Green Function's Properties and Existence Theorems for Nonlinear Singular-Delay Differential Equations

Abstract

In this paper, we are dealing with singular fractional differential equations (DEs) having delay and U-p (p-Laplacian operator). In our problem, we Contemplate two fractional order differential operators that is Riemann-Liouville and Caputo's with fractional integral and fractional differential initial boundary conditions.The SFDE is given by {D-gamma[U*(p)[D(kappa)x(t)]] + Q(t)zeta(1)(t, x(t - e*)) = 0, T-0(1)-gamma(U-p*[D(kappa)x(t)]]t=0 = 0 =T02-gamma(Up*[D kappa x(t)]]vertical bar t=0, D-delta* x(1) = 0, x(1) = x'(0), x((k)) (0) = 0 for k = 2, 3, ..., n-1, zeta 1 is a continuous function and singular at t and x(t) for some values of t 2 [0; 1]. The operator D-gamma is Riemann{Liouville fractional derivative while D delta*;D-kappa stand for Caputo fractional derivatives and delta*, gamma is an element of(1, 2], n - 1 < kappa <= n; where n >= 3. For the study of the EUS, fixed point approach is followed in this paper and an application is given to explain the findings.