Open access peer-reviewed chapter

# Multiple Solutions for Some Classes Integro-Dynamic Equations on Time Scales

Written By

Svetlin G. Georgiev

Submitted: 09 September 2020 Reviewed: 22 December 2020 Published: 08 September 2021

DOI: 10.5772/intechopen.95621

From the Edited Volume

## Recent Developments in the Solution of Nonlinear Differential Equations

Edited by Bruno Carpentieri

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## Abstract

In this chapter we study a class of second-order integro-dynamic equations on time scales. A new topological approach is applied to prove the existence of at least two non-negative solutions. The arguments are based upon a recent theoretical result.

### Keywords

• integro-dynamic equations
• time scale
• BVP
• existence
• positive solution
• fixed point
• cone
• sum of operators

## 1. Introduction

Many problems arising in applied mathematics and mathematical physics can be modeled as differential equations, integral equations and integro-differential equations.

Integral and integro-differential equations can be solved using the Adomian decomposition method (ADM) [1, 2], Galerkin method [3], rationalized Haar functions method [4], homotopy perturbation method (HPM) [5, 6] and variational iteration method (VIM) [7]. ADM can be applied for linear and nonlinear problems and it is a method that represents the solution of the considered problems in the form of Adomian polynomials. Rationalized Haar functions and Galerkin methods are numerical methods that can be applied in different ways for the solutions of integral and integro-differential equations. VIM is an analytical method and can be used for different classes linear and nonlinear problems. HPM is a semi-analytical method for solving of linear and nonlinear differential, integral and integro-differential equations.

In recent years, time scales and time scale analogous of some well-known differential equations, integral equations and integro-differential equations have taken prominent attention. The new derivative, proposed by Stefan Hilger in [8], gives the ordinary derivative if the time scale is the set of the real numbers and the forward difference operator if the time scale is the set of the integers. Thus, the need for obtaining separate results for discrete and continuous cases is avoided by using the time scales calculus.

This chapter outlines an application of a new approach for investigations of integro-differential equations and integro-dynamic equations on time scales. The approach is based on a new theoretical result. Let T be a time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let also, a,bT, a<b. In this chapter we will investigate the following second-order integro-dynamic equation

xΔ2t=atktsfsxsxΔsΔs,tab,E1

subject to the boundary conditions

xa=α,xσ2b=β,E2

where

(H1)kCrdaσ2b×aσ2b, α,βR, α0.

(H2)fCaσ2b×R2 and

fsuva1sup1+a2svp2+a3s,saσ2b,u,vR,E3

ajCrdaσ2b, j1,2,3, are non-negative functions, p1,p20.

We will investigate the BVP (1), (2) for existence of non-negative solutions. Our main result in this chapter is as follows.

Theorem 1.1.SupposeH1-H2. Then the BVP(1), (2)has at least two non-negative solutions.

Linear integro-dynamic equations of arbitrary order on time scales are investigated in [9] using ADM. Nonlinear integro-dynamic equations of second order on time scales are studied in [10] using the series solution method. Asymptotic behavior of non-oscillatory solutions of a class of nonlinear second order integro-dynamic equations on time scales is considered in [11].

The chapter is organized as follows. In the next section, we will give some basic definitions and facts by time scale calculus. In Section 3, we give some auxiliary results which will be used for the proof of our main result. In Section 4, we will prove our main result. In Section 5, we will give an example. Conclusion is given in Section 6.

## 2. Time scales revisited

Time scales calculus originates from the pioneering work of Hilger [8] in which the author aimed to unify discrete and continuous analysis. Time scales have gained much attention recently. This section is devoted to a brief introduction of some basic notions and concepts on time scales. For detailed introduction to time scale calculus we refer the reader to the books [12, 13].

Definition 2.1.A time scaleTis an arbitrary nonempty closed subset of the real numbers.

Definition 2.2.

1. The operatorσ:TTgiven by

σt=infsT:s>tfortTE4

will be called the forward jump operator.

