Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 68, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLUTION ESTIMATES AND STABILITY TESTS FOR NONLINEAR DELAY INTEGRO-DIFFERENTIAL EQUATIONS SANDRA PINELAS, OSMAN TUNÇ Abstract. In this article, we examine various qualitative features of solutions of a nonlinear delay integro-differential equation. We prove three new theorems which include sufficient conditions on asymptotic stability (AS), integrability, and boundedness of solutions, using a suitable Lyapunov-Krasovskii functional. We present examples that show applications of our results. 1. Introduction According to the literature, Volterra’s work [39] on elasticity was a starting point of theory on delay integro-differential equations (DIDEs). It was found that for some substances, the magnetic or electric polarization depends not only on the electromagnetic field at that moment, but also on the electromagnetic state of the matter at earlier instants. This and other scientific and engineering problems been modeled with DIDEs. For example, population dynamics, biological applications, genetics, noise term phenomenon, competition between tumor cells and immune system, artificial neural networks. and RLC circuits have been modeled as IDEs in [4, 6, 5, 10, 18, 19, 20, 22, 25, 39, 42]. In the previous five decades, qualitative properties of solutions of first order IDEs and functional DEs have been discussed, see for example the references in this article. However, there are only a few works on second order IDEs, see [1, 7, 9, 15, 16, 29, 30, 33, 45]. In this work, we consider the second order DIDE ẍ+ m∑ i=1 fi(t, x, ẋ) + n∑ i=1 gi(x(t− τi)) + g(x, ẋ) + h(x) = p(t, x, ẋ) + l∑ i=1 ∫ t t−τi Ci(t, s)qi(s, ẋ(s)) ds . (1.1) 2020 Mathematics Subject Classification. 34D05, 34K20, 45J05. Key words and phrases. Lyapunov- Krasovskii functions; delay integro-differential equation; qualitative properties. ©2022. This work is licensed under a CC BY 4.0 license. Submitted August 1, 2022. Published October 5, 2022. 1 2 S. PINELAS, O. TUNÇ EJDE-2022/68 As a next step, we transform (1.1) into the system dx dt = y, dy dt = − m∑ i=1 fi(t, x, y)− n∑ i=1 gi(x)− g(x, y)− h(x) + n∑ i=1 ∫ t t−τi g′i(x(s))y(s) ds+ p(t, x, y) + l∑ i=1 ∫ t t−τi Ci(t, s)qi(s, y(s)) ds, (1.2) where x ∈ R, t ∈ [−τ,∞), τi > 0 are constant delays, τ = max{τ1, . . ., τn}, l ≤ n, l, m,n ∈ N, fi, p ∈ C(R+ × R2,R), fi(t, x, 0) = 0, g ∈ C(R2,R), g(x, 0) = 0, gi ∈ C1(R,R), gi(0) = 0, h ∈ C(R,R), h(0) = 0, Ci ∈ C([−τ,∞) × [−τ,∞),R), qi ∈ C([−τ,∞) × R,R) and qi(s, 0) = 0, (i = 1, . . . , l). This continuity condition allows the existence of solutions to (1.1). In addition, through this paper, it is assumed the existence and continuity of the derivatives g′i(x) = dgi dx , i = 1, 2, . . . , n. Throughout this article x and y denote x(t) and y(t), respectively. It is seen that nonlinear system (1.2) has multiple kernels and delays. In par- ticular, the mathematical models given as (1.1) and its modified version are useful for researchers working on ecology problems, population dynamics, artificial neural networks, and so forth. Berezansky