Proceedings of the 6th International Conference on Differential Equations and Dynamical Systems, (2009) 162–166 DCDIS A Supplement, Copyright c  2009 Watam Press NONLOCAL INITIAL VALUE PROBLEM FOR FIRST-ORDER DYNAMIC EQUATIONS ON TIME SCALES Douglas R. Anderson 1 and Abdelkader Boucherif 2 1 Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562 USA, Email: andersod@cord.edu 2 Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia, Email: aboucher@kfupm.edu.sa Abstract. In this study, conditions for the existence of at least one solution to a nonlinear first-order nonlocal initial value problem on time scales are discussed. The results extend previous work in the continuous case to the discrete, quantum, and general time scales setting, and are based on the Leray-Schauder fixed point theorem. Keywords. Time scales; Nonlinear dynamic equations; Leray- Schauder fixed point theorem; Initial value problems; Existence. AMS (MOS) subject classification: 34B05, 39A10. 1 Introduction We are interested in the first-order nonlocal time-scale initial value problem      x Δ (t)= f ( t, x σ (t) ) , t ∈ (a, b) T , x(a)+ m X j=1 γ j x(t j )=0 (1.1) where m ≥ 1 and the points t j ∈ T κ for j ∈{1, 2,...,m} with a ≤ t 1 ≤···≤ t m <b; 1+ m X j=1 γ j 6=0, γ j ∈ R,j ∈{1,...,m}; (1.2) the function f :[a, b] T × R → R is continuous and satisfies |f (t, x)|≤ ( w ( t, |x| ) : t ∈ [a, t m ] T p(t)q ( |x| ) : t ∈ [t m ,b] T , (1.3) where w :[a, t m ] T × R + → R + is integrable and non- decreasing in its second argument; p :[t m ,b] T → R + is right-dense continuous; and q : R + → R + is nondecreas- ing with 1/q integrable on R + . Moreover, we assume that there exists R 0 > 0 such that η>R 0 implies 1 η Z tm a w(s, η)Δs< 1 A (1.4) and Z b tm p(s)Δs< Z ∞ R * 0 dη q(η) , (1.5) where α :=  1+ ∑ m j=1 γ j  -1 , A := 1 + |α| ∑ m j=1 |γ j |, and R * 0 := A R tm a w(s, R 0 )Δs. Problem (1.1) extends to general time scales the spe- cial case T = R; see Boucherif and Precup [5]. There has of late been interest in first-order problems on time scales. Anderson [1], Cabada and Vivero [6], Dai and Tis- dell [7], Otero-Espinar and Vivero [9], Sun [11], Sun and Li [12], and Tian and Ge [13] all recently consider first- order boundary value problems on time scales, but none of them consider the nonlocal problem. For more general information concerning dynamic equations on time scales, introduced by Aulbach and Hilger [2] and Hilger [8], see the excellent text by Bohner and Peterson [3]. It is straightforward to check that problem (1.1) is equivalent to the following integral equation in C[a, b] T x(t)= Z t a f ( s, x σ (s) ) Δs -α m X j=1 γ j Z tj a f ( s, x σ (s) ) Δs. This can be viewed as a fixed point problem in C[a, b] T for the completely continuous operator L : C[a, b] T → C[a, b] T given by Lx(t)= Z t a f ( s, x σ (s) ) Δs - α m X j=1 γ j Z tj a f ( s, x σ (s) ) Δs. Notice that L appears as a sum of two integral operators, one, say of Fredholm type, whose values depend only on 162