2. Main Results Here by using Lemma 1, we establish some new results in the form of integrodifferential inequalities instead of integral inequality.

Theorem 2. Let x(t), x•(t), f(t), and g(t) be nonnegative real valued continuous functions defined for R+=[0,∞). Let c1 and c2 be positive constants. If(6)x•2t≤c1+∫0tfsx•2sdsc2+∫0tgsx•2sds,for all t∈R+, then(7)x•t≤1Htc1c2Qt,where c1c2≥1.

H ( t ) , R(t), and Q(t) are defined as in (2), (3), and (4), respectively, for all t∈R+.

Proof. Define a function z2(t) by the right-hand side of (6), such that(8)z2t=c1+∫0tfsx•2sdsc2+∫0tgsx•2sds,where(9)z20=c1c2.From (6) and (8), we get(10)x•2t≤z2t.By differentiating (8) and using the fact that(11)x•t≤ztwe observe(12)2ztz′t≤c1gt+c2ftz2t+ft∫0tgσdσ+gt∫0tfσdσz4tor(13)2z-3tz′t-c1gt+c2ftz-2t≤ft∫0tgσdσ+gt∫0tfσdσ.Let(14)vt=z-2t.Differentiating (14) with respect to x, we get(15)v′t=-2z-3tz′t,where(16)v0=c1c2-1.By substituting (14) and (15) in (13), we have(17)v′t+c1gt+c2ftvt≥-Rt.Inequality (17) implies the estimation for v(t) and by using (16), we observe that(18)vt≥c1c2-1Q-1t1-c1c2∫0tRsQsdsds,where R(t) and Q(t) are defined as in (3) and (4) and by applying (11) and (14) it is noticed that(19)x•t≤1Htc1c2Qt,where H(t) is defined as in (2). This completes the proof.

Theorem 3. Let x(t), x•(t), f(t), g(t), c1, and c2 be defined as in Theorem 2 for R+=[0,∞). If(20)x•2t≤c1+∫0tfsx•sdsc2+∫0tgsx•2sds,for all t∈R+, then(21)x•t≤1Htc1c2Qt,where c1c2≥1 and(22)Ht=1-c1c22∫0tRsQsds>0,(23)Qt=exp12∫0tc1gσ+c2fσdσfor all t∈R+.

Proof. Define a function z2(t) by the right-hand side of (20), such that(24)z2t=c1+∫0tfsx•sdsc2+∫0tgsx•2sds,where(25)z20=c1c2.From (20) and (24), we get(26)x•2t≤z2t.By differentiating (24) and since z(t) is monotone nondecreasing function for t∈R+ and using the fact that(27)x•t≤ztwe observe that(28)2ztz′t≤c1gt+c2ftz2t+ft∫0tgσdσ+gt∫0tfσdσz3tor(29)2z-2tz′t-c1gt+c2ftz-1t≤ft∫0tgσdσ+gt∫0tfσdσ.Let(30)vt=z-1t;v0=c1c2-1/2.By repeating the same steps from (14)–(18) in (29) with suitable modifications, the estimation for v(t) implies(31)vt≥c1c2-1/2Q-1t1-c1c22∫0tRsQsdsds.From (27) and (30) in (31), we get(32)x•t≤1Htc1c2Qtfor all t∈R+, where R(t), H(t), and Q(t) are defined as in (3), (22), and (23), respectively. This completes the proof.

Theorem 4. Let x(t), x•(t), f(t), g(t), c1, and c2 be defined as in Theorem 2 for R+=[0,∞). If(33)x•pt≤c1+∫0tfsx•psdsc2+∫0tgsx•sds,for all t∈R+, then(34)x•t≤1Htc1c21/pQt,where c1c2≥1 and p>0.(35)Ht=1-c1c21/pp∫0tRsQsds>0,(36)Qt=exp1p∫0tc1gσ+c2fσdσfor all t∈R+.

