Application of Reproducing Kernel Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations

and Applied Analysis 3 2. Several Reproducing Kernel Spaces In this section, several reproducing kernels needed are constructed in order to solve 1.1 and 1.2 using RKHSmethod. Before the construction, we utilize the reproducing kernel concept. Throughout this paper C is the set of complex numbers, L2 a, b {u | ∫b a u2 x dx < ∞}, l2 {A | ∑∞i 1 Ai 2 < ∞}, and the superscript n in u n t denotes the n-th derivative of u t . Definition 2.1 see 18 . Let E be a nonempty abstract set. A function K : E × E → C is a reproducing kernel of the Hilbert spaceH if 1 for each t ∈ E, K ·, t ∈ H, 2 for each t ∈ E and φ ∈ H, 〈φ · , K ·, t 〉 φ t . The last condition is called “the reproducing property”: the value of the function φ at the point t is reproducing by the inner product of φ with K ·, t . A Hilbert space which possesses a reproducing kernel is called a RKHS 18 . Next, we first construct the space W2 2 a, b in which every function satisfies the initial condition 1.2 and then utilize the space W1 2 a, b . Definition 2.2 see 30 . W2 2 a, b {u : u, u′ are absolutely continuous on a, b , u, u′, u′′ ∈ L2 a, b , and u a 0}. The inner product and the norm inW2 2 a, b are defined, respectively, by 〈u, v〉W2 2 u a v a u ′ a v′ a ∫b a u′′ ( y ) v′′ ( y ) dy 2.1 and ‖u‖W2 2 √ 〈u, u〉W2 2 , where u, v ∈ W 2 2 a, b . Definition 2.3 see 23 . W1 2 a, b {u : u is absolutely continuous on a, b and u, u′ ∈ L2 a, b }. The inner product and the norm inW1 2 a, b are defined, respectively, by 〈u, v〉W1 2 ∫b a u t v t u ′ t v′ t dt and ‖u‖W1 2 √ 〈u, u〉W1 2 , where u, v ∈ W 1 2 a, b . In 23 , the authors have proved that the space W1 2 a, b is a complete reproducing kernel space and its reproducing kernel function is given by Tx ( y ) 1 2 sinh b − a [ cosh ( x y − b − a) cosh(∣∣x − y∣∣ − b a)]. 2.2 From the definition of the reproducing kernel spaces W1 2 a, b and W 2 2 a, b , we get W 1 2 a, b ⊃ W2 2 a, b . The Hilbert space W2 2 a, b is called a reproducing kernel if for each fixed x ∈ a, b and any u y ∈ W2 2 a, b , there exist K x, y ∈ W2 2 a, b simply Kx y and y ∈ a, b such that 〈u y , Kx y 〉W2 2 u x . The next theorem formulates the reproducing kernel function Kx y . 4 Abstract and Applied Analysis Theorem 2.4. The Hilbert spaceW2 2 a, b is a reproducing kernel and its reproducing kernel function Kx y can be written as