2. The operator ρ:TT defined by

ρt=supsT:s<tfortTE5

will be called the backward jump operator.

3. The function μ:T0 defined by

μt=σttfortTE6

will be called the graininess function.

We set

infØ=supT,supØ=infT.E7

Observe that σtt for any tT and ρtt for any tT. Below, suppose that T is a time scale with forward jump operator and backward jump operator σ and ρ, respectively.

Definition 2.3.We define the set

Tκ=T\ρsupTsupTifsupT<Totherwise.E8

Using the forward and backward jump operators, one can classify the elements of a time scale.

Definition 2.4.The pointtTis said to be

1. right-scattered if σt>t.

2. right-dense if t<supT and σt=t.

3. left-scattered if ρt<t.

4. left-dense if t>infT and ρt=t.

5. isolated if it is left-scattered and right-scattered at the same time.

6. dense if it is left-dense and right-dense at the same time.

Definition 2.5.Letf:TRbe a given function andtTκ. The delta or Hilger derivative offattwill be called the numberfΔt, provided that it exists, if for anyε>0there is a neighborhoodUoft,U=tδt+δTfor someδ>0, such that

fσtfsfΔtσtsεσtsforallsU.E9

IffΔtexists for anytTκ, then we say thatfis delta or Hilger differentiable inTκ. The functionfΔ:TRwill be called the delta derivative or Hilger derivative, shortly derivative, offinTκ.

Remark 2.6.The delta derivative coincides with the classical derivative in the case whenT=R.

Note that the delta derivative is well defined.

Theorem 2.7.Letf:TRbe a given function andtTκ.

1. The function f is continuous at t, if it is differentiable at t.

2. The function f is differentiable at t and

fΔt=fσtftμt,E10

if f is continuous at t and t is tight-scattered.

1. Let t is right-dense. Then the function f is differentiable at t if and only if the limit

limstftfstsE11

exists as a finite number. In this case, we have

fΔt=limstftfsts.E12

1. We have

fσt=ft+μtfΔt,E13

if f is differentiable at t.

Definition 2.8.Letf:TtoRis a given function.

1. We say thatfis regulated if its right-sided limits exist (finite) at all right-dense points inTand its left-sided limits exist (finite) at all left-dense points inT.

2. We say thatfis pre-differentiable with region of differentiationDif

1. it is continuous,

2. DTκ,

3. Tκ\Dis countable and contains no right-scattered elements ofT,

4. fis differentiable at eachtD.

To define indefinite integral and Cauchy integral on time scale we have a need of the following basic result.

Theorem 2.9.Lett0T,x0R,f:TκRbe a given regulated function. Then there exists unique functionFthat is pre-differentiable and

FΔt=ftforanytD,Ft0=x0.E14

Definition 2.10.

1. Letf:TRis a regulated function. Then any functionFin Theorem 2.9. is said to be a pre-antiderivative of the functionfand the indefinite integral of the regulated functionfis defined by

ftΔt=Ft+c.E15

Herecis an arbitrary constant. Define the Cauchy integral as follows

τsftΔt=FsFτforallτ,sT.E16

2. A function F:TR is said to be an antiderivative of the function f:TR if

FΔt=ftholdsforalltTκ.E17

Definition 2.11.Letf:TRbe a given function. If it is continuous at right-dense points inTand its left-sided limits exist (finite) at left-dense points inT, then we say thatfis rd-continuous. WithCrdTwe will denote the set of all rd-continuous functionsf:TRand withCrd1Twe will denote the set of all functionsf:TRthat are differentiable and whose derivative are rd-continuous.

We will note that if f is rd-continuous, then it is regulated Below, we will list some of the properties of the Cauchy integral.

Theorem 2.12.Leta,b,cT,αRandf,gCrdT. Then we have the following.

