AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 912874 10.1155/2013/912874 912874 Research Article Some New Nonlinear Weakly Singular Inequalities and Applications to Volterra-Type Difference Equation http://orcid.org/0000-0001-9380-4558 Zheng Kelong 1 Feng Wenqiang 2 http://orcid.org/0000-0003-1760-4065 Guo Chunxiang 3 Diblík Josef 1 School of Science Southwest University of Science and Technology Mianyang, Sichuan 621010 China swust.edu.cn 2 Department of Mathematics University of Tennessee, Knoxville TN 37996-0612 USA tennessee.edu 3 School of Business Sichuan University Chengdu, Sichuan 610064 China scu.edu.cn 2013 31 10 2013 2013 03 08 2013 21 09 2013 25 09 2013 2013 Copyright © 2013 Kelong Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known weakly singular inequalities and can be used in the analysis of nonlinear Volterra-type difference equations with weakly singular kernel. An application to the upper bound of solutions of a nonlinear difference equation is also presented.

1. Introduction

The discrete version of the well-known Gronwall-Bellman inequality is an important tool in the development of the theory of difference equations as well as the analysis of the numerical schemes of differential equations. A great deal of interest has been given to these inequalities, and many results on their generalizations have been found; for example, see . Among them, one of the fundamental cases is Pachpatte’s result  for the difference inequality: (1)u(n)a(n)+s=0n-1f(s)u(s).

In particular, due to the study of the behavior and numerical solutions for the singular integral equations, some discrete weakly singular integral inequalities also have drawn more and more attention . Dixon and McKee  investigated the convergence of discretization methods for the Volterra integral and integrodifferential equations, by using the following inequality: (2)xiψi+Mh1-αj=0i-1xj(i-j)α,i=1,2,,N,hhhhhn>0,Nh=T.

Henry  presented a linear integral inequality with weakly kernel: (3)x(t)a(t)+0t(t-s)β-1b(s)x(s)ds to investigate some qualitative properties for a parabolic equation. The corresponding discrete version was discussed by Slodic˘ka . But he studied the case τk=τ, that is, the case of constant differences. Furthermore, the first formulation of the inequality with a nonlinearity and τk nonconstant was studied in , in which the general nonlinear discrete case as follows: (4)xnan+k=0n-1(tn-tk)β-1τkbkω(xk) was considered. However, his results are based on the so-called “(q) condition”: (1) ω satisfies e-qt[ω(u)]qR(t)ω(e-qt)uq; (2) there exists c>0 such that ane-τtnc. Recently, a new nonlinear difference inequality: (5)xnαan+k=0n-1(tn-tk)β-1τkbkxkλ was discussed by Yang et al. . For other new weakly singular inequalities, lots of work can be found, for example, in  and references therein.

In this paper, we investigate the new nonlinear weakly singular inequality: (6)xnan+k=0n-1(tn-tk)β-1τkbkω(xk), where 0<β1, t0=0, τk=tk+1-tk, supkτk=τ, and limttk=. Compared to the existing result, our result does not need the so-called “(q) condition” proposed in  and can be used to obtain pointwise explicit bounds on solutions for a class of more general weakly singular inequalities of Volterra type. Finally, we also present an application to Volterra-type difference equation with weakly singular kernel.

2. Preliminaries

Let be the set of real numbers, +=(0,), and ={0,1,2,}. C(X,Y) denotes the collection of continuous functions from the set X to the set Y. As usual, the empty sum is taken to be 0.

Lemma 1 (Discrete Jensen inequality, [<xref ref-type="bibr" rid="B11">11</xref>]).

Let A1,A2,,An be nonnegative real numbers, and let r>1 be a real number. Then, (7)(A1+A2++An)rnr-1(A1r+A2r++Anr).

Lemma 2 (Discrete Hölder inequality, [<xref ref-type="bibr" rid="B11">11</xref>]).

Let ai,bi  (i=1,2,,n) be nonnegative real numbers, and let p,q be positive numbers such that (1/p)+(1/q)=1 (or p=1,q=). Then, (8)i=1naibi(i=1naip)1/p(i=1nbiq)1/q.

