This **Hammerstein** **integral** equation cannot be solved exactly. The purpose of this paper is to consider the application of Adomian’s decomposition method which has been shown to be particularly well suited for the solution of **Hammerstein** **integral** **equations** (see [5]). An advantage of this method is that it produces an analytic approximation to the solution, i.e. a function deﬁned on [0, 1], rather than approx- imate numerical values at a discrete set of points.

Within the past years or so, methods for approximating solutions of equation (.) when A is an accretive-type operator have become a ﬂourishing area of research for nu- merous mathematicians. Numerous convergence theorems have been published in var- ious Banach spaces and under various continuity assumptions. Many important results have been proved, thanks to geometric properties of Banach spaces developed from the mid-s to the early s. The theory of approximation of solutions of the equation when A is of the accretive-type reached a level of maturity appropriate for an examina- tion of its central themes. This resulted in the publication of several monographs which presented in-depth coverage of the main ideas, concepts, and most important results on iterative algorithms for appropriation of ﬁxed points of nonexpansive and pseudocon- tractive mappings and their generalizations, approximation of zeros of accretive-type op- erators; iterative algorithms for solutions of **Hammerstein** **integral** **equations** involving accretive-type mappings; iterative approximation of common ﬁxed points (and common zeros) of families of these mappings; solutions of equilibrium problems; and so on (see, e.g., Agarwal et al. []; Berinde []; Chidume []; Reich []; Censor and Reich []; William and Shahzad [], and the references therein). Typical of the results proved for solutions of equation (.) is the following theorem.

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In this paper, I present a hybrid of rationalized Haar functions method for solving nonlinear mixed **Hammerstein** **integral** **equations**. Several numerical methods for approximating the solution of **Hammerstein** **integral** **equations** are known. For Fredholm-**Hammerstein** **integral** **equations**, the classical method of successive approximations was introduced in [1]. A variation of the Nystrom method was presented in [2]. A collocation type method was developed in [3]. In [4], Brunner applied a collocation-type method to nonlinear Volterra- **Hammerstein** **integral** **equations** and integro-differential **equations**, and discussed its connection with the iterated collocation method. Guoqiang [5] introduced and discussed the asymptotic error expansion of a collocation- type method for Volterra- **Hammerstein** **integral** **equations**.

It is obvious that if an iterative algorithm can be developed for the approximation of solu- tions of equation of **Hammerstein**-type (1.2), this will certainly be a welcome complement to the Galerkin approximation method. Attempts had been made to approximate solutions of **equations** of **Hammerstein**-type using Mann-type (see e.g., Mann [25]) iteration scheme. However, the results obtained were not satisfactory (see [16]). The recurrence formulas used in these attempts, even in real Hilbert spaces, involved K −1 which is required to be strongly monotone when K is, and this, apart from limiting the class of mappings to which such iterative schemes are applicable, is also not convenient in any possible applications.

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The STHW are applied to solve nonlinear Volterra- **Hammerstein** **integral** **equations**. This method is simple. It uses only six pieces of information from the past and evaluates the driving function only six per step. However, the STHW method is very practical for computational purpose since considerable computational effort is required to improve accuracy. From the Figures 1-6, STHW fit well to these types of problems. This STHW provided a momentum for advancing numerical methods for solving nonlinear Volterra- **Hammerstein** **equations**.

As mentioned previously, numerical solutions of the Fred- holm and Volterra **integral** **equations** can be approximated by the Galerkin method, the collocation method, the quadrature method such as Nystrom (see, e.g., [1], [2] and the references cited within). In 2014, Shekarabi et al. [3] proposed the Petrov-Galerkin method for solving the Stochastic Volterra **Integral** **Equations**. In 2015, Khumalo and Mamba [4] pre- sented the numerical method for solving the Volterra **integral** **equations** using the collocation and quadrature methods. For these techniques, the **integral** **equations** are converted into a system of linear **equations**. However, the computational cost is usually high since the matrix is large and dense,

Implicit function theorems are an important tool in nonlinear analysis. They have significant applications in the theory of nonlinear **integral** **equations**. One of the most important results is the classic Hildebrandt-Graves theorem. The main assumption in all its formulations is some diﬀerentiability requirement. Applying this theorem to various types of **Hammerstein** **integral** **equations** in Banach spaces, it turned out that the hypothesis of existence and continuity of the derivative of the operators related to the studied equation is too restrictive. In 1 it is introduced an interesting linearization property for parameter dependent operators in Banach spaces. Moreover, it is proved a generalization of the Hildebrandt-Graves theorem which implies easily the second averaging theorem of Bogoljubov for ordinary diﬀerential **equations** on the real line.

Let E be a nonempty closed uniformly convex and 2-uniformly smooth Banach space with dual E ∗ . We construct some implicit and explicit algorithms for solving the equation 0 ∈ AJu in the Banach space E, where A : E ∗ → E is a monotone mapping and J : E → E ∗ is the normalized duality map which plays an indispensable role in this research paper. The advantages of the algorithm are that the resolvent operator is not involved, which makes the iteration simple for computation; moreover, the zero point problem of monotone map- pings is extended from Hilbert spaces to Banach spaces. The proposed algorithms con- verge strongly to a zero of the composed mapping AJ under concise parameter conditions. In addition, the main result is applied to approximate the minimizer of a proper convex function and the solution of **Hammerstein** **integral** **equations**. To some extent, our results extend and unify some results considered in Xu [12], Zegeye [1], Chidume and Idu [2], Chidume [3, 35], and Ibarakia and Takahashi [22].