1. abft+gtΔt=abftΔt+abgtΔt,

2. abαftΔt=αabftΔt,

3. abftΔt=baftΔt,

4. abftΔt=acftΔt+cbftΔt,

5. abfσtgΔtΔt=fgbfgaabfΔtgtΔt,

6. abftgΔtΔt=fgbfgaabfΔtgσtΔt,

7. aaftΔt=0,

8. If ftgt on ab, then

abftΔtabgtΔt,E18

9. If ft0 for all at<b, then abftΔt0.

Let

Gts=σsaσ2btσ2ba,σst,taσ2bσsσ2ba,ts,taσ2b,saσb.E19

We have

Gtsσ2ba,taσ2b,saσb.E20

In [12], it is proved that G is the Green function for the BVP

xΔ2=0,xa=xσ2b=0.E21

## 3. Auxiliary results

Let X be a real Banach space.

Definition 3.1.A mappingK:XXthat is continuous and maps bounded sets into relatively compact sets will be called completely continuous.

The concept for k-set contraction is related to that of the Kuratowski measure of noncompactness which we recall for completeness.

Definition 3.2.Suppose thatΩXis the class of all bounded sets ofX. The functionα:ΩX0that is defined in the following manner

αY=infδ>0:Y=j=1mYjanddiamYjδj{1m},E22

where diamYj=supxyX:xyYj is the diameter of Yj, j1m, is said to be Kuratowski measure of noncompactness.

For the main properties of measure of noncompactness we refer the reader to [14].

Definition 3.3.If the mappingK:XXis continuous and bounded and there exists a nonnegative constantksuch that

αKYY,E23

for any bounded set YX, then we say that it is a k-set contraction.

Note that any completely continuous mapping K:XX is a 0-set contraction (see [15]).

Definition 3.4.Suppose thatXandYare real Banach spaces. Then the mapK:XYis called expansive if there exists a constanth>1for which

KxKyYhxyXE24

for any x,yX.

Definition 3.5.A closed, convex setPinXis said to be cone if.

1. αxPfor anyα0and for anyxP,

2. x,xPimpliesx=0.

Denote P=P\0,

Pr1=uP:u<r1,E25
Pr1,r2=uP:r1<u<r2

for positive constants r1,r2 such that 0<r1r2. The following result will be used to prove our main result. We refer the reader to [16, 17] for more details.

Theorem 3.6.LetPbe a cone in a Banach spaceE. LetΩbe a subset ofP,0Ωand0<r<L<Rare real constants. Let also,T:ΩEis an expansive operator with a constanth>1,S:PR¯Eis ak-set contraction with0k<h1andSPR¯ITΩ. Assume thatPr,LΩØ,PL,RΩØand there exist anu0Psuch thatTxλu0Pfor allλ0andxPrΩ+λu0and the following conditions hold:

1. Sxxλu0, xPr, λ0,

2. Sx+T0h1x and Tx+Fxx, xPLΩ,

3. Sxxλu0, xPR, λ0.

ThenT+Shas at least two fixed pointsx1Pr,LΩ,x2PL,RΩ, i.e.,

r<x1<L<x2<R.E26

Let

A1=ασ2b+α+2βmaxaσ2bσ2ba,A2=maxtsaσ2b×aσ2bkts,A3=maxmaxsaσ2bajsj=1,2,3,A4=maxσ2ba2σ2ba,E27

and

ϕt=ασ2bβa+βαtσ2ba,taσ2b.E28

Then

ϕtασ2b+βmaxaσ2b+β+αmaxaσ2bσ2ba=A1,taσ2b.E29

Suppose that E=Crd1aσ2b is endowed with the norm

x=maxmaxtaσ2bxtmaxtaσ2bxΔt,E30

provided it exists. Next two lemmas give integral representations of the solutions of the BVP (1), (2).

Lemma 3.7.IfxEis a solution to the integral equation

then x is a solution to the BVP (1), (2).