Furthermore, take p=q=2; then, one gets the discrete Cauchy-Schwarz inequality.

Lemma 3.

Suppose that ω(u)C(+,+) is nondecreasing. Let an,cn be nonnegative and nondecreasing in n. If yn is nonnegative such that (9)ynan+cnk=0n-1bkω(yk),n. Then, (10)ynΩ-1[Ω(an)+cnk=0n-1bk],0nM, where Ω(v)=v0v(1/ω(s))ds,vv0, Ω-1 is the inverse function of Ω, and M is defined by (11)M=sup{i:Ω(ai)+cik=0i-1bkDom(Ω-1)}.

3. Main Results

Assume that

(A1) an,bn are nonnegative functions for n, respectively;

(A2) ω(u)C(+,+) is nondecreasing and ω(0)=0.

Define a~n=max0kn,kak and τ=max0kn-1,kτk, where τk is the variable time step.

Theorem 4.

Under assumptions (A1) and (A2), if xn is nonnegative such that (6), then

for  0<β1/2, letting p=1+β and q=(1+β)/β, one has (12)xn[Ω-1(k=0n-1Ω(2q-1a~nq)+2q-1τ1-(q/p)β2×Kq/p(β)eqτtnk=0n-1e-qτtkbkq)]1/q, for 0nN1, where Ω(u)=u0u(1/ωq(s1/q))  ds,uu00, Ω-1 is the inverse function of Ω, (13)K(β)=(1+β)-β2Γ(β2), and N1 is the largest integer number such that (14)Ω(2q-1a~nq)+2q-1τ1-(q/p)β2Kq/p(β)eqτtn×k=0n-1e-qτtkbkqDom(Ω-1);

for 1/2<β1, letting p=2 and q=2, one has (15)xn[Ω-1(Ω(2a~n2)+B(β)τ2-2βe2τtnk=0n-1e-2τtkbk2)]1/2, for 0nN2, where Ω(u)  =u0u(1/ω2(s1/2))ds,  uu00, (16)B(β)=41-βΓ(2β-1),β>12, and N2 is the largest integer number such that (17)Ω(2a~n2)+B(β)τ2-2βe2τtnk=0n-1e-2τtkbk2Dom(Ω-1).

Proof.

By definition of a~n and assumption (A1), a~n is nonnegative and nondecreasing and a~nan. It follows from (6) that (18)xna~n+k=0n-1(tn-tk)β-1τkbkω(xk).

If 0<β1/2, using Lemma 2 with the indices p=1+β,q=(1+β)/β for (18), we get (19)xna~n+k=0n-1(tn-tk)β-1τk1/pτk1/qeτtke-τtkbkω(xk)a~n+τ1/qk=0n-1(tn-tk)β-1τk1/peτtke-τtkbkω(xk  )a~n+τ1/q[k=0n-1(tn-tk)p(β-1)τkepτtk]1/p×[k=0n-1e-qτtkbkqωq(xk)]1/q. By Lemma 1, the inequality above yields (20)xnq2q-1a~nq+2q-1τ[k=0n-1(tn-tk)p(β-1)τkepτtk]q/p×[k=0n-1e-qτtkbkqωq(xk)]. Consider that (21)k=0n-1(tn-tk)p(β-1)τkepτtk0tn(tn-s)p(β-1)epτsds=epτtn0tnηp(β-1)e-pτηdη,=epτtn(pτ)1+p(β-1)0pτtnσp(β-1)e-σdσK(β)τ-β2epτtn, where K(β)=(1+β)-β2Γ(β2) and Γ(z)=0uz-1e-udu,(Rez>0) is the well-known G-function. Thus, we have (22)xnq2q-1a~nq+2q-1τ1-(q/p)β2×Kq/p(β)eqτtnk=0n-1e-qτtkbkqωq(xk). Let vn=xnq, An=2q-1a~nq, and Cn=2q-1τ1-(q/p)β2Kq/p(β)eqτtn. Obviously, An,Cn are nondecreasing for n and ωq(vk1/q) satisfies the assumption (A2). Equation (22) can be rewritten as (23)vnAn+Cnk=0n-1e-qτtkbkqωq(vk1/q), which is similar to inequality (9). Using Lemma 3, from (23), we have (24)vnΩ-1[(Ω(An)+Cnk=0n-1e-qτtkbkq)], for 0nN1, where N1 is the largest integer number such that (25)Ω(An)+Cnk=0n-1e-qτtkbkqDom(Ω-1). Therefore, by xn=vn1/q, (12) holds for 0nN1.

