In this paper, we are concerned with the existence of solutions of the following nonlocal

*p*-Laplacian dynamic equation on a time scale

$\mathbb{T}$:

$-{\left({\varphi}_{p}({u}^{\mathrm{\u25b3}}(t))\right)}^{\mathrm{\nabla}}=\frac{\lambda a(t)f(u(t))}{{({\int}_{0}^{T}f(u(s))\mathrm{\nabla}s)}^{k}},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in {(0,T)}_{\mathbb{T}},$

(1.1)

with integral initial value

$\begin{array}{r}u(0)={\int}_{0}^{T}g(s)u(s)\mathrm{\nabla}s,\\ {u}^{\mathrm{\u25b3}}(0)=A,\end{array}$

(1.2)

where ${\varphi}_{p}(\cdot )$ is the *p*-Laplace operator defined by ${\varphi}_{p}(s)={|s|}^{p-2}s$, $p>1$, ${\varphi}_{p}^{-1}={\varphi}_{q}$ with *q* the Hölder conjugate of *p*, *i.e.*, $\frac{1}{p}+\frac{1}{q}=1$, $\lambda >0$, $k>0$, $f:{[0,T]}_{\mathbb{T}}\u27f6{\mathbb{R}}^{+\ast}$ is continuous (${\mathbb{R}}^{+\ast}$ denotes positive real numbers), $a:{[0,T]}_{\mathbb{T}}\u27f6{\mathbb{R}}^{+}$ is left dense continuous, $g(s)\in {L}^{1}({[0,T]}_{\mathbb{T}})$ and *A* is a real constant.

This model arises in ohmic heating phenomena, which occur in shear bands of metals which are deformed at high strain rates [1, 2], in the theory of gravitational equilibrium of polytropic stars [3], in the investigation of the fully turbulent behavior of real flows, using invariant measures for the Euler equation [4], in modeling aggregation of cells via interaction with a chemical substance (chemotaxis) [5]. For the one-dimensional case, problems with the nonlocal initial condition appear in the investigation of diffusion phenomena for a small amount of gas in a transparent tube [6, 7]; nonlocal initial value problems in higher dimension are important from the point of view of their practical applications to modeling and investigating of pollution processes in rivers and seas, which are caused by sew-age [8].

The study of dynamic equations on time scales has led to some important applications [9–11], and an amount of literature has been devoted to the study the existence of solutions of second-order nonlinear boundary value problems (*e.g.*, see [12–18]).

Motivated by the above works, in this paper, we study the existence of solutions to Problem (1.1), (1.2). Compared with the works mentioned above, this article has the following new features: firstly, the main technique used in this paper is the topological degree method; secondly, Problem (1.1), (1.2) involves the integral initial condition.

The paper is organized as follows. We introduce some necessary definitions and lemmas in the rest of this section. In Section 2, we provide some necessary preliminaries, and in Section 3, the main results are stated and proved.

**Definition 1.1** For

$t<sup\mathbb{T}$ and

$r>inf\mathbb{T}$, define the forward jump operator

*σ* and the backward jump operator

*ρ*, respectively,

$\sigma (t)=inf\{\tau \in \mathbb{T}\mid \tau >t\}\in \mathbb{T},\phantom{\rule{2em}{0ex}}\rho (r)=sup\{\tau \in \mathbb{T}\mid \tau <r\}\in \mathbb{T}$

for all $t,r\in \mathbb{T}$. If $\sigma (t)>t$, *t* is said to be right scattered, and if $\rho (r)<r$, *r* is said to be left scattered. If $\sigma (t)=t$, *t* is said to be right dense, and if $\rho (r)=r$, *r* is said to be left dense. If $\mathbb{T}$ has a right scattered minimum *m*, define
; otherwise, set
. If $\mathbb{T}$ has a left scattered maximum *M*, define
; otherwise, set
.

**Definition 1.2** For

$x:\mathbb{T}\u27f6\mathbb{R}$ and

, we define the delta derivative of

$x(t)$,

${x}^{\mathrm{\u25b3}}(t)$, to be the number (when it exists) with the property that for any

$\epsilon >0$, there is a neighborhood

*U* of

*t* such that

$|[x(\sigma (t))-x(s)]-{x}^{\mathrm{\u25b3}}(t)[\sigma (t)-s]|<\epsilon |\sigma (t)-s|$

for all

$s\in U$. For

$x:\mathbb{T}\u27f6\mathbb{R}$ and

, we define the nabla derivative of

$x(t)$,

${x}^{\mathrm{\nabla}}(t)$, to be the number (when it exists) with the property that for any

$\epsilon >0$, there is a neighborhood

*V* of

*t* such that

$|[x(\rho (t))-x(s)]-{x}^{\mathrm{\nabla}}(t)[\rho (t)-s]|<\epsilon |\rho (t)-s|$

for all $s\in V$.

**Definition 1.3** If

${F}^{\mathrm{\u25b3}}(t)=f(t)$, then we define the delta integral by

${\int}_{a}^{t}f(s)\mathrm{\u25b3}s=F(t)-F(a).$

If

${\mathrm{\Phi}}^{\mathrm{\nabla}}(t)=f(t)$, then we define the nabla integral by

${\int}_{a}^{t}f(s)\mathrm{\nabla}s=\mathrm{\Phi}(t)-\mathrm{\Phi}(a).$

Throughout this paper, we assume that $\mathbb{T}$ is a nonempty closed subset of ℝ with $0\in {\mathbb{T}}_{k}$, $T\in {\mathbb{T}}^{k}$.

**Lemma 1.1** (Alternative theorem)

*Suppose that* *X* *is a Banach space and* *A* *is a completely continuous operator from* *X* *to X*.

*Then for any* $\lambda \ne 0$,

*only one of the following statements holds*:

- (i)
*For any* $y\in X$,

*there exists a unique* $x\in X$,

*such that*

- (ii)
*There exists an* $x\in X$,

$x\ne 0$,

*such that*