The author gives sufficient conditions for the existence of a solution of the integral equation in the space \(L^ p_ n(0,w)\) \[ (1)\quad d/dx\int^{w}_{0}S(x,t)y(t)dt=g(x), \] where \(w>0\), S(x,t) is an \(n\times n\)-matrix of the form \([K_{i,j}(w_ ix-w_ jt)]^ n_{i,j=1}\), where \(K_{i,j}\in L^ q(-w_ jw,w_ iw)\), \(w_ k>0\), \(1/p+1/q=1\). The resolvent operator T is described. Equation (1) contains as a special case e.g. the equation \(\mu f(x)+\int_{\Delta}k(x- t)f(t)dt=h(x),\) \(x\in \Delta\), where \(\Delta\) is a system of nonoverlapping intervals \([a_ k,b_ k]\), \(k=1,2,...,n\), which plays an important role in elasticity theory (contact problems). Applications to some physical problems are given, as well.