If 1/2<β1, applying Cauchy-Schwarz inequality for (18), that is, p=q=2, we get (26)xna~n+k=0n-1(tn-tk)β-1τk1/2τk1/2eτtke-τtkbkω(xk)a~n+τ1/2k=0n-1(tn-tk)β-1τk1/2eτtke-τtkbkω(xk)a~n+τ1/2[k=0n-1(tn-tk)2(β-1)τke2τtk]1/2×[k=0n-1e-2τtkbk2ω2(xk)]1/2. By Lemma 1, the inequality above yields (27)xn22a~n2+2τ[k=0n-1(tn-tk)2(β-1)τke2τtk]×[k=0n-1e-2τtkbk2ω2(xk)]. Because (28)k=0n-1(tn-tk)2(β-1)τke2τtk0tn(tn-s)2(β-1)e2τsds=e2τtn(2τ)2β-102τtnσ2(β-1)e-σdσ12B(β)τ1-2βe2τtn, where B(β)=41-βΓ(2β-1),β>1/2, it follows from (27) that (29)xn22a~n2+B(β)τ2-2βe2τtn[k=0n-1e-2τtkbk2ω2(xk)]. Let vn=xn2, An=2a~n2, and Cn=B(β)τ2-2βe2τtn. Similarly, An,Cn also are nondecreasing for n and ω2(vk1/2) also satisfies the assumption (A2). Equation (29) can be rewritten as (30)vnAn+Cn(k=0n-1e-2τtkbk2ω2(vk1/2)), which also is similar to inequality (9). Using Lemma 3, from (30), we have (31)vn[Ω-1(Ω(An)+Cnk=0n-1e-2τtkbk2)], for 0nN2, and N2 is the largest integer number such that (32)Ω(An)+Cnk=0n-1e-2τtkbk2Dom(Ω-1). Clearly, by xn=vn1/2, (15) also holds for 0nN2.

Remark 5.

Here, we note that the most significant work in the study of weakly singular inequalities is Medved˘’s method, originally presented in the paper  and also applied in the paper . But his result holds under the assumption “ω(u) satisfies the condition (q),” that is, “e-qt[ω(u)]qR(t)ω(e-qtuq), where R(t) is a continuous, nonnegative function.” In our result, the condition (q) is eliminated.

Corollary 6.

Under assumptions (A1) and (A2), let ν>0, μ>0(ν>μ). If xn is nonnegative such that (33)xnνan+k=0n-1(tn-tk)β-1τkbkxkμ, then

if 0<β1/2, let p=1+β and q=(1+β)/β, and one gets (34)xn[k=0n-1(2q-1a~nq)(ν-μ)/ν+ν-μν2q-1τ1-(q/p)β2×Kq/p(β)eqτtnk=0n-1e-qτtkbkq]1/(ν-μ)q for n0, where K(β) is defined as in Theorem 4;

if 1/2<β1, let p=q=2, and one gets (35)xn[k=0n-1(2a~n2)(ν-μ)/ν+ν-μνB(β)τ2-2β×e2τtnk=0n-1e-2τtkbk2]1/2(ν-μ), for n0, where B(β) is defined as in Theorem 4

Proof.

Let zn=xnν, then xn=zn1/ν and xnμ=znμ/ν. From (33), we have (36)znan+k=0n-1(tn-tk)β-1τkbkzkμ/ν.

