d e r the p rob lem of the approx imate solution of a s y s t e m of non l inea r H a m m e r s t e i n type in tegra l equa t ions , us ing a va r i an t [1] of SokoLov's me thod [2] of a v e r a g i n g funct ional c o r r e c t i o n s . 1. A lgo r i t hm of Approx ima te Solution. Cons ide r the s y s t e m of non l inea r H a m m e r s t e i n t y p e in tegra l equa t ions

Strongly Nonlinear Equations of Hammerstein Type

Strongly Nonlinear Equations of Hammerstein Type

On the Variational Method for the Existence of Solutions of Nonlinear Equations of Hammerstein Type

On the Variational Method for the Existence of Solutions of Nonlinear Equations of Hammerstein Type

On the variational method for the existence of solutions of nonlinear equations of Hammerstein type

Let X X be a real Banach space and X ∗ {X^ \ast } its conjugate Banach space. Let A A be an unbounded monotone linear mapping from X X to X ∗ {X^ \ast } and N N a potential mapping from X ∗ {X^ \ast } to X X . In this paper we establish the existence of a solution of the equation u + A N u = v u + ANu = v for a given v v in X ∗ {X^ \ast } using variational method. Our method consists in using a splitting of A A via an auxiliary Hilbert space and solving an equivalent equation in this auxiliary Hilbert space. In §2, we prove the same result in the case when X X is a Hilbert space using the natural splitting of A A in terms of its square root. We do this to compare and contrast the proofs in the two cases.

New Existence Theorems for Nonlinear Equations of Hammerstein Type

New Existence Theorems for Nonlinear Equations of Hammerstein Type

New existence theorems for nonlinear equations of Hammerstein type.

Let X X be a real Banach space, X ∗ {X^ \ast } its dual, A A a linear map of X X into X ∗ {X^ \ast } and N N a nonlinear map of X ∗ {X^ \ast } into X X . Using the recent results of Browder and Gupta, Brezis, and Petryshyn, in this paper we study the abstract Hammerstein equation, w + A N w = 0 w + ANw = 0 . Assuming suitable growth conditions on N N , new existence results are obtained under the following conditions on X , A X,A and N N . In §1: X X is reflexive, A A bounded with f ( x ) = ( A x , x ) f(x) = (Ax,x) weakly lower semicontinuous, N N bounded and of type ( M ) (\text {M} ) . In §2: X X is a general space, A A angle-bounded, N N pseudo-monotone. In §3: X X is weakly complete, A A strictly (strongly) monotone, N N bounded (unbounded) and of type ( M ) (\text {M} ) . In §4: X X is a general space, A A is monotone and symmetric, N N is potential. In §5: X X is reflexive and with Schauder basis, X ∗ {X^ \ast } strictly convex, N N quasibounded and either monotone, or bounded and pseudo-monotone, or bounded and of type ( M ) (\text {M} ) .