Solution of the dual integral equations system of special kind (Q2755259)

The author considers the system of dual integral equations NEWLINE\[NEWLINE\begin{cases} \int\limits_{0}^{\infty}(A_1(\lambda)X_{1k}(\lambda)+ A_2(\lambda)X_{2k}(\lambda))J_{k-1}(\lambda r) d\lambda=\nu_{1k}(r),\\ \int\limits_{0}^{\infty}(-A_1(\lambda)X_{1k}(\lambda)+ A_2(\lambda)X_{2k}(\lambda))J_{k+1}(\lambda r) d\lambda=\nu_{2k}(r), \end{cases} \quad 0\leq r\leq a;NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{cases}\int\limits_{0}^{\infty}(X_{1k}(\lambda)+ X_{2k}(\lambda))J_{k-1}(\lambda r)\lambda d\lambda=0,\\ \int\limits_{0}^{\infty}(-X_{1k}(\lambda)+ X_{2k}(\lambda))J_{k+1}(\lambda r)\lambda d\lambda=0,\\ \end{cases} \quad a< r<+\infty, NEWLINE\]NEWLINE \(k=1,2,3,\ldots\), where \(\nu_{jk}(\lambda):[0,a]\to R\) are bounded continuous functions; \(A_{j}(\lambda)=(1-g_{j}(\lambda))\), \(g_{j}:(0,+\infty)\to [0,1], j=1,2 \) are given continuous functions such that \(\int_{0}^{+\infty}|g_{j}(\lambda)|d\lambda<\infty\), \(\int_{0}^{+\infty}|g_{j}(\lambda)|\lambda d\lambda<\infty\); \(J_{k}(\lambda)\) is a Bessel function of the first kind; \(X_{jk}(\lambda)\) are unknown continuous functions with the bounded variation on any finite interval from \((0,+\infty)\) such that NEWLINE\[NEWLINE\int_{0}^{+\infty}|X_{jk}(\lambda)|\lambda^{1/2} d\lambda<\infty,\quad \int_{0}^{+\infty}|X_{jk}(\lambda)|\lambda^{3/2} d\lambda<\infty.NEWLINE\]NEWLINE The given system is reduced to two Fredholm integral equations of the second kind with the kernel NEWLINE\[NEWLINEK_{jk}(x,t)=\sqrt{xt}\int_{0}^{+\infty}g_{j}(\lambda) J_{k-1/2}(\lambda x)J_{k-1/2}(\lambda t)\lambda d\lambda,\quad j=1,2,\quad k=1,2,\ldots.NEWLINE\]NEWLINE Then the problem of existence and uniqueness of solution of the given system is studied.