Clearly, ω(zk)=zkμ/ν satisfies the assumption (A2). According to the definition of Ω in Theorem 4, for 0<β1/2, letting u0=0, we have (37)Ω(u)=u0u1ωq(s1/q)ds=0udssμ/ν=νν-μu(ν-μ)/ν,(38)Ω-1(u)=(ν-μνu)ν/(ν-μ),              Dom(Ω-1)=[0,).

It can be seen easily from (38) that N1=. Substituting (37) and (38) into (12), we get (39)zn[k=0n-1(2q-1a~nq)(ν-μ)/ν+ν-μν2q-1τ1-(q/p)β23×Kq/p(β)eqτtnk=0n-1e-qτtkbkq]ν/(ν-μ)q.

In view of xn=zn1/ν, we can obtain (34). For the case that 1/2<β1, in fact, Ω and Ω-1 are the same as (37) and (38), respectively. So, it follows from (37), (38), and (15) that (40)xn[k=0n-1(2a~n2)(ν-μ)/ν+ν-μνB(β)τ2-2β×e2τtnk=0n-1e-2τtkbk2]1/2(ν-μ)  , for n>0.

Remark 7.

In , Yang et al. investigated inequality (33), under the assumption that an is nondecreasing. Clearly, our result does not need such condition, and we get a more concise formula.

Remark 8.

Letting ν=2 and μ=1, we can get the interesting Henry version of the Ou-Iang-Pachpatte-type difference inequality . Thus, our result is a more general discrete analogue for such inequality.

Corollary 9.

Under assumptions (A1) and (A2), if xn is nonnegative such that (41)xnan+k=0n-1(tn-tk)β-1τkbkxk, then

if 0<β1/2, let p=1+β and q=(1+β)/β, and one gets (42)xn2(q-1)/qa~nexp(k=0n-12(q-1)/qτ1-(q/p)β2Kq/p(β)×eqτtnk=0n-1e-qτtkbkq), for n0, where K(β) is defined as in Theorem 4;

if 1/2<β1, let p=q=2, and one gets (43)xn2a~nexp(12B(β)τ2-2βe2τtnk=0n-1e-2τtkbk2), for n0, where B(β) is defined as in Theorem 4.

Proof.

In (41), ω(u)=u also satisfies the assumption (A2). Thus, we have (44)Ω(u)=u0udss=lnuu0,Ω-1(u)=u0exp(u),Dom(Ω-1)=[0,).

Similarly to the computation in Corollary 6, the estimates (42) and (43) hold, respectively.

4. Application

In this section, we apply our results to discuss the upper bound of solution of a Volterra type difference equation with weakly singular kernel.

Consider the following the inequality: (45)xn1+k=0n-1(tn-tk)-1/2τkxk.

Obviously, (45) is the special case of inequality (6), then we get (46)an=1,β=12,ω=u.

Thus, we can take p=1+β=3/2 and q=(1+β)/β=3; then, q/p=2. Moreover, (47)a~n=1,K(β)=(1+β)-β2Γ(β2)=(32)-1/4Γ(14),Ω(u)=0udss=2u,Ω-1(u)=u24.

According to Theorem 4, we obtain (48)xn[k=0n-1Ω-1(Ω(2q-1a~nq)+2q-1τ1-(q/p)β2)×Kq/p(β)eqτtnk=0n-1e-qτtkbkq]1/q=[k=0n-1Ω-1(Ω(4)+4τ1/2(32)-1/2)×Γ2(14)e3τtnk=0n-1e-3τtkbk3]1/3=[Ω-1(4+436τ1/2Γ2(14)e3τtnk=0n-1e-3τtkbk3)]1/3=4-1/3(4+436τ1/2Γ2(14)e3τtnk=0n-1e-3τtkbk3)2/3 for n>0, which indicates that we get the upper bound of xn.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Doctoral Program Research Funds of Southwest University of Science and Technology (no. 11zx7129) and the Fundamental Research Funds for the Central Universities (no. skqy201324). The work is also supported by the Applied Basic Research Project of Sichuan Province (no. 2013JY0096).

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