Proof. Define a function zp(t) by the right-hand side of (33), such that(37)zpt=c1+∫0tfsx•psdsc2+∫0tgsx•sds,where(38)zp0=c1c2.From (33) and (37), we get(39)x•pt≤zptor(40)x•t≤zt.By differentiating (37) and since z(t) is monotone nondecreasing function for t∈R+, we observe that(41)pzp-1tz′t≤c1gt+c2ftzpt+ft∫0tgσdσ+gt∫0tfσdσzp+1tor(42)pz-2tz′t-c1gt+c2ftz-1t≤ft∫0tgσdσ+gt∫0tfσdσ.Let(43)vt=z-1t;v0=c1c2-1/p.By repeating the same steps from (14)–(18) in (42) with suitable modifications, the estimation for v(t) implies(44)vt≥c1c2-1/pQ-1t1-c1c21/pp∫0tRsQsdsds.From (40) and (43) in (44), we get(45)x•t≤1Htc1c21/pQt,for all t∈R+, where R(t), H(t), and Q(t) are defined as in (3), (35), and (36), respectively. This completes the proof.

Theorem 5. Let x(t), x•(t), f(t), g(t), c1, and c2 be defined as in Theorem 2 for R+=[0,∞). If(46)x•pt≤c1+∫0tfsx•t+x•psdsc2+∫0tgsx•sds,for all t∈R+, then(47)x•t≤1Htc1c21/pQt,where c1c2≥1 and p>0.(48)Ht=1-c1c21/pp∫0tRsQsds>0,(49)Qt=exp1p∫0tc1gσ+2c2fσ+fσ∫0sgηdη+gσ∫0sfηdηdσfor all t∈R+.

Proof. Define a function zp(t) by the right-hand side of (46), such that(50)zpt=c1+∫0tfsx•t+x•psdsc2+∫0tgsx•sds,where(51)zp0=c1c2.From (46) and (50), we get(52)x•pt≤zptor(53)x•t≤zt.By differentiating (50) and since z(t) is monotone nondecreasing function for t∈R+, we observe that(54)pzp-1tz′t≤c1gσ+2c2fσ+fσ∫0sgηdη+gσ∫0sfηdηzpt+Rtzp+1tor(55)pz-2tz′t-c1gσ+2c2fσ+fσ∫0sgηdη+gσ∫0sfηdηz-1t≤Rt.Let(56)vt=z-1t;v0=c1c2-1/p.By repeating the same steps from (14)–(18) in (55) with suitable modifications, the estimation for v(t) implies(57)vt≥c1c2-1/pQ-1t1-c1c21/pp∫0tRsQsdsds.From (53) and (56) in (57), we get(58)x•t≤1Htc1c21/pQt,for all t∈R+, where R(t), H(t), and Q(t) are defined as in (3), (48), and (49), respectively. This completes the proof.

Theorem 6. Let x(t), x•(t), f(t), g(t), c1, and c2 be defined as in Theorem 2 for R+=[0,∞). If(59)x•pt≤c1+∫0tfsx•qsdsc2+∫0tgsx•sds,for all t∈R+, then(60)x•t≤1Qt1/p-qc1c2p-q/p+p-qp∫0tc1fs+c2gsQsds1/p-q,where c1c2≥1, p>q≥1, and p-q≥1.(61)Qt=exp-p-qp∫0tRsdsfor all t∈R+.

Proof. Define a function zp(t) by the right-hand side of (59), such that(62)zpt=c1+∫0tfsx•qsdsc2+∫0tgsx•sds,where(63)zp0=c1c2.From (59) and (62), we get(64)x•pt≤zptor(65)x•t≤zt.By differentiating (62) and since z(t) is monotone nondecreasing function for t∈R+, we observe that(66)pzp-1tz′t≤c1gt+c2ftzqt+ft∫0tgσdσ+gt∫0tfσdσz1+qtor(67)pzp-q-1tz′t-Rtzp-qt≤c1gt+c2ft.Let(68)vt=zp-qt;v0=c1c2p-q/p.By repeating the same steps from (14)–(18) in (67) with suitable modifications, the estimation for v(t) implies(69)vt≤1Qt1/p-qc1c2p-q/p+p-qp∫0tc1fs+c2gsQsds1/p-q.From (65) and (68) in (69), we get(70)x•t≤1Qt1/p-qc1c2p-q/p+p-qp∫0tc1fs+c2gsQsds1/p-q,for all t∈R+, where R(t) and Q(t) are defined as in (3) and (61), respectively. This completes the proof.

3. Application As an application, the explicit bounds of some of the integral inequalities can be found by some examples.

Example 1. Let us consider the explicit bound on the solution of the nonlinear integrodifferential equation(71)x•2t≤1+∫0tfsx•2sds1+∫0tgsx•sds,where x•(s) is a nonnegative real valued continuous function and every solution of x•(s) of (71) exists for R+.

By using the application of Theorem 4 to (71), we observe that(72)x•t≤1HtQt,where(73)Rt=∫0tds+∫0tds=2t,(74)Ht=1-12∫0tRsQsds>0=1-12∫0t2sesds=1-∫0tsesds=-tet+et,(75)Qt=exp12∫0t2 ds=exp∫0tds=et.Therefore the right-hand side of (74) provides the bound of the solution of (75) of known quantities(76)x•t≤etet+tet≤11-tfor 0≤t<1.

Example 2. Let us consider the nonlinear integrodifferential equation of the form(77)x•pt≤1+∫0tfsx•t+x•2sds1+∫0tgsx•sds,where x•(s) is a nonnegative real valued continuous function and every solution of x•(s) of (77) exists for R+.

By using the application of Theorem 5 to (77), we observe that(78)x•t≤1HtQt,where(79)Rt=∫0tds+∫0tds=2t,(80)Ht=1-12∫0tRsQsds>0=1-12∫0t2se3/2s+1/2s2ds=1-∫0tse3/2s+1/2s2ds,(81)Qt=exp12∫0t1+2+∫0tdσ+∫0tdσds=exp123t+t2=e3/2t+1/2t2.Therefore the right-hand side of (80) provides the bound of the solution of (77) of known quantities(82)x•t≤e3/2t+1/2t21-∫0tse3/2s+1/2s2dsfor 0≤t<1.

Example 3. Now let us consider the boundedness and asymptotic behaviour of the solutions of nonlinear Volterra integrodifferential inequality of the form(83)x•pt=a1t+∫0tAt-sx•qsdsa2t+∫0tBt-sx•sds;x•(t) is nonnegative real valued continuous function defined on R+ and a1, a2, A, B are real valued continuous function defined on R+.

We assume that every solution of x•(t) in (81) exists on R+, and p and q are defined as in Theorem 6. Define the following hypotheses on the function of (81) as(84)a1t≤c1,a2t≤c2.Also(85)At-s≤M1f1s,(86)Bt-s≤N1g1s,(87)Dt=1Qt1/p-qc1c2p-q/p+p-qp∫0tc1gs+c2fsQsds1/p-q<∞for all 0≤s≤t, s,t∈R+. M, N, c1, c2 are nonnegative real constants and f1, g1 are nonnegative real valued continuous function defined on R+.

Proof. For the boundedness of the solution of nonlinear integrodifferential equation (83), let us suppose that the hypotheses (84), (85), and (86) are satisfied and let x•(t) be a solution of (83); then we observe that(88)x•pt≤a1t+∫0tAt-sx•qsdsa2t+∫0tBt-sx•sds.Replacing f by M1f1 and g by N1g1 and applying the same proof with some modifications of Theorem 6 in (88) and with R(t) and Q(t) being the same as defined in Theorem 6, we noticed that every solution of x•(t) of (88) that exists on R+ is bounded; that is,(89)x•pt≤Dt.For the asymptotic behaviour of the solution of nonlinear integrodifferential equation (83), assume the following hypotheses(90)a1t≤c1e-μt,a2t≤c2e-μt.Also(91)At-s≤M1f1se-μt-2s,Bt-s≤N1g1se-μt-2sare satisfied. Let x•(t) be a solution of (83); then(92)x•pt≤e-2μtc1+∫0tM1f1sx•qse2μsdsc2+∫0tN1g1sx•se2μsds.Let zm(t) be x•pte2μt for m>0 in (92) and by applying the same proof with some changes of Theorem 6 in (92), we get(93)zt≤Dt⟹x•t≤Dte-2μt.Therefore the solution x•(t) of (83) is asymptotically stable.