About nonlinear integro-differential Volterra and Fredholm equations

Two nonlinear problems in terms of abstract operator equations of the form ${Bx=f}$ are investigated in this paper. In the first problem the operator $B$ contains a linear differential operator $A$, the Volterra operator $K$ with kernel of convolution type and the inner product of vectors ${g(x) \Phi(u)}$ with nonlinear bounded functionals $\Phi$. The first problem is given by equation ${Bu(x)=A u(x)–Ku(x)–g(x)\Phi(u)=f(x)}$ with boundary condition ${D(B)=D(A)}$. In the second problem the operator $B$ contains a linear differential operator $A$ and the inner product of vectors ${g(x)F(A u)}$ with nonlinear bounded on ${C[a, b]}$ functionals $F$, where ${F(A u)}$ denotes the nonlinear Fredholm integral. The second problem is presented by equation ${Bu=A u–gF(A u)=f}$ with boundary condition ${D(B)=D(A)}$.

A direct method for exact solutions of nonlinear integro-differential Volterra and Fredholm equations is proposed. Specifically, the three theorems about existing exact solutions are proved in this paper.

The first theorem is mean that for nonzero constant $\alpha$ ${_0}$ Volterra integro-differential equation ${A u(x)–Ku(x)=0}$ is reducing to Volterra integral equation and has a unique zero solution. During it the operator ${A–K}$ is closed and continuously invertible. Also, if the functions ${u(t)}$, ${g(t)}$ and ${f(t)}$ are of exponential order $\alpha$ then nonhomogeneous equation ${A u(x)–Ku(x)=f(x)}$ for each ${f(x)}$ has a unique solution, shown in this paper.

The second theorem is mean that for the first investigated problem with an injective operator ${A–K}$, for ${f(x)}$ and ${g(x)}$ from ${C[a, b]}$, the exact solution is given by equation ${u=(A–K)^{–1}f +(A–K)^{–1}gb*}$ for every vector ${b^*= \Phi(u)}$, that solves nonlinear algebraic (transcendental) system of $n$ equations ${b= \Phi((A–K)^{–1}f +(A–K)^{–1}gb)}$. And if the last algebraic system has no solution, then investigated prob- lem also has no solution.

The third theorem is means that exact solution of the second investigated problem is given by ${u=A^{– 1}(f+gd^*)}$ for every vector ${d^*=F(A u)}$, that solves nonlinear algebraic (transcendental) system of $n$ equations ${d=F(f + gd)}$. In this case we have same property – if the last algebraic system has no solution, then investigated problem also has no solution.

Two particular examples for each considered problem are shown for illustration of exact solutions giving by perform the suggested in this paper methods. In the first example was considered integro-differential Volterra and Fredholm equation and in the second case was considered equation with nonlinear Fredholm integral.