New qualitative criteria for solutions of Volterra integro-differential equations

Abstract In this paper, we consider certain non-linear scalar Volterra integro-differential equations and Volterra integro-differential systems of first order. We investigate the boundedness, stability, uniformly asymptotic stability, integrability and square integrability of solutions to the scalar equations and the system considered. The technique used to prove the results of the paper is based on the second method of Lyapunov. From the obtained results, we extend and improve some related results that can be found in the literature.


Introduction
The theory of linear and non-linear Volterra integrodifferential equations and systems provides important mathematical models for many real-world phenomena in science and engineering. Therefore, it is very important during investigations which are related to sciences and engineering to have information about the qualitative properties of solutions of linear and non-linear Volterra integro-differential equations and systems without solving them. However, relatively few kinds of Volterra integro-differential equations and systems can be solved explicitly. Therefore, during scientific investigations, researchers need to find analytical methods which allow them to study the qualitative behaviour of solutions of linear and non-linear Volterra integro-differential equations and systems without solving them. The main theories, techniques and methods in the literature used to investigate the qualitative behaviour of paths of solutions of Volterra integro-differential equations and systems, without needing to find their analytical solutions, include the second method of Lyapunov, continuation methods, coincidence degree theory, perturbation theory, fixed point method or theory, iterative techniques and the variation of constants formula.
Furthermore, investigations on stability, boundedness, asymptotic stability, uniform asymptotic stability, integrability and square integrability of solutions, the existence of periodic solutions, etc., of linear and non-linear Volterra integro-differential equations and systems have great importance in many scientific fields such as atomic energy, biology, bi-dimensional gravity chemistry, control theory, differential geometry, economy, engineering techniques, fluid mechanics, information theory, Jacobi fields, medicine, population dynamics, physics and many others (e.g. Corduneanu, 1977;Staffans, 1988;Lakshmikantham and Rama Mohana Rao, 1995;Burton, 2005;Wazwaz, 2011). Through these sources, readers can find a lot of detailed information on the theory and applications of integral and integro-differential equations in some of these scientific fields.
For more information on the theory and applications of linear and non-linear scalar Volterra integrodifferential equations and Volterra integro-differential systems, the results obtained in the literature on the various qualitative behaviours of solutions of several kinds of these integro-differential equations and sys- Lakshmikantham and Rama Mohana Rao (1995) and Wazwaz (2011), and the paper by Staffans (1988).
For the reasons mentioned above, the qualitative behaviour of Volterra integro-differential equations and systems require further investigation.
We now summarize some works which are will be useful for readers of this paper. Burton and Mahfoud (1983) considered the following linear and homogeneous scalar Volterra integro-differential equations: respectively. Burton and Mahfoud (1983) investigated various kinds of stability and some relations between them, and they also proved their results by means of suitable Lyapunov functionals. Burton and Mahfoud (1983) also proved the following three theorems for Equations (1) and (2), respectively.
Theorem C (Burton and Mahfoud, 1983). Suppose there exists a constant a > 0 such that Then Equation (4) is stable if and only if AðtÞ<0: In this paper, motivated by Volterra integro-differential Equation (1), at the first stage, we consider a nonlinear Volterra integro-differential equation of the form where t 2 I; I ¼ ½0; 1Þ; x 2 R; p : R ! R þ ; R þ ¼ ð0; 1Þ; h : R ! R with hð0Þ ¼ 0; q : I Â R ! R are continuous functions for the arguments displayed explicitly, and CðtÀsÞ is a continuous function for 0 s t<1: In the next stage, in the same way, motivated by linear Volterra integro-differential Equation (2), we consider the following non-linear Volterra integrodifferential equation of the form are continuous functions for the arguments shown explicitly, such that the function r is differentiable, and C is a continuous function for 0 s t<1: Let us define a function r 1 by: Then, in view of the non-linear Volterra integrodifferential Equation (4) and the former expression, we can clearly write that: Burton and Mahfoud (1983) extended their results from the scalar case to the n-dimensional linear Volterra integro-differential system: Burton and Mahfoud (1983) established necessary and sufficient conditions which guarantee the stability of the zero solution of the Volterra integro-differential system (5).
In this paper, in the third stage, motivated by the Volterra integro-differential system (5), we consider the Volterra integro-differential system of the form: where t ! 0; x 2 R n ; A is an n Â n-continuous matrix function for 0 t<1; C is an n Â n-continuous matrix function for 0 s t<1; D is a continuous n-dimensional function for t 2 I; and H; Hð0Þ ¼ 0; is a continuous n-dimensional function for x 2 R n : Finally, we discuss the uniform asymptotic stability of the zero solution and square integrability of solutions of the Volterra integro-differential system (6).
In brief, to summarize the above information, it follows that, motivated by the papers, books and monographs mentioned above, and by the paper of Burton and Mahfoud (1983) in particular, the aim of this paper is to prove new five theorems on the stability, boundedness, uniform asymptotic stability and integrability of solutions of scalar Volterra integro-differential Equations (3) and (4) and uniform asymptotic stability of the zero solution and square integrability of solutions of the Volterra integro-differential system (6).
When we compare the results of Burton and Mahfoud (1983) and the results of this paper, it can be clearly seen that the equations and results of this paper improve and extend the equations and results of Burton and Mahfoud (1983), and they make a contribution to the related literature. In addition, when we look at the literature mentioned above, it can be seen that in the most of the papers, sufficient conditions are obtained for stability, boundedness, uniform asymptotic stability and integrability of solutions, instead of necessary and sufficient conditions. In this paper, for a result to be given, necessary and sufficient conditions on the stability of the solutions are given. Furthermore, the integrability of solutions was not discussed by Burton and Mahfoud (1983). In this paper, the integrability and square integrability of the solutions are also studied. This case makes a contribution to the results of Burton and Mahfoud's (1983) Theorems A, B and C. Finally, the results and assumptions presented here are different from those that can be found in the literature. These are the contributions of this paper to the literature.
For convenience, in what follows, we write x instead of xðtÞ:

