Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms

where φ is either h(x) or p(t), u is either x or t, and all roots of the restoring term h(x) are isolated. It should be noted that there exist many papers dealing with boundedness of solutions to certain nonlinear differential equations of third and fourth order in the literature [1–15]. For nonlinear differential equations of fourth order, Afuwape and Adesina [1] used the frequency-domain approach to discuss the stability and periodicity of solutions, while Tunç and Tiryaki [11, 12] used intrinsic method to study the boundedness and stability of solutions. On the same time, Tunç [13– 15] used Lyapunov’s secondmethod to investigate the stability and boundedness properties of solutions of certain fourth order nonlinear differential equations. Further, other papers in this connection include those of Andres [2], Ogundare [6], and Omeike [7, 8], where the Cauchy formula was applied to evaluate the boundedness of solutions to certain third and fourth order nonlinear differential equations with oscillatory restoring and forcing terms. The aim of this work is to extend and improve the previous studies and make some contributions to the literature since there are only a few papers on the boundedness of solutions of fourth order differential equations with oscillatory restoring and forcing terms (see [6–8]). It should be noted that the equation considered here, (1), includes and extends that of Ogundare [6] and Omeike [7, 8].

It should be noted that there exist many papers dealing with boundedness of solutions to certain nonlinear differential equations of third and fourth order in the literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].For nonlinear differential equations of fourth order, Afuwape and Adesina [1] used the frequency-domain approach to discuss the stability and periodicity of solutions, while Tunc ¸and Tiryaki [11,12] used intrinsic method to study the boundedness and stability of solutions.On the same time, Tunc ¸ [13][14][15] used Lyapunov's second method to investigate the stability and boundedness properties of solutions of certain fourth order nonlinear differential equations.Further, other papers in this connection include those of Andres [2], Ogundare [6], and Omeike [7,8], where the Cauchy formula was applied to evaluate the boundedness of solutions to certain third and fourth order nonlinear differential equations with oscillatory restoring and forcing terms.
The aim of this work is to extend and improve the previous studies and make some contributions to the literature since there are only a few papers on the boundedness of solutions of fourth order differential equations with oscillatory restoring and forcing terms (see [6][7][8]).It should be noted that the equation considered here, (1), includes and extends that of Ogundare [6] and Omeike [7,8].

Preliminary Results
We need the following lemmas in the proof of our main result.Lemma 1.One assumes that there exist positive constants , , , , and , ( 2 > 4) such that the following conditions hold for all  ∈  and  ≥ 0: Note that the constants , , and  satisfy the conditions ensuring that the auxiliary equation has negative real roots.
Proof.Substituting  :=   , we get from (1) that with the solutions of the form where  is an arbitrary constant and   is a great enough number.Let us assume that the assumptions (4) and ( 5) hold.Thus, by the conditions of Lemma but also lim sup This completes the proof of Lemma 1.
Lemma 2. In addition to the assumptions of Lemma 1, one assumes that the following conditions hold: where   is a suitable constant.Then, every bounded solution () of (1) either satisfies the relation or there exists a root  of ℎ() such that (() − ) oscillates.
Proof.Let () be a fixed bounded solution of (1).Substituting this solution into (1) and integrating the result from   to  (  -a great enough number), we obtain the following: By noting the assumption (ii) of Lemma 1 and the boundedness of solution (), it follows that there exists a constant   for  ≥   such that Now let us assume that () does not converge to any root  of ℎ(), that is, lim sup and simultaneously, ℎ ( ()) ≥ 0 or ℎ ( ()) ≤ 0 for  ≥   . Then, evidently is a composed monotone function with a finite or infinite limit for  → ∞.Since (13) implies that the "divergent case" can be disregarded, then it follows from (15) which is a contradiction to (17).Thus, the estimates ( 14) and (18) imply that lim sup However, according to the assertion of Lemma 1, this case is impossible, and that is why (() − ) necessarily oscillates.The remaining part of Lemma 2 is followed from the assertion where  ≥ 2 is a natural number and  = 1, . . ., ( − 1).This completes the proof.
Lemma 3. In addition to the assumptions of Lemma 2, one assumes that the following conditions hold: Hence, by the boundedness of   (),   (),   (), and   (), it follows that there exists a constant  such that lim sup which according to (24), gives the following estimates: or lim sup which is a contradiction to lim sup  → ∞ |()| > 0. This completes the proof of Lemma 3.
We now give the main result of this paper.
Let  1 ≥  0 be the last point with ( 1 ) =   (-even) and  2 >  1 the first point with ( 2 ) =  +1 .If we integrate (1) from  1 to ,  1 ≤  ≤  2 , we come to Since  > 1, then all the conditions of the theorem are satisfied; thus, all solutions () of the above differential equation and their derivatives up to order three are bounded, and for each of them, there exists a root  of ℎ(()) such that (() − ) oscillates.
where  > 0 is an arbitrary small constant, a contradiction to ( 2 ) =  +1 .The remaining part of the theorem follows from Lemma 3; therefore, we omit the details of the proof.The proof is complete.