Proof. Since G is the Green function of the BVP (3) and ϕΔ2t=0, taσ2b, we get

xΔ2t=atktsfsxsxΔsΔs,taσ2b,E32

and

xa=ϕa=ασ2bβa+βαaσ2ba=α,xσ2b=ϕσ2b=ασ2bβa+βασ2bσ2ba=β.E33

Thus, x is a solution to the BVP (1), (2). This completes the proof.□

For xE, define the operator

F1xt=attσs(xsaσbGss1as1ks1s2fs2xs2xΔs2Δs2Δs1ϕs)Δs,taσ2b.E34

Lemma 3.8.IfxEis a solution to the integral equation

F1xt=0,taσ2b,E35

then x is a solution to the BVP (1), (2).

Proof. We have

0=F1xΔt=at(xsaσbGss1as1ks1s2fs2xs2xΔs2Δs2Δs1ϕs)Δs,taσ2b,E36

whereupon

taσ2b. Hence and Lemma 3.7, we conclude that x is a solution to the BVP (1), (2). This completes the proof.□

Now, we will give an estimate of the norm of the operator F1.

Lemma 3.9.IfxEandxcfor some positive constantc, then

F1xA4c+σ2baσba2A2A3cp1+cp2+1+A1.E38

Proof. We have

F1xtattσsxs+aσbGss1as1ks1s2fs2xs2xΔs2Δs2Δs1+ϕsΔsattσsc+σ2baaσbas1A2a1s2xs2p1+a2s2xΔs2p2+a3s2)Δs2Δs1+A1ΔsA4c+σ2baσba2A2A3cp1+cp2+1+A1,E39

taσ2b, and

F1xΔtatxs+aσbGss1as1ks1s2fs2xs2xΔs2Δs2Δs1+ϕs)Δsatc+σ2baaσbas1A2a1s2xs2p1+a2s2xΔs2p2+a3s2Δs2Δs1+A1ΔsA4c+σ2baσba2A2A3cp1+cp2+1+A1,E40

taσ2b. Thus,

F1xA4c+σ2baσba2A2A3cp1+cp2+1+A1.E41

This completes the proof.□

Below, suppose

(H3) Suppose that the positive constants A, m, ε, r1, L1, R1 and R satisfy the following conditions

r1<L120<L1<R1,m045,ε>1,R1<εL120,E42
AA4R1+σ2baσba2A2A3R1p1+R1p2+1+A1<L120,E43
AA4L1+σ2baσba2A2A3L1p1+L1p2+1+A1<45mL1.E44

In the next section, we will give an example for constants A, m, ε, r1, L1, R1 and R that satisfy H3. For xE, define the operator

Fxt=AF1xt,taσ2b.E45

By Lemma 3.9, we get the following result.

Lemma 3.10.IfxEandxcfor some positive constantc, then

FxAA4c+σ2baσba2A2A3cp1+cp2+1+A1.E46

Lemma 3.11.IfxEis a solution to the integral equation

0=L15+Fxt,taσ2b,E47

then it is a solution to the BVP (1), (2).

Proof. We have

0=FxΔ2t=AF1xΔ2t,taσ2b,E48

whereupon

taσ2b, and

taσ2b. Now, the assertion follows from Lemma 3.7. This completes the proof.□

## 4. Proof of the Main result

Let

P˜={uE:u0on[t0,)}.E51

With P we will denote the set of all equi-continuous families in P˜. For vE, define the operators

Tvt=1+vtεL110,Svt=εFvtmεvtεL110,E52

tt0. Note that any fixed point vE of the operator T+S is a solution to the IVP (1). Define

Pr1=vP:v<r1,PL1=vP:v<L1,PR1=vP:v<R1,Pr1,L1=vP:r1<v<L1,PL1,R1=vP:L1<v<R1,R2=R1+AmA4R1+σ2baσba2A2A3R1p1+R1p2+1+A1+L15m,Ω=PR2=vP:v<R2.E53

1. For v1,v2Ω, we have

Tv1Tv2=1+v1v2,E54

whereupon T:ΩE is an expansive operator with a constant 1+>1.