Qualitative criteria for solutions
Let q t; x ð Þ 0

A. Hypotheses
We assume that the following hypotheses hold. ðA1Þ There exists a positive constant c such that hð0Þ ¼ 0; jhðxÞj cjxj for x 2 R; x 6 ¼ 0 ðA2Þ There exists a positive constant d such that pðtÞÀc Ð 1 t jCðu À tÞjdu > d for t 2 I ðA3Þ jqðt; xÞj rðtÞjxj for t 2 I and x 2 R where rðtÞ is a non-negative and continuous function for all t 2 I and r 2 L 1 ð0; 1Þ: Theorem 1. If hypotheses ðA1Þ and ðA2Þ hold, then all solutions of scalar Volterra integro-differential equation (3) are bounded and the zero solution of Volterra integro-differential equation (3) is uniformly asymptotically stable, and xðtÞ 2 L 1 ½0; 1Þ; where L 1 ½0; 1Þ is the space of all Lebesgue integrable functions on ½0; 1Þ: Proof. We define a Lyapunov function W 0 ¼ W 0 ðt; xðtÞÞ by Hence, it is clear that W 0 ðt; 0Þ ¼ 0 and W 0 ðt; xÞ ! jxj Differentiating the Lyapunov function W 0 with respect to t and using hypotheses ðA1Þ and ðA2Þ; we obtain This last inequality completes the proof of Theorem 1. That is, the zero solution of Volterra integro-differential Equation (3) is uniformly asymptotic stable.
For the next step, in view of the above information, since W 0 0 Àdjxj we can write Integrating the former inequality from zero t 0 to t; we can find This last estimate implies the boundedness of all solutions of Volterra integro-differential Equation (3).
Finally, integrating the inequality from t 0 to t; we have It is known that the Lyapunov function W 0 ðt; xðtÞÞ is positive definite and a decreasing function. Then, it is obvious that Therefore, we can conclude that xðtÞ 2 L 1 ½0; 1Þ: That is, the solution xðtÞ is integrable on the interval ½0; 1Þ: For the next result, let q t; x ð Þ 6 ¼ 0 Theorem 2. If hypotheses ðA1Þ-ðA3Þ are satisfied, then all solutions of Volterra integro-differential equation (3)  By integrating the former inequality on the interval ½t 0 ; t and using the Gronwall inequality, we have Then, in view of the definition of the Lyapunov function W 0 ðt; xðtÞÞ and the last inequality, we have Since rðtÞ 2 L 1 ½0; 1Þ; then we can say Hence, we obtain jxj L 0 : Thus, we can conclude the desired result. That is, all solutions are bounded. This is the end of the proof of Theorem 2.