et al. [9] studied the following qualitative properties of solutions to second order functional differential equations (FDEs): existence of solutions, oscil- lation and non-oscillation, exponential stability, and instability. These equations in- clude delay differential equations, integro-differential equations and equations with distributed delay. In particular, Berezansky et al. [9] considered the following linear FDEs with variable delays: ẍ(t) + 2m∑ i=1 pi(t)x(t− τi(t)) = f(t), ẍ(t) + 2m∑ i=1 pi(t)x(t− τi(t)) + n∑ j=1 qj(t)x(t− θj(t)) = f(t), ẍ(t) + m∑ i=1 ai(t)x(t− τi(t))− m∑ i=1 bi(t)x(t− θi(t)) = f(t). Next we outline the contributions of this article. To the best of our information, the movements of orbits to (1.1) have not investigated in the literature; therefore, we present a novel work. Second order DIDEs with multiple kernels and delays have many applications in engineering [10, 18, 19, 20, 22, 23, 25], but fundamental properties of their solutions are rarely investigated. Therefore, investigating second order DIDEs is also a desirable feature of our work Finally, the results of this article have suitable conditions for applications, because of functional w defined by (2.1) below. The techniques and results here are different from those in [9]. The rest of this article is arranged as follows: Section 2 presents two theorems about stability and integrability results. Section 3 includes a numerical example as applications of the stability and integrability results in Section 2. Section 4 includes Theorem 4.1 which addresses the boundedness of solutions. Section 5 includes a EJDE-2022/68 SOLUTION ESTIMATES AND STABILITY TESTS 3 numerical example as an application of the boundedness result of Section 5. Section 6 presents the conclusions from this article. 2. Stability and integrability results We use the following assumptions for proving the results of this article. (A1) gi(0) = 0, gi(x) x ≥ bi, x 6= 0, h(0) = 0, h(x) x ≥ h0, x 6= 0, |g′i(x)| ≤ αi for all x ∈ R, where, bi > 0, h0 > 0, αi > 0, bi, h0, αi ∈ R, for i = 1, 2, . . . , n; (A2) fi(t, x, 0) = 0, yfi(t, x, y) ≥ fi0y2, y 6= 0, i = 1, 2, . . . ,m, g(x, 0) = 0, yg(x, y) ≥ g0y 2, y 6= 0, for all x, y ∈ R, |qi(s, y(s))| ≤ ri|y(s)|, |Ci(t, s)| ≤ di, where ri > 0, di > 0, ri, qi ∈ R for i = 1, 2, . . . , l; and the positive constants fi0, g0, αi, di, ri satisfy m∑ i=1 fi0 + g0 − 2−1τ ( n∑ i=1 αi + l∑ i=1 (αi + 2diri) + n∑ i=l+1 αi ) ≥ σ, where σ > 0, σ ∈ R; (A3) |p(t, x, y)| ≤ |p0(t)||y|, for all t ∈ R+, x, y ∈ R,∫ ∞ 0 |p0(t)|dt <∞. Theorem 2.1. If (A1) and (A2) hold and p(t, x, y) ≡ 0, then the trivial solution of (1.2) is asymptotically stable. Proof. As an auxiliary tool to prove this theorem, we define the Lyapunov-Krasovskii functional W (·) = W (t, xt, yt) = 2 n∑ i=1 ∫ x 0 gi(s) ds+ 2 ∫ x 0 h(s) ds+ y2 + n∑ i=1 γi ∫ 0 −τi ∫ t t+s y2(θ) dθ ds, (2.1) where γ1, . . . , γn are positive constants to be determined later. We have W (t, xt, yt) = 2 ∫ x 0 g1(s) s sds+ · · ·+ 2 ∫ x 0 gn(s) s sds+ 2 ∫ x 0 h(s) s sds+ y2 + n∑ i=1 γi ∫ 0 −τi ∫ t t+s y2(θ) dθ ds. 4 S. PINELAS, O. TUNÇ EJDE-2022/68 Using condition (A1), we obtain W (t, xt, yt) ≥ (b1 + b2 + . . .+ bn + h0)x2 + y2. (2.2) The derivative of W (t, xt, yt) along the trajectories of (1.2) gives W ′(·) = 2g1(x)y + 2g2(x)y + . . .+ 2gn(x)y + 2h(x)y + 2yy′ + n∑ i=1 (γiτi)y 2 − n∑ i=1 (γi ∫ t t−τi y2(s)) ds = 2y n∑ i=1 gi(x) + 2y [ − m∑ i=1 fi(t, x, y)− n∑ i=1 gi(x)− g(x, y)− h(x) ] + 2h(x)y + 2y n∑ i=1 ∫ t t−τi g′i(x(s))y(s) ds + 2y l∑ i=1 ∫ t t−τi Ci(t, s)qi(s, y(s)) ds+ n∑ i=1 (γiτi)y 2 − n∑ i=1 (γi ∫ t t−τi y2(s)) ds = −2y m∑ i=1 fi(t, x, y)− 2yg(x, y) + 2y n∑ i=1 ∫ t t−τi g′i(x(s))y(s) ds + 2y l∑ i=1 ∫ t t−τi Ci(t, s)qi(s, y(s)) ds+ n∑ i=1 (γiτi)y 2 − n∑ i=1 γi ∫ t t−τi y2(s) ds. Using conditions (A1), (A2) and doing elementary calculations, we obtain 2y ∫ t t−τi g′i(x(s))y(s) ds ≤ 2|y(t)| ∫ t t−τi |g′i(x(s))||y(s)|ds ≤ αi ∫ t t−τi (y2(t) + y2(s)) ds = αiτiy 2 + αi ∫ t t−τi y2(s) ds, for i = 1, 2, . . . n; and 2y ∫ t t−τi Ci(t, s)qi(s, y(s)) ds ≤ 2|y| ∫ t t−τi |Ci(t, s)||qi(s, y(s))|ds ≤ 2diri|y| ∫ t t−τi |y(s)|ds ≤ diri ∫ t t−τi (y2(t) + y2(s)) ds = diriτiy 2 + diri ∫ t t−τi y2(s) ds, EJDE-2022/68 SOLUTION ESTIMATES AND STABILITY TESTS 5 for i = 1, 2, . . . , l. Hence, W ′(·) ≤ −2y m∑ i=1 fi(t, x, y)− 2yg(x, y) + [ n∑ i=1 (αiτi) + l∑ i=1 (diriτi) + n∑ i=1 (γiτi) ] y2 + (α1 + d1r1 − γ1) ∫ t t−τ1 y2(s) ds+ (α2 + d2r2 − γ2) ∫ t t−τ2 y2(s) ds + (αl + dlrl − γl) ∫ t t−τl y2(s) ds+ . . .+ (αn − γn) ∫ t t−τn y2(s) ds. Let γ1 = α1 + d1r1, γ2 = α2 + d2r2, . . . , γl = αl + dlrl, . . . , γn = αn. Then using condition (A2), we obtain W ′(·) ≤ −2y m∑ i=1 fi(t, x, y)− 2yg(x, y) + [ n∑ i=1 (αiτi) + l∑ i=1 (diriτi) + l∑ i=1 (αi + diri) τi + n∑ i=l+1 (αiτi) ] y2 ≤ −2y2 m∑ i=1 fi0 − 2g0y 2 + [ n∑ i=1 (αiτi) + l∑ i=1 (diriτi) + l∑ i=1 (αi + diri) τi + n∑ i=l+1 (αiτi) ] y2. Let τ = max{τ1, τ2, . . . , τn}. Then W ′(·) ≤ −2 [ m∑ i=1 fi0 + g0 − 2−1τ ( n∑ i=1 αi + l∑ i=1 (αi + 2diri) + n∑ i=l+1 αi )] y2 ≤ −σy2 < 0, y 6= 0, provided that τ < 2 ∑m i=1 fi0 + 2g0∑n i=1 αi + ∑l i=1 (αi + 2diri) + ∑n i=l+1 αi = σ. In addition, it can be shown that the only invariant set in W ′(·) = 0 is {0, 0} (see, Hale [18]). Then, the trivial solution of system of(1.2) is asymptotically stable. � Theorem 2.2. If (A1), (A2) hold and p(t, x, y) ≡ 0, then the squares of the deriv- ative of solutions x(t) of (1.2) are Lebesgue integrable. Proof. From Theorem 2.1, we have that W ′(t, xt, yt) ≤ −σy2 < 0, y 6= 0. Integrating we obtain W (t, xt, yt)−W (t0, φ(t0), ψ(t0)) ≤ −σ ∫ t t0 y2(s) ds. Then ∫ ∞ t0 y2(s) ds ≤ σ−1W (t0, φ(t0), ψ(t0))− σ−1W (t, xt, yt) ≤ K, where K = σ−1W (t0, φ(t0), ψ(t0)). � 6 S. PINELAS, O. TUNÇ EJDE-2022/68 3. Numerical applications of stability and integrability results Example 3.1. Let p(·) ≡ 0. As a particular case of (1.1), we consider the nonlinear second order DIDE with multiple kernels and delays, d2x dt2 + ( t+ x2 + ( dx dt )2 + 25 )dx dt + 17 dx dt + 2x + x7 + 2x(t− 4−1) + 2x(t− 8−1) = ∫ t t− 1 4 1 1 + t4 + s2 x′(s) [1 + (x′(s)) 2 ][1 + exp(s2)] ds + ∫ t t− 1 8 1 1 + t6 + s2 x′(s) [1 + (x′(s)) 2 ][1 + exp(s4)] ds. (3.1) This equation can be transformed into the system dx dt = y, dy dt = − ( t+ x2 + y2 + 25 ) y − 17y − 6x− x7 + 2 ∫ t t− 1 4 y(s) ds+ 2 ∫ t t− 1 8 y(s) ds = ∫ t t− 1 4 1 1 + t4 + s2 y(s) [1 + y2(s)][1 + exp(s2)] ds + ∫ t t− 1 8 1 1 + t6 + s2 y(s) [1 + y2(s)][1 + exp(s4)] ds, t ≥ 1 8 . (3.2) Hence, comparing (1.2) and (3.2) gives the relations f1(t, x, y) = ( t+ x2 + y2 + 25 ) y, f1(t, x, 0) = 0, f1(t, x, y)y = ( t+ x2 + y2 + 25 ) y2 ≥ 25y2, f10 = 25, y 6= 0; g(x, y) = 17y, g(x, 0) = 0, g(x, y)y = 17y2 ≥ 16y2, g0 = 16, y 6= 0; g1(x) = 2x, g1(0) = 0, g1(x) x = 2 > 1 = b1, x 6= 0, g′1(x) = 2, |g′1(x)| = 2 < 3 = α1; g2(x) = 2x, g2(0) = 0, g2(x) x = 2 > 1 = b2, x 6= 0; g′2(x) = 2, |g′2(x)| = 2 < 3 = α2; h(x) = 2x+ x7, h(0) = 0, h(x) x = 2 + x6 ≥ 2 = h0, x 6= 0,∫ t t−τ1 C1(t, s)q1(s, y(s)) ds = ∫ t t− 1 4 1 1 + t4 + s2 y(s) [1 + y2(s)][1 + exp(s2)] ds, EJDE-2022/68 SOLUTION ESTIMATES AND STABILITY TESTS 7 C1(t, s) = 1 1 + t4 + s2 , |C1(t, s)| = 1 1 + t4 + s2 ≤ 1 = d1, q1(s, y(s)) = y(s) [1 + y2(s)][1 + exp(s2)] , |q1(s, y(s))| = |y(s)| [1 + y2(s)][1 + exp(s2)] ≤ |y(s)|, r1 = 1;∫ t t−τ2 C2(t, s)q2(s, y(s)) ds = ∫ t t− 1 8 1 1 + t6 + s2 y(s) [1 + y2(s)][1 + exp(s4)] ds, C2(t, s) = 1 1 + t6 + s2 , |C2(t, s)| = 1 1 + t6 + s2 ≤ 1 = d2, q2(s, y(s)) = y(s) [1 + y2(s)][1 + exp(s4)] , |q2(s, y(s))| = |y(s)| [1 + y2(s)][1 + exp(s4)] ≤ |y(s)|, r2 = 1; τ = max{4−1, 8−1} = 4−1; [f10 + g0 − 2−1τ(α1 + α2 + d1r1 + d2r2)] = [25 + 16− 8−1(3 + 3 + 1 + 1)] = 40 > 39 = σ > 0. Hence, when p(t, x, y) ≡ 0, the conditions of Theorems 2.1 and 2.2 are fulfilled. Therefore their results hold. 0.125 50.125 100.125 150.125 200.125 250.125 time(s) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x (t ) Figure 1. Trajectories of the solution x of (3.1), which shows the asymptotic stability and integrability of the solutions depending on various values of initial function. 4. Boundedness result Theorem 4.1. If (A1)–(A3) hold, then the solution (x(t), y(t)) of system (1.2) are bounded. Proof. From (A1)–(A3) and some calculations, we obtain W ′(t, xt, yt) ≤ 2yp(t, x, y) ≤ 2|y| |p(t, x, y)| 8 S. PINELAS, O. TUNÇ EJDE-2022/68 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time(s) -40 -30 -20 -10 0 10 20 30 40 y (t ) Figure 2. Trajectories of the solution y of (3.1), which shows the asymptotic stability and integrability of the solutions depending on various values of initial function. ≤ 2|p0(t)|y2 ≤ 2|p0(t)|W (t, xt, yt). Hence, W ′(t, xt, yt) W (t, xt, yt) ≤ 2|p0(t)|. Integrating this inequality, we obtain W (t, xt, yt) ≤W (t0, φt0 , ψt0) exp(2 ∫ t t0 |p0(s)|ds) ≤W (t0, φt0 , ψt0) exp(2 ∫ ∞ t0 |p0(s)|ds) ≤M0. Hence, in view of (2.2) and the last inequality above, we derive that (b1 + b2 + . . .+ bn + h0)x2 + y2 ≤W (t, xt, yt) ≤M0. Then (b1 + b2 + . . .+ bn + h0)x2 + y2 ≤M0. Thus, |x(t)| ≤ ( M0∑n i=1 bi + h0 )1/2 , |y(t)| ≤ √ M0 for all t ≥ t0. These inequalities verify that the solution (x(t), y(t)) of (1.2) are bounded. � EJDE-2022/68 SOLUTION ESTIMATES AND STABILITY TESTS 9 5. Numerical application of the bounded result Example 5.1. Let p(·) 6= 0. As a particular case of (1.1), we consider the nonlinear second order DIDE with multiple kernels and delays, d2x dt2 + ( t+ x2 + ( dx dt )2 + 25 )dx dt + 17 dx dt + 2x + x7 + 2x(t− 4−1) + 2x(t− 8−1) = ∫ t t− 1 4 1 1 + t4 + s2 x′(s) [1 + (x′(s)) 2 ][1 + exp(s2)] ds + ∫ t t− 1 8 1 1 + t6 + s2 x′(s) [1 + (x′(s)) 2 ][1 + exp(s4)] ds + x′ exp(t) 1 + exp(2t) + exp(x2 + (x′) 2 ) . (5.1) This equation can be transformed into the system dx dt = y, dy dt = − ( t+ x2 + y2 + 25 ) y − 17y − 6x− x7 + 2 ∫ t t− 1 4 y(s) ds+ 2 ∫ t t− 1 8 y(s) ds = ∫ t t− 1 4 1 1 + t4 + s2 y(s) [1 + y2(s)][1 + exp(s2)] ds + ∫ t t− 1 8 1 1 + t6 + s2 y(s) [1 + y2(s)][1 + exp(s4)] ds + y exp(t) 1 + exp(2t) + exp(x2 + y2) . (5.2) All the data in Example 3.1 hold for (5.2). We need only to consider the function p(t, x, y). Hence, we derive that |p(t, x, y)| = |y| exp(t) 1 + exp(2t) + exp(x2 + y2) ≤ |y| exp(t) 1 + exp(2t) ≤ |p0(t)||y|, |p0(t)| = exp(t) 1 + exp(2t) ,∫ ∞ 0 |p0(t)|dt = ∫ ∞ 0 exp(t) 1 + exp(2t) dt = π 4 <∞. Thus, the conditions of Theorem 4.1 hold. Then, all solutions of (5.2) are bounded. 6. Conclusion In this article, a class of nonlinear DIDEs of second order with multiple ker- nels and delays has been considered. Three new results have been given on the behaviors of solutions of considered equations. New numerical applications related to the obtained results have been given. The aim of this paper is to do the new contributions to the theory of DIDEs of higher order. 10 S. PINELAS, O. TUNÇ EJDE-2022/68 0.125 5.125 10.125 15.125 20.125 25.125 30.125 35.125 40.125 45.125 50.125 time(s) -1.5 -1 -0.5 0 0.5 1 1.5 x (t ) Figure 3. Trajectories of the solution x(t) of (5.1), which shows the boundedness of the solutions depending on various values of initial function. 0.125 5.125 10.125 15.125 20.125 25.125 30.125 35.125 40.125 45.125 50.125 time(s) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 y (t ) Figure 4. Trajectories of the solution y(t) of (5.1), which shows the boundedness of the solutions depending on various values of initial function. Acknowledgements. The authors would like to thank the anonymous referee and the handling editor for many useful comments and suggestions, leading to a substantial improvement of the presentation of this article. References [1] A. A. Adeyanju, A. T. Ademola, B. S. 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Yassine; Stability of global bounded solutions to a nonautonomous nonlinear second order integro-differential equation. Z. Anal. Anwend. 37 (2018), no. 1, 83–99. [46] Z. D. Zhang; Asymptotic stability of Volterra integro-differential equations. (in Chinese) J. Harbin Inst. Tech. 1990, no. 4, 11–19. [47] W. Zhuang; Existence and uniqueness of solutions of nonlinear integro-differential equations of Volterra type in a Banach space. Appl. Anal. 22 (1986), no. 2, 157–166. Sandra Pinelas CINAMIL - Centro de Investigação da Academia Militar, Academia Militar, Amadora, Portugal Email address: sandra.pinelas@gmail.com Osman Tunç Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil University, Campus, 65080 Van, Turkey Email address: osmantunc89@gmail.com 1. Introduction 2. Stability and integrability results 3. Numerical applications of stability and integrability results 4. Boundedness result 5. Numerical application of the bounded result 6. Conclusion Acknowledgements References