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Over the last couple of decades, wavelets have been studied extensively and have emerged as a powerful computational tool for attaining numerical solutions for a wide range of prob- lems including **integral**, algebraic, differential, partial-differential, functional-delay and integro- differential **equations**. Wavelets are calculated as continuously oscillatory functions and pos- sess attractive features: zero-mean, fast decay, short life, time-frequency representation, multi- resolution, etc. Wavelets have the ability to detect information at different scales and at different locations throughout a computational domain. Wavelets can provide a basis set in which the ba- sis functions are constructed by dilating and translating a fixed function known as the mother wavelet. The wavelet method allows the creation of very fast algorithms when compared with the algorithms ordinarily used. Wavelets are considerably useful for solving Fredholm cum Volterra **Hammerstein** **integral** **equations** and provide accurate solutions.

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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

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T t . (39) Apply other methods of discretization to (34), in par- ticular, the quadrature (cub ature) proce sses for the case of homog eneou s **integral** equatio ns and cha nge of deri- vates by t heir differe nce analog ues in di fferential equa- tions. We obtain app roximate problems t o find approxi- mate the eigenva lues and eigenvectors of matrix opera- tor-functions in the form

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15. El-Sayed, A.M.A. and Hashem, H.H.G. Integrable and continuous solu- tions of a nonlinear quadratic **integral** equation, EJQTDE, 25 (2008) 1-10. 16. El-Sayed, A.M.A. and Hashem, H.H.G. Monotonic positive solution of nonlinear quadratic **Hammerstein** and Urysohn functional **integral** equa- tions, Commentationes Math., 48(2) (2008) 199-207.

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Dual **integral** **equations** with trigonometric kernel are reinvestigated here for a solution. The behaviour of one of the integrals at the end points of the interval complementary to the one in which it is deﬁned plays the key role in determin- ing the solution of the dual **integral** **equations**. The solution of the dual **integral** **equations** is then applied to ﬁnd an exact solution of the water wave scattering problems.

1 < α of 1.1 is discussed in 9. It turns out to be useful to study Bihari type inequalities with abstract Lebesgue **integral**. It is motivated proceeding in this direction as follows. We can get new facts about the nature of Bihari type inequalities even in the finite dimensional environment; the results can be applied in the study of certain new classes of diﬀerential and **integral** **equations** see 7–11.

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The differential and **integral** inequalities occupy a very privileged position in the theory of differential and **integral** **equations**. On the basis of various motivations, in the recent years nonlinear **integral** inequalities have received considerable attention because of the important applications to a variety of problems in diverse fields of nonlinear differential and **integral** e- quations. Some **integral** inequalities for differential and **integral** **equations** are established by Gronwall [6], Bellman [2] and Pachpatte [8, 9] which provide explicit bounds on solutions

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We propose a reformulation of the boundary **integral** **equations** for the Helmholtz equation in a domain in terms of incoming and outgoing boundary waves. We obtain transfer operator descriptions which are exact and thus incorporate features such as diffraction and evanescent coupling; these effects are absent in the well-known semiclassical transfer operators in the sense of Bogomolny. It has long been established that transfer operators are equivalent to the boundary **integral** approach within semiclassical approximation. Exact treatments have been restricted to specific boundary conditions (such as Dirichlet or Neumann). The approach we propose is independent of the boundary conditions, and in fact allows one to decouple entirely the problem of propagating waves across the interior from the problem of reflecting waves at the boundary. As an application, we show how the decomposition may be used to calculate Goos–H¨anchen shifts of ray dynamics in billiards with variable boundary conditions and for dielectric cavities.

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In this paper, we investigated on functional **Hammerstein** integro-differential equa- tions of fractional order. Here we also presented an approximate method to solve these **equations**. We proved convergence and stability of the method, too. At the end, we gave some numerical examples, which show the accuracy of the method.

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Spaˇ cek’s and Hanˇ s’s theorems to multivalued contraction mappings. Random fixed point theorems with an application to Random differential **equations** in Banach spaces are obtained by Itoh [8]. Sehgal and Waters [17] had obtained several random fixed point theorems including random analogue of the classical results due to Rothe [13]. In recent past, several fixed point theorems including Kannan type [10] Chatterjeea [5] and Zamfirescu type [20] have been generalized in stochastic version (see for detail in Joshi and Bose [9], Saha et al. ([14], [15]).

3. The solution of **integral** **equations**. In this section, we apply the algorithms de- scribed in Section 2 to some problems of **integral** **equations**. Most of the problems discussed were solved using Taylor-type method in [8]. The decomposition method is an alternative method for solving these **equations**. Whenever appropriate, we will note the comparison.

In this paper, we establish the existence of solutions to systems of **Hammerstein** **integral** inclusions under mixed monotonicity type conditions. Existence of solutions to systems of diﬀerential inclusions with initial value condition or periodic boundary value condition are also obtained. Our results rely on ﬁxed point theorems for multivalued weak contractions on complete metric spaces endowed with a graph.

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