1. Introduction

The purpose of this paper is to study the solvability of the following equation: with the initial condition and the integral conditions where is an unknown function, , , and are given functions supposed to be sufficiently regular, while and are suitably defined functions satisfying certain conditions to be specified later and is a positive constant.

Since 1930, various classical types of initial boundary value problems have been investigated by many authors using Rothe time-discretization method; see, for instance, the monographs by Rektorys [1] and Kačur [2] and references cited therein. The linear case of our problem, that is, , appears, for instance, in the modelling of the quasistatic flexure of a thermoelastic rod (see [3]) and has been studied, firstly, by the second author with a more general second-order parabolic equation or a 2m-parabolic equation in [3–5] by means of the energy-integrals method and, secondly, by the two authors via the Rothe method [6–8]. For other models, we refer the reader, for instance, to [9–12], and references therein.

The paper is organized as follows. In Section 2, we transform problem (1.1)–(1.3) to an equivalent one with homogeneous integral conditions, namely, problem (2.3). Then, we specify notations and assumptions on data before stating the precise sense of the desired solution. In Section 3, by the Rothe discretization in time method, we construct approximate solutions to problem (2.3). Some a priori estimates for the approximations are derived in Section 4, while Section 5 is devoted to establish the existence and uniqueness of the solution.

2. Preliminaries, Notation, and Main Result

It is convenient at the beginning to reduce problem (1.1)–(1.3) with inhomogeneous integral conditions to an equivalent one with homogeneous conditions. For this, we introduce a new unknown function by setting

where

Then, the function is seen to be the solution of the following problem:

where Hence, instead of looking for the function we search for the function The solution of problem (1.1)–(1.3) will be simply given by the formula

We introduce the function spaces, which we need in our investigation. Let and be the standard function spaces. We denote by the linear space of continuous functions with compact support in . Since such functions are Lebesgue integrable, we can define on the bilinear form given by where

The bilinear form (2.5) is considered as a scalar product on for which is not complete.

Definition 2.1. We denote by a completion of for the scalar product (2.5), which is denoted by , called the Bouziani space or the space of square integrable primitive functions on By the norm of function from we understand the nonnegative number where denotes the norm of in

For , we have the elementary inequality

We denote by the space of functions which are square integrable in the Bochner sense, with the scalar product

Since the space is a Hilbert space, it can be shown that is a Hilbert space as well. The set of all continuous abstract functions in equipped with the norm

is denoted . Let be the set which we define as follows:

Since is the null space of the continuous linear mapping : , , it is a closed linear subspace of , consequently is a Hilbert space endowed with the inner product . Strong or weak convergence is denoted by or , respectively. The letter will stand for a generic positive constant which may be different in the same discussion.

Lemma 2.2 (Gronwall's lemma). Let , , be real integrable functions on the interval If then
In particular, if is a constant and is nondecreasing, then
Let be a sequence of real nonnegative numbers satisfying where, , and are positive constants, such that . Then takes place for all

In the sequel, we make the following assumptions.

Functions and are Lipschitz continuous, that is,

The mapping is Lipschitz continuous in both variables, that is,

for all , , and satisfies

for all and all , where and are positive constants.

and

We will be concerned with a weak solution in the following sense.

Definition 2.3. A function is called a weak solution to problem (2.3) if the following conditions are satisfied:(i),(ii) is strongly differentiable a.e. in and ,(iii) in ,(iv)the identity holds for all and a.e. .

To close this section, we announce the main result of the paper.

Theorem 2.4. Under assumptions , problem (2.3) admits a unique weak solution , in the sense of Definition (2.3).

3. Construction of an Approximate Solution

In order to solve problem (2.3) by the Rothe method, we proceed as follows. Let be a positive integer, we divide the time interval into subintervals , , where and . Then, for each problem (2.3) may be approximated by the following recurrent sequence of time-discretized problems. Successively, for , we look for functions such that starting from

where , , . For this, multiplying for all (3.1) by and integrating over , we get

Noting that, using a standard integration by parts, we have

Carrying out some integrations by parts and invoking (3.6), we obtain for each term in (3.5) and for the last one

By virtue of (3.7) and (3.8), (3.5) becomes or

Let and be two functions defined by

It is easy to see that the bilinear form is continuous on and V-elliptic, and the form is continuous for each . Then, Lax-Milgram lemma guarantees the existence and uniqueness of , for all .

4. A Priori Estimates

Lemma 4.1. There exists such that, for all and all , the solution of the discretized problem (3.1)–(3.4) satisfies the estimates

Proof. Testing the difference (3.9)-(3.9) with , taking into account assumptions and the Cauchy-Schwarz inequality, we obtain where Multiplying the left-hand side of the last inequality with and adding the terme we get Applying the last inequality recursively, it follows that or, by virtue of Lemma 2.2 there exists such that where and so our proof is complete.

We address now the question of convergence and existence.

5. Convergence and Existence

Now let us introduce the Rothe function obtained from the functions by piecewise linear interpolation with respect to time

as well the step functions , , , and defined as follows:

Corollary 5.1. There exist such that the estimates hold for all .

Proof. For the inequalities (5.5), (5.6), and (5.7) see [6, Corollary  4.2.], whereas for the last inequality, assumption and estimate (4.1) guarantee the desired result.

Proposition 5.2. The sequence converges in the norm of the space to some function and the error estimate takes place for all .

Proof. By virtue of (5.2), (5.3), and (5.4) the variational equation (3.9) may be rewritten in the form for a.e. In view of (5.10), using (5.6) and (5.8) with the fact that we obtain Now, for , being two positive integers, testing the difference (5.10)-(5.10) with which is in with the help of the Cauchy-Schwarz inequality and taking into account that and, by virtue of (5.12) we obtain after some rearrangements To derive the required result, we need to estimate the third and the last terms in the right-hand side, for this, let be arbitrary but fixed in without loss of generality we can suppose that there exist three positive integers and , such that Hence, using (5.4) we can write By virtue of assumption and the fact that , there exist such that Therefore, recalling assumptions , and having in mind that we estimate from where, we derive for all Taking the supremum with respect to from to in the right-hand side, invoking the fact that and estimate (5.7), we obtain so that Let , from assumption it follows that Ignoring the second term in the left-hand side of (5.14) which is clearly positive and using estimates (5.5), (5.7), (5.21), and (5.22) yield Integrating this inequality with respect to time from 0 to and invoking the fact that we get whence Accordingly, by Gronwall's lemma we obtain consequently takes place for all . This implies that is a Cauchy sequence in the Banach space , and hence it converges in the norm of this latter to some function . Besides, passing to the limit in (5.27), we obtain the desired error estimate, which finishes the proof.