B. Hypotheses
We assume that the following hypotheses hold. ðC1Þ There exists a positive constant l such that wð0Þ ¼ 0; jwðxÞj ljxj for x 2 R; x 6 ¼ 0 ðC2Þ Ð 1 0 jCðt; sÞjds<1 and Ð 1 t jCðu; tÞjdu<1 ðC3Þ There exist positive constants l and b such that for t 2 I and x 2 R; x 6 ¼ 0 Theorem 3. If hypotheses ðC1Þ-ðC3Þ are satisfied, then all solutions of scalar Volterra integro-differential equation (4) are stable if and only if pðtÞr 1 ðxÞ>0: Proof. ð)Þ : First, we suppose that pðtÞr 1 ðxÞ>0: We define a Lyapunov function W 1 ¼ W 1 ðt; xðtÞÞ by If the hypotheses of Theorem 3 hold, then it is obvious that W 1 ðt; 0Þ ¼ 0 and W 1 ðt; xÞ ! x 2 Differentiating the Lyapunov function W 1 ðt; xðtÞÞ with respect to t; subject to the hypotheses of Theorem 3 and some elementary inequalities, we can obtain Conversely, we now suppose that pðtÞr 1 ðxÞ<0: We define a Lyapunov function W 2 ¼ W 2 ðt; xðtÞÞ by Differentiating the Lyapunov function W 2 with respect to t and taking into consideration hypotheses ðC1Þ-ðC3Þ, then an easy calculation implies that Integrating the above estimate from zero t 0 to t; we obtain x 2 s ð Þds Given any t 0 ! 0 and any d > 0; it can be found any continuous function / : ½0; t 0 ! R with j/ðtÞj<d and W 2 ðt 0 ; /ð:ÞÞ > 0 such that if xðtÞ ¼ xðt; t 0 ; /Þ is a solution of Volterra integro-differential Equation (4), then we can obtain If t ! 1; then jxðtÞj ! 1: This is a contradiction. This result completes the proof of Theorem 3.

C. Hypotheses
We suppose the following hypotheses hold: ðH1Þ Hð0Þ ¼ 0; jHðxÞj djxj; where d>0 and d 2 R: Let AðtÞ be an n Â n-real, symmetric, bounded and continuous matrix for all t 2 I: There exists an n Â n-constant symmetric matrix B such that x T ½AðtÞB þ BAðtÞx Àd 0 jxj 2 for all t 2 I; x 2 R n ; Ð t 0 jCðt; sÞjds is defined and bounded, and Ð 1 t jCðu; tÞjdu<1: Theorem 4. If hypotheses ðH1Þ-ðH3Þ hold, then the zero solution of Volterra integro-differential system (6) is uniformly asymptotic stable and all solutions of this system are square integrable.
Proof. We define a Lyapunov function W 3 ¼ W 3 ðt; xðtÞÞ by By a similar discussion as in Burton and Mahfoud (1983), the positive definite of the Lyapunov function W 3 can be easily shown. We omit the details of this discussion.
In the light of hypotheses ðH1Þ-ðH3Þ, the time derivative of the Lyapunov function W 3 with respect to Volterra integro-differential system (6) leads to  When we look of the first and the last terms of the former inequalities, then Thus, in view of the above discussion, we can conclude that the zero solution of Volterra integro-differential system (6) is uniformly asymptotic stable.
In addition, integrating the last inequality, that is, W 0 3 ðt; xðtÞÞ ÀMjxj 2 from t 1 to t; we find From the above discussion, it can be seen that the function W 3 ðt; xðtÞÞ is positive definite and a decreasing function. Therefore, we can choose As a result of the above inequalities, it follows that ð t t 1 jx s ð Þj 2 ds M À1 N Thus, we can conclude that the solution xðtÞ of Volterra integro-differential system (6) is square integrable, that is, xðtÞ 2 L 2 ½0; 1Þ; where L 2 ½0; 1Þ is the space of all Lebesgue square-integrable functions on ½0; 1Þ: This is the end of the proof of Theorem 4. Finally, let D t ð Þ 6 ¼ 0 Theorem 5. Let hypotheses ðH1Þ-ðH4Þ hold. Then, all solutions of Volterra integro-differential equation (6) are bounded.
Proof. We benefit from the Lyapunov function W 3 ð:Þ ¼ W 3 ðt; xðtÞÞ: In light of the assumptions of ðH1ÞÀðH4Þ; we can find W 0 : ð Þ ÀMjxj 2 þ 2jD t ð ÞjjBjjxj 2 The rest of the proof is similar to the proof of Theorem 2. We omit the rest of the proof.

Conclusion
In this paper, we have presented sufficient conditions for stability, boundedness, uniformly asymptotic stability, integrability and square integrability of solutions of a few scalar non-linear Volterra integrodifferential equations and a Volterra integro-differential system. The technique of the proof is based on the second method of Lyapunov. The obtained results include and improve upon some results that can be found in the literature (Burton and Mahfoud, 1983, Theorems A, B and C).