Integral Equations Arising in Boundary Value Problems at Resonance

Seppo Seikkala and Dmitri Vorobiev

Chapter 35 in Integral Methods in Science and Engineering, 2002, pp 221-226 from Springer

Abstract: Abstract Let $$ L:{H^2}\left( {0,1} \right) \to {L_2}\left( {0,1} \right) $$ be a formally self-adjoint second-order linear differential operator $$Lu = (pu\prime )\prime + qu,$$ , where $$ p \in {C^2}\left[ {0,1} \right],p(x) \ne 0,0 \leqslant x \leqslant 1 $$ , and $$ q \in C\left[ {0,1} \right] $$ . We shall consider the boundary value problem 1 $$\begin{array}{*{20}{c}} {Lu(t) = f(t,u(t),u\prime (t))} & {a.e. in J = [0,1],} \\ {{{B}_{i}}u = {{d}_{i}},} & {i = 1,2,} \\ \end{array}$$ where $$ f:J \times {R^2} \to R $$ is square integrable for every $$ u \in {H^2}\left( {0,1} \right) $$ and the boundary conditions are either separated, or mixed, that is, $${{\begin{array}{*{20}{c}} {{{B}_{1}}u = {{a}_{0}}u(0) + {{b}_{0}}u\prime (0) = {{d}_{1}},} & {{{B}_{2}}u = {{a}_{1}}u(1) + {{b}_{1}}u\prime (1) = d} \\ \end{array} }_{2}},$$ , or $${{\begin{array}{*{20}{c}} {{{B}_{1}}u = {{a}_{0}}u(0) + {{b}_{0}}u(1) = {{d}_{1}},} & {{{B}_{2}}u = {{a}_{1}}u\prime (0) + {{b}_{1}}u\prime (1) = d} \\ \end{array} }_{2}}.$$

Date: 2002
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DOI: 10.1007/978-1-4612-0111-3_35

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