Abstract:

In this paper,it is investigated the following with Cauchy's kernel
Lu(t)=Σ_j=0^m[a₁(t)u^(j)(t)+(1/(πj))∫(K₁(t,τ)u^(j)(τ)/(τ-1))dτ]=(1/(πi))∫((φ[t,τ,U(τ),λ])/(τ-t))dτ.(1)
u^(j)(t₀)=u₀^j (t₀∈Γ,j=0,…,m-1) (1).
Where Γ is a simple Liapunov's closed path, u (t) is a nonknown function u(t)={u(t), u(t),…, u⁽ⁿ⁾ (t)}, u₀^j are some real or complex number. Ref.[1]-[s] applied the interpolation and topological method for the equation (1). By using the Liapunov's method,Ref、[6],[7],studied the solution of equation (1).

Key words: nonlinear singular integral, solution of differential equation, extension of solution