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Abstract We study the existence of continuous solution of a delay functional integral equation of Volterra-Stieltjes type. The continuous dependence of the unique solution will be also proved .where 𝑔𝑖∶ [0,𝑇 ]×[0,𝑇 ] → 𝑅 is nondecreasing in the second argument and 𝑑𝑠 indicates the integration w.r. to s . Here the existence of at least or exact one continuous solution 𝑥 ∈ 𝐶[0,𝑇 ] of (1) will be proved. The continues dependence of the unique solution 𝑥 ∈ 𝐶[0,𝑇 ] of (1) on the delay function 𝜑(𝑡) and the function 𝑓2 will be proved.integration w.r. to s. Here the existence of at least or exact one continuous solution 𝑥 ∈ 𝐶[0,𝑇 ] of (2) will be proved. The existence of at least or exact one absolutely continuous solution 𝑥 ∈ 𝐴𝐶[0,𝑇 ] of (3) and (4) will be proved. The continues dependence of the unique solution 𝑥 ∈ 𝐶[0,𝑇 ] of (2) and the unique solution 𝑥 ∈ 𝐴𝐶[0,𝑇 ] of (3) and (4) on the delay function 𝜑(𝑡) and the function 𝑓2 will be proved. ( See Appendix B). |