1. For vP¯R1, we get

SvεFv+v+εL110εAA4R1+σ2baσba2A2A3R1p1+R1p2+1+A1+mR1+L110.E55

Therefore SP¯R1 is uniformly bounded. Since S:P¯R1E is continuous, we have that SP¯R1 is equi-continuous. Consequently S:P¯R1E is a 0-set contraction.

1. Let v1P¯R1. Set

v2=v1+1mFv1+L15m.E56

Note that by the second inequality of H3 and by Lemma 3.10, it follows that εFv1+εL150 on t0. We have v20 on t0 and

v2v1+1mFv1+L15mR1+AmA4R1+σ2baσba2A2A3R1p1+R1p2+1+A1+L15m=R2.E57

Therefore v2Ω and

εmv2=εmv1εFv1εL110εL110E58

or

ITv2=εmv2+εL110=Sv1.E59

Consequently SP¯R1ITΩ.

1. Suppose that there exists an v0P such that Tvλv0P for all λ0, vPr1Ω+λv0 and Sv=vλv0 for some λ0 and for some vPr1. Then

r1vλv0=SvSvt=εFvt+εmvt+εL110εL120,tt0,E60

because by the second inequality of H3 and by Lemma 3.10, it follows that εFv+εL1200 on t0.

1. Let vPL1. Then

Sv+T0=εFv+mεv+εL15εFv+mv+L15εAA4L1+(σ2ba)(σba)2A2A3L1p1+L1p2+1+A1+m+15L1εL1=εv.E61

Note that in the last inequality we have used the third inequality of H3.

1. Now, assume that vPL1Ω is such that

v=Tv+Sv,E62

whereupon

Fv+L150ont0.E63

Since vPL, we have that v0 on t0 and by the second inequality of H3 and by Lemma 3.10, it follows that Fv+L15>Fv+L1200 on t0.This is a contradiction.

1. Suppose that there exists an v0P such that Tvλv0P for all λ0, vPR1, vPR1Ω+λv0 and Sv=vλv0 for some λ0 and for some vPR1. Then

R1vλv0=SvSvt=εFvt+εmvt+εL110εL120,tt0,E64

Therefore all conditions of Theorem 3.6 hold. Hence, the IVP (1) has at least two solutions u1 and u2 so that

r1<u1<L1<u2<R1.E65

## 5. An example

In this section we will illustrate our main result with an example. Firstly, we will give an example for the constants A, m, ε, r1, L1 and R1 that satisfy the hypothesis H3. Let T=Z, a=0, b=10, α=β=1,

a1s=a2s=a3s=13,fsuv=11+v4,kss1=s12,E66

s012, s1011, and

r1=1,L1=100,R1=200,ε=1010,p1=p2=0,m=12,A=11050.E67

Then

A1=12+12312=4,A2=144,A3=13,A4=144E68

and

AA4R1+σ2baσba2A2A3R1p1+R1p2+1+A1=11050144200+123144133+4<5=L120,AA4L1+σ2baσba2A2A3R1p1+R1p2+1+A1=11050144100+123144133+4<3105=45mL120,E69

i.e., H1-H3 hold. Consequently the BVP

xΔ2t=0ts211+xΔs4Δs,t010,x0=x12=1,E70

has at least two non-negative solutions.

## 6. Conclusion

In this chapter we introduce a class of BVPs for a class second-order integro-dynamic equations on time scales. We give some integral representations of the solutions of the considered BVP. We apply a new multiple fixed point theorem and we prove that the considered BVP has at least two nontrivial solutions. The approach in this chapter can be applied for investigations of IVPs and BVPs for dynamic equations and integro-dynamic equations of arbitrary order on time scales.

AMS Subject Classification: 39 A 10, 39 A 99

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Written By

Svetlin G. Georgiev

Submitted: 09 September 2020 Reviewed: 22 December 2020 Published